3.4.63 \(\int \frac {1}{(d+e x)^{5/2} (b x+c x^2)^3} \, dx\)

Optimal. Leaf size=470 \[ -\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)}-\frac {\left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}+\frac {c^{9/2} \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}+\frac {c x (2 c d-b e) \left (-7 b^2 e^2-12 b c d e+12 c^2 d^2\right )+b (c d-b e) \left (-7 b^2 e^2-3 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac {e (2 c d-b e) \left (-35 b^4 e^4+10 b^3 c d e^3+2 b^2 c^2 d^2 e^2-24 b c^3 d^3 e+12 c^4 d^4\right )}{4 b^4 d^4 \sqrt {d+e x} (c d-b e)^4}+\frac {e \left (-35 b^4 e^4+45 b^3 c d e^3+27 b^2 c^2 d^2 e^2-144 b c^3 d^3 e+72 c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3} \]

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Rubi [A]  time = 0.88, antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {740, 822, 828, 826, 1166, 208} \begin {gather*} \frac {c x (2 c d-b e) \left (-7 b^2 e^2-12 b c d e+12 c^2 d^2\right )+b (c d-b e) \left (-7 b^2 e^2-3 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)^2}+\frac {e (2 c d-b e) \left (2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4-24 b c^3 d^3 e+12 c^4 d^4\right )}{4 b^4 d^4 \sqrt {d+e x} (c d-b e)^4}+\frac {e \left (27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4-144 b c^3 d^3 e+72 c^4 d^4\right )}{12 b^4 d^3 (d+e x)^{3/2} (c d-b e)^3}+\frac {c^{9/2} \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}-\frac {\left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}-\frac {c x (2 c d-b e)+b (c d-b e)}{2 b^2 d \left (b x+c x^2\right )^2 (d+e x)^{3/2} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(e*(72*c^4*d^4 - 144*b*c^3*d^3*e + 27*b^2*c^2*d^2*e^2 + 45*b^3*c*d*e^3 - 35*b^4*e^4))/(12*b^4*d^3*(c*d - b*e)^
3*(d + e*x)^(3/2)) + (e*(2*c*d - b*e)*(12*c^4*d^4 - 24*b*c^3*d^3*e + 2*b^2*c^2*d^2*e^2 + 10*b^3*c*d*e^3 - 35*b
^4*e^4))/(4*b^4*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(2*b^2*d*(c*d - b*e)*(d
+ e*x)^(3/2)*(b*x + c*x^2)^2) + (b*(c*d - b*e)*(12*c^2*d^2 - 3*b*c*d*e - 7*b^2*e^2) + c*(2*c*d - b*e)*(12*c^2
*d^2 - 12*b*c*d*e - 7*b^2*e^2)*x)/(4*b^4*d^2*(c*d - b*e)^2*(d + e*x)^(3/2)*(b*x + c*x^2)) - ((48*c^2*d^2 + 60*
b*c*d*e + 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2)) + (c^(9/2)*(48*c^2*d^2 - 156*b*c*d*e + 1
43*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(9/2))

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x
)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*
c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2
*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -
b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,
c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||
IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 828

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((
e*f - d*g)*(d + e*x)^(m + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/(c*d^2 - b*d*e + a*e^2), Int[((d
+ e*x)^(m + 1)*Simp[c*d*f - f*b*e + a*e*g - c*(e*f - d*g)*x, x])/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && FractionQ[m] && LtQ[m, -1]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^3} \, dx &=-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+\frac {9}{2} c e (2 c d-b e) x}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx}{2 b^2 d (c d-b e)}\\ &=-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^2 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )+\frac {5}{4} c e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^3 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )+\frac {1}{4} c e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {1}{4} (c d-b e)^4 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )+\frac {1}{4} c e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^4 (c d-b e)^4}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{4} e (c d-b e)^4 \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )-\frac {1}{4} c d e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )+\frac {1}{4} c e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{b^4 d^4 (c d-b e)^4}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}+\frac {\left (c \left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 d^4}-\frac {\left (c^5 \left (48 c^2 d^2-156 b c d e+143 b^2 e^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {b e}{2}+\frac {1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt {d+e x}\right )}{4 b^5 (c d-b e)^4}\\ &=\frac {e \left (72 c^4 d^4-144 b c^3 d^3 e+27 b^2 c^2 d^2 e^2+45 b^3 c d e^3-35 b^4 e^4\right )}{12 b^4 d^3 (c d-b e)^3 (d+e x)^{3/2}}+\frac {e (2 c d-b e) \left (12 c^4 d^4-24 b c^3 d^3 e+2 b^2 c^2 d^2 e^2+10 b^3 c d e^3-35 b^4 e^4\right )}{4 b^4 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) (d+e x)^{3/2} \left (b x+c x^2\right )^2}+\frac {b (c d-b e) \left (12 c^2 d^2-3 b c d e-7 b^2 e^2\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-7 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 (d+e x)^{3/2} \left (b x+c x^2\right )}-\frac {\left (48 c^2 d^2+60 b c d e+35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}}+\frac {c^{9/2} \left (48 c^2 d^2-156 b c d e+143 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.33, size = 301, normalized size = 0.64 \begin {gather*} \frac {6 b^4 d^2 (b e-c d)^3+3 b^3 d x (c d-b e)^3 (7 b e+8 c d)-x^2 \left (3 b^2 c d \left (7 b^2 e^2+3 b c d e-12 c^2 d^2\right ) (c d-b e)^2+(b+c x) \left (3 b c d (b e-c d) \left (7 b^3 e^3-2 b^2 c d e^2-36 b c^2 d^2 e+24 c^3 d^3\right )-(b+c x) \left ((c d-b e)^3 \left (35 b^2 e^2+60 b c d e+48 c^2 d^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {e x}{d}+1\right )-c^3 d^3 \left (143 b^2 e^2-156 b c d e+48 c^2 d^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c (d+e x)}{c d-b e}\right )\right )\right )\right )}{12 b^5 d^3 x^2 (b+c x)^2 (d+e x)^{3/2} (c d-b e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(6*b^4*d^2*(-(c*d) + b*e)^3 + 3*b^3*d*(c*d - b*e)^3*(8*c*d + 7*b*e)*x - x^2*(3*b^2*c*d*(c*d - b*e)^2*(-12*c^2*
d^2 + 3*b*c*d*e + 7*b^2*e^2) + (b + c*x)*(3*b*c*d*(-(c*d) + b*e)*(24*c^3*d^3 - 36*b*c^2*d^2*e - 2*b^2*c*d*e^2
+ 7*b^3*e^3) - (b + c*x)*(-(c^3*d^3*(48*c^2*d^2 - 156*b*c*d*e + 143*b^2*e^2)*Hypergeometric2F1[-3/2, 1, -1/2,
(c*(d + e*x))/(c*d - b*e)]) + (c*d - b*e)^3*(48*c^2*d^2 + 60*b*c*d*e + 35*b^2*e^2)*Hypergeometric2F1[-3/2, 1,
-1/2, 1 + (e*x)/d]))))/(12*b^5*d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*(d + e*x)^(3/2))

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IntegrateAlgebraic [A]  time = 1.51, size = 858, normalized size = 1.83 \begin {gather*} \frac {-72 c^7 (d+e x)^2 d^8+216 c^7 (d+e x)^3 d^7+288 b c^6 e (d+e x)^2 d^7-8 b^4 c^3 e^4 d^6-216 c^7 (d+e x)^4 d^6-756 b c^6 e (d+e x)^3 d^6-381 b^2 c^5 e^2 (d+e x)^2 d^6+24 b^5 c^2 e^5 d^5+72 c^7 (d+e x)^5 d^5+648 b c^6 e (d+e x)^4 d^5+822 b^2 c^5 e^2 (d+e x)^3 d^5+135 b^3 c^4 e^3 (d+e x)^2 d^5-112 b^4 c^3 e^4 (d+e x) d^5-24 b^6 c e^6 d^4-180 b c^6 e (d+e x)^5 d^4-525 b^2 c^5 e^2 (d+e x)^4 d^4-165 b^3 c^4 e^3 (d+e x)^3 d^4+768 b^4 c^3 e^4 (d+e x)^2 d^4+280 b^5 c^2 e^5 (d+e x) d^4+8 b^7 e^7 d^3+84 b^2 c^5 e^2 (d+e x)^5 d^3-30 b^3 c^4 e^3 (d+e x)^4 d^3-1372 b^4 c^3 e^4 (d+e x)^3 d^3-1425 b^5 c^2 e^5 (d+e x)^2 d^3-224 b^6 c e^6 (d+e x) d^3+54 b^3 c^4 e^3 (d+e x)^5 d^2+988 b^4 c^3 e^4 (d+e x)^4 d^2+1845 b^5 c^2 e^5 (d+e x)^3 d^2+862 b^6 c e^6 (d+e x)^2 d^2+56 b^7 e^7 (d+e x) d^2-240 b^4 c^3 e^4 (d+e x)^5 d-865 b^5 c^2 e^5 (d+e x)^4 d-800 b^6 c e^6 (d+e x)^3 d-175 b^7 e^7 (d+e x)^2 d+105 b^5 c^2 e^5 (d+e x)^5+210 b^6 c e^6 (d+e x)^4+105 b^7 e^7 (d+e x)^3}{12 b^4 d^4 e (b e-c d)^4 x^2 (d+e x)^{3/2} (-c d+b e+c (d+e x))^2}+\frac {\left (48 d^2 c^{13/2}-156 b d e c^{11/2}+143 b^2 e^2 c^{9/2}\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {b e-c d} \sqrt {d+e x}}{c d-b e}\right )}{4 b^5 (b e-c d)^{9/2}}+\frac {\left (-48 c^2 d^2-60 b c e d-35 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^(5/2)*(b*x + c*x^2)^3),x]

[Out]

(-8*b^4*c^3*d^6*e^4 + 24*b^5*c^2*d^5*e^5 - 24*b^6*c*d^4*e^6 + 8*b^7*d^3*e^7 - 112*b^4*c^3*d^5*e^4*(d + e*x) +
280*b^5*c^2*d^4*e^5*(d + e*x) - 224*b^6*c*d^3*e^6*(d + e*x) + 56*b^7*d^2*e^7*(d + e*x) - 72*c^7*d^8*(d + e*x)^
2 + 288*b*c^6*d^7*e*(d + e*x)^2 - 381*b^2*c^5*d^6*e^2*(d + e*x)^2 + 135*b^3*c^4*d^5*e^3*(d + e*x)^2 + 768*b^4*
c^3*d^4*e^4*(d + e*x)^2 - 1425*b^5*c^2*d^3*e^5*(d + e*x)^2 + 862*b^6*c*d^2*e^6*(d + e*x)^2 - 175*b^7*d*e^7*(d
+ e*x)^2 + 216*c^7*d^7*(d + e*x)^3 - 756*b*c^6*d^6*e*(d + e*x)^3 + 822*b^2*c^5*d^5*e^2*(d + e*x)^3 - 165*b^3*c
^4*d^4*e^3*(d + e*x)^3 - 1372*b^4*c^3*d^3*e^4*(d + e*x)^3 + 1845*b^5*c^2*d^2*e^5*(d + e*x)^3 - 800*b^6*c*d*e^6
*(d + e*x)^3 + 105*b^7*e^7*(d + e*x)^3 - 216*c^7*d^6*(d + e*x)^4 + 648*b*c^6*d^5*e*(d + e*x)^4 - 525*b^2*c^5*d
^4*e^2*(d + e*x)^4 - 30*b^3*c^4*d^3*e^3*(d + e*x)^4 + 988*b^4*c^3*d^2*e^4*(d + e*x)^4 - 865*b^5*c^2*d*e^5*(d +
e*x)^4 + 210*b^6*c*e^6*(d + e*x)^4 + 72*c^7*d^5*(d + e*x)^5 - 180*b*c^6*d^4*e*(d + e*x)^5 + 84*b^2*c^5*d^3*e^
2*(d + e*x)^5 + 54*b^3*c^4*d^2*e^3*(d + e*x)^5 - 240*b^4*c^3*d*e^4*(d + e*x)^5 + 105*b^5*c^2*e^5*(d + e*x)^5)/
(12*b^4*d^4*e*(-(c*d) + b*e)^4*x^2*(d + e*x)^(3/2)*(-(c*d) + b*e + c*(d + e*x))^2) + ((48*c^(13/2)*d^2 - 156*b
*c^(11/2)*d*e + 143*b^2*c^(9/2)*e^2)*ArcTan[(Sqrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(4*b^5*(-
(c*d) + b*e)^(9/2)) + ((-48*c^2*d^2 - 60*b*c*d*e - 35*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(9/2))

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fricas [B]  time = 13.81, size = 6969, normalized size = 14.83

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

[1/24*(3*((48*c^8*d^7*e^2 - 156*b*c^7*d^6*e^3 + 143*b^2*c^6*d^5*e^4)*x^6 + 2*(48*c^8*d^8*e - 108*b*c^7*d^7*e^2
- 13*b^2*c^6*d^6*e^3 + 143*b^3*c^5*d^5*e^4)*x^5 + (48*c^8*d^9 + 36*b*c^7*d^8*e - 433*b^2*c^6*d^7*e^2 + 416*b^
3*c^5*d^6*e^3 + 143*b^4*c^4*d^5*e^4)*x^4 + 2*(48*b*c^7*d^9 - 108*b^2*c^6*d^8*e - 13*b^3*c^5*d^7*e^2 + 143*b^4*
c^4*d^6*e^3)*x^3 + (48*b^2*c^6*d^9 - 156*b^3*c^5*d^8*e + 143*b^4*c^4*d^7*e^2)*x^2)*sqrt(c/(c*d - b*e))*log((c*
e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 3*((48*c^8*d^6*e^2 - 132*b*c
^7*d^5*e^3 + 83*b^2*c^6*d^4*e^4 + 28*b^3*c^5*d^3*e^5 + 18*b^4*c^4*d^2*e^6 - 80*b^5*c^3*d*e^7 + 35*b^6*c^2*e^8)
*x^6 + 2*(48*c^8*d^7*e - 84*b*c^7*d^6*e^2 - 49*b^2*c^6*d^5*e^3 + 111*b^3*c^5*d^4*e^4 + 46*b^4*c^4*d^3*e^5 - 62
*b^5*c^3*d^2*e^6 - 45*b^6*c^2*d*e^7 + 35*b^7*c*e^8)*x^5 + (48*c^8*d^8 + 60*b*c^7*d^7*e - 397*b^2*c^6*d^6*e^2 +
228*b^3*c^5*d^5*e^3 + 213*b^4*c^4*d^4*e^4 + 20*b^5*c^3*d^3*e^5 - 267*b^6*c^2*d^2*e^6 + 60*b^7*c*d*e^7 + 35*b^
8*e^8)*x^4 + 2*(48*b*c^7*d^8 - 84*b^2*c^6*d^7*e - 49*b^3*c^5*d^6*e^2 + 111*b^4*c^4*d^5*e^3 + 46*b^5*c^3*d^4*e^
4 - 62*b^6*c^2*d^3*e^5 - 45*b^7*c*d^2*e^6 + 35*b^8*d*e^7)*x^3 + (48*b^2*c^6*d^8 - 132*b^3*c^5*d^7*e + 83*b^4*c
^4*d^6*e^2 + 28*b^5*c^3*d^5*e^3 + 18*b^6*c^2*d^4*e^4 - 80*b^7*c*d^3*e^5 + 35*b^8*d^2*e^6)*x^2)*sqrt(d)*log((e*
x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(6*b^4*c^4*d^8 - 24*b^5*c^3*d^7*e + 36*b^6*c^2*d^6*e^2 - 24*b^7*c*d^
5*e^3 + 6*b^8*d^4*e^4 - 3*(24*b*c^7*d^6*e^2 - 60*b^2*c^6*d^5*e^3 + 28*b^3*c^5*d^4*e^4 + 18*b^4*c^4*d^3*e^5 - 8
0*b^5*c^3*d^2*e^6 + 35*b^6*c^2*d*e^7)*x^5 - (144*b*c^7*d^7*e - 252*b^2*c^6*d^6*e^2 - 105*b^3*c^5*d^5*e^3 + 240
*b^4*c^4*d^4*e^4 - 212*b^5*c^3*d^3*e^5 - 340*b^6*c^2*d^2*e^6 + 210*b^7*c*d*e^7)*x^4 - (72*b*c^7*d^8 + 36*b^2*c
^6*d^7*e - 438*b^3*c^5*d^6*e^2 + 255*b^4*c^4*d^5*e^3 + 180*b^5*c^3*d^4*e^4 - 565*b^6*c^2*d^3*e^5 + 40*b^7*c*d^
2*e^6 + 105*b^8*d*e^7)*x^3 - (108*b^2*c^6*d^8 - 225*b^3*c^5*d^7*e + 180*b^5*c^3*d^5*e^3 - 30*b^6*c^2*d^4*e^4 -
278*b^7*c*d^3*e^5 + 140*b^8*d^2*e^6)*x^2 - 3*(8*b^3*c^5*d^8 - 25*b^4*c^4*d^7*e + 20*b^5*c^3*d^6*e^2 + 10*b^6*
c^2*d^5*e^3 - 20*b^7*c*d^4*e^4 + 7*b^8*d^3*e^5)*x)*sqrt(e*x + d))/((b^5*c^6*d^9*e^2 - 4*b^6*c^5*d^8*e^3 + 6*b^
7*c^4*d^7*e^4 - 4*b^8*c^3*d^6*e^5 + b^9*c^2*d^5*e^6)*x^6 + 2*(b^5*c^6*d^10*e - 3*b^6*c^5*d^9*e^2 + 2*b^7*c^4*d
^8*e^3 + 2*b^8*c^3*d^7*e^4 - 3*b^9*c^2*d^6*e^5 + b^10*c*d^5*e^6)*x^5 + (b^5*c^6*d^11 - 9*b^7*c^4*d^9*e^2 + 16*
b^8*c^3*d^8*e^3 - 9*b^9*c^2*d^7*e^4 + b^11*d^5*e^6)*x^4 + 2*(b^6*c^5*d^11 - 3*b^7*c^4*d^10*e + 2*b^8*c^3*d^9*e
^2 + 2*b^9*c^2*d^8*e^3 - 3*b^10*c*d^7*e^4 + b^11*d^6*e^5)*x^3 + (b^7*c^4*d^11 - 4*b^8*c^3*d^10*e + 6*b^9*c^2*d
^9*e^2 - 4*b^10*c*d^8*e^3 + b^11*d^7*e^4)*x^2), 1/24*(6*((48*c^8*d^7*e^2 - 156*b*c^7*d^6*e^3 + 143*b^2*c^6*d^5
*e^4)*x^6 + 2*(48*c^8*d^8*e - 108*b*c^7*d^7*e^2 - 13*b^2*c^6*d^6*e^3 + 143*b^3*c^5*d^5*e^4)*x^5 + (48*c^8*d^9
+ 36*b*c^7*d^8*e - 433*b^2*c^6*d^7*e^2 + 416*b^3*c^5*d^6*e^3 + 143*b^4*c^4*d^5*e^4)*x^4 + 2*(48*b*c^7*d^9 - 10
8*b^2*c^6*d^8*e - 13*b^3*c^5*d^7*e^2 + 143*b^4*c^4*d^6*e^3)*x^3 + (48*b^2*c^6*d^9 - 156*b^3*c^5*d^8*e + 143*b^
4*c^4*d^7*e^2)*x^2)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d))
+ 3*((48*c^8*d^6*e^2 - 132*b*c^7*d^5*e^3 + 83*b^2*c^6*d^4*e^4 + 28*b^3*c^5*d^3*e^5 + 18*b^4*c^4*d^2*e^6 - 80*
b^5*c^3*d*e^7 + 35*b^6*c^2*e^8)*x^6 + 2*(48*c^8*d^7*e - 84*b*c^7*d^6*e^2 - 49*b^2*c^6*d^5*e^3 + 111*b^3*c^5*d^
4*e^4 + 46*b^4*c^4*d^3*e^5 - 62*b^5*c^3*d^2*e^6 - 45*b^6*c^2*d*e^7 + 35*b^7*c*e^8)*x^5 + (48*c^8*d^8 + 60*b*c^
7*d^7*e - 397*b^2*c^6*d^6*e^2 + 228*b^3*c^5*d^5*e^3 + 213*b^4*c^4*d^4*e^4 + 20*b^5*c^3*d^3*e^5 - 267*b^6*c^2*d
^2*e^6 + 60*b^7*c*d*e^7 + 35*b^8*e^8)*x^4 + 2*(48*b*c^7*d^8 - 84*b^2*c^6*d^7*e - 49*b^3*c^5*d^6*e^2 + 111*b^4*
c^4*d^5*e^3 + 46*b^5*c^3*d^4*e^4 - 62*b^6*c^2*d^3*e^5 - 45*b^7*c*d^2*e^6 + 35*b^8*d*e^7)*x^3 + (48*b^2*c^6*d^8
- 132*b^3*c^5*d^7*e + 83*b^4*c^4*d^6*e^2 + 28*b^5*c^3*d^5*e^3 + 18*b^6*c^2*d^4*e^4 - 80*b^7*c*d^3*e^5 + 35*b^
8*d^2*e^6)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(6*b^4*c^4*d^8 - 24*b^5*c^3*d^7*e + 3
6*b^6*c^2*d^6*e^2 - 24*b^7*c*d^5*e^3 + 6*b^8*d^4*e^4 - 3*(24*b*c^7*d^6*e^2 - 60*b^2*c^6*d^5*e^3 + 28*b^3*c^5*d
^4*e^4 + 18*b^4*c^4*d^3*e^5 - 80*b^5*c^3*d^2*e^6 + 35*b^6*c^2*d*e^7)*x^5 - (144*b*c^7*d^7*e - 252*b^2*c^6*d^6*
e^2 - 105*b^3*c^5*d^5*e^3 + 240*b^4*c^4*d^4*e^4 - 212*b^5*c^3*d^3*e^5 - 340*b^6*c^2*d^2*e^6 + 210*b^7*c*d*e^7)
*x^4 - (72*b*c^7*d^8 + 36*b^2*c^6*d^7*e - 438*b^3*c^5*d^6*e^2 + 255*b^4*c^4*d^5*e^3 + 180*b^5*c^3*d^4*e^4 - 56
5*b^6*c^2*d^3*e^5 + 40*b^7*c*d^2*e^6 + 105*b^8*d*e^7)*x^3 - (108*b^2*c^6*d^8 - 225*b^3*c^5*d^7*e + 180*b^5*c^3
*d^5*e^3 - 30*b^6*c^2*d^4*e^4 - 278*b^7*c*d^3*e^5 + 140*b^8*d^2*e^6)*x^2 - 3*(8*b^3*c^5*d^8 - 25*b^4*c^4*d^7*e
+ 20*b^5*c^3*d^6*e^2 + 10*b^6*c^2*d^5*e^3 - 20*b^7*c*d^4*e^4 + 7*b^8*d^3*e^5)*x)*sqrt(e*x + d))/((b^5*c^6*d^9
*e^2 - 4*b^6*c^5*d^8*e^3 + 6*b^7*c^4*d^7*e^4 - 4*b^8*c^3*d^6*e^5 + b^9*c^2*d^5*e^6)*x^6 + 2*(b^5*c^6*d^10*e -
3*b^6*c^5*d^9*e^2 + 2*b^7*c^4*d^8*e^3 + 2*b^8*c^3*d^7*e^4 - 3*b^9*c^2*d^6*e^5 + b^10*c*d^5*e^6)*x^5 + (b^5*c^6
*d^11 - 9*b^7*c^4*d^9*e^2 + 16*b^8*c^3*d^8*e^3 - 9*b^9*c^2*d^7*e^4 + b^11*d^5*e^6)*x^4 + 2*(b^6*c^5*d^11 - 3*b
^7*c^4*d^10*e + 2*b^8*c^3*d^9*e^2 + 2*b^9*c^2*d^8*e^3 - 3*b^10*c*d^7*e^4 + b^11*d^6*e^5)*x^3 + (b^7*c^4*d^11 -
4*b^8*c^3*d^10*e + 6*b^9*c^2*d^9*e^2 - 4*b^10*c*d^8*e^3 + b^11*d^7*e^4)*x^2), 1/24*(6*((48*c^8*d^6*e^2 - 132*
b*c^7*d^5*e^3 + 83*b^2*c^6*d^4*e^4 + 28*b^3*c^5*d^3*e^5 + 18*b^4*c^4*d^2*e^6 - 80*b^5*c^3*d*e^7 + 35*b^6*c^2*e
^8)*x^6 + 2*(48*c^8*d^7*e - 84*b*c^7*d^6*e^2 - 49*b^2*c^6*d^5*e^3 + 111*b^3*c^5*d^4*e^4 + 46*b^4*c^4*d^3*e^5 -
62*b^5*c^3*d^2*e^6 - 45*b^6*c^2*d*e^7 + 35*b^7*c*e^8)*x^5 + (48*c^8*d^8 + 60*b*c^7*d^7*e - 397*b^2*c^6*d^6*e^
2 + 228*b^3*c^5*d^5*e^3 + 213*b^4*c^4*d^4*e^4 + 20*b^5*c^3*d^3*e^5 - 267*b^6*c^2*d^2*e^6 + 60*b^7*c*d*e^7 + 35
*b^8*e^8)*x^4 + 2*(48*b*c^7*d^8 - 84*b^2*c^6*d^7*e - 49*b^3*c^5*d^6*e^2 + 111*b^4*c^4*d^5*e^3 + 46*b^5*c^3*d^4
*e^4 - 62*b^6*c^2*d^3*e^5 - 45*b^7*c*d^2*e^6 + 35*b^8*d*e^7)*x^3 + (48*b^2*c^6*d^8 - 132*b^3*c^5*d^7*e + 83*b^
4*c^4*d^6*e^2 + 28*b^5*c^3*d^5*e^3 + 18*b^6*c^2*d^4*e^4 - 80*b^7*c*d^3*e^5 + 35*b^8*d^2*e^6)*x^2)*sqrt(-d)*arc
tan(sqrt(e*x + d)*sqrt(-d)/d) + 3*((48*c^8*d^7*e^2 - 156*b*c^7*d^6*e^3 + 143*b^2*c^6*d^5*e^4)*x^6 + 2*(48*c^8*
d^8*e - 108*b*c^7*d^7*e^2 - 13*b^2*c^6*d^6*e^3 + 143*b^3*c^5*d^5*e^4)*x^5 + (48*c^8*d^9 + 36*b*c^7*d^8*e - 433
*b^2*c^6*d^7*e^2 + 416*b^3*c^5*d^6*e^3 + 143*b^4*c^4*d^5*e^4)*x^4 + 2*(48*b*c^7*d^9 - 108*b^2*c^6*d^8*e - 13*b
^3*c^5*d^7*e^2 + 143*b^4*c^4*d^6*e^3)*x^3 + (48*b^2*c^6*d^9 - 156*b^3*c^5*d^8*e + 143*b^4*c^4*d^7*e^2)*x^2)*sq
rt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 2*(
6*b^4*c^4*d^8 - 24*b^5*c^3*d^7*e + 36*b^6*c^2*d^6*e^2 - 24*b^7*c*d^5*e^3 + 6*b^8*d^4*e^4 - 3*(24*b*c^7*d^6*e^2
- 60*b^2*c^6*d^5*e^3 + 28*b^3*c^5*d^4*e^4 + 18*b^4*c^4*d^3*e^5 - 80*b^5*c^3*d^2*e^6 + 35*b^6*c^2*d*e^7)*x^5 -
(144*b*c^7*d^7*e - 252*b^2*c^6*d^6*e^2 - 105*b^3*c^5*d^5*e^3 + 240*b^4*c^4*d^4*e^4 - 212*b^5*c^3*d^3*e^5 - 34
0*b^6*c^2*d^2*e^6 + 210*b^7*c*d*e^7)*x^4 - (72*b*c^7*d^8 + 36*b^2*c^6*d^7*e - 438*b^3*c^5*d^6*e^2 + 255*b^4*c^
4*d^5*e^3 + 180*b^5*c^3*d^4*e^4 - 565*b^6*c^2*d^3*e^5 + 40*b^7*c*d^2*e^6 + 105*b^8*d*e^7)*x^3 - (108*b^2*c^6*d
^8 - 225*b^3*c^5*d^7*e + 180*b^5*c^3*d^5*e^3 - 30*b^6*c^2*d^4*e^4 - 278*b^7*c*d^3*e^5 + 140*b^8*d^2*e^6)*x^2 -
3*(8*b^3*c^5*d^8 - 25*b^4*c^4*d^7*e + 20*b^5*c^3*d^6*e^2 + 10*b^6*c^2*d^5*e^3 - 20*b^7*c*d^4*e^4 + 7*b^8*d^3*
e^5)*x)*sqrt(e*x + d))/((b^5*c^6*d^9*e^2 - 4*b^6*c^5*d^8*e^3 + 6*b^7*c^4*d^7*e^4 - 4*b^8*c^3*d^6*e^5 + b^9*c^2
*d^5*e^6)*x^6 + 2*(b^5*c^6*d^10*e - 3*b^6*c^5*d^9*e^2 + 2*b^7*c^4*d^8*e^3 + 2*b^8*c^3*d^7*e^4 - 3*b^9*c^2*d^6*
e^5 + b^10*c*d^5*e^6)*x^5 + (b^5*c^6*d^11 - 9*b^7*c^4*d^9*e^2 + 16*b^8*c^3*d^8*e^3 - 9*b^9*c^2*d^7*e^4 + b^11*
d^5*e^6)*x^4 + 2*(b^6*c^5*d^11 - 3*b^7*c^4*d^10*e + 2*b^8*c^3*d^9*e^2 + 2*b^9*c^2*d^8*e^3 - 3*b^10*c*d^7*e^4 +
b^11*d^6*e^5)*x^3 + (b^7*c^4*d^11 - 4*b^8*c^3*d^10*e + 6*b^9*c^2*d^9*e^2 - 4*b^10*c*d^8*e^3 + b^11*d^7*e^4)*x
^2), 1/12*(3*((48*c^8*d^7*e^2 - 156*b*c^7*d^6*e^3 + 143*b^2*c^6*d^5*e^4)*x^6 + 2*(48*c^8*d^8*e - 108*b*c^7*d^7
*e^2 - 13*b^2*c^6*d^6*e^3 + 143*b^3*c^5*d^5*e^4)*x^5 + (48*c^8*d^9 + 36*b*c^7*d^8*e - 433*b^2*c^6*d^7*e^2 + 41
6*b^3*c^5*d^6*e^3 + 143*b^4*c^4*d^5*e^4)*x^4 + 2*(48*b*c^7*d^9 - 108*b^2*c^6*d^8*e - 13*b^3*c^5*d^7*e^2 + 143*
b^4*c^4*d^6*e^3)*x^3 + (48*b^2*c^6*d^9 - 156*b^3*c^5*d^8*e + 143*b^4*c^4*d^7*e^2)*x^2)*sqrt(-c/(c*d - b*e))*ar
ctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e))/(c*e*x + c*d)) + 3*((48*c^8*d^6*e^2 - 132*b*c^7*d^5*e^3 +
83*b^2*c^6*d^4*e^4 + 28*b^3*c^5*d^3*e^5 + 18*b^4*c^4*d^2*e^6 - 80*b^5*c^3*d*e^7 + 35*b^6*c^2*e^8)*x^6 + 2*(48
*c^8*d^7*e - 84*b*c^7*d^6*e^2 - 49*b^2*c^6*d^5*e^3 + 111*b^3*c^5*d^4*e^4 + 46*b^4*c^4*d^3*e^5 - 62*b^5*c^3*d^2
*e^6 - 45*b^6*c^2*d*e^7 + 35*b^7*c*e^8)*x^5 + (48*c^8*d^8 + 60*b*c^7*d^7*e - 397*b^2*c^6*d^6*e^2 + 228*b^3*c^5
*d^5*e^3 + 213*b^4*c^4*d^4*e^4 + 20*b^5*c^3*d^3*e^5 - 267*b^6*c^2*d^2*e^6 + 60*b^7*c*d*e^7 + 35*b^8*e^8)*x^4 +
2*(48*b*c^7*d^8 - 84*b^2*c^6*d^7*e - 49*b^3*c^5*d^6*e^2 + 111*b^4*c^4*d^5*e^3 + 46*b^5*c^3*d^4*e^4 - 62*b^6*c
^2*d^3*e^5 - 45*b^7*c*d^2*e^6 + 35*b^8*d*e^7)*x^3 + (48*b^2*c^6*d^8 - 132*b^3*c^5*d^7*e + 83*b^4*c^4*d^6*e^2 +
28*b^5*c^3*d^5*e^3 + 18*b^6*c^2*d^4*e^4 - 80*b^7*c*d^3*e^5 + 35*b^8*d^2*e^6)*x^2)*sqrt(-d)*arctan(sqrt(e*x +
d)*sqrt(-d)/d) - (6*b^4*c^4*d^8 - 24*b^5*c^3*d^7*e + 36*b^6*c^2*d^6*e^2 - 24*b^7*c*d^5*e^3 + 6*b^8*d^4*e^4 - 3
*(24*b*c^7*d^6*e^2 - 60*b^2*c^6*d^5*e^3 + 28*b^3*c^5*d^4*e^4 + 18*b^4*c^4*d^3*e^5 - 80*b^5*c^3*d^2*e^6 + 35*b^
6*c^2*d*e^7)*x^5 - (144*b*c^7*d^7*e - 252*b^2*c^6*d^6*e^2 - 105*b^3*c^5*d^5*e^3 + 240*b^4*c^4*d^4*e^4 - 212*b^
5*c^3*d^3*e^5 - 340*b^6*c^2*d^2*e^6 + 210*b^7*c*d*e^7)*x^4 - (72*b*c^7*d^8 + 36*b^2*c^6*d^7*e - 438*b^3*c^5*d^
6*e^2 + 255*b^4*c^4*d^5*e^3 + 180*b^5*c^3*d^4*e^4 - 565*b^6*c^2*d^3*e^5 + 40*b^7*c*d^2*e^6 + 105*b^8*d*e^7)*x^
3 - (108*b^2*c^6*d^8 - 225*b^3*c^5*d^7*e + 180*b^5*c^3*d^5*e^3 - 30*b^6*c^2*d^4*e^4 - 278*b^7*c*d^3*e^5 + 140*
b^8*d^2*e^6)*x^2 - 3*(8*b^3*c^5*d^8 - 25*b^4*c^4*d^7*e + 20*b^5*c^3*d^6*e^2 + 10*b^6*c^2*d^5*e^3 - 20*b^7*c*d^
4*e^4 + 7*b^8*d^3*e^5)*x)*sqrt(e*x + d))/((b^5*c^6*d^9*e^2 - 4*b^6*c^5*d^8*e^3 + 6*b^7*c^4*d^7*e^4 - 4*b^8*c^3
*d^6*e^5 + b^9*c^2*d^5*e^6)*x^6 + 2*(b^5*c^6*d^10*e - 3*b^6*c^5*d^9*e^2 + 2*b^7*c^4*d^8*e^3 + 2*b^8*c^3*d^7*e^
4 - 3*b^9*c^2*d^6*e^5 + b^10*c*d^5*e^6)*x^5 + (b^5*c^6*d^11 - 9*b^7*c^4*d^9*e^2 + 16*b^8*c^3*d^8*e^3 - 9*b^9*c
^2*d^7*e^4 + b^11*d^5*e^6)*x^4 + 2*(b^6*c^5*d^11 - 3*b^7*c^4*d^10*e + 2*b^8*c^3*d^9*e^2 + 2*b^9*c^2*d^8*e^3 -
3*b^10*c*d^7*e^4 + b^11*d^6*e^5)*x^3 + (b^7*c^4*d^11 - 4*b^8*c^3*d^10*e + 6*b^9*c^2*d^9*e^2 - 4*b^10*c*d^8*e^3
+ b^11*d^7*e^4)*x^2)]

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giac [B]  time = 0.73, size = 942, normalized size = 2.00 \begin {gather*} -\frac {{\left (48 \, c^{7} d^{2} - 156 \, b c^{6} d e + 143 \, b^{2} c^{5} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c^{4} d^{4} - 4 \, b^{6} c^{3} d^{3} e + 6 \, b^{7} c^{2} d^{2} e^{2} - 4 \, b^{8} c d e^{3} + b^{9} e^{4}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (18 \, {\left (x e + d\right )} c d e^{5} + c d^{2} e^{5} - 9 \, {\left (x e + d\right )} b e^{6} - b d e^{6}\right )}}{3 \, {\left (c^{4} d^{8} - 4 \, b c^{3} d^{7} e + 6 \, b^{2} c^{2} d^{6} e^{2} - 4 \, b^{3} c d^{5} e^{3} + b^{4} d^{4} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{7} d^{5} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{7} d^{6} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{7} d^{7} e - 24 \, \sqrt {x e + d} c^{7} d^{8} e - 60 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{6} d^{4} e^{2} + 216 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{6} d^{5} e^{2} - 252 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{6} d^{6} e^{2} + 96 \, \sqrt {x e + d} b c^{6} d^{7} e^{2} + 28 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{5} d^{3} e^{3} - 175 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{5} d^{4} e^{3} + 274 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{5} d^{5} e^{3} - 127 \, \sqrt {x e + d} b^{2} c^{5} d^{6} e^{3} + 18 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} c^{4} d^{2} e^{4} - 10 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c^{4} d^{3} e^{4} - 55 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c^{4} d^{4} e^{4} + 45 \, \sqrt {x e + d} b^{3} c^{4} d^{5} e^{4} - 32 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} c^{3} d e^{5} + 140 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} c^{3} d^{2} e^{5} - 180 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} c^{3} d^{3} e^{5} + 80 \, \sqrt {x e + d} b^{4} c^{3} d^{4} e^{5} + 11 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{5} c^{2} e^{6} - 99 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} c^{2} d e^{6} + 199 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} c^{2} d^{2} e^{6} - 123 \, \sqrt {x e + d} b^{5} c^{2} d^{3} e^{6} + 22 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{6} c e^{7} - 80 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} c d e^{7} + 66 \, \sqrt {x e + d} b^{6} c d^{2} e^{7} + 11 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{7} e^{8} - 13 \, \sqrt {x e + d} b^{7} d e^{8}}{4 \, {\left (b^{4} c^{4} d^{8} - 4 \, b^{5} c^{3} d^{7} e + 6 \, b^{6} c^{2} d^{6} e^{2} - 4 \, b^{7} c d^{5} e^{3} + b^{8} d^{4} e^{4}\right )} {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )}^{2}} + \frac {{\left (48 \, c^{2} d^{2} + 60 \, b c d e + 35 \, b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

-1/4*(48*c^7*d^2 - 156*b*c^6*d*e + 143*b^2*c^5*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^4*d^4
- 4*b^6*c^3*d^3*e + 6*b^7*c^2*d^2*e^2 - 4*b^8*c*d*e^3 + b^9*e^4)*sqrt(-c^2*d + b*c*e)) - 2/3*(18*(x*e + d)*c*
d*e^5 + c*d^2*e^5 - 9*(x*e + d)*b*e^6 - b*d*e^6)/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e
^3 + b^4*d^4*e^4)*(x*e + d)^(3/2)) + 1/4*(24*(x*e + d)^(7/2)*c^7*d^5*e - 72*(x*e + d)^(5/2)*c^7*d^6*e + 72*(x*
e + d)^(3/2)*c^7*d^7*e - 24*sqrt(x*e + d)*c^7*d^8*e - 60*(x*e + d)^(7/2)*b*c^6*d^4*e^2 + 216*(x*e + d)^(5/2)*b
*c^6*d^5*e^2 - 252*(x*e + d)^(3/2)*b*c^6*d^6*e^2 + 96*sqrt(x*e + d)*b*c^6*d^7*e^2 + 28*(x*e + d)^(7/2)*b^2*c^5
*d^3*e^3 - 175*(x*e + d)^(5/2)*b^2*c^5*d^4*e^3 + 274*(x*e + d)^(3/2)*b^2*c^5*d^5*e^3 - 127*sqrt(x*e + d)*b^2*c
^5*d^6*e^3 + 18*(x*e + d)^(7/2)*b^3*c^4*d^2*e^4 - 10*(x*e + d)^(5/2)*b^3*c^4*d^3*e^4 - 55*(x*e + d)^(3/2)*b^3*
c^4*d^4*e^4 + 45*sqrt(x*e + d)*b^3*c^4*d^5*e^4 - 32*(x*e + d)^(7/2)*b^4*c^3*d*e^5 + 140*(x*e + d)^(5/2)*b^4*c^
3*d^2*e^5 - 180*(x*e + d)^(3/2)*b^4*c^3*d^3*e^5 + 80*sqrt(x*e + d)*b^4*c^3*d^4*e^5 + 11*(x*e + d)^(7/2)*b^5*c^
2*e^6 - 99*(x*e + d)^(5/2)*b^5*c^2*d*e^6 + 199*(x*e + d)^(3/2)*b^5*c^2*d^2*e^6 - 123*sqrt(x*e + d)*b^5*c^2*d^3
*e^6 + 22*(x*e + d)^(5/2)*b^6*c*e^7 - 80*(x*e + d)^(3/2)*b^6*c*d*e^7 + 66*sqrt(x*e + d)*b^6*c*d^2*e^7 + 11*(x*
e + d)^(3/2)*b^7*e^8 - 13*sqrt(x*e + d)*b^7*d*e^8)/((b^4*c^4*d^8 - 4*b^5*c^3*d^7*e + 6*b^6*c^2*d^6*e^2 - 4*b^7
*c*d^5*e^3 + b^8*d^4*e^4)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2) + 1/4*(48*c^2*d
^2 + 60*b*c*d*e + 35*b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)*d^4)

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maple [A]  time = 0.09, size = 582, normalized size = 1.24 \begin {gather*} -\frac {25 \sqrt {e x +d}\, c^{5} e^{3}}{4 \left (b e -c d \right )^{4} \left (c e x +b e \right )^{2} b^{2}}+\frac {37 \sqrt {e x +d}\, c^{6} d \,e^{2}}{4 \left (b e -c d \right )^{4} \left (c e x +b e \right )^{2} b^{3}}-\frac {3 \sqrt {e x +d}\, c^{7} d^{2} e}{\left (b e -c d \right )^{4} \left (c e x +b e \right )^{2} b^{4}}-\frac {23 \left (e x +d \right )^{\frac {3}{2}} c^{6} e^{2}}{4 \left (b e -c d \right )^{4} \left (c e x +b e \right )^{2} b^{3}}-\frac {143 c^{5} e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 \left (b e -c d \right )^{4} \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c^{7} d e}{\left (b e -c d \right )^{4} \left (c e x +b e \right )^{2} b^{4}}+\frac {39 c^{6} d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} \sqrt {\left (b e -c d \right ) c}\, b^{4}}-\frac {12 c^{7} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{4} \sqrt {\left (b e -c d \right ) c}\, b^{5}}+\frac {6 b \,e^{6}}{\left (b e -c d \right )^{4} \sqrt {e x +d}\, d^{4}}-\frac {12 c \,e^{5}}{\left (b e -c d \right )^{4} \sqrt {e x +d}\, d^{3}}+\frac {2 e^{5}}{3 \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}} d^{3}}-\frac {35 e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} d^{\frac {9}{2}}}-\frac {15 c e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{4} d^{\frac {7}{2}}}-\frac {12 c^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{5} d^{\frac {5}{2}}}-\frac {13 \sqrt {e x +d}}{4 b^{3} d^{3} x^{2}}-\frac {3 \sqrt {e x +d}\, c}{b^{4} d^{2} e \,x^{2}}+\frac {11 \left (e x +d \right )^{\frac {3}{2}}}{4 b^{3} d^{4} x^{2}}+\frac {3 \left (e x +d \right )^{\frac {3}{2}} c}{b^{4} d^{3} e \,x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x)

[Out]

-23/4*e^2*c^6/b^3/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)+3*e*c^7/b^4/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*
d-25/4*e^3*c^5/b^2/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)+37/4*e^2*c^6/b^3/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^
(1/2)*d-3*e*c^7/b^4/(b*e-c*d)^4/(c*e*x+b*e)^2*(e*x+d)^(1/2)*d^2-143/4*e^2*c^5/b^3/(b*e-c*d)^4/((b*e-c*d)*c)^(1
/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)+39*e*c^6/b^4/(b*e-c*d)^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1
/2)/((b*e-c*d)*c)^(1/2)*c)*d-12*c^7/b^5/(b*e-c*d)^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/
2)*c)*d^2+11/4/d^4/b^3/x^2*(e*x+d)^(3/2)+3/e/d^3/b^4/x^2*(e*x+d)^(3/2)*c-13/4/d^3/b^3/x^2*(e*x+d)^(1/2)-3/e/d^
2/b^4/x^2*(e*x+d)^(1/2)*c-35/4*e^2/d^(9/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))-15*e/d^(7/2)/b^4*arctanh((e*x+d)
^(1/2)/d^(1/2))*c-12/d^(5/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*c^2+6*e^6/(b*e-c*d)^4/d^4/(e*x+d)^(1/2)*b-12*e
^5/(b*e-c*d)^4/d^3/(e*x+d)^(1/2)*c+2/3*e^5/(b*e-c*d)^3/d^3/(e*x+d)^(3/2)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 4.82, size = 11876, normalized size = 25.27

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^3*(d + e*x)^(5/2)),x)

[Out]

- ((2*e^5)/(3*(c*d^2 - b*d*e)) - (14*e^5*(b*e - 2*c*d)*(d + e*x))/(3*(c*d^2 - b*d*e)^2) - (e*(d + e*x)^5*(24*c
^7*d^5 + 35*b^5*c^2*e^5 - 80*b^4*c^3*d*e^4 + 28*b^2*c^5*d^3*e^2 + 18*b^3*c^4*d^2*e^3 - 60*b*c^6*d^4*e))/(4*b^4
*(c*d^2 - b*d*e)^4) + (e*(d + e*x)^4*(216*c^7*d^6 - 210*b^6*c*e^6 + 865*b^5*c^2*d*e^5 + 525*b^2*c^5*d^4*e^2 +
30*b^3*c^4*d^3*e^3 - 988*b^4*c^3*d^2*e^4 - 648*b*c^6*d^5*e))/(12*b^4*(c*d^2 - b*d*e)^4) + (e*(d + e*x)^2*(72*c
^6*d^6 - 175*b^6*e^6 + 165*b^2*c^4*d^4*e^2 + 30*b^3*c^3*d^3*e^3 - 738*b^4*c^2*d^2*e^4 - 216*b*c^5*d^5*e + 687*
b^5*c*d*e^5))/(12*b^4*(c*d^2 - b*d*e)^3) - (e*(d + e*x)^3*(105*b^7*e^7 + 216*c^7*d^7 + 822*b^2*c^5*d^5*e^2 - 1
65*b^3*c^4*d^4*e^3 - 1372*b^4*c^3*d^3*e^4 + 1845*b^5*c^2*d^2*e^5 - 756*b*c^6*d^6*e - 800*b^6*c*d*e^6))/(12*b^4
*(c*d^2 - b*d*e)^4))/(c^2*(d + e*x)^(11/2) - (4*c^2*d - 2*b*c*e)*(d + e*x)^(9/2) - (d + e*x)^(5/2)*(4*c^2*d^3
+ 2*b^2*d*e^2 - 6*b*c*d^2*e) + (d + e*x)^(7/2)*(b^2*e^2 + 6*c^2*d^2 - 6*b*c*d*e) + (d + e*x)^(3/2)*(c^2*d^4 +
b^2*d^2*e^2 - 2*b*c*d^3*e)) - (atan(((((d + e*x)^(1/2)*(589824*b^12*c^27*d^36*e^2 - 10616832*b^13*c^26*d^35*e^
3 + 89518080*b^14*c^25*d^34*e^4 - 468971520*b^15*c^24*d^33*e^5 + 1707439360*b^16*c^23*d^32*e^6 - 4579446784*b^
17*c^22*d^31*e^7 + 9364822016*b^18*c^21*d^30*e^8 - 14937190400*b^19*c^20*d^29*e^9 + 18936107520*b^20*c^19*d^28
*e^10 - 19535324160*b^21*c^18*d^27*e^11 + 17074641408*b^22*c^17*d^26*e^12 - 13484230656*b^23*c^16*d^25*e^13 +
10265639040*b^24*c^15*d^24*e^14 - 7643066880*b^25*c^14*d^23*e^15 + 5421597440*b^26*c^13*d^22*e^16 - 3708136960
*b^27*c^12*d^21*e^17 + 2608529792*b^28*c^11*d^20*e^18 - 1894041600*b^29*c^10*d^19*e^19 + 1274465280*b^30*c^9*d
^18*e^20 - 707773440*b^31*c^8*d^17*e^21 + 301648512*b^32*c^7*d^16*e^22 - 93688320*b^33*c^6*d^15*e^23 + 1993088
0*b^34*c^5*d^14*e^24 - 2598400*b^35*c^4*d^13*e^25 + 156800*b^36*c^3*d^12*e^26) + ((35*b^2*e^2 + 48*c^2*d^2 + 6
0*b*c*d*e)*(24576*b^18*c^24*d^38*e^3 - 466944*b^19*c^23*d^37*e^4 + 4185088*b^20*c^22*d^36*e^5 - 23500800*b^21*
c^21*d^35*e^6 + 92710912*b^22*c^20*d^34*e^7 - 273566720*b^23*c^19*d^33*e^8 + 629578752*b^24*c^18*d^32*e^9 - 11
69833984*b^25*c^17*d^31*e^10 + 1818910720*b^26*c^16*d^30*e^11 - 2465058816*b^27*c^15*d^29*e^12 + 3031169024*b^
28*c^14*d^28*e^13 - 3457871872*b^29*c^13*d^27*e^14 + 3626348544*b^30*c^12*d^26*e^15 - 3385559040*b^31*c^11*d^2
5*e^16 + 2714064896*b^32*c^10*d^24*e^17 - 1813512192*b^33*c^9*d^23*e^18 + 986251264*b^34*c^8*d^22*e^19 - 42681
5488*b^35*c^7*d^21*e^20 + 143109120*b^36*c^6*d^20*e^21 - 35796992*b^37*c^5*d^19*e^22 + 6285312*b^38*c^4*d^18*e
^23 - 691200*b^39*c^3*d^17*e^24 + 35840*b^40*c^2*d^16*e^25 - ((d + e*x)^(1/2)*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*
c*d*e)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20
*d^38*e^5 + 88719360*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 158760
9600*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^
13*d^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^1
5 + 1270087680*b^36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^3
9*c^6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*
b^43*c^2*d^20*e^23))/(8*b^5*(d^9)^(1/2))))/(8*b^5*(d^9)^(1/2)))*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*1i)/(8*
b^5*(d^9)^(1/2)) + (((d + e*x)^(1/2)*(589824*b^12*c^27*d^36*e^2 - 10616832*b^13*c^26*d^35*e^3 + 89518080*b^14*
c^25*d^34*e^4 - 468971520*b^15*c^24*d^33*e^5 + 1707439360*b^16*c^23*d^32*e^6 - 4579446784*b^17*c^22*d^31*e^7 +
9364822016*b^18*c^21*d^30*e^8 - 14937190400*b^19*c^20*d^29*e^9 + 18936107520*b^20*c^19*d^28*e^10 - 1953532416
0*b^21*c^18*d^27*e^11 + 17074641408*b^22*c^17*d^26*e^12 - 13484230656*b^23*c^16*d^25*e^13 + 10265639040*b^24*c
^15*d^24*e^14 - 7643066880*b^25*c^14*d^23*e^15 + 5421597440*b^26*c^13*d^22*e^16 - 3708136960*b^27*c^12*d^21*e^
17 + 2608529792*b^28*c^11*d^20*e^18 - 1894041600*b^29*c^10*d^19*e^19 + 1274465280*b^30*c^9*d^18*e^20 - 7077734
40*b^31*c^8*d^17*e^21 + 301648512*b^32*c^7*d^16*e^22 - 93688320*b^33*c^6*d^15*e^23 + 19930880*b^34*c^5*d^14*e^
24 - 2598400*b^35*c^4*d^13*e^25 + 156800*b^36*c^3*d^12*e^26) - ((35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*(24576*
b^18*c^24*d^38*e^3 - 466944*b^19*c^23*d^37*e^4 + 4185088*b^20*c^22*d^36*e^5 - 23500800*b^21*c^21*d^35*e^6 + 92
710912*b^22*c^20*d^34*e^7 - 273566720*b^23*c^19*d^33*e^8 + 629578752*b^24*c^18*d^32*e^9 - 1169833984*b^25*c^17
*d^31*e^10 + 1818910720*b^26*c^16*d^30*e^11 - 2465058816*b^27*c^15*d^29*e^12 + 3031169024*b^28*c^14*d^28*e^13
- 3457871872*b^29*c^13*d^27*e^14 + 3626348544*b^30*c^12*d^26*e^15 - 3385559040*b^31*c^11*d^25*e^16 + 271406489
6*b^32*c^10*d^24*e^17 - 1813512192*b^33*c^9*d^23*e^18 + 986251264*b^34*c^8*d^22*e^19 - 426815488*b^35*c^7*d^21
*e^20 + 143109120*b^36*c^6*d^20*e^21 - 35796992*b^37*c^5*d^19*e^22 + 6285312*b^38*c^4*d^18*e^23 - 691200*b^39*
c^3*d^17*e^24 + 35840*b^40*c^2*d^16*e^25 + ((d + e*x)^(1/2)*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*(16384*b^22
*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 887193
60*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^3
4*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 426
5377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^3
6*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 +
12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23
))/(8*b^5*(d^9)^(1/2))))/(8*b^5*(d^9)^(1/2)))*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*1i)/(8*b^5*(d^9)^(1/2)))/
((((d + e*x)^(1/2)*(589824*b^12*c^27*d^36*e^2 - 10616832*b^13*c^26*d^35*e^3 + 89518080*b^14*c^25*d^34*e^4 - 46
8971520*b^15*c^24*d^33*e^5 + 1707439360*b^16*c^23*d^32*e^6 - 4579446784*b^17*c^22*d^31*e^7 + 9364822016*b^18*c
^21*d^30*e^8 - 14937190400*b^19*c^20*d^29*e^9 + 18936107520*b^20*c^19*d^28*e^10 - 19535324160*b^21*c^18*d^27*e
^11 + 17074641408*b^22*c^17*d^26*e^12 - 13484230656*b^23*c^16*d^25*e^13 + 10265639040*b^24*c^15*d^24*e^14 - 76
43066880*b^25*c^14*d^23*e^15 + 5421597440*b^26*c^13*d^22*e^16 - 3708136960*b^27*c^12*d^21*e^17 + 2608529792*b^
28*c^11*d^20*e^18 - 1894041600*b^29*c^10*d^19*e^19 + 1274465280*b^30*c^9*d^18*e^20 - 707773440*b^31*c^8*d^17*e
^21 + 301648512*b^32*c^7*d^16*e^22 - 93688320*b^33*c^6*d^15*e^23 + 19930880*b^34*c^5*d^14*e^24 - 2598400*b^35*
c^4*d^13*e^25 + 156800*b^36*c^3*d^12*e^26) - ((35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*(24576*b^18*c^24*d^38*e^3
- 466944*b^19*c^23*d^37*e^4 + 4185088*b^20*c^22*d^36*e^5 - 23500800*b^21*c^21*d^35*e^6 + 92710912*b^22*c^20*d
^34*e^7 - 273566720*b^23*c^19*d^33*e^8 + 629578752*b^24*c^18*d^32*e^9 - 1169833984*b^25*c^17*d^31*e^10 + 18189
10720*b^26*c^16*d^30*e^11 - 2465058816*b^27*c^15*d^29*e^12 + 3031169024*b^28*c^14*d^28*e^13 - 3457871872*b^29*
c^13*d^27*e^14 + 3626348544*b^30*c^12*d^26*e^15 - 3385559040*b^31*c^11*d^25*e^16 + 2714064896*b^32*c^10*d^24*e
^17 - 1813512192*b^33*c^9*d^23*e^18 + 986251264*b^34*c^8*d^22*e^19 - 426815488*b^35*c^7*d^21*e^20 + 143109120*
b^36*c^6*d^20*e^21 - 35796992*b^37*c^5*d^19*e^22 + 6285312*b^38*c^4*d^18*e^23 - 691200*b^39*c^3*d^17*e^24 + 35
840*b^40*c^2*d^16*e^25 + ((d + e*x)^(1/2)*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*(16384*b^22*c^23*d^41*e^2 - 3
35872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^37*
e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 2698936320
*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^12*
d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16 -
571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c^5*
d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23))/(8*b^5*(d^9)^(1
/2))))/(8*b^5*(d^9)^(1/2)))*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e))/(8*b^5*(d^9)^(1/2)) - (((d + e*x)^(1/2)*(5
89824*b^12*c^27*d^36*e^2 - 10616832*b^13*c^26*d^35*e^3 + 89518080*b^14*c^25*d^34*e^4 - 468971520*b^15*c^24*d^3
3*e^5 + 1707439360*b^16*c^23*d^32*e^6 - 4579446784*b^17*c^22*d^31*e^7 + 9364822016*b^18*c^21*d^30*e^8 - 149371
90400*b^19*c^20*d^29*e^9 + 18936107520*b^20*c^19*d^28*e^10 - 19535324160*b^21*c^18*d^27*e^11 + 17074641408*b^2
2*c^17*d^26*e^12 - 13484230656*b^23*c^16*d^25*e^13 + 10265639040*b^24*c^15*d^24*e^14 - 7643066880*b^25*c^14*d^
23*e^15 + 5421597440*b^26*c^13*d^22*e^16 - 3708136960*b^27*c^12*d^21*e^17 + 2608529792*b^28*c^11*d^20*e^18 - 1
894041600*b^29*c^10*d^19*e^19 + 1274465280*b^30*c^9*d^18*e^20 - 707773440*b^31*c^8*d^17*e^21 + 301648512*b^32*
c^7*d^16*e^22 - 93688320*b^33*c^6*d^15*e^23 + 19930880*b^34*c^5*d^14*e^24 - 2598400*b^35*c^4*d^13*e^25 + 15680
0*b^36*c^3*d^12*e^26) + ((35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*(24576*b^18*c^24*d^38*e^3 - 466944*b^19*c^23*d
^37*e^4 + 4185088*b^20*c^22*d^36*e^5 - 23500800*b^21*c^21*d^35*e^6 + 92710912*b^22*c^20*d^34*e^7 - 273566720*b
^23*c^19*d^33*e^8 + 629578752*b^24*c^18*d^32*e^9 - 1169833984*b^25*c^17*d^31*e^10 + 1818910720*b^26*c^16*d^30*
e^11 - 2465058816*b^27*c^15*d^29*e^12 + 3031169024*b^28*c^14*d^28*e^13 - 3457871872*b^29*c^13*d^27*e^14 + 3626
348544*b^30*c^12*d^26*e^15 - 3385559040*b^31*c^11*d^25*e^16 + 2714064896*b^32*c^10*d^24*e^17 - 1813512192*b^33
*c^9*d^23*e^18 + 986251264*b^34*c^8*d^22*e^19 - 426815488*b^35*c^7*d^21*e^20 + 143109120*b^36*c^6*d^20*e^21 -
35796992*b^37*c^5*d^19*e^22 + 6285312*b^38*c^4*d^18*e^23 - 691200*b^39*c^3*d^17*e^24 + 35840*b^40*c^2*d^16*e^2
5 - ((d + e*x)^(1/2)*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*
e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^37*e^6 - 293707776*b^27*
c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^33*e^10
- 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 343982080
0*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^
26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 1884160*b
^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23))/(8*b^5*(d^9)^(1/2))))/(8*b^5*(d^9)^(
1/2)))*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e))/(8*b^5*(d^9)^(1/2)) + 1769472*b^8*c^28*d^33*e^3 - 29196288*b^9*
c^27*d^32*e^4 + 222621696*b^10*c^26*d^31*e^5 - 1037076480*b^11*c^25*d^30*e^6 + 3281114880*b^12*c^24*d^29*e^7 -
7384738176*b^13*c^23*d^28*e^8 + 11940731264*b^14*c^22*d^27*e^9 - 13391621568*b^15*c^21*d^26*e^10 + 8822378240
*b^16*c^20*d^25*e^11 + 174168800*b^17*c^19*d^24*e^12 - 7908536064*b^18*c^18*d^23*e^13 + 10270788736*b^19*c^17*
d^22*e^14 - 8868525952*b^20*c^16*d^21*e^15 + 8022944160*b^21*c^15*d^20*e^16 - 9013107840*b^22*c^14*d^19*e^17 +
9481058368*b^23*c^13*d^18*e^18 - 7612941312*b^24*c^12*d^17*e^19 + 4396193824*b^25*c^11*d^16*e^20 - 1772817920
*b^26*c^10*d^15*e^21 + 475772160*b^27*c^9*d^14*e^22 - 76585600*b^28*c^8*d^13*e^23 + 5605600*b^29*c^7*d^12*e^24
))*(35*b^2*e^2 + 48*c^2*d^2 + 60*b*c*d*e)*1i)/(4*b^5*(d^9)^(1/2)) - (atan((((-c^9*(b*e - c*d)^9)^(1/2)*((d + e
*x)^(1/2)*(589824*b^12*c^27*d^36*e^2 - 10616832*b^13*c^26*d^35*e^3 + 89518080*b^14*c^25*d^34*e^4 - 468971520*b
^15*c^24*d^33*e^5 + 1707439360*b^16*c^23*d^32*e^6 - 4579446784*b^17*c^22*d^31*e^7 + 9364822016*b^18*c^21*d^30*
e^8 - 14937190400*b^19*c^20*d^29*e^9 + 18936107520*b^20*c^19*d^28*e^10 - 19535324160*b^21*c^18*d^27*e^11 + 170
74641408*b^22*c^17*d^26*e^12 - 13484230656*b^23*c^16*d^25*e^13 + 10265639040*b^24*c^15*d^24*e^14 - 7643066880*
b^25*c^14*d^23*e^15 + 5421597440*b^26*c^13*d^22*e^16 - 3708136960*b^27*c^12*d^21*e^17 + 2608529792*b^28*c^11*d
^20*e^18 - 1894041600*b^29*c^10*d^19*e^19 + 1274465280*b^30*c^9*d^18*e^20 - 707773440*b^31*c^8*d^17*e^21 + 301
648512*b^32*c^7*d^16*e^22 - 93688320*b^33*c^6*d^15*e^23 + 19930880*b^34*c^5*d^14*e^24 - 2598400*b^35*c^4*d^13*
e^25 + 156800*b^36*c^3*d^12*e^26) + ((-c^9*(b*e - c*d)^9)^(1/2)*(143*b^2*e^2 + 48*c^2*d^2 - 156*b*c*d*e)*(2457
6*b^18*c^24*d^38*e^3 - 466944*b^19*c^23*d^37*e^4 + 4185088*b^20*c^22*d^36*e^5 - 23500800*b^21*c^21*d^35*e^6 +
92710912*b^22*c^20*d^34*e^7 - 273566720*b^23*c^19*d^33*e^8 + 629578752*b^24*c^18*d^32*e^9 - 1169833984*b^25*c^
17*d^31*e^10 + 1818910720*b^26*c^16*d^30*e^11 - 2465058816*b^27*c^15*d^29*e^12 + 3031169024*b^28*c^14*d^28*e^1
3 - 3457871872*b^29*c^13*d^27*e^14 + 3626348544*b^30*c^12*d^26*e^15 - 3385559040*b^31*c^11*d^25*e^16 + 2714064
896*b^32*c^10*d^24*e^17 - 1813512192*b^33*c^9*d^23*e^18 + 986251264*b^34*c^8*d^22*e^19 - 426815488*b^35*c^7*d^
21*e^20 + 143109120*b^36*c^6*d^20*e^21 - 35796992*b^37*c^5*d^19*e^22 + 6285312*b^38*c^4*d^18*e^23 - 691200*b^3
9*c^3*d^17*e^24 + 35840*b^40*c^2*d^16*e^25 - ((-c^9*(b*e - c*d)^9)^(1/2)*(d + e*x)^(1/2)*(143*b^2*e^2 + 48*c^2
*d^2 - 156*b*c*d*e)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234
240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35
*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 44029
70624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*
c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 -
58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21
*e^22 - 8192*b^43*c^2*d^20*e^23))/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c
^6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c
*d*e^8))))/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c^6*d^6*e^3 - 126*b^9*c^
5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c*d*e^8)))*(143*b^2*e^2
+ 48*c^2*d^2 - 156*b*c*d*e)*1i)/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c^6
*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c*d
*e^8)) + ((-c^9*(b*e - c*d)^9)^(1/2)*((d + e*x)^(1/2)*(589824*b^12*c^27*d^36*e^2 - 10616832*b^13*c^26*d^35*e^3
+ 89518080*b^14*c^25*d^34*e^4 - 468971520*b^15*c^24*d^33*e^5 + 1707439360*b^16*c^23*d^32*e^6 - 4579446784*b^1
7*c^22*d^31*e^7 + 9364822016*b^18*c^21*d^30*e^8 - 14937190400*b^19*c^20*d^29*e^9 + 18936107520*b^20*c^19*d^28*
e^10 - 19535324160*b^21*c^18*d^27*e^11 + 17074641408*b^22*c^17*d^26*e^12 - 13484230656*b^23*c^16*d^25*e^13 + 1
0265639040*b^24*c^15*d^24*e^14 - 7643066880*b^25*c^14*d^23*e^15 + 5421597440*b^26*c^13*d^22*e^16 - 3708136960*
b^27*c^12*d^21*e^17 + 2608529792*b^28*c^11*d^20*e^18 - 1894041600*b^29*c^10*d^19*e^19 + 1274465280*b^30*c^9*d^
18*e^20 - 707773440*b^31*c^8*d^17*e^21 + 301648512*b^32*c^7*d^16*e^22 - 93688320*b^33*c^6*d^15*e^23 + 19930880
*b^34*c^5*d^14*e^24 - 2598400*b^35*c^4*d^13*e^25 + 156800*b^36*c^3*d^12*e^26) - ((-c^9*(b*e - c*d)^9)^(1/2)*(1
43*b^2*e^2 + 48*c^2*d^2 - 156*b*c*d*e)*(24576*b^18*c^24*d^38*e^3 - 466944*b^19*c^23*d^37*e^4 + 4185088*b^20*c^
22*d^36*e^5 - 23500800*b^21*c^21*d^35*e^6 + 92710912*b^22*c^20*d^34*e^7 - 273566720*b^23*c^19*d^33*e^8 + 62957
8752*b^24*c^18*d^32*e^9 - 1169833984*b^25*c^17*d^31*e^10 + 1818910720*b^26*c^16*d^30*e^11 - 2465058816*b^27*c^
15*d^29*e^12 + 3031169024*b^28*c^14*d^28*e^13 - 3457871872*b^29*c^13*d^27*e^14 + 3626348544*b^30*c^12*d^26*e^1
5 - 3385559040*b^31*c^11*d^25*e^16 + 2714064896*b^32*c^10*d^24*e^17 - 1813512192*b^33*c^9*d^23*e^18 + 98625126
4*b^34*c^8*d^22*e^19 - 426815488*b^35*c^7*d^21*e^20 + 143109120*b^36*c^6*d^20*e^21 - 35796992*b^37*c^5*d^19*e^
22 + 6285312*b^38*c^4*d^18*e^23 - 691200*b^39*c^3*d^17*e^24 + 35840*b^40*c^2*d^16*e^25 + ((-c^9*(b*e - c*d)^9)
^(1/2)*(d + e*x)^(1/2)*(143*b^2*e^2 + 48*c^2*d^2 - 156*b*c*d*e)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d
^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^37*e^6 - 293707776*b
^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^33*e
^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 34398
20800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16 - 571539456*b^37*c^
8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 18841
60*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23))/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b
^6*c^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c^6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*
c^3*d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c*d*e^8))))/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*c
^7*d^7*e^2 + 84*b^8*c^6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*c
^2*d^2*e^7 - 9*b^13*c*d*e^8)))*(143*b^2*e^2 + 48*c^2*d^2 - 156*b*c*d*e)*1i)/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6
*c^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c^6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^
3*d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c*d*e^8)))/(1769472*b^8*c^28*d^33*e^3 - 29196288*b^9*c^27*d^32*e^4 +
222621696*b^10*c^26*d^31*e^5 - 1037076480*b^11*c^25*d^30*e^6 + 3281114880*b^12*c^24*d^29*e^7 - 7384738176*b^13
*c^23*d^28*e^8 + 11940731264*b^14*c^22*d^27*e^9 - 13391621568*b^15*c^21*d^26*e^10 + 8822378240*b^16*c^20*d^25*
e^11 + 174168800*b^17*c^19*d^24*e^12 - 7908536064*b^18*c^18*d^23*e^13 + 10270788736*b^19*c^17*d^22*e^14 - 8868
525952*b^20*c^16*d^21*e^15 + 8022944160*b^21*c^15*d^20*e^16 - 9013107840*b^22*c^14*d^19*e^17 + 9481058368*b^23
*c^13*d^18*e^18 - 7612941312*b^24*c^12*d^17*e^19 + 4396193824*b^25*c^11*d^16*e^20 - 1772817920*b^26*c^10*d^15*
e^21 + 475772160*b^27*c^9*d^14*e^22 - 76585600*b^28*c^8*d^13*e^23 + 5605600*b^29*c^7*d^12*e^24 - ((-c^9*(b*e -
c*d)^9)^(1/2)*((d + e*x)^(1/2)*(589824*b^12*c^27*d^36*e^2 - 10616832*b^13*c^26*d^35*e^3 + 89518080*b^14*c^25*
d^34*e^4 - 468971520*b^15*c^24*d^33*e^5 + 1707439360*b^16*c^23*d^32*e^6 - 4579446784*b^17*c^22*d^31*e^7 + 9364
822016*b^18*c^21*d^30*e^8 - 14937190400*b^19*c^20*d^29*e^9 + 18936107520*b^20*c^19*d^28*e^10 - 19535324160*b^2
1*c^18*d^27*e^11 + 17074641408*b^22*c^17*d^26*e^12 - 13484230656*b^23*c^16*d^25*e^13 + 10265639040*b^24*c^15*d
^24*e^14 - 7643066880*b^25*c^14*d^23*e^15 + 5421597440*b^26*c^13*d^22*e^16 - 3708136960*b^27*c^12*d^21*e^17 +
2608529792*b^28*c^11*d^20*e^18 - 1894041600*b^29*c^10*d^19*e^19 + 1274465280*b^30*c^9*d^18*e^20 - 707773440*b^
31*c^8*d^17*e^21 + 301648512*b^32*c^7*d^16*e^22 - 93688320*b^33*c^6*d^15*e^23 + 19930880*b^34*c^5*d^14*e^24 -
2598400*b^35*c^4*d^13*e^25 + 156800*b^36*c^3*d^12*e^26) + ((-c^9*(b*e - c*d)^9)^(1/2)*(143*b^2*e^2 + 48*c^2*d^
2 - 156*b*c*d*e)*(24576*b^18*c^24*d^38*e^3 - 466944*b^19*c^23*d^37*e^4 + 4185088*b^20*c^22*d^36*e^5 - 23500800
*b^21*c^21*d^35*e^6 + 92710912*b^22*c^20*d^34*e^7 - 273566720*b^23*c^19*d^33*e^8 + 629578752*b^24*c^18*d^32*e^
9 - 1169833984*b^25*c^17*d^31*e^10 + 1818910720*b^26*c^16*d^30*e^11 - 2465058816*b^27*c^15*d^29*e^12 + 3031169
024*b^28*c^14*d^28*e^13 - 3457871872*b^29*c^13*d^27*e^14 + 3626348544*b^30*c^12*d^26*e^15 - 3385559040*b^31*c^
11*d^25*e^16 + 2714064896*b^32*c^10*d^24*e^17 - 1813512192*b^33*c^9*d^23*e^18 + 986251264*b^34*c^8*d^22*e^19 -
426815488*b^35*c^7*d^21*e^20 + 143109120*b^36*c^6*d^20*e^21 - 35796992*b^37*c^5*d^19*e^22 + 6285312*b^38*c^4*
d^18*e^23 - 691200*b^39*c^3*d^17*e^24 + 35840*b^40*c^2*d^16*e^25 - ((-c^9*(b*e - c*d)^9)^(1/2)*(d + e*x)^(1/2)
*(143*b^2*e^2 + 48*c^2*d^2 - 156*b*c*d*e)*(16384*b^22*c^23*d^41*e^2 - 335872*b^23*c^22*d^40*e^3 + 3276800*b^24
*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^37*e^6 - 293707776*b^27*c^18*d^36*e^7 + 76
2052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 2698936320*b^30*c^15*d^33*e^10 - 3783802880*b^31*
c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^12*d^30*e^13 + 3439820800*b^34*c^11*d^29*e
^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16 - 571539456*b^37*c^8*d^26*e^17 + 20638924
8*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c^5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21
+ 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23))/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*
c^7*d^7*e^2 + 84*b^8*c^6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*
c^2*d^2*e^7 - 9*b^13*c*d*e^8))))/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c^
6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c*
d*e^8)))*(143*b^2*e^2 + 48*c^2*d^2 - 156*b*c*d*e))/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*c^7*d
^7*e^2 + 84*b^8*c^6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*c^2*d
^2*e^7 - 9*b^13*c*d*e^8)) + ((-c^9*(b*e - c*d)^9)^(1/2)*((d + e*x)^(1/2)*(589824*b^12*c^27*d^36*e^2 - 10616832
*b^13*c^26*d^35*e^3 + 89518080*b^14*c^25*d^34*e^4 - 468971520*b^15*c^24*d^33*e^5 + 1707439360*b^16*c^23*d^32*e
^6 - 4579446784*b^17*c^22*d^31*e^7 + 9364822016*b^18*c^21*d^30*e^8 - 14937190400*b^19*c^20*d^29*e^9 + 18936107
520*b^20*c^19*d^28*e^10 - 19535324160*b^21*c^18*d^27*e^11 + 17074641408*b^22*c^17*d^26*e^12 - 13484230656*b^23
*c^16*d^25*e^13 + 10265639040*b^24*c^15*d^24*e^14 - 7643066880*b^25*c^14*d^23*e^15 + 5421597440*b^26*c^13*d^22
*e^16 - 3708136960*b^27*c^12*d^21*e^17 + 2608529792*b^28*c^11*d^20*e^18 - 1894041600*b^29*c^10*d^19*e^19 + 127
4465280*b^30*c^9*d^18*e^20 - 707773440*b^31*c^8*d^17*e^21 + 301648512*b^32*c^7*d^16*e^22 - 93688320*b^33*c^6*d
^15*e^23 + 19930880*b^34*c^5*d^14*e^24 - 2598400*b^35*c^4*d^13*e^25 + 156800*b^36*c^3*d^12*e^26) - ((-c^9*(b*e
- c*d)^9)^(1/2)*(143*b^2*e^2 + 48*c^2*d^2 - 156*b*c*d*e)*(24576*b^18*c^24*d^38*e^3 - 466944*b^19*c^23*d^37*e^
4 + 4185088*b^20*c^22*d^36*e^5 - 23500800*b^21*c^21*d^35*e^6 + 92710912*b^22*c^20*d^34*e^7 - 273566720*b^23*c^
19*d^33*e^8 + 629578752*b^24*c^18*d^32*e^9 - 1169833984*b^25*c^17*d^31*e^10 + 1818910720*b^26*c^16*d^30*e^11 -
2465058816*b^27*c^15*d^29*e^12 + 3031169024*b^28*c^14*d^28*e^13 - 3457871872*b^29*c^13*d^27*e^14 + 3626348544
*b^30*c^12*d^26*e^15 - 3385559040*b^31*c^11*d^25*e^16 + 2714064896*b^32*c^10*d^24*e^17 - 1813512192*b^33*c^9*d
^23*e^18 + 986251264*b^34*c^8*d^22*e^19 - 426815488*b^35*c^7*d^21*e^20 + 143109120*b^36*c^6*d^20*e^21 - 357969
92*b^37*c^5*d^19*e^22 + 6285312*b^38*c^4*d^18*e^23 - 691200*b^39*c^3*d^17*e^24 + 35840*b^40*c^2*d^16*e^25 + ((
-c^9*(b*e - c*d)^9)^(1/2)*(d + e*x)^(1/2)*(143*b^2*e^2 + 48*c^2*d^2 - 156*b*c*d*e)*(16384*b^22*c^23*d^41*e^2 -
335872*b^23*c^22*d^40*e^3 + 3276800*b^24*c^21*d^39*e^4 - 20234240*b^25*c^20*d^38*e^5 + 88719360*b^26*c^19*d^3
7*e^6 - 293707776*b^27*c^18*d^36*e^7 + 762052608*b^28*c^17*d^35*e^8 - 1587609600*b^29*c^16*d^34*e^9 + 26989363
20*b^30*c^15*d^33*e^10 - 3783802880*b^31*c^14*d^32*e^11 + 4402970624*b^32*c^13*d^31*e^12 - 4265377792*b^33*c^1
2*d^30*e^13 + 3439820800*b^34*c^11*d^29*e^14 - 2302033920*b^35*c^10*d^28*e^15 + 1270087680*b^36*c^9*d^27*e^16
- 571539456*b^37*c^8*d^26*e^17 + 206389248*b^38*c^7*d^25*e^18 - 58368000*b^39*c^6*d^24*e^19 + 12451840*b^40*c^
5*d^23*e^20 - 1884160*b^41*c^4*d^22*e^21 + 180224*b^42*c^3*d^21*e^22 - 8192*b^43*c^2*d^20*e^23))/(8*(b^14*e^9
- b^5*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c^6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4
*d^4*e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c*d*e^8))))/(8*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6*c
^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c^6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^3*
d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c*d*e^8)))*(143*b^2*e^2 + 48*c^2*d^2 - 156*b*c*d*e))/(8*(b^14*e^9 - b^5
*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c^6*d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*
e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c*d*e^8))))*(-c^9*(b*e - c*d)^9)^(1/2)*(143*b^2*e^2 +
48*c^2*d^2 - 156*b*c*d*e)*1i)/(4*(b^14*e^9 - b^5*c^9*d^9 + 9*b^6*c^8*d^8*e - 36*b^7*c^7*d^7*e^2 + 84*b^8*c^6*
d^6*e^3 - 126*b^9*c^5*d^5*e^4 + 126*b^10*c^4*d^4*e^5 - 84*b^11*c^3*d^3*e^6 + 36*b^12*c^2*d^2*e^7 - 9*b^13*c*d*
e^8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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