3.4.55 \(\int \frac {1}{(d+e x)^{7/2} (b x+c x^2)^2} \, dx\)

Optimal. Leaf size=349 \[ -\frac {c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}+\frac {(7 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{9/2}}-\frac {e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}-\frac {e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)}-\frac {e \left (7 b^4 e^4-24 b^3 c d e^3+26 b^2 c^2 d^2 e^2-4 b c^3 d^3 e+2 c^4 d^4\right )}{b^2 d^4 \sqrt {d+e x} (c d-b e)^4} \]

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Rubi [A]  time = 0.74, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {740, 828, 826, 1166, 208} \begin {gather*} -\frac {e \left (26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4-4 b c^3 d^3 e+2 c^4 d^4\right )}{b^2 d^4 \sqrt {d+e x} (c d-b e)^4}-\frac {e (2 c d-b e) \left (7 b^2 e^2-3 b c d e+3 c^2 d^2\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac {e \left (7 b^2 e^2-10 b c d e+10 c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}-\frac {c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}+\frac {(7 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{9/2}}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]

[Out]

-(e*(10*c^2*d^2 - 10*b*c*d*e + 7*b^2*e^2))/(5*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2)) - (e*(2*c*d - b*e)*(3*c^2
*d^2 - 3*b*c*d*e + 7*b^2*e^2))/(3*b^2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*c^4*d^4 - 4*b*c^3*d^3*e + 26*
b^2*c^2*d^2*e^2 - 24*b^3*c*d*e^3 + 7*b^4*e^4))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (b*(c*d - b*e) + c*(2*c
*d - b*e)*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(5/2)*(b*x + c*x^2)) + ((4*c*d + 7*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d
]])/(b^3*d^(9/2)) - (c^(9/2)*(4*c*d - 11*b*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(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 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)^{7/2} \left (b x+c x^2\right )^2} \, dx &=-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e) (4 c d+7 b e)+\frac {7}{2} c e (2 c d-b e) x}{(d+e x)^{7/2} \left (b x+c x^2\right )} \, dx}{b^2 d (c d-b e)}\\ &=-\frac {e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e)^2 (4 c d+7 b e)+\frac {1}{2} c e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{b^2 d^2 (c d-b e)^2}\\ &=-\frac {e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 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}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e)^3 (4 c d+7 b e)+\frac {1}{2} c e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{b^2 d^3 (c d-b e)^3}\\ &=-\frac {e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {\int \frac {\frac {1}{2} (c d-b e)^4 (4 c d+7 b e)+\frac {1}{2} c e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right ) x}{\sqrt {d+e x} \left (b x+c x^2\right )} \, dx}{b^2 d^4 (c d-b e)^4}\\ &=-\frac {e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac {2 \operatorname {Subst}\left (\int \frac {\frac {1}{2} e (c d-b e)^4 (4 c d+7 b e)-\frac {1}{2} c d e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )+\frac {1}{2} c e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 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^2 d^4 (c d-b e)^4}\\ &=-\frac {e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}+\frac {\left (c^5 (4 c d-11 b e)\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 )}{b^3 (c d-b e)^4}-\frac {(c (4 c d+7 b e)) \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 )}{b^3 d^4}\\ &=-\frac {e \left (10 c^2 d^2-10 b c d e+7 b^2 e^2\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac {e (2 c d-b e) \left (3 c^2 d^2-3 b c d e+7 b^2 e^2\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac {e \left (2 c^4 d^4-4 b c^3 d^3 e+26 b^2 c^2 d^2 e^2-24 b^3 c d e^3+7 b^4 e^4\right )}{b^2 d^4 (c d-b e)^4 \sqrt {d+e x}}-\frac {b (c d-b e)+c (2 c d-b e) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}+\frac {(4 c d+7 b e) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{9/2}}-\frac {c^{9/2} (4 c d-11 b e) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}\\ \end {align*}

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Mathematica [C]  time = 0.12, size = 171, normalized size = 0.49 \begin {gather*} \frac {c^2 d^2 x (b+c x) (4 c d-11 b e) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {c (d+e x)}{c d-b e}\right )-(c d-b e) \left (x (b+c x) \left (-7 b^2 e^2+3 b c d e+4 c^2 d^2\right ) \, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {e x}{d}+1\right )-5 b d \left (b^2 e+b c (e x-d)-2 c^2 d x\right )\right )}{5 b^3 d^2 x (b+c x) (d+e x)^{5/2} (c d-b e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]

[Out]

(c^2*d^2*(4*c*d - 11*b*e)*x*(b + c*x)*Hypergeometric2F1[-5/2, 1, -3/2, (c*(d + e*x))/(c*d - b*e)] - (c*d - b*e
)*(-5*b*d*(b^2*e - 2*c^2*d*x + b*c*(-d + e*x)) + (4*c^2*d^2 + 3*b*c*d*e - 7*b^2*e^2)*x*(b + c*x)*Hypergeometri
c2F1[-5/2, 1, -3/2, 1 + (e*x)/d]))/(5*b^3*d^2*(c*d - b*e)^2*x*(b + c*x)*(d + e*x)^(5/2))

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IntegrateAlgebraic [A]  time = 1.44, size = 550, normalized size = 1.58 \begin {gather*} \frac {\left (11 b c^{9/2} e-4 c^{11/2} d\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x} \sqrt {b e-c d}}{c d-b e}\right )}{b^3 (b e-c d)^{9/2}}+\frac {(7 b e+4 c d) \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{9/2}}+\frac {6 b^5 d^3 e^5+14 b^5 d^2 e^5 (d+e x)+70 b^5 d e^5 (d+e x)^2-105 b^5 e^5 (d+e x)^3-18 b^4 c d^4 e^4-56 b^4 c d^3 e^4 (d+e x)-296 b^4 c d^2 e^4 (d+e x)^2+535 b^4 c d e^4 (d+e x)^3-105 b^4 c e^4 (d+e x)^4+18 b^3 c^2 d^5 e^3+70 b^3 c^2 d^4 e^3 (d+e x)+452 b^3 c^2 d^3 e^3 (d+e x)^2-990 b^3 c^2 d^2 e^3 (d+e x)^3+360 b^3 c^2 d e^3 (d+e x)^4-6 b^2 c^3 d^6 e^2-28 b^2 c^3 d^5 e^2 (d+e x)-226 b^2 c^3 d^4 e^2 (d+e x)^2+710 b^2 c^3 d^3 e^2 (d+e x)^3-390 b^2 c^3 d^2 e^2 (d+e x)^4-75 b c^4 d^4 e (d+e x)^3+60 b c^4 d^3 e (d+e x)^4+30 c^5 d^5 (d+e x)^3-30 c^5 d^4 (d+e x)^4}{15 b^2 d^4 x (d+e x)^{5/2} (b e-c d)^4 (b e+c (d+e x)-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]

[Out]

(-6*b^2*c^3*d^6*e^2 + 18*b^3*c^2*d^5*e^3 - 18*b^4*c*d^4*e^4 + 6*b^5*d^3*e^5 - 28*b^2*c^3*d^5*e^2*(d + e*x) + 7
0*b^3*c^2*d^4*e^3*(d + e*x) - 56*b^4*c*d^3*e^4*(d + e*x) + 14*b^5*d^2*e^5*(d + e*x) - 226*b^2*c^3*d^4*e^2*(d +
e*x)^2 + 452*b^3*c^2*d^3*e^3*(d + e*x)^2 - 296*b^4*c*d^2*e^4*(d + e*x)^2 + 70*b^5*d*e^5*(d + e*x)^2 + 30*c^5*
d^5*(d + e*x)^3 - 75*b*c^4*d^4*e*(d + e*x)^3 + 710*b^2*c^3*d^3*e^2*(d + e*x)^3 - 990*b^3*c^2*d^2*e^3*(d + e*x)
^3 + 535*b^4*c*d*e^4*(d + e*x)^3 - 105*b^5*e^5*(d + e*x)^3 - 30*c^5*d^4*(d + e*x)^4 + 60*b*c^4*d^3*e*(d + e*x)
^4 - 390*b^2*c^3*d^2*e^2*(d + e*x)^4 + 360*b^3*c^2*d*e^3*(d + e*x)^4 - 105*b^4*c*e^4*(d + e*x)^4)/(15*b^2*d^4*
(-(c*d) + b*e)^4*x*(d + e*x)^(5/2)*(-(c*d) + b*e + c*(d + e*x))) + ((-4*c^(11/2)*d + 11*b*c^(9/2)*e)*ArcTan[(S
qrt[c]*Sqrt[-(c*d) + b*e]*Sqrt[d + e*x])/(c*d - b*e)])/(b^3*(-(c*d) + b*e)^(9/2)) + ((4*c*d + 7*b*e)*ArcTanh[S
qrt[d + e*x]/Sqrt[d]])/(b^3*d^(9/2))

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fricas [B]  time = 6.90, size = 5752, normalized size = 16.48

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

[-1/30*(15*((4*c^6*d^6*e^3 - 11*b*c^5*d^5*e^4)*x^5 + (12*c^6*d^7*e^2 - 29*b*c^5*d^6*e^3 - 11*b^2*c^4*d^5*e^4)*
x^4 + 3*(4*c^6*d^8*e - 7*b*c^5*d^7*e^2 - 11*b^2*c^4*d^6*e^3)*x^3 + (4*c^6*d^9 + b*c^5*d^8*e - 33*b^2*c^4*d^7*e
^2)*x^2 + (4*b*c^5*d^9 - 11*b^2*c^4*d^8*e)*x)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqr
t(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) - 15*((4*c^6*d^5*e^3 - 9*b*c^5*d^4*e^4 - 4*b^2*c^4*d^3*e^5 + 26*b^3
*c^3*d^2*e^6 - 24*b^4*c^2*d*e^7 + 7*b^5*c*e^8)*x^5 + (12*c^6*d^6*e^2 - 23*b*c^5*d^5*e^3 - 21*b^2*c^4*d^4*e^4 +
74*b^3*c^3*d^3*e^5 - 46*b^4*c^2*d^2*e^6 - 3*b^5*c*d*e^7 + 7*b^6*e^8)*x^4 + 3*(4*c^6*d^7*e - 5*b*c^5*d^6*e^2 -
13*b^2*c^4*d^5*e^3 + 22*b^3*c^3*d^4*e^4 + 2*b^4*c^2*d^3*e^5 - 17*b^5*c*d^2*e^6 + 7*b^6*d*e^7)*x^3 + (4*c^6*d^
8 + 3*b*c^5*d^7*e - 31*b^2*c^4*d^6*e^2 + 14*b^3*c^3*d^5*e^3 + 54*b^4*c^2*d^4*e^4 - 65*b^5*c*d^3*e^5 + 21*b^6*d
^2*e^6)*x^2 + (4*b*c^5*d^8 - 9*b^2*c^4*d^7*e - 4*b^3*c^3*d^6*e^2 + 26*b^4*c^2*d^5*e^3 - 24*b^5*c*d^4*e^4 + 7*b
^6*d^3*e^5)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(15*b^2*c^4*d^8 - 60*b^3*c^3*d^7*e + 9
0*b^4*c^2*d^6*e^2 - 60*b^5*c*d^5*e^3 + 15*b^6*d^4*e^4 + 15*(2*b*c^5*d^5*e^3 - 4*b^2*c^4*d^4*e^4 + 26*b^3*c^3*d
^3*e^5 - 24*b^4*c^2*d^2*e^6 + 7*b^5*c*d*e^7)*x^4 + 5*(18*b*c^5*d^6*e^2 - 33*b^2*c^4*d^5*e^3 + 170*b^3*c^3*d^4*
e^4 - 90*b^4*c^2*d^3*e^5 - 23*b^5*c*d^2*e^6 + 21*b^6*d*e^7)*x^3 + (90*b*c^5*d^7*e - 135*b^2*c^4*d^6*e^2 + 436*
b^3*c^3*d^5*e^3 + 358*b^4*c^2*d^4*e^4 - 679*b^5*c*d^3*e^5 + 245*b^6*d^2*e^6)*x^2 + (30*b*c^5*d^8 - 15*b^2*c^4*
d^7*e - 90*b^3*c^3*d^6*e^2 + 556*b^4*c^2*d^5*e^3 - 537*b^5*c*d^4*e^4 + 161*b^6*d^3*e^5)*x)*sqrt(e*x + d))/((b^
3*c^5*d^9*e^3 - 4*b^4*c^4*d^8*e^4 + 6*b^5*c^3*d^7*e^5 - 4*b^6*c^2*d^6*e^6 + b^7*c*d^5*e^7)*x^5 + (3*b^3*c^5*d^
10*e^2 - 11*b^4*c^4*d^9*e^3 + 14*b^5*c^3*d^8*e^4 - 6*b^6*c^2*d^7*e^5 - b^7*c*d^6*e^6 + b^8*d^5*e^7)*x^4 + 3*(b
^3*c^5*d^11*e - 3*b^4*c^4*d^10*e^2 + 2*b^5*c^3*d^9*e^3 + 2*b^6*c^2*d^8*e^4 - 3*b^7*c*d^7*e^5 + b^8*d^6*e^6)*x^
3 + (b^3*c^5*d^12 - b^4*c^4*d^11*e - 6*b^5*c^3*d^10*e^2 + 14*b^6*c^2*d^9*e^3 - 11*b^7*c*d^8*e^4 + 3*b^8*d^7*e^
5)*x^2 + (b^4*c^4*d^12 - 4*b^5*c^3*d^11*e + 6*b^6*c^2*d^10*e^2 - 4*b^7*c*d^9*e^3 + b^8*d^8*e^4)*x), -1/30*(30*
((4*c^6*d^6*e^3 - 11*b*c^5*d^5*e^4)*x^5 + (12*c^6*d^7*e^2 - 29*b*c^5*d^6*e^3 - 11*b^2*c^4*d^5*e^4)*x^4 + 3*(4*
c^6*d^8*e - 7*b*c^5*d^7*e^2 - 11*b^2*c^4*d^6*e^3)*x^3 + (4*c^6*d^9 + b*c^5*d^8*e - 33*b^2*c^4*d^7*e^2)*x^2 + (
4*b*c^5*d^9 - 11*b^2*c^4*d^8*e)*x)*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)) - 15*((4*c^6*d^5*e^3 - 9*b*c^5*d^4*e^4 - 4*b^2*c^4*d^3*e^5 + 26*b^3*c^3*d^2*e^6 - 24*b^4*c^2*d
*e^7 + 7*b^5*c*e^8)*x^5 + (12*c^6*d^6*e^2 - 23*b*c^5*d^5*e^3 - 21*b^2*c^4*d^4*e^4 + 74*b^3*c^3*d^3*e^5 - 46*b^
4*c^2*d^2*e^6 - 3*b^5*c*d*e^7 + 7*b^6*e^8)*x^4 + 3*(4*c^6*d^7*e - 5*b*c^5*d^6*e^2 - 13*b^2*c^4*d^5*e^3 + 22*b^
3*c^3*d^4*e^4 + 2*b^4*c^2*d^3*e^5 - 17*b^5*c*d^2*e^6 + 7*b^6*d*e^7)*x^3 + (4*c^6*d^8 + 3*b*c^5*d^7*e - 31*b^2*
c^4*d^6*e^2 + 14*b^3*c^3*d^5*e^3 + 54*b^4*c^2*d^4*e^4 - 65*b^5*c*d^3*e^5 + 21*b^6*d^2*e^6)*x^2 + (4*b*c^5*d^8
- 9*b^2*c^4*d^7*e - 4*b^3*c^3*d^6*e^2 + 26*b^4*c^2*d^5*e^3 - 24*b^5*c*d^4*e^4 + 7*b^6*d^3*e^5)*x)*sqrt(d)*log(
(e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(15*b^2*c^4*d^8 - 60*b^3*c^3*d^7*e + 90*b^4*c^2*d^6*e^2 - 60*b^5*
c*d^5*e^3 + 15*b^6*d^4*e^4 + 15*(2*b*c^5*d^5*e^3 - 4*b^2*c^4*d^4*e^4 + 26*b^3*c^3*d^3*e^5 - 24*b^4*c^2*d^2*e^6
+ 7*b^5*c*d*e^7)*x^4 + 5*(18*b*c^5*d^6*e^2 - 33*b^2*c^4*d^5*e^3 + 170*b^3*c^3*d^4*e^4 - 90*b^4*c^2*d^3*e^5 -
23*b^5*c*d^2*e^6 + 21*b^6*d*e^7)*x^3 + (90*b*c^5*d^7*e - 135*b^2*c^4*d^6*e^2 + 436*b^3*c^3*d^5*e^3 + 358*b^4*c
^2*d^4*e^4 - 679*b^5*c*d^3*e^5 + 245*b^6*d^2*e^6)*x^2 + (30*b*c^5*d^8 - 15*b^2*c^4*d^7*e - 90*b^3*c^3*d^6*e^2
+ 556*b^4*c^2*d^5*e^3 - 537*b^5*c*d^4*e^4 + 161*b^6*d^3*e^5)*x)*sqrt(e*x + d))/((b^3*c^5*d^9*e^3 - 4*b^4*c^4*d
^8*e^4 + 6*b^5*c^3*d^7*e^5 - 4*b^6*c^2*d^6*e^6 + b^7*c*d^5*e^7)*x^5 + (3*b^3*c^5*d^10*e^2 - 11*b^4*c^4*d^9*e^3
+ 14*b^5*c^3*d^8*e^4 - 6*b^6*c^2*d^7*e^5 - b^7*c*d^6*e^6 + b^8*d^5*e^7)*x^4 + 3*(b^3*c^5*d^11*e - 3*b^4*c^4*d
^10*e^2 + 2*b^5*c^3*d^9*e^3 + 2*b^6*c^2*d^8*e^4 - 3*b^7*c*d^7*e^5 + b^8*d^6*e^6)*x^3 + (b^3*c^5*d^12 - b^4*c^4
*d^11*e - 6*b^5*c^3*d^10*e^2 + 14*b^6*c^2*d^9*e^3 - 11*b^7*c*d^8*e^4 + 3*b^8*d^7*e^5)*x^2 + (b^4*c^4*d^12 - 4*
b^5*c^3*d^11*e + 6*b^6*c^2*d^10*e^2 - 4*b^7*c*d^9*e^3 + b^8*d^8*e^4)*x), -1/30*(30*((4*c^6*d^5*e^3 - 9*b*c^5*d
^4*e^4 - 4*b^2*c^4*d^3*e^5 + 26*b^3*c^3*d^2*e^6 - 24*b^4*c^2*d*e^7 + 7*b^5*c*e^8)*x^5 + (12*c^6*d^6*e^2 - 23*b
*c^5*d^5*e^3 - 21*b^2*c^4*d^4*e^4 + 74*b^3*c^3*d^3*e^5 - 46*b^4*c^2*d^2*e^6 - 3*b^5*c*d*e^7 + 7*b^6*e^8)*x^4 +
3*(4*c^6*d^7*e - 5*b*c^5*d^6*e^2 - 13*b^2*c^4*d^5*e^3 + 22*b^3*c^3*d^4*e^4 + 2*b^4*c^2*d^3*e^5 - 17*b^5*c*d^2
*e^6 + 7*b^6*d*e^7)*x^3 + (4*c^6*d^8 + 3*b*c^5*d^7*e - 31*b^2*c^4*d^6*e^2 + 14*b^3*c^3*d^5*e^3 + 54*b^4*c^2*d^
4*e^4 - 65*b^5*c*d^3*e^5 + 21*b^6*d^2*e^6)*x^2 + (4*b*c^5*d^8 - 9*b^2*c^4*d^7*e - 4*b^3*c^3*d^6*e^2 + 26*b^4*c
^2*d^5*e^3 - 24*b^5*c*d^4*e^4 + 7*b^6*d^3*e^5)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + 15*((4*c^6*d^6*e
^3 - 11*b*c^5*d^5*e^4)*x^5 + (12*c^6*d^7*e^2 - 29*b*c^5*d^6*e^3 - 11*b^2*c^4*d^5*e^4)*x^4 + 3*(4*c^6*d^8*e - 7
*b*c^5*d^7*e^2 - 11*b^2*c^4*d^6*e^3)*x^3 + (4*c^6*d^9 + b*c^5*d^8*e - 33*b^2*c^4*d^7*e^2)*x^2 + (4*b*c^5*d^9 -
11*b^2*c^4*d^8*e)*x)*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)) + 2*(15*b^2*c^4*d^8 - 60*b^3*c^3*d^7*e + 90*b^4*c^2*d^6*e^2 - 60*b^5*c*d^5*e^3 + 15*b^6*d^4
*e^4 + 15*(2*b*c^5*d^5*e^3 - 4*b^2*c^4*d^4*e^4 + 26*b^3*c^3*d^3*e^5 - 24*b^4*c^2*d^2*e^6 + 7*b^5*c*d*e^7)*x^4
+ 5*(18*b*c^5*d^6*e^2 - 33*b^2*c^4*d^5*e^3 + 170*b^3*c^3*d^4*e^4 - 90*b^4*c^2*d^3*e^5 - 23*b^5*c*d^2*e^6 + 21*
b^6*d*e^7)*x^3 + (90*b*c^5*d^7*e - 135*b^2*c^4*d^6*e^2 + 436*b^3*c^3*d^5*e^3 + 358*b^4*c^2*d^4*e^4 - 679*b^5*c
*d^3*e^5 + 245*b^6*d^2*e^6)*x^2 + (30*b*c^5*d^8 - 15*b^2*c^4*d^7*e - 90*b^3*c^3*d^6*e^2 + 556*b^4*c^2*d^5*e^3
- 537*b^5*c*d^4*e^4 + 161*b^6*d^3*e^5)*x)*sqrt(e*x + d))/((b^3*c^5*d^9*e^3 - 4*b^4*c^4*d^8*e^4 + 6*b^5*c^3*d^7
*e^5 - 4*b^6*c^2*d^6*e^6 + b^7*c*d^5*e^7)*x^5 + (3*b^3*c^5*d^10*e^2 - 11*b^4*c^4*d^9*e^3 + 14*b^5*c^3*d^8*e^4
- 6*b^6*c^2*d^7*e^5 - b^7*c*d^6*e^6 + b^8*d^5*e^7)*x^4 + 3*(b^3*c^5*d^11*e - 3*b^4*c^4*d^10*e^2 + 2*b^5*c^3*d^
9*e^3 + 2*b^6*c^2*d^8*e^4 - 3*b^7*c*d^7*e^5 + b^8*d^6*e^6)*x^3 + (b^3*c^5*d^12 - b^4*c^4*d^11*e - 6*b^5*c^3*d^
10*e^2 + 14*b^6*c^2*d^9*e^3 - 11*b^7*c*d^8*e^4 + 3*b^8*d^7*e^5)*x^2 + (b^4*c^4*d^12 - 4*b^5*c^3*d^11*e + 6*b^6
*c^2*d^10*e^2 - 4*b^7*c*d^9*e^3 + b^8*d^8*e^4)*x), -1/15*(15*((4*c^6*d^6*e^3 - 11*b*c^5*d^5*e^4)*x^5 + (12*c^6
*d^7*e^2 - 29*b*c^5*d^6*e^3 - 11*b^2*c^4*d^5*e^4)*x^4 + 3*(4*c^6*d^8*e - 7*b*c^5*d^7*e^2 - 11*b^2*c^4*d^6*e^3)
*x^3 + (4*c^6*d^9 + b*c^5*d^8*e - 33*b^2*c^4*d^7*e^2)*x^2 + (4*b*c^5*d^9 - 11*b^2*c^4*d^8*e)*x)*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)) + 15*((4*c^6*d^5*e^3 - 9*b*c^5*d^
4*e^4 - 4*b^2*c^4*d^3*e^5 + 26*b^3*c^3*d^2*e^6 - 24*b^4*c^2*d*e^7 + 7*b^5*c*e^8)*x^5 + (12*c^6*d^6*e^2 - 23*b*
c^5*d^5*e^3 - 21*b^2*c^4*d^4*e^4 + 74*b^3*c^3*d^3*e^5 - 46*b^4*c^2*d^2*e^6 - 3*b^5*c*d*e^7 + 7*b^6*e^8)*x^4 +
3*(4*c^6*d^7*e - 5*b*c^5*d^6*e^2 - 13*b^2*c^4*d^5*e^3 + 22*b^3*c^3*d^4*e^4 + 2*b^4*c^2*d^3*e^5 - 17*b^5*c*d^2*
e^6 + 7*b^6*d*e^7)*x^3 + (4*c^6*d^8 + 3*b*c^5*d^7*e - 31*b^2*c^4*d^6*e^2 + 14*b^3*c^3*d^5*e^3 + 54*b^4*c^2*d^4
*e^4 - 65*b^5*c*d^3*e^5 + 21*b^6*d^2*e^6)*x^2 + (4*b*c^5*d^8 - 9*b^2*c^4*d^7*e - 4*b^3*c^3*d^6*e^2 + 26*b^4*c^
2*d^5*e^3 - 24*b^5*c*d^4*e^4 + 7*b^6*d^3*e^5)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + (15*b^2*c^4*d^8 -
60*b^3*c^3*d^7*e + 90*b^4*c^2*d^6*e^2 - 60*b^5*c*d^5*e^3 + 15*b^6*d^4*e^4 + 15*(2*b*c^5*d^5*e^3 - 4*b^2*c^4*d
^4*e^4 + 26*b^3*c^3*d^3*e^5 - 24*b^4*c^2*d^2*e^6 + 7*b^5*c*d*e^7)*x^4 + 5*(18*b*c^5*d^6*e^2 - 33*b^2*c^4*d^5*e
^3 + 170*b^3*c^3*d^4*e^4 - 90*b^4*c^2*d^3*e^5 - 23*b^5*c*d^2*e^6 + 21*b^6*d*e^7)*x^3 + (90*b*c^5*d^7*e - 135*b
^2*c^4*d^6*e^2 + 436*b^3*c^3*d^5*e^3 + 358*b^4*c^2*d^4*e^4 - 679*b^5*c*d^3*e^5 + 245*b^6*d^2*e^6)*x^2 + (30*b*
c^5*d^8 - 15*b^2*c^4*d^7*e - 90*b^3*c^3*d^6*e^2 + 556*b^4*c^2*d^5*e^3 - 537*b^5*c*d^4*e^4 + 161*b^6*d^3*e^5)*x
)*sqrt(e*x + d))/((b^3*c^5*d^9*e^3 - 4*b^4*c^4*d^8*e^4 + 6*b^5*c^3*d^7*e^5 - 4*b^6*c^2*d^6*e^6 + b^7*c*d^5*e^7
)*x^5 + (3*b^3*c^5*d^10*e^2 - 11*b^4*c^4*d^9*e^3 + 14*b^5*c^3*d^8*e^4 - 6*b^6*c^2*d^7*e^5 - b^7*c*d^6*e^6 + b^
8*d^5*e^7)*x^4 + 3*(b^3*c^5*d^11*e - 3*b^4*c^4*d^10*e^2 + 2*b^5*c^3*d^9*e^3 + 2*b^6*c^2*d^8*e^4 - 3*b^7*c*d^7*
e^5 + b^8*d^6*e^6)*x^3 + (b^3*c^5*d^12 - b^4*c^4*d^11*e - 6*b^5*c^3*d^10*e^2 + 14*b^6*c^2*d^9*e^3 - 11*b^7*c*d
^8*e^4 + 3*b^8*d^7*e^5)*x^2 + (b^4*c^4*d^12 - 4*b^5*c^3*d^11*e + 6*b^6*c^2*d^10*e^2 - 4*b^7*c*d^9*e^3 + b^8*d^
8*e^4)*x)]

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giac [B]  time = 0.31, size = 643, normalized size = 1.84 \begin {gather*} \frac {{\left (4 \, c^{6} d - 11 \, b c^{5} e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{4} d^{4} - 4 \, b^{4} c^{3} d^{3} e + 6 \, b^{5} c^{2} d^{2} e^{2} - 4 \, b^{6} c d e^{3} + b^{7} e^{4}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (x e + d\right )}^{\frac {3}{2}} c^{5} d^{4} e - 2 \, \sqrt {x e + d} c^{5} d^{5} e - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} b c^{4} d^{3} e^{2} + 5 \, \sqrt {x e + d} b c^{4} d^{4} e^{2} + 6 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{2} c^{3} d^{2} e^{3} - 10 \, \sqrt {x e + d} b^{2} c^{3} d^{3} e^{3} - 4 \, {\left (x e + d\right )}^{\frac {3}{2}} b^{3} c^{2} d e^{4} + 10 \, \sqrt {x e + d} b^{3} c^{2} d^{2} e^{4} + {\left (x e + d\right )}^{\frac {3}{2}} b^{4} c e^{5} - 5 \, \sqrt {x e + d} b^{4} c d e^{5} + \sqrt {x e + d} b^{5} e^{6}}{{\left (b^{2} c^{4} d^{8} - 4 \, b^{3} c^{3} d^{7} e + 6 \, b^{4} c^{2} d^{6} e^{2} - 4 \, b^{5} c d^{5} e^{3} + b^{6} 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 )}} - \frac {2 \, {\left (150 \, {\left (x e + d\right )}^{2} c^{2} d^{2} e^{3} + 20 \, {\left (x e + d\right )} c^{2} d^{3} e^{3} + 3 \, c^{2} d^{4} e^{3} - 150 \, {\left (x e + d\right )}^{2} b c d e^{4} - 30 \, {\left (x e + d\right )} b c d^{2} e^{4} - 6 \, b c d^{3} e^{4} + 45 \, {\left (x e + d\right )}^{2} b^{2} e^{5} + 10 \, {\left (x e + d\right )} b^{2} d e^{5} + 3 \, b^{2} d^{2} e^{5}\right )}}{15 \, {\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 {5}{2}}} - \frac {{\left (4 \, c d + 7 \, b e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

(4*c^6*d - 11*b*c^5*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^4*d^4 - 4*b^4*c^3*d^3*e + 6*b^5*c^
2*d^2*e^2 - 4*b^6*c*d*e^3 + b^7*e^4)*sqrt(-c^2*d + b*c*e)) - (2*(x*e + d)^(3/2)*c^5*d^4*e - 2*sqrt(x*e + d)*c^
5*d^5*e - 4*(x*e + d)^(3/2)*b*c^4*d^3*e^2 + 5*sqrt(x*e + d)*b*c^4*d^4*e^2 + 6*(x*e + d)^(3/2)*b^2*c^3*d^2*e^3
- 10*sqrt(x*e + d)*b^2*c^3*d^3*e^3 - 4*(x*e + d)^(3/2)*b^3*c^2*d*e^4 + 10*sqrt(x*e + d)*b^3*c^2*d^2*e^4 + (x*e
+ d)^(3/2)*b^4*c*e^5 - 5*sqrt(x*e + d)*b^4*c*d*e^5 + sqrt(x*e + d)*b^5*e^6)/((b^2*c^4*d^8 - 4*b^3*c^3*d^7*e +
6*b^4*c^2*d^6*e^2 - 4*b^5*c*d^5*e^3 + b^6*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)) - 2/15*(150*(x*e + d)^2*c^2*d^2*e^3 + 20*(x*e + d)*c^2*d^3*e^3 + 3*c^2*d^4*e^3 - 150*(x*e + d)^2*b*c*
d*e^4 - 30*(x*e + d)*b*c*d^2*e^4 - 6*b*c*d^3*e^4 + 45*(x*e + d)^2*b^2*e^5 + 10*(x*e + d)*b^2*d*e^5 + 3*b^2*d^2
*e^5)/((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)^(5/2)) - (4*c*d
+ 7*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)*d^4)

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maple [A]  time = 0.07, size = 364, normalized size = 1.04 \begin {gather*} -\frac {11 c^{5} 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^{2}}+\frac {4 c^{6} d \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^{3}}-\frac {\sqrt {e x +d}\, c^{5} e}{\left (b e -c d \right )^{4} \left (c e x +b e \right ) b^{2}}-\frac {6 b^{2} e^{5}}{\left (b e -c d \right )^{4} \sqrt {e x +d}\, d^{4}}+\frac {20 b c \,e^{4}}{\left (b e -c d \right )^{4} \sqrt {e x +d}\, d^{3}}-\frac {20 c^{2} e^{3}}{\left (b e -c d \right )^{4} \sqrt {e x +d}\, d^{2}}-\frac {4 b \,e^{4}}{3 \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}} d^{3}}+\frac {8 c \,e^{3}}{3 \left (b e -c d \right )^{3} \left (e x +d \right )^{\frac {3}{2}} d^{2}}-\frac {2 e^{3}}{5 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {5}{2}} d^{2}}+\frac {7 e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} d^{\frac {9}{2}}}+\frac {4 c \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3} d^{\frac {7}{2}}}-\frac {\sqrt {e x +d}}{b^{2} d^{4} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(7/2)/(c*x^2+b*x)^2,x)

[Out]

-e*c^5/b^2/(b*e-c*d)^4*(e*x+d)^(1/2)/(c*e*x+b*e)-11*e*c^5/b^2/(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)+4*c^6/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)*d-1/b^2/d^4*(e*x+d)^(1/2)/x+7*e/b^2/d^(9/2)*arctanh((e*x+d)^(1/2)/d^(1/2))+4/b^3/d^(7/2)*arctanh((e*x+d)^(
1/2)/d^(1/2))*c-2/5*e^3/(b*e-c*d)^2/d^2/(e*x+d)^(5/2)-4/3*e^4/(b*e-c*d)^3/d^3/(e*x+d)^(3/2)*b+8/3*e^3/(b*e-c*d
)^3/d^2/(e*x+d)^(3/2)*c-6*e^5/(b*e-c*d)^4/d^4/(e*x+d)^(1/2)*b^2+20*e^4/(b*e-c*d)^4/d^3/(e*x+d)^(1/2)*b*c-20*e^
3/(b*e-c*d)^4/d^2/(e*x+d)^(1/2)*c^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)^(7/2)/(c*x^2+b*x)^2,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 = 3.82, size = 7254, normalized size = 20.79

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x + c*x^2)^2*(d + e*x)^(7/2)),x)

[Out]

(atan((((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*((d + e*x)^(1/2)*(1088*b^7*c^24*d^33*e^3 - 64*b^6*c^25*d^3
4*e^2 - 8404*b^8*c^23*d^32*e^4 + 38720*b^9*c^22*d^31*e^5 - 116512*b^10*c^21*d^30*e^6 + 230912*b^11*c^20*d^29*e
^7 - 267432*b^12*c^19*d^28*e^8 + 38544*b^13*c^18*d^27*e^9 + 473880*b^14*c^17*d^26*e^10 - 851136*b^15*c^16*d^25
*e^11 + 393646*b^16*c^15*d^24*e^12 + 1207368*b^17*c^14*d^23*e^13 - 3343724*b^18*c^13*d^22*e^14 + 4835160*b^19*
c^12*d^21*e^15 - 4903382*b^20*c^11*d^20*e^16 + 3751968*b^21*c^10*d^19*e^17 - 2217072*b^22*c^9*d^18*e^18 + 1013
232*b^23*c^8*d^17*e^19 - 353210*b^24*c^7*d^16*e^20 + 91080*b^25*c^6*d^15*e^21 - 16412*b^26*c^5*d^14*e^22 + 184
8*b^27*c^4*d^13*e^23 - 98*b^28*c^3*d^12*e^24) - ((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*(8*b^10*c^23*d^37
*e^3 - 148*b^11*c^22*d^36*e^4 + 1160*b^12*c^21*d^35*e^5 - 4760*b^13*c^20*d^34*e^6 + 8036*b^14*c^19*d^33*e^7 +
21868*b^15*c^18*d^32*e^8 - 194304*b^16*c^17*d^31*e^9 + 709280*b^17*c^16*d^30*e^10 - 1744160*b^18*c^15*d^29*e^1
1 + 3218072*b^19*c^14*d^28*e^12 - 4654832*b^20*c^13*d^27*e^13 + 5394480*b^21*c^12*d^26*e^14 - 5063240*b^22*c^1
1*d^25*e^15 + 3863800*b^23*c^10*d^24*e^16 - 2393152*b^24*c^9*d^23*e^17 + 1194528*b^25*c^8*d^22*e^18 - 474056*b
^26*c^7*d^21*e^19 + 146300*b^27*c^6*d^20*e^20 - 33880*b^28*c^5*d^19*e^21 + 5544*b^29*c^4*d^18*e^22 - 572*b^30*
c^3*d^17*e^23 + 28*b^31*c^2*d^16*e^24 - ((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*
c^23*d^41*e^2 - 328*b^13*c^22*d^40*e^3 + 3200*b^14*c^21*d^39*e^4 - 19760*b^15*c^20*d^38*e^5 + 86640*b^16*c^19*
d^37*e^6 - 286824*b^17*c^18*d^36*e^7 + 744192*b^18*c^17*d^35*e^8 - 1550400*b^19*c^16*d^34*e^9 + 2635680*b^20*c
^15*d^33*e^10 - 3695120*b^21*c^14*d^32*e^11 + 4299776*b^22*c^13*d^31*e^12 - 4165408*b^23*c^12*d^30*e^13 + 3359
200*b^24*c^11*d^29*e^14 - 2248080*b^25*c^10*d^28*e^15 + 1240320*b^26*c^9*d^27*e^16 - 558144*b^27*c^8*d^26*e^17
+ 201552*b^28*c^7*d^25*e^18 - 57000*b^29*c^6*d^24*e^19 + 12160*b^30*c^5*d^23*e^20 - 1840*b^31*c^4*d^22*e^21 +
176*b^32*c^3*d^21*e^22 - 8*b^33*c^2*d^20*e^23))/(2*(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7
*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b^7*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e
^7 - 9*b^11*c*d*e^8))))/(2*(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3
- 126*b^7*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8)))*1i
)/(2*(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b^7*c^5*d^5*e^4
+ 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8)) + ((-c^9*(b*e - c*d)^9)^(
1/2)*(11*b*e - 4*c*d)*((d + e*x)^(1/2)*(1088*b^7*c^24*d^33*e^3 - 64*b^6*c^25*d^34*e^2 - 8404*b^8*c^23*d^32*e^4
+ 38720*b^9*c^22*d^31*e^5 - 116512*b^10*c^21*d^30*e^6 + 230912*b^11*c^20*d^29*e^7 - 267432*b^12*c^19*d^28*e^8
+ 38544*b^13*c^18*d^27*e^9 + 473880*b^14*c^17*d^26*e^10 - 851136*b^15*c^16*d^25*e^11 + 393646*b^16*c^15*d^24*
e^12 + 1207368*b^17*c^14*d^23*e^13 - 3343724*b^18*c^13*d^22*e^14 + 4835160*b^19*c^12*d^21*e^15 - 4903382*b^20*
c^11*d^20*e^16 + 3751968*b^21*c^10*d^19*e^17 - 2217072*b^22*c^9*d^18*e^18 + 1013232*b^23*c^8*d^17*e^19 - 35321
0*b^24*c^7*d^16*e^20 + 91080*b^25*c^6*d^15*e^21 - 16412*b^26*c^5*d^14*e^22 + 1848*b^27*c^4*d^13*e^23 - 98*b^28
*c^3*d^12*e^24) + ((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*(8*b^10*c^23*d^37*e^3 - 148*b^11*c^22*d^36*e^4
+ 1160*b^12*c^21*d^35*e^5 - 4760*b^13*c^20*d^34*e^6 + 8036*b^14*c^19*d^33*e^7 + 21868*b^15*c^18*d^32*e^8 - 194
304*b^16*c^17*d^31*e^9 + 709280*b^17*c^16*d^30*e^10 - 1744160*b^18*c^15*d^29*e^11 + 3218072*b^19*c^14*d^28*e^1
2 - 4654832*b^20*c^13*d^27*e^13 + 5394480*b^21*c^12*d^26*e^14 - 5063240*b^22*c^11*d^25*e^15 + 3863800*b^23*c^1
0*d^24*e^16 - 2393152*b^24*c^9*d^23*e^17 + 1194528*b^25*c^8*d^22*e^18 - 474056*b^26*c^7*d^21*e^19 + 146300*b^2
7*c^6*d^20*e^20 - 33880*b^28*c^5*d^19*e^21 + 5544*b^29*c^4*d^18*e^22 - 572*b^30*c^3*d^17*e^23 + 28*b^31*c^2*d^
16*e^24 + ((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*c^23*d^41*e^2 - 328*b^13*c^22*
d^40*e^3 + 3200*b^14*c^21*d^39*e^4 - 19760*b^15*c^20*d^38*e^5 + 86640*b^16*c^19*d^37*e^6 - 286824*b^17*c^18*d^
36*e^7 + 744192*b^18*c^17*d^35*e^8 - 1550400*b^19*c^16*d^34*e^9 + 2635680*b^20*c^15*d^33*e^10 - 3695120*b^21*c
^14*d^32*e^11 + 4299776*b^22*c^13*d^31*e^12 - 4165408*b^23*c^12*d^30*e^13 + 3359200*b^24*c^11*d^29*e^14 - 2248
080*b^25*c^10*d^28*e^15 + 1240320*b^26*c^9*d^27*e^16 - 558144*b^27*c^8*d^26*e^17 + 201552*b^28*c^7*d^25*e^18 -
57000*b^29*c^6*d^24*e^19 + 12160*b^30*c^5*d^23*e^20 - 1840*b^31*c^4*d^22*e^21 + 176*b^32*c^3*d^21*e^22 - 8*b^
33*c^2*d^20*e^23))/(2*(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3 - 12
6*b^7*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8))))/(2*(b^
12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b^7*c^5*d^5*e^4 + 126*b
^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8)))*1i)/(2*(b^12*e^9 - b^3*c^9*d^9 +
9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b^7*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^
9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8)))/(((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*((d + e*
x)^(1/2)*(1088*b^7*c^24*d^33*e^3 - 64*b^6*c^25*d^34*e^2 - 8404*b^8*c^23*d^32*e^4 + 38720*b^9*c^22*d^31*e^5 - 1
16512*b^10*c^21*d^30*e^6 + 230912*b^11*c^20*d^29*e^7 - 267432*b^12*c^19*d^28*e^8 + 38544*b^13*c^18*d^27*e^9 +
473880*b^14*c^17*d^26*e^10 - 851136*b^15*c^16*d^25*e^11 + 393646*b^16*c^15*d^24*e^12 + 1207368*b^17*c^14*d^23*
e^13 - 3343724*b^18*c^13*d^22*e^14 + 4835160*b^19*c^12*d^21*e^15 - 4903382*b^20*c^11*d^20*e^16 + 3751968*b^21*
c^10*d^19*e^17 - 2217072*b^22*c^9*d^18*e^18 + 1013232*b^23*c^8*d^17*e^19 - 353210*b^24*c^7*d^16*e^20 + 91080*b
^25*c^6*d^15*e^21 - 16412*b^26*c^5*d^14*e^22 + 1848*b^27*c^4*d^13*e^23 - 98*b^28*c^3*d^12*e^24) + ((-c^9*(b*e
- c*d)^9)^(1/2)*(11*b*e - 4*c*d)*(8*b^10*c^23*d^37*e^3 - 148*b^11*c^22*d^36*e^4 + 1160*b^12*c^21*d^35*e^5 - 47
60*b^13*c^20*d^34*e^6 + 8036*b^14*c^19*d^33*e^7 + 21868*b^15*c^18*d^32*e^8 - 194304*b^16*c^17*d^31*e^9 + 70928
0*b^17*c^16*d^30*e^10 - 1744160*b^18*c^15*d^29*e^11 + 3218072*b^19*c^14*d^28*e^12 - 4654832*b^20*c^13*d^27*e^1
3 + 5394480*b^21*c^12*d^26*e^14 - 5063240*b^22*c^11*d^25*e^15 + 3863800*b^23*c^10*d^24*e^16 - 2393152*b^24*c^9
*d^23*e^17 + 1194528*b^25*c^8*d^22*e^18 - 474056*b^26*c^7*d^21*e^19 + 146300*b^27*c^6*d^20*e^20 - 33880*b^28*c
^5*d^19*e^21 + 5544*b^29*c^4*d^18*e^22 - 572*b^30*c^3*d^17*e^23 + 28*b^31*c^2*d^16*e^24 + ((-c^9*(b*e - c*d)^9
)^(1/2)*(11*b*e - 4*c*d)*(d + e*x)^(1/2)*(16*b^12*c^23*d^41*e^2 - 328*b^13*c^22*d^40*e^3 + 3200*b^14*c^21*d^39
*e^4 - 19760*b^15*c^20*d^38*e^5 + 86640*b^16*c^19*d^37*e^6 - 286824*b^17*c^18*d^36*e^7 + 744192*b^18*c^17*d^35
*e^8 - 1550400*b^19*c^16*d^34*e^9 + 2635680*b^20*c^15*d^33*e^10 - 3695120*b^21*c^14*d^32*e^11 + 4299776*b^22*c
^13*d^31*e^12 - 4165408*b^23*c^12*d^30*e^13 + 3359200*b^24*c^11*d^29*e^14 - 2248080*b^25*c^10*d^28*e^15 + 1240
320*b^26*c^9*d^27*e^16 - 558144*b^27*c^8*d^26*e^17 + 201552*b^28*c^7*d^25*e^18 - 57000*b^29*c^6*d^24*e^19 + 12
160*b^30*c^5*d^23*e^20 - 1840*b^31*c^4*d^22*e^21 + 176*b^32*c^3*d^21*e^22 - 8*b^33*c^2*d^20*e^23))/(2*(b^12*e^
9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b^7*c^5*d^5*e^4 + 126*b^8*c^
4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8))))/(2*(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c
^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b^7*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^
3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8))))/(2*(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7
*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b^7*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e
^7 - 9*b^11*c*d*e^8)) - ((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*((d + e*x)^(1/2)*(1088*b^7*c^24*d^33*e^3
- 64*b^6*c^25*d^34*e^2 - 8404*b^8*c^23*d^32*e^4 + 38720*b^9*c^22*d^31*e^5 - 116512*b^10*c^21*d^30*e^6 + 230912
*b^11*c^20*d^29*e^7 - 267432*b^12*c^19*d^28*e^8 + 38544*b^13*c^18*d^27*e^9 + 473880*b^14*c^17*d^26*e^10 - 8511
36*b^15*c^16*d^25*e^11 + 393646*b^16*c^15*d^24*e^12 + 1207368*b^17*c^14*d^23*e^13 - 3343724*b^18*c^13*d^22*e^1
4 + 4835160*b^19*c^12*d^21*e^15 - 4903382*b^20*c^11*d^20*e^16 + 3751968*b^21*c^10*d^19*e^17 - 2217072*b^22*c^9
*d^18*e^18 + 1013232*b^23*c^8*d^17*e^19 - 353210*b^24*c^7*d^16*e^20 + 91080*b^25*c^6*d^15*e^21 - 16412*b^26*c^
5*d^14*e^22 + 1848*b^27*c^4*d^13*e^23 - 98*b^28*c^3*d^12*e^24) - ((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*
(8*b^10*c^23*d^37*e^3 - 148*b^11*c^22*d^36*e^4 + 1160*b^12*c^21*d^35*e^5 - 4760*b^13*c^20*d^34*e^6 + 8036*b^14
*c^19*d^33*e^7 + 21868*b^15*c^18*d^32*e^8 - 194304*b^16*c^17*d^31*e^9 + 709280*b^17*c^16*d^30*e^10 - 1744160*b
^18*c^15*d^29*e^11 + 3218072*b^19*c^14*d^28*e^12 - 4654832*b^20*c^13*d^27*e^13 + 5394480*b^21*c^12*d^26*e^14 -
5063240*b^22*c^11*d^25*e^15 + 3863800*b^23*c^10*d^24*e^16 - 2393152*b^24*c^9*d^23*e^17 + 1194528*b^25*c^8*d^2
2*e^18 - 474056*b^26*c^7*d^21*e^19 + 146300*b^27*c^6*d^20*e^20 - 33880*b^28*c^5*d^19*e^21 + 5544*b^29*c^4*d^18
*e^22 - 572*b^30*c^3*d^17*e^23 + 28*b^31*c^2*d^16*e^24 - ((-c^9*(b*e - c*d)^9)^(1/2)*(11*b*e - 4*c*d)*(d + e*x
)^(1/2)*(16*b^12*c^23*d^41*e^2 - 328*b^13*c^22*d^40*e^3 + 3200*b^14*c^21*d^39*e^4 - 19760*b^15*c^20*d^38*e^5 +
86640*b^16*c^19*d^37*e^6 - 286824*b^17*c^18*d^36*e^7 + 744192*b^18*c^17*d^35*e^8 - 1550400*b^19*c^16*d^34*e^9
+ 2635680*b^20*c^15*d^33*e^10 - 3695120*b^21*c^14*d^32*e^11 + 4299776*b^22*c^13*d^31*e^12 - 4165408*b^23*c^12
*d^30*e^13 + 3359200*b^24*c^11*d^29*e^14 - 2248080*b^25*c^10*d^28*e^15 + 1240320*b^26*c^9*d^27*e^16 - 558144*b
^27*c^8*d^26*e^17 + 201552*b^28*c^7*d^25*e^18 - 57000*b^29*c^6*d^24*e^19 + 12160*b^30*c^5*d^23*e^20 - 1840*b^3
1*c^4*d^22*e^21 + 176*b^32*c^3*d^21*e^22 - 8*b^33*c^2*d^20*e^23))/(2*(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e
- 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b^7*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 +
36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8))))/(2*(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 8
4*b^6*c^6*d^6*e^3 - 126*b^7*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b
^11*c*d*e^8))))/(2*(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b
^7*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8)) - 64*b^4*c^
25*d^30*e^3 + 960*b^5*c^24*d^29*e^4 - 7180*b^6*c^23*d^28*e^5 + 35560*b^7*c^22*d^27*e^6 - 125748*b^8*c^21*d^26*
e^7 + 314496*b^9*c^20*d^25*e^8 - 508886*b^10*c^19*d^24*e^9 + 326832*b^11*c^18*d^23*e^10 + 760408*b^12*c^17*d^2
2*e^11 - 2806584*b^13*c^16*d^21*e^12 + 4917990*b^14*c^15*d^20*e^13 - 5803448*b^15*c^14*d^19*e^14 + 4974956*b^1
6*c^13*d^18*e^15 - 3162096*b^17*c^12*d^17*e^16 + 1483782*b^18*c^11*d^16*e^17 - 501472*b^19*c^10*d^15*e^18 + 11
5824*b^20*c^9*d^14*e^19 - 16408*b^21*c^8*d^13*e^20 + 1078*b^22*c^7*d^12*e^21))*(-c^9*(b*e - c*d)^9)^(1/2)*(11*
b*e - 4*c*d)*1i)/(b^12*e^9 - b^3*c^9*d^9 + 9*b^4*c^8*d^8*e - 36*b^5*c^7*d^7*e^2 + 84*b^6*c^6*d^6*e^3 - 126*b^7
*c^5*d^5*e^4 + 126*b^8*c^4*d^4*e^5 - 84*b^9*c^3*d^3*e^6 + 36*b^10*c^2*d^2*e^7 - 9*b^11*c*d*e^8) - ((2*e^3)/(5*
(c*d^2 - b*d*e)) + (2*e^3*(d + e*x)^2*(35*b^2*e^2 + 113*c^2*d^2 - 113*b*c*d*e))/(15*(c*d^2 - b*d*e)^3) - (14*e
^3*(b*e - 2*c*d)*(d + e*x))/(15*(c*d^2 - b*d*e)^2) + (e*(d + e*x)^4*(2*c^5*d^4 + 7*b^4*c*e^4 - 24*b^3*c^2*d*e^
3 + 26*b^2*c^3*d^2*e^2 - 4*b*c^4*d^3*e))/(b^2*(c*d^2 - b*d*e)^4) + (e*(b*e - 2*c*d)*(d + e*x)^3*(21*b^4*e^4 +
3*c^4*d^4 + 68*b^2*c^2*d^2*e^2 - 6*b*c^3*d^3*e - 65*b^3*c*d*e^3))/(3*b^2*(c*d^2 - b*d*e)^4))/(c*(d + e*x)^(9/2
) + (c*d^2 - b*d*e)*(d + e*x)^(5/2) + (b*e - 2*c*d)*(d + e*x)^(7/2)) - (atan((b^24*d^20*e^24*(d + e*x)^(1/2)*3
43i - b^23*c*d^21*e^23*(d + e*x)^(1/2)*6615i - b^3*c^21*d^41*e^3*(d + e*x)^(1/2)*924i + b^4*c^20*d^40*e^4*(d +
e*x)^(1/2)*13167i - b^5*c^19*d^39*e^5*(d + e*x)^(1/2)*83160i + b^6*c^18*d^38*e^6*(d + e*x)^(1/2)*298914i - b^
7*c^17*d^37*e^7*(d + e*x)^(1/2)*627066i + b^8*c^16*d^36*e^8*(d + e*x)^(1/2)*548163i + b^9*c^15*d^35*e^9*(d + e
*x)^(1/2)*953260i - b^10*c^14*d^34*e^10*(d + e*x)^(1/2)*4260564i + b^11*c^13*d^33*e^11*(d + e*x)^(1/2)*7526715
i - b^12*c^12*d^32*e^12*(d + e*x)^(1/2)*7070070i + b^13*c^11*d^31*e^13*(d + e*x)^(1/2)*735546i + b^14*c^10*d^3
0*e^14*(d + e*x)^(1/2)*9071172i - b^15*c^9*d^29*e^15*(d + e*x)^(1/2)*16762207i + b^16*c^8*d^28*e^16*(d + e*x)^
(1/2)*18354798i - b^17*c^7*d^27*e^17*(d + e*x)^(1/2)*14431032i + b^18*c^6*d^26*e^18*(d + e*x)^(1/2)*8573180i -
b^19*c^5*d^25*e^19*(d + e*x)^(1/2)*3893967i + b^20*c^4*d^24*e^20*(d + e*x)^(1/2)*1340031i - b^21*c^3*d^23*e^2
1*(d + e*x)^(1/2)*339702i + b^22*c^2*d^22*e^22*(d + e*x)^(1/2)*60018i)/(d^9*(d^9)^(1/2)*(d^9*(d^9*(d^9*(13167*
b^4*c^20*e^4 - 924*b^3*c^21*d*e^3) + 735546*b^13*c^11*e^13 - 7070070*b^12*c^12*d*e^12 - 83160*b^5*c^19*d^8*e^5
+ 298914*b^6*c^18*d^7*e^6 - 627066*b^7*c^17*d^6*e^7 + 548163*b^8*c^16*d^5*e^8 + 953260*b^9*c^15*d^4*e^9 - 426
0564*b^10*c^14*d^3*e^10 + 7526715*b^11*c^13*d^2*e^11) + 60018*b^22*c^2*e^22 - 339702*b^21*c^3*d*e^21 + 9071172
*b^14*c^10*d^8*e^14 - 16762207*b^15*c^9*d^7*e^15 + 18354798*b^16*c^8*d^6*e^16 - 14431032*b^17*c^7*d^5*e^17 + 8
573180*b^18*c^6*d^4*e^18 - 3893967*b^19*c^5*d^3*e^19 + 1340031*b^20*c^4*d^2*e^20) + 343*b^24*d^7*e^24 - 6615*b
^23*c*d^8*e^23)))*(7*b*e + 4*c*d)*1i)/(b^3*(d^9)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (b + c x\right )^{2} \left (d + e x\right )^{\frac {7}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(7/2)/(c*x**2+b*x)**2,x)

[Out]

Integral(1/(x**2*(b + c*x)**2*(d + e*x)**(7/2)), x)

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