3.4.84 \(\int \frac {1}{(d+e x)^{3/2} (b x+c x^2)^3} \, dx\) [384]

Optimal. Leaf size=370 \[ \frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{2 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}-\frac {3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{7/2}}+\frac {3 c^{7/2} \left (16 c^2 d^2-44 b c d e+33 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}} \]

[Out]

-3/4*(5*b^2*e^2+12*b*c*d*e+16*c^2*d^2)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^(7/2)+3/4*c^(7/2)*(33*b^2*e^2-44*b
*c*d*e+16*c^2*d^2)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^5/(-b*e+c*d)^(7/2)+3/4*e*(-b^2*e^2-b*c*d*
e+c^2*d^2)*(5*b^2*e^2-8*b*c*d*e+8*c^2*d^2)/b^4/d^3/(-b*e+c*d)^3/(e*x+d)^(1/2)+1/2*(-b*(-b*e+c*d)-c*(-b*e+2*c*d
)*x)/b^2/d/(-b*e+c*d)/(c*x^2+b*x)^2/(e*x+d)^(1/2)+1/4*(b*(5*b^3*e^3-17*b*c^2*d^2*e+12*c^3*d^3)+c*(-b*e+2*c*d)*
(-5*b^2*e^2-12*b*c*d*e+12*c^2*d^2)*x)/b^4/d^2/(-b*e+c*d)^2/(c*x^2+b*x)/(e*x+d)^(1/2)

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Rubi [A]
time = 0.41, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {754, 836, 842, 840, 1180, 214} \begin {gather*} -\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 \sqrt {d+e x} (c d-b e)}-\frac {3 \left (5 b^2 e^2+12 b c d e+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{7/2}}+\frac {3 c^{7/2} \left (33 b^2 e^2-44 b c d e+16 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)^{7/2}}+\frac {3 e \left (-b^2 e^2-b c d e+c^2 d^2\right ) \left (5 b^2 e^2-8 b c d e+8 c^2 d^2\right )}{4 b^4 d^3 \sqrt {d+e x} (c d-b e)^3}+\frac {b \left (5 b^3 e^3-17 b c^2 d^2 e+12 c^3 d^3\right )+c x (2 c d-b e) \left (-5 b^2 e^2-12 b c d e+12 c^2 d^2\right )}{4 b^4 d^2 \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*e*(c^2*d^2 - b*c*d*e - b^2*e^2)*(8*c^2*d^2 - 8*b*c*d*e + 5*b^2*e^2))/(4*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]
) - (b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(2*b^2*d*(c*d - b*e)*Sqrt[d + e*x]*(b*x + c*x^2)^2) + (b*(12*c^3*d^3 -
17*b*c^2*d^2*e + 5*b^3*e^3) + c*(2*c*d - b*e)*(12*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)*x)/(4*b^4*d^2*(c*d - b*e)
^2*Sqrt[d + e*x]*(b*x + c*x^2)) - (3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*
b^5*d^(7/2)) + (3*c^(7/2)*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*
e]])/(4*b^5*(c*d - b*e)^(7/2))

Rule 214

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

Rule 754

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 836

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 840

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 842

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 1180

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)^{3/2} \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) \sqrt {d+e x} \left (b x+c x^2\right )^2}-\frac {\int \frac {\frac {1}{2} \left (12 c^2 d^2-5 b c d e-5 b^2 e^2\right )+\frac {7}{2} c e (2 c d-b e) x}{(d+e x)^{3/2} \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) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} (c d-b e)^2 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )+\frac {3}{4} c e (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{2 b^4 d^2 (c d-b e)^2}\\ &=\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \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) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\int \frac {\frac {3}{4} (c d-b e)^3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )+\frac {3}{4} c e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 d^3 (c d-b e)^3}\\ &=\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \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) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\text {Subst}\left (\int \frac {-\frac {3}{4} c d e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )+\frac {3}{4} e (c d-b e)^3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )+\frac {3}{4} c e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\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^3 (c d-b e)^3}\\ &=\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \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) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}+\frac {\left (3 c \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right )\right ) \text {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^3}-\frac {\left (3 c^4 \left (16 c^2 d^2-44 b c d e+33 b^2 e^2\right )\right ) \text {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)^3}\\ &=\frac {3 e \left (c^2 d^2-b c d e-b^2 e^2\right ) \left (8 c^2 d^2-8 b c d e+5 b^2 e^2\right )}{4 b^4 d^3 (c d-b e)^3 \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) \sqrt {d+e x} \left (b x+c x^2\right )^2}+\frac {b \left (12 c^3 d^3-17 b c^2 d^2 e+5 b^3 e^3\right )+c (2 c d-b e) \left (12 c^2 d^2-12 b c d e-5 b^2 e^2\right ) x}{4 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \left (b x+c x^2\right )}-\frac {3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{7/2}}+\frac {3 c^{7/2} \left (16 c^2 d^2-44 b c d e+33 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)^{7/2}}\\ \end {align*}

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Mathematica [A]
time = 1.47, size = 408, normalized size = 1.10 \begin {gather*} \frac {\frac {b \left (24 c^6 d^4 x^3 (d+e x)+b^6 e^3 \left (2 d^2-5 d e x-15 e^2 x^2\right )-12 b c^5 d^3 x^2 \left (-3 d^2+d e x+4 e^2 x^2\right )+b^2 c^4 d^2 x \left (8 d^3-65 d^2 e x-58 d e^2 x^2+15 e^3 x^3\right )-b^5 c e^2 \left (6 d^3-7 d^2 e x+d e^2 x^2+30 e^3 x^3\right )+b^4 c^2 e \left (6 d^4+9 d^3 e x+23 d^2 e^2 x^2+13 d e^3 x^3-15 e^4 x^4\right )+b^3 c^3 d \left (-2 d^4-19 d^3 e x+7 d^2 e^2 x^2+33 d e^3 x^3+9 e^4 x^4\right )\right )}{d^3 (c d-b e)^3 x^2 (b+c x)^2 \sqrt {d+e x}}+\frac {3 c^{7/2} \left (16 c^2 d^2-44 b c d e+33 b^2 e^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}-\frac {3 \left (16 c^2 d^2+12 b c d e+5 b^2 e^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{7/2}}}{4 b^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((b*(24*c^6*d^4*x^3*(d + e*x) + b^6*e^3*(2*d^2 - 5*d*e*x - 15*e^2*x^2) - 12*b*c^5*d^3*x^2*(-3*d^2 + d*e*x + 4*
e^2*x^2) + b^2*c^4*d^2*x*(8*d^3 - 65*d^2*e*x - 58*d*e^2*x^2 + 15*e^3*x^3) - b^5*c*e^2*(6*d^3 - 7*d^2*e*x + d*e
^2*x^2 + 30*e^3*x^3) + b^4*c^2*e*(6*d^4 + 9*d^3*e*x + 23*d^2*e^2*x^2 + 13*d*e^3*x^3 - 15*e^4*x^4) + b^3*c^3*d*
(-2*d^4 - 19*d^3*e*x + 7*d^2*e^2*x^2 + 33*d*e^3*x^3 + 9*e^4*x^4)))/(d^3*(c*d - b*e)^3*x^2*(b + c*x)^2*Sqrt[d +
e*x]) + (3*c^(7/2)*(16*c^2*d^2 - 44*b*c*d*e + 33*b^2*e^2)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])
/(-(c*d) + b*e)^(7/2) - (3*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(7/2))/(4*b
^5)

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Maple [A]
time = 0.53, size = 293, normalized size = 0.79

method	result	size
derivativedivides	\(2 e^{5} \left (-\frac {\frac {-\frac {e b \left (7 b e +12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9}{8} b^{2} d \,e^{2}+\frac {3}{2} b c \,d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (5 b^{2} e^{2}+12 b c d e +16 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{b^{5} d^{3} e^{5}}+\frac {1}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {c^{4} \left (\frac {\left (\frac {19}{8} b^{2} e^{2} c -\frac {3}{2} d b e \,c^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {3 b e \left (7 b^{2} e^{2}-11 b c d e +4 d^{2} c^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (33 b^{2} e^{2}-44 b c d e +16 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5} \left (b e -c d \right )^{3}}\right )\)	\(293\)
default	\(2 e^{5} \left (-\frac {\frac {-\frac {e b \left (7 b e +12 c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8}+\left (\frac {9}{8} b^{2} d \,e^{2}+\frac {3}{2} b c \,d^{2} e \right ) \sqrt {e x +d}}{e^{2} x^{2}}+\frac {3 \left (5 b^{2} e^{2}+12 b c d e +16 d^{2} c^{2}\right ) \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 \sqrt {d}}}{b^{5} d^{3} e^{5}}+\frac {1}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {c^{4} \left (\frac {\left (\frac {19}{8} b^{2} e^{2} c -\frac {3}{2} d b e \,c^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\frac {3 b e \left (7 b^{2} e^{2}-11 b c d e +4 d^{2} c^{2}\right ) \sqrt {e x +d}}{8}}{\left (c \left (e x +d \right )+b e -c d \right )^{2}}+\frac {3 \left (33 b^{2} e^{2}-44 b c d e +16 d^{2} c^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{8 \sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} e^{5} \left (b e -c d \right )^{3}}\right )\)	\(293\)
risch	\(-\frac {\sqrt {e x +d}\, \left (-7 b e x -12 c d x +2 b d \right )}{4 d^{3} b^{4} x^{2}}+\frac {19 e^{2} c^{5} \left (e x +d \right )^{\frac {3}{2}}}{4 b^{3} \left (b e -c d \right )^{3} \left (c e x +b e \right )^{2}}-\frac {3 d e \,c^{6} \left (e x +d \right )^{\frac {3}{2}}}{b^{4} \left (b e -c d \right )^{3} \left (c e x +b e \right )^{2}}+\frac {21 e^{3} c^{4} \sqrt {e x +d}}{4 b^{2} \left (b e -c d \right )^{3} \left (c e x +b e \right )^{2}}-\frac {33 d \,e^{2} c^{5} \sqrt {e x +d}}{4 b^{3} \left (b e -c d \right )^{3} \left (c e x +b e \right )^{2}}+\frac {3 d^{2} e \,c^{6} \sqrt {e x +d}}{b^{4} \left (b e -c d \right )^{3} \left (c e x +b e \right )^{2}}+\frac {99 e^{2} c^{4} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{4 b^{3} \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}}-\frac {33 d e \,c^{5} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{4} \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}}+\frac {12 d^{2} c^{6} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (b e -c d \right ) c}}\right )}{b^{5} \left (b e -c d \right )^{3} \sqrt {\left (b e -c d \right ) c}}+\frac {2 e^{5}}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}-\frac {15 e^{2} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{4 b^{3} d^{\frac {7}{2}}}-\frac {9 e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c}{b^{4} d^{\frac {5}{2}}}-\frac {12 \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) c^{2}}{b^{5} d^{\frac {3}{2}}}\)	\(483\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^5*(-1/b^5/d^3/e^5*((-1/8*e*b*(7*b*e+12*c*d)*(e*x+d)^(3/2)+(9/8*b^2*d*e^2+3/2*b*c*d^2*e)*(e*x+d)^(1/2))/e^2
/x^2+3/8*(5*b^2*e^2+12*b*c*d*e+16*c^2*d^2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))+1/d^3/(b*e-c*d)^3/(e*x+d)^(
1/2)+c^4/b^5/e^5/(b*e-c*d)^3*(((19/8*b^2*e^2*c-3/2*d*b*e*c^2)*(e*x+d)^(3/2)+3/8*b*e*(7*b^2*e^2-11*b*c*d*e+4*c^
2*d^2)*(e*x+d)^(1/2))/(c*(e*x+d)+b*e-c*d)^2+3/8*(33*b^2*e^2-44*b*c*d*e+16*c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan(
c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))))

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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)^(3/2)/(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(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1237 vs. \(2 (352) = 704\).
time = 12.30, size = 4984, normalized size = 13.47 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(3*(16*c^7*d^7*x^4 + 32*b*c^6*d^7*x^3 + 16*b^2*c^5*d^7*x^2 + 33*(b^2*c^5*d^4*x^5 + 2*b^3*c^4*d^4*x^4 + b
^4*c^3*d^4*x^3)*e^3 - 11*(4*b*c^6*d^5*x^5 + 5*b^2*c^5*d^5*x^4 - 2*b^3*c^4*d^5*x^3 - 3*b^4*c^3*d^5*x^2)*e^2 + 4
*(4*c^7*d^6*x^5 - 3*b*c^6*d^6*x^4 - 18*b^2*c^5*d^6*x^3 - 11*b^3*c^4*d^6*x^2)*e)*sqrt(c/(c*d - b*e))*log((2*c*d
- 2*(c*d - b*e)*sqrt(x*e + d)*sqrt(c/(c*d - b*e)) + (c*x - b)*e)/(c*x + b)) - 3*(16*c^7*d^6*x^4 + 32*b*c^6*d^
6*x^3 + 16*b^2*c^5*d^6*x^2 - 5*(b^5*c^2*x^5 + 2*b^6*c*x^4 + b^7*x^3)*e^6 + (3*b^4*c^3*d*x^5 + b^5*c^2*d*x^4 -
7*b^6*c*d*x^3 - 5*b^7*d*x^2)*e^5 + (5*b^3*c^4*d^2*x^5 + 13*b^4*c^3*d^2*x^4 + 11*b^5*c^2*d^2*x^3 + 3*b^6*c*d^2*
x^2)*e^4 + (17*b^2*c^5*d^3*x^5 + 39*b^3*c^4*d^3*x^4 + 27*b^4*c^3*d^3*x^3 + 5*b^5*c^2*d^3*x^2)*e^3 - (36*b*c^6*
d^4*x^5 + 55*b^2*c^5*d^4*x^4 + 2*b^3*c^4*d^4*x^3 - 17*b^4*c^3*d^4*x^2)*e^2 + 4*(4*c^7*d^5*x^5 - b*c^6*d^5*x^4
- 14*b^2*c^5*d^5*x^3 - 9*b^3*c^4*d^5*x^2)*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) - 2*(24*b*c^
6*d^6*x^3 + 36*b^2*c^5*d^6*x^2 + 8*b^3*c^4*d^6*x - 2*b^4*c^3*d^6 - 15*(b^5*c^2*d*x^4 + 2*b^6*c*d*x^3 + b^7*d*x
^2)*e^5 + (9*b^4*c^3*d^2*x^4 + 13*b^5*c^2*d^2*x^3 - b^6*c*d^2*x^2 - 5*b^7*d^2*x)*e^4 + (15*b^3*c^4*d^3*x^4 + 3
3*b^4*c^3*d^3*x^3 + 23*b^5*c^2*d^3*x^2 + 7*b^6*c*d^3*x + 2*b^7*d^3)*e^3 - (48*b^2*c^5*d^4*x^4 + 58*b^3*c^4*d^4
*x^3 - 7*b^4*c^3*d^4*x^2 - 9*b^5*c^2*d^4*x + 6*b^6*c*d^4)*e^2 + (24*b*c^6*d^5*x^4 - 12*b^2*c^5*d^5*x^3 - 65*b^
3*c^4*d^5*x^2 - 19*b^4*c^3*d^5*x + 6*b^5*c^2*d^5)*e)*sqrt(x*e + d))/(b^5*c^5*d^8*x^4 + 2*b^6*c^4*d^8*x^3 + b^7
*c^3*d^8*x^2 - (b^8*c^2*d^4*x^5 + 2*b^9*c*d^4*x^4 + b^10*d^4*x^3)*e^4 + (3*b^7*c^3*d^5*x^5 + 5*b^8*c^2*d^5*x^4
+ b^9*c*d^5*x^3 - b^10*d^5*x^2)*e^3 - 3*(b^6*c^4*d^6*x^5 + b^7*c^3*d^6*x^4 - b^8*c^2*d^6*x^3 - b^9*c*d^6*x^2)
*e^2 + (b^5*c^5*d^7*x^5 - b^6*c^4*d^7*x^4 - 5*b^7*c^3*d^7*x^3 - 3*b^8*c^2*d^7*x^2)*e), 1/8*(6*(16*c^7*d^7*x^4
+ 32*b*c^6*d^7*x^3 + 16*b^2*c^5*d^7*x^2 + 33*(b^2*c^5*d^4*x^5 + 2*b^3*c^4*d^4*x^4 + b^4*c^3*d^4*x^3)*e^3 - 11*
(4*b*c^6*d^5*x^5 + 5*b^2*c^5*d^5*x^4 - 2*b^3*c^4*d^5*x^3 - 3*b^4*c^3*d^5*x^2)*e^2 + 4*(4*c^7*d^6*x^5 - 3*b*c^6
*d^6*x^4 - 18*b^2*c^5*d^6*x^3 - 11*b^3*c^4*d^6*x^2)*e)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(x*e + d)*
sqrt(-c/(c*d - b*e))/(c*x*e + c*d)) + 3*(16*c^7*d^6*x^4 + 32*b*c^6*d^6*x^3 + 16*b^2*c^5*d^6*x^2 - 5*(b^5*c^2*x
^5 + 2*b^6*c*x^4 + b^7*x^3)*e^6 + (3*b^4*c^3*d*x^5 + b^5*c^2*d*x^4 - 7*b^6*c*d*x^3 - 5*b^7*d*x^2)*e^5 + (5*b^3
*c^4*d^2*x^5 + 13*b^4*c^3*d^2*x^4 + 11*b^5*c^2*d^2*x^3 + 3*b^6*c*d^2*x^2)*e^4 + (17*b^2*c^5*d^3*x^5 + 39*b^3*c
^4*d^3*x^4 + 27*b^4*c^3*d^3*x^3 + 5*b^5*c^2*d^3*x^2)*e^3 - (36*b*c^6*d^4*x^5 + 55*b^2*c^5*d^4*x^4 + 2*b^3*c^4*
d^4*x^3 - 17*b^4*c^3*d^4*x^2)*e^2 + 4*(4*c^7*d^5*x^5 - b*c^6*d^5*x^4 - 14*b^2*c^5*d^5*x^3 - 9*b^3*c^4*d^5*x^2)
*e)*sqrt(d)*log((x*e - 2*sqrt(x*e + d)*sqrt(d) + 2*d)/x) + 2*(24*b*c^6*d^6*x^3 + 36*b^2*c^5*d^6*x^2 + 8*b^3*c^
4*d^6*x - 2*b^4*c^3*d^6 - 15*(b^5*c^2*d*x^4 + 2*b^6*c*d*x^3 + b^7*d*x^2)*e^5 + (9*b^4*c^3*d^2*x^4 + 13*b^5*c^2
*d^2*x^3 - b^6*c*d^2*x^2 - 5*b^7*d^2*x)*e^4 + (15*b^3*c^4*d^3*x^4 + 33*b^4*c^3*d^3*x^3 + 23*b^5*c^2*d^3*x^2 +
7*b^6*c*d^3*x + 2*b^7*d^3)*e^3 - (48*b^2*c^5*d^4*x^4 + 58*b^3*c^4*d^4*x^3 - 7*b^4*c^3*d^4*x^2 - 9*b^5*c^2*d^4*
x + 6*b^6*c*d^4)*e^2 + (24*b*c^6*d^5*x^4 - 12*b^2*c^5*d^5*x^3 - 65*b^3*c^4*d^5*x^2 - 19*b^4*c^3*d^5*x + 6*b^5*
c^2*d^5)*e)*sqrt(x*e + d))/(b^5*c^5*d^8*x^4 + 2*b^6*c^4*d^8*x^3 + b^7*c^3*d^8*x^2 - (b^8*c^2*d^4*x^5 + 2*b^9*c
*d^4*x^4 + b^10*d^4*x^3)*e^4 + (3*b^7*c^3*d^5*x^5 + 5*b^8*c^2*d^5*x^4 + b^9*c*d^5*x^3 - b^10*d^5*x^2)*e^3 - 3*
(b^6*c^4*d^6*x^5 + b^7*c^3*d^6*x^4 - b^8*c^2*d^6*x^3 - b^9*c*d^6*x^2)*e^2 + (b^5*c^5*d^7*x^5 - b^6*c^4*d^7*x^4
- 5*b^7*c^3*d^7*x^3 - 3*b^8*c^2*d^7*x^2)*e), 1/8*(6*(16*c^7*d^6*x^4 + 32*b*c^6*d^6*x^3 + 16*b^2*c^5*d^6*x^2 -
5*(b^5*c^2*x^5 + 2*b^6*c*x^4 + b^7*x^3)*e^6 + (3*b^4*c^3*d*x^5 + b^5*c^2*d*x^4 - 7*b^6*c*d*x^3 - 5*b^7*d*x^2)
*e^5 + (5*b^3*c^4*d^2*x^5 + 13*b^4*c^3*d^2*x^4 + 11*b^5*c^2*d^2*x^3 + 3*b^6*c*d^2*x^2)*e^4 + (17*b^2*c^5*d^3*x
^5 + 39*b^3*c^4*d^3*x^4 + 27*b^4*c^3*d^3*x^3 + 5*b^5*c^2*d^3*x^2)*e^3 - (36*b*c^6*d^4*x^5 + 55*b^2*c^5*d^4*x^4
+ 2*b^3*c^4*d^4*x^3 - 17*b^4*c^3*d^4*x^2)*e^2 + 4*(4*c^7*d^5*x^5 - b*c^6*d^5*x^4 - 14*b^2*c^5*d^5*x^3 - 9*b^3
*c^4*d^5*x^2)*e)*sqrt(-d)*arctan(sqrt(x*e + d)*sqrt(-d)/d) - 3*(16*c^7*d^7*x^4 + 32*b*c^6*d^7*x^3 + 16*b^2*c^5
*d^7*x^2 + 33*(b^2*c^5*d^4*x^5 + 2*b^3*c^4*d^4*x^4 + b^4*c^3*d^4*x^3)*e^3 - 11*(4*b*c^6*d^5*x^5 + 5*b^2*c^5*d^
5*x^4 - 2*b^3*c^4*d^5*x^3 - 3*b^4*c^3*d^5*x^2)*e^2 + 4*(4*c^7*d^6*x^5 - 3*b*c^6*d^6*x^4 - 18*b^2*c^5*d^6*x^3 -
11*b^3*c^4*d^6*x^2)*e)*sqrt(c/(c*d - b*e))*log((2*c*d - 2*(c*d - b*e)*sqrt(x*e + d)*sqrt(c/(c*d - b*e)) + (c*
x - b)*e)/(c*x + b)) + 2*(24*b*c^6*d^6*x^3 + 36*b^2*c^5*d^6*x^2 + 8*b^3*c^4*d^6*x - 2*b^4*c^3*d^6 - 15*(b^5*c^
2*d*x^4 + 2*b^6*c*d*x^3 + b^7*d*x^2)*e^5 + (9*b^4*c^3*d^2*x^4 + 13*b^5*c^2*d^2*x^3 - b^6*c*d^2*x^2 - 5*b^7*d^2
*x)*e^4 + (15*b^3*c^4*d^3*x^4 + 33*b^4*c^3*d^3*x^3 + 23*b^5*c^2*d^3*x^2 + 7*b^6*c*d^3*x + 2*b^7*d^3)*e^3 - (48
*b^2*c^5*d^4*x^4 + 58*b^3*c^4*d^4*x^3 - 7*b^4*c...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \left (b + c x\right )^{3} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 787 vs. \(2 (352) = 704\).
time = 1.75, size = 787, normalized size = 2.13 \begin {gather*} -\frac {3 \, {\left (16 \, c^{6} d^{2} - 44 \, b c^{5} d e + 33 \, b^{2} c^{4} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{4 \, {\left (b^{5} c^{3} d^{3} - 3 \, b^{6} c^{2} d^{2} e + 3 \, b^{7} c d e^{2} - b^{8} e^{3}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, e^{5}}{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \sqrt {x e + d}} + \frac {24 \, {\left (x e + d\right )}^{\frac {7}{2}} c^{6} d^{4} e - 72 \, {\left (x e + d\right )}^{\frac {5}{2}} c^{6} d^{5} e + 72 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{6} d^{6} e - 24 \, \sqrt {x e + d} c^{6} d^{7} e - 48 \, {\left (x e + d\right )}^{\frac {7}{2}} b c^{5} d^{3} e^{2} + 180 \, {\left (x e + d\right )}^{\frac {5}{2}} b c^{5} d^{4} e^{2} - 216 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{5} d^{5} e^{2} + 84 \, \sqrt {x e + d} b c^{5} d^{6} e^{2} + 15 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{2} c^{4} d^{2} e^{3} - 118 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{2} c^{4} d^{3} e^{3} + 199 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{4} d^{4} e^{3} - 96 \, \sqrt {x e + d} b^{2} c^{4} d^{5} e^{3} + 9 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{3} c^{3} d e^{4} - 3 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{3} c^{3} d^{2} e^{4} - 38 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c^{3} d^{3} e^{4} + 30 \, \sqrt {x e + d} b^{3} c^{3} d^{4} e^{4} - 7 \, {\left (x e + d\right )}^{\frac {7}{2}} b^{4} c^{2} e^{5} + 41 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{4} c^{2} d e^{5} - 58 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{4} c^{2} d^{2} e^{5} + 30 \, \sqrt {x e + d} b^{4} c^{2} d^{3} e^{5} - 14 \, {\left (x e + d\right )}^{\frac {5}{2}} b^{5} c e^{6} + 41 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{5} c d e^{6} - 33 \, \sqrt {x e + d} b^{5} c d^{2} e^{6} - 7 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{6} e^{7} + 9 \, \sqrt {x e + d} b^{6} d e^{7}}{4 \, {\left (b^{4} c^{3} d^{6} - 3 \, b^{5} c^{2} d^{5} e + 3 \, b^{6} c d^{4} e^{2} - b^{7} d^{3} e^{3}\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 {3 \, {\left (16 \, c^{2} d^{2} + 12 \, b c d e + 5 \, b^{2} e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{4 \, b^{5} \sqrt {-d} d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/4*(16*c^6*d^2 - 44*b*c^5*d*e + 33*b^2*c^4*e^2)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^5*c^3*d^3 -
3*b^6*c^2*d^2*e + 3*b^7*c*d*e^2 - b^8*e^3)*sqrt(-c^2*d + b*c*e)) - 2*e^5/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*
d^4*e^2 - b^3*d^3*e^3)*sqrt(x*e + d)) + 1/4*(24*(x*e + d)^(7/2)*c^6*d^4*e - 72*(x*e + d)^(5/2)*c^6*d^5*e + 72*
(x*e + d)^(3/2)*c^6*d^6*e - 24*sqrt(x*e + d)*c^6*d^7*e - 48*(x*e + d)^(7/2)*b*c^5*d^3*e^2 + 180*(x*e + d)^(5/2
)*b*c^5*d^4*e^2 - 216*(x*e + d)^(3/2)*b*c^5*d^5*e^2 + 84*sqrt(x*e + d)*b*c^5*d^6*e^2 + 15*(x*e + d)^(7/2)*b^2*
c^4*d^2*e^3 - 118*(x*e + d)^(5/2)*b^2*c^4*d^3*e^3 + 199*(x*e + d)^(3/2)*b^2*c^4*d^4*e^3 - 96*sqrt(x*e + d)*b^2
*c^4*d^5*e^3 + 9*(x*e + d)^(7/2)*b^3*c^3*d*e^4 - 3*(x*e + d)^(5/2)*b^3*c^3*d^2*e^4 - 38*(x*e + d)^(3/2)*b^3*c^
3*d^3*e^4 + 30*sqrt(x*e + d)*b^3*c^3*d^4*e^4 - 7*(x*e + d)^(7/2)*b^4*c^2*e^5 + 41*(x*e + d)^(5/2)*b^4*c^2*d*e^
5 - 58*(x*e + d)^(3/2)*b^4*c^2*d^2*e^5 + 30*sqrt(x*e + d)*b^4*c^2*d^3*e^5 - 14*(x*e + d)^(5/2)*b^5*c*e^6 + 41*
(x*e + d)^(3/2)*b^5*c*d*e^6 - 33*sqrt(x*e + d)*b^5*c*d^2*e^6 - 7*(x*e + d)^(3/2)*b^6*e^7 + 9*sqrt(x*e + d)*b^6
*d*e^7)/((b^4*c^3*d^6 - 3*b^5*c^2*d^5*e + 3*b^6*c*d^4*e^2 - b^7*d^3*e^3)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*
d^2 + (x*e + d)*b*e - b*d*e)^2) + 3/4*(16*c^2*d^2 + 12*b*c*d*e + 5*b^2*e^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^
5*sqrt(-d)*d^3)

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Mupad [B]
time = 4.07, size = 2500, normalized size = 6.76 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((2*e^5)/(c*d^2 - b*d*e) + (e*(d + e*x)^2*(15*b^6*e^6 - 72*c^6*d^6 - 199*b^2*c^4*d^4*e^2 + 38*b^3*c^3*d^3*e^
3 + 106*b^4*c^2*d^2*e^4 + 216*b*c^5*d^5*e - 89*b^5*c*d*e^5))/(4*b^4*(c*d^2 - b*d*e)^3) + (e*(d + e*x)*(25*b^5*
e^5 + 24*c^5*d^5 + 36*b^2*c^3*d^3*e^2 + 6*b^3*c^2*d^2*e^3 - 60*b*c^4*d^4*e - 56*b^4*c*d*e^4))/(4*b^4*(c*d^2 -
b*d*e)^2) - (3*e*(d + e*x)^4*(8*c^6*d^4 - 5*b^4*c^2*e^4 + 3*b^3*c^3*d*e^3 + 5*b^2*c^4*d^2*e^2 - 16*b*c^5*d^3*e
))/(4*b^4*(c*d^2 - b*d*e)^3) + (e*(d + e*x)^3*(72*c^6*d^5 + 30*b^5*c*e^5 - 73*b^4*c^2*d*e^4 + 118*b^2*c^4*d^3*
e^2 + 3*b^3*c^3*d^2*e^3 - 180*b*c^5*d^4*e))/(4*b^4*(c*d^2 - b*d*e)^3))/(c^2*(d + e*x)^(9/2) - (4*c^2*d - 2*b*c
*e)*(d + e*x)^(7/2) - (d + e*x)^(3/2)*(4*c^2*d^3 + 2*b^2*d*e^2 - 6*b*c*d^2*e) + (d + e*x)^(5/2)*(b^2*e^2 + 6*c
^2*d^2 - 6*b*c*d*e) + (d + e*x)^(1/2)*(c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e)) - (atan((((-c^7*(b*e - c*d)^7)^(1
/2)*((d + e*x)^(1/2)*(589824*b^12*c^22*d^28*e^2 - 8257536*b^13*c^21*d^27*e^3 + 53342208*b^14*c^20*d^26*e^4 - 2
10382848*b^15*c^19*d^25*e^5 + 564860160*b^16*c^18*d^24*e^6 - 1089838080*b^17*c^17*d^23*e^7 + 1555380864*b^18*c
^16*d^22*e^8 - 1667850624*b^19*c^15*d^21*e^9 + 1358257536*b^20*c^14*d^20*e^10 - 855642240*b^21*c^13*d^19*e^11
+ 438185088*b^22*c^12*d^18*e^12 - 201386880*b^23*c^11*d^17*e^13 + 90100224*b^24*c^10*d^16*e^14 - 37986048*b^25
*c^9*d^15*e^15 + 15108480*b^26*c^8*d^14*e^16 - 6844032*b^27*c^7*d^13*e^17 + 3399552*b^28*c^6*d^12*e^18 - 13006
08*b^29*c^5*d^11*e^19 + 293760*b^30*c^4*d^10*e^20 - 28800*b^31*c^3*d^9*e^21) - (3*(-c^7*(b*e - c*d)^7)^(1/2)*(
33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*(356352*b^19*c^18*d^28*e^4 - 24576*b^18*c^19*d^29*e^3 - 2396160*b^20*c^1
7*d^27*e^5 + 9897984*b^21*c^16*d^26*e^6 - 28065792*b^22*c^15*d^25*e^7 + 57891840*b^23*c^14*d^24*e^8 - 90071040
*b^24*c^13*d^23*e^9 + 108810240*b^25*c^12*d^22*e^10 - 105566208*b^26*c^11*d^21*e^11 + 86406144*b^27*c^10*d^20*
e^12 - 63393792*b^28*c^9*d^19*e^13 + 43075584*b^29*c^8*d^18*e^14 - 26173440*b^30*c^7*d^17*e^15 + 13108224*b^31
*c^6*d^16*e^16 - 4964352*b^32*c^5*d^15*e^17 + 1302528*b^33*c^4*d^14*e^18 - 208896*b^34*c^3*d^13*e^19 + 15360*b
^35*c^2*d^12*e^20 + (3*(-c^7*(b*e - c*d)^7)^(1/2)*(d + e*x)^(1/2)*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*(1638
4*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 8314880*b^25*c^15*d^28*e^5 + 2
6091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e^8 - 146432000*b^29*c^11*
d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^32*c^8*d^21*e^12 - 469647
36*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 1105920*b^36*c^4*d^17*e^16
- 139264*b^37*c^3*d^16*e^17 + 8192*b^38*c^2*d^15*e^18))/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*
c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6))))/(8*(b^12*e^7
- b^5*c^7*d^7 + 7*b^6*c^6*d^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d
^2*e^5 - 7*b^11*c*d*e^6)))*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*3i)/(8*(b^12*e^7 - b^5*c^7*d^7 + 7*b^6*c^6*d
^6*e - 21*b^7*c^5*d^5*e^2 + 35*b^8*c^4*d^4*e^3 - 35*b^9*c^3*d^3*e^4 + 21*b^10*c^2*d^2*e^5 - 7*b^11*c*d*e^6)) +
((-c^7*(b*e - c*d)^7)^(1/2)*((d + e*x)^(1/2)*(589824*b^12*c^22*d^28*e^2 - 8257536*b^13*c^21*d^27*e^3 + 533422
08*b^14*c^20*d^26*e^4 - 210382848*b^15*c^19*d^25*e^5 + 564860160*b^16*c^18*d^24*e^6 - 1089838080*b^17*c^17*d^2
3*e^7 + 1555380864*b^18*c^16*d^22*e^8 - 1667850624*b^19*c^15*d^21*e^9 + 1358257536*b^20*c^14*d^20*e^10 - 85564
2240*b^21*c^13*d^19*e^11 + 438185088*b^22*c^12*d^18*e^12 - 201386880*b^23*c^11*d^17*e^13 + 90100224*b^24*c^10*
d^16*e^14 - 37986048*b^25*c^9*d^15*e^15 + 15108480*b^26*c^8*d^14*e^16 - 6844032*b^27*c^7*d^13*e^17 + 3399552*b
^28*c^6*d^12*e^18 - 1300608*b^29*c^5*d^11*e^19 + 293760*b^30*c^4*d^10*e^20 - 28800*b^31*c^3*d^9*e^21) - (3*(-c
^7*(b*e - c*d)^7)^(1/2)*(33*b^2*e^2 + 16*c^2*d^2 - 44*b*c*d*e)*(24576*b^18*c^19*d^29*e^3 - 356352*b^19*c^18*d^
28*e^4 + 2396160*b^20*c^17*d^27*e^5 - 9897984*b^21*c^16*d^26*e^6 + 28065792*b^22*c^15*d^25*e^7 - 57891840*b^23
*c^14*d^24*e^8 + 90071040*b^24*c^13*d^23*e^9 - 108810240*b^25*c^12*d^22*e^10 + 105566208*b^26*c^11*d^21*e^11 -
86406144*b^27*c^10*d^20*e^12 + 63393792*b^28*c^9*d^19*e^13 - 43075584*b^29*c^8*d^18*e^14 + 26173440*b^30*c^7*
d^17*e^15 - 13108224*b^31*c^6*d^16*e^16 + 4964352*b^32*c^5*d^15*e^17 - 1302528*b^33*c^4*d^14*e^18 + 208896*b^3
4*c^3*d^13*e^19 - 15360*b^35*c^2*d^12*e^20 + (3*(-c^7*(b*e - c*d)^7)^(1/2)*(d + e*x)^(1/2)*(33*b^2*e^2 + 16*c^
2*d^2 - 44*b*c*d*e)*(16384*b^22*c^18*d^31*e^2 - 253952*b^23*c^17*d^30*e^3 + 1843200*b^24*c^16*d^29*e^4 - 83148
80*b^25*c^15*d^28*e^5 + 26091520*b^26*c^14*d^27*e^6 - 60383232*b^27*c^13*d^26*e^7 + 106602496*b^28*c^12*d^25*e
^8 - 146432000*b^29*c^11*d^24*e^9 + 158146560*b^30*c^10*d^23*e^10 - 134717440*b^31*c^9*d^22*e^11 + 90202112*b^
32*c^8*d^21*e^12 - 46964736*b^33*c^7*d^20*e^13 + 18636800*b^34*c^6*d^19*e^14 - 5447680*b^35*c^5*d^18*e^15 + 11
05920*b^36*c^4*d^17*e^16 - 139264*b^37*c^3*d^16...

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