3.18.80 \(\int \frac {(a+b x) (a^2+2 a b x+b^2 x^2)^{5/2}}{(d+e x)^8} \, dx\)

Optimal. Leaf size=41 \[ \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (d+e x)^7 (b d-a e)} \]

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Rubi [A]  time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {767} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (d+e x)^7 (b d-a e)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^8,x]

[Out]

(a^2 + 2*a*b*x + b^2*x^2)^(7/2)/(7*(b*d - a*e)*(d + e*x)^7)

Rule 767

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

Rubi steps

\begin {align*} \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^8} \, dx &=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{7/2}}{7 (b d-a e) (d+e x)^7}\\ \end {align*}

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Mathematica [B]  time = 0.11, size = 289, normalized size = 7.05 \begin {gather*} -\frac {\sqrt {(a+b x)^2} \left (a^6 e^6+a^5 b e^5 (d+7 e x)+a^4 b^2 e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a^3 b^3 e^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+a^2 b^4 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+a b^5 e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+b^6 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )\right )}{7 e^7 (a+b x) (d+e x)^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^8,x]

[Out]

-1/7*(Sqrt[(a + b*x)^2]*(a^6*e^6 + a^5*b*e^5*(d + 7*e*x) + a^4*b^2*e^4*(d^2 + 7*d*e*x + 21*e^2*x^2) + a^3*b^3*
e^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + a^2*b^4*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x
^3 + 35*e^4*x^4) + a*b^5*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + b
^6*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35*d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6)))/(e^7*(a +
b*x)*(d + e*x)^7)

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IntegrateAlgebraic [F]  time = 180.03, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((a + b*x)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2))/(d + e*x)^8,x]

[Out]

$Aborted

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fricas [B]  time = 0.42, size = 398, normalized size = 9.71 \begin {gather*} -\frac {7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \, {\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="fricas")

[Out]

-1/7*(7*b^6*e^6*x^6 + b^6*d^6 + a*b^5*d^5*e + a^2*b^4*d^4*e^2 + a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + a^5*b*d*e^
5 + a^6*e^6 + 21*(b^6*d*e^5 + a*b^5*e^6)*x^5 + 35*(b^6*d^2*e^4 + a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 35*(b^6*d^3*
e^3 + a*b^5*d^2*e^4 + a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 21*(b^6*d^4*e^2 + a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4 + a
^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 7*(b^6*d^5*e + a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3 + a^3*b^3*d^2*e^4 + a^4*b^2*d
*e^5 + a^5*b*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e
^9*x^2 + 7*d^6*e^8*x + d^7*e^7)

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giac [B]  time = 0.19, size = 514, normalized size = 12.54 \begin {gather*} -\frac {{\left (7 \, b^{6} x^{6} e^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{6} d x^{5} e^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{6} d^{2} x^{4} e^{4} \mathrm {sgn}\left (b x + a\right ) + 35 \, b^{6} d^{3} x^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, b^{6} d^{4} x^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 7 \, b^{6} d^{5} x e \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, a b^{5} x^{5} e^{6} \mathrm {sgn}\left (b x + a\right ) + 35 \, a b^{5} d x^{4} e^{5} \mathrm {sgn}\left (b x + a\right ) + 35 \, a b^{5} d^{2} x^{3} e^{4} \mathrm {sgn}\left (b x + a\right ) + 21 \, a b^{5} d^{3} x^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 7 \, a b^{5} d^{4} x e^{2} \mathrm {sgn}\left (b x + a\right ) + a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} x^{4} e^{6} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{2} b^{4} d x^{3} e^{5} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{2} b^{4} d^{2} x^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{2} b^{4} d^{3} x e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 35 \, a^{3} b^{3} x^{3} e^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{3} b^{3} d x^{2} e^{5} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{3} b^{3} d^{2} x e^{4} \mathrm {sgn}\left (b x + a\right ) + a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 21 \, a^{4} b^{2} x^{2} e^{6} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{4} b^{2} d x e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 7 \, a^{5} b x e^{6} \mathrm {sgn}\left (b x + a\right ) + a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )} e^{\left (-7\right )}}{7 \, {\left (x e + d\right )}^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/7*(7*b^6*x^6*e^6*sgn(b*x + a) + 21*b^6*d*x^5*e^5*sgn(b*x + a) + 35*b^6*d^2*x^4*e^4*sgn(b*x + a) + 35*b^6*d^
3*x^3*e^3*sgn(b*x + a) + 21*b^6*d^4*x^2*e^2*sgn(b*x + a) + 7*b^6*d^5*x*e*sgn(b*x + a) + b^6*d^6*sgn(b*x + a) +
21*a*b^5*x^5*e^6*sgn(b*x + a) + 35*a*b^5*d*x^4*e^5*sgn(b*x + a) + 35*a*b^5*d^2*x^3*e^4*sgn(b*x + a) + 21*a*b^
5*d^3*x^2*e^3*sgn(b*x + a) + 7*a*b^5*d^4*x*e^2*sgn(b*x + a) + a*b^5*d^5*e*sgn(b*x + a) + 35*a^2*b^4*x^4*e^6*sg
n(b*x + a) + 35*a^2*b^4*d*x^3*e^5*sgn(b*x + a) + 21*a^2*b^4*d^2*x^2*e^4*sgn(b*x + a) + 7*a^2*b^4*d^3*x*e^3*sgn
(b*x + a) + a^2*b^4*d^4*e^2*sgn(b*x + a) + 35*a^3*b^3*x^3*e^6*sgn(b*x + a) + 21*a^3*b^3*d*x^2*e^5*sgn(b*x + a)
+ 7*a^3*b^3*d^2*x*e^4*sgn(b*x + a) + a^3*b^3*d^3*e^3*sgn(b*x + a) + 21*a^4*b^2*x^2*e^6*sgn(b*x + a) + 7*a^4*b
^2*d*x*e^5*sgn(b*x + a) + a^4*b^2*d^2*e^4*sgn(b*x + a) + 7*a^5*b*x*e^6*sgn(b*x + a) + a^5*b*d*e^5*sgn(b*x + a)
+ a^6*e^6*sgn(b*x + a))*e^(-7)/(x*e + d)^7

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maple [B]  time = 0.05, size = 386, normalized size = 9.41 \begin {gather*} -\frac {\left (7 b^{6} e^{6} x^{6}+21 a \,b^{5} e^{6} x^{5}+21 b^{6} d \,e^{5} x^{5}+35 a^{2} b^{4} e^{6} x^{4}+35 a \,b^{5} d \,e^{5} x^{4}+35 b^{6} d^{2} e^{4} x^{4}+35 a^{3} b^{3} e^{6} x^{3}+35 a^{2} b^{4} d \,e^{5} x^{3}+35 a \,b^{5} d^{2} e^{4} x^{3}+35 b^{6} d^{3} e^{3} x^{3}+21 a^{4} b^{2} e^{6} x^{2}+21 a^{3} b^{3} d \,e^{5} x^{2}+21 a^{2} b^{4} d^{2} e^{4} x^{2}+21 a \,b^{5} d^{3} e^{3} x^{2}+21 b^{6} d^{4} e^{2} x^{2}+7 a^{5} b \,e^{6} x +7 a^{4} b^{2} d \,e^{5} x +7 a^{3} b^{3} d^{2} e^{4} x +7 a^{2} b^{4} d^{3} e^{3} x +7 a \,b^{5} d^{4} e^{2} x +7 b^{6} d^{5} e x +a^{6} e^{6}+a^{5} b d \,e^{5}+a^{4} b^{2} d^{2} e^{4}+a^{3} b^{3} d^{3} e^{3}+a^{2} b^{4} d^{4} e^{2}+a \,b^{5} d^{5} e +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{7 \left (e x +d \right )^{7} \left (b x +a \right )^{5} e^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,x)

[Out]

-1/7*(7*b^6*e^6*x^6+21*a*b^5*e^6*x^5+21*b^6*d*e^5*x^5+35*a^2*b^4*e^6*x^4+35*a*b^5*d*e^5*x^4+35*b^6*d^2*e^4*x^4
+35*a^3*b^3*e^6*x^3+35*a^2*b^4*d*e^5*x^3+35*a*b^5*d^2*e^4*x^3+35*b^6*d^3*e^3*x^3+21*a^4*b^2*e^6*x^2+21*a^3*b^3
*d*e^5*x^2+21*a^2*b^4*d^2*e^4*x^2+21*a*b^5*d^3*e^3*x^2+21*b^6*d^4*e^2*x^2+7*a^5*b*e^6*x+7*a^4*b^2*d*e^5*x+7*a^
3*b^3*d^2*e^4*x+7*a^2*b^4*d^3*e^3*x+7*a*b^5*d^4*e^2*x+7*b^6*d^5*e*x+a^6*e^6+a^5*b*d*e^5+a^4*b^2*d^2*e^4+a^3*b^
3*d^3*e^3+a^2*b^4*d^4*e^2+a*b^5*d^5*e+b^6*d^6)*((b*x+a)^2)^(5/2)/(e*x+d)^7/e^7/(b*x+a)^5

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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((b*x+a)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/(e*x+d)^8,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(a*e-b*d>0)', see `assume?` for
more details)Is a*e-b*d zero or nonzero?

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mupad [B]  time = 2.38, size = 1010, normalized size = 24.63 \begin {gather*} \frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{6\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{6\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{6\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{6\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{6\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{6\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^6}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{3\,e^7}+\frac {d\,\left (\frac {b^6\,d}{3\,e^6}-\frac {2\,b^5\,\left (3\,a\,e-2\,b\,d\right )}{3\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^3}-\frac {\left (\frac {a^6}{7\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {6\,a\,b^5}{7\,e}-\frac {b^6\,d}{7\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{7\,e}\right )}{e}+\frac {20\,a^3\,b^3}{7\,e}\right )}{e}-\frac {15\,a^4\,b^2}{7\,e}\right )}{e}+\frac {6\,a^5\,b}{7\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^7}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{5\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{5\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{5\,e^4}-\frac {2\,b^5\,\left (3\,a\,e-b\,d\right )}{5\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{5\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^5}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{2\,e^7}+\frac {b^6\,d}{2\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^2}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{4\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{4\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{4\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{4\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^4}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{e^7\,\left (a+b\,x\right )\,\left (d+e\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*x)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2))/(d + e*x)^8,x)

[Out]

(((b^6*d^5 - 6*a^5*b*e^5 + 15*a^4*b^2*d*e^4 + 15*a^2*b^4*d^3*e^2 - 20*a^3*b^3*d^2*e^3 - 6*a*b^5*d^4*e)/(6*e^7)
+ (d*((b^6*d^4*e + 15*a^4*b^2*e^5 - 6*a*b^5*d^3*e^2 - 20*a^3*b^3*d*e^4 + 15*a^2*b^4*d^2*e^3)/(6*e^7) - (d*((2
0*a^3*b^3*e^5 - b^6*d^3*e^2 + 6*a*b^5*d^2*e^3 - 15*a^2*b^4*d*e^4)/(6*e^7) - (d*((d*((b^6*d)/(6*e^3) - (b^5*(6*
a*e - b*d))/(6*e^3)))/e + (b^4*(15*a^2*e^2 + b^2*d^2 - 6*a*b*d*e))/(6*e^4)))/e))/e))/e)*(a^2 + b^2*x^2 + 2*a*b
*x)^(1/2))/((a + b*x)*(d + e*x)^6) - (((10*b^6*d^2 + 15*a^2*b^4*e^2 - 24*a*b^5*d*e)/(3*e^7) + (d*((b^6*d)/(3*e
^6) - (2*b^5*(3*a*e - 2*b*d))/(3*e^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^3) - ((a^6/(7
*e) - (d*((d*((d*((d*((d*((6*a*b^5)/(7*e) - (b^6*d)/(7*e^2)))/e - (15*a^2*b^4)/(7*e)))/e + (20*a^3*b^3)/(7*e))
)/e - (15*a^4*b^2)/(7*e)))/e + (6*a^5*b)/(7*e)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^7) -
(((5*b^6*d^4 + 15*a^4*b^2*e^4 - 40*a^3*b^3*d*e^3 + 45*a^2*b^4*d^2*e^2 - 24*a*b^5*d^3*e)/(5*e^7) + (d*((4*b^6*
d^3*e - 20*a^3*b^3*e^4 - 18*a*b^5*d^2*e^2 + 30*a^2*b^4*d*e^3)/(5*e^7) + (d*((d*((b^6*d)/(5*e^4) - (2*b^5*(3*a*
e - b*d))/(5*e^4)))/e + (3*b^4*(5*a^2*e^2 + b^2*d^2 - 4*a*b*d*e))/(5*e^5)))/e))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(
1/2))/((a + b*x)*(d + e*x)^5) + (((5*b^6*d - 6*a*b^5*e)/(2*e^7) + (b^6*d)/(2*e^7))*(a^2 + b^2*x^2 + 2*a*b*x)^(
1/2))/((a + b*x)*(d + e*x)^2) + (((10*b^6*d^3 - 20*a^3*b^3*e^3 + 45*a^2*b^4*d*e^2 - 36*a*b^5*d^2*e)/(4*e^7) +
(d*((d*((b^6*d)/(4*e^5) - (3*b^5*(2*a*e - b*d))/(4*e^5)))/e + (3*b^4*(5*a^2*e^2 + 2*b^2*d^2 - 6*a*b*d*e))/(4*e
^6)))/e)*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/((a + b*x)*(d + e*x)^4) - (b^6*(a^2 + b^2*x^2 + 2*a*b*x)^(1/2))/(e^7
*(a + b*x)*(d + e*x))

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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((b*x+a)*(b**2*x**2+2*a*b*x+a**2)**(5/2)/(e*x+d)**8,x)

[Out]

Timed out

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