3.1483 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^5} \, dx\)

Optimal. Leaf size=155 \[ -\frac{b^5 x (5 b d-6 a e)}{e^6}+\frac{15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac{20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac{15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac{2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac{(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac{b^6 x^2}{2 e^5} \]

[Out]

-((b^5*(5*b*d - 6*a*e)*x)/e^6) + (b^6*x^2)/(2*e^5) - (b*d - a*e)^6/(4*e^7*(d + e
*x)^4) + (2*b*(b*d - a*e)^5)/(e^7*(d + e*x)^3) - (15*b^2*(b*d - a*e)^4)/(2*e^7*(
d + e*x)^2) + (20*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)) + (15*b^4*(b*d - a*e)^2*Log
[d + e*x])/e^7

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Rubi [A]  time = 0.37691, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{b^5 x (5 b d-6 a e)}{e^6}+\frac{15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac{20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac{15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac{2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac{(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac{b^6 x^2}{2 e^5} \]

Antiderivative was successfully verified.

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

[Out]

-((b^5*(5*b*d - 6*a*e)*x)/e^6) + (b^6*x^2)/(2*e^5) - (b*d - a*e)^6/(4*e^7*(d + e
*x)^4) + (2*b*(b*d - a*e)^5)/(e^7*(d + e*x)^3) - (15*b^2*(b*d - a*e)^4)/(2*e^7*(
d + e*x)^2) + (20*b^3*(b*d - a*e)^3)/(e^7*(d + e*x)) + (15*b^4*(b*d - a*e)^2*Log
[d + e*x])/e^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b^{6} \int x\, dx}{e^{5}} + \frac{15 b^{4} \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{7}} - \frac{20 b^{3} \left (a e - b d\right )^{3}}{e^{7} \left (d + e x\right )} - \frac{15 b^{2} \left (a e - b d\right )^{4}}{2 e^{7} \left (d + e x\right )^{2}} - \frac{2 b \left (a e - b d\right )^{5}}{e^{7} \left (d + e x\right )^{3}} - \frac{\left (a e - b d\right )^{6}}{4 e^{7} \left (d + e x\right )^{4}} + \frac{\left (6 a e - 5 b d\right ) \int b^{11}\, dx}{b^{6} e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**5,x)

[Out]

b**6*Integral(x, x)/e**5 + 15*b**4*(a*e - b*d)**2*log(d + e*x)/e**7 - 20*b**3*(a
*e - b*d)**3/(e**7*(d + e*x)) - 15*b**2*(a*e - b*d)**4/(2*e**7*(d + e*x)**2) - 2
*b*(a*e - b*d)**5/(e**7*(d + e*x)**3) - (a*e - b*d)**6/(4*e**7*(d + e*x)**4) + (
6*a*e - 5*b*d)*Integral(b**11, x)/(b**6*e**6)

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Mathematica [A]  time = 0.23702, size = 301, normalized size = 1.94 \[ -\frac{a^6 e^6+2 a^5 b e^5 (d+4 e x)+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a^2 b^4 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+2 a b^5 e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-60 b^4 (d+e x)^4 (b d-a e)^2 \log (d+e x)+b^6 \left (-\left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )}{4 e^7 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(a^6*e^6 + 2*a^5*b*e^5*(d + 4*e*x) + 5*a^4*b^2*e^4*(d^2 + 4*d*e*x + 6*e^2*x^2)
+ 20*a^3*b^3*e^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 5*a^2*b^4*d*e^2*(
25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + 2*a*b^5*e*(77*d^5 + 248*d^4*
e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) - b^6*(57*d^
6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4*x^4 - 12*d*e^5*x
^5 + 2*e^6*x^6) - 60*b^4*(b*d - a*e)^2*(d + e*x)^4*Log[d + e*x])/(4*e^7*(d + e*x
)^4)

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Maple [B]  time = 0.016, size = 498, normalized size = 3.2 \[ -{\frac{{a}^{6}}{4\,e \left ( ex+d \right ) ^{4}}}-30\,{\frac{{b}^{5}\ln \left ( ex+d \right ) da}{{e}^{6}}}+{\frac{3\,d{a}^{5}b}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}+20\,{\frac{{d}^{3}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+60\,{\frac{d{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-60\,{\frac{{d}^{2}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}-10\,{\frac{{d}^{4}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+5\,{\frac{{d}^{3}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{d}^{4}{a}^{2}{b}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{{b}^{6}{x}^{2}}{2\,{e}^{5}}}+{\frac{3\,{d}^{5}a{b}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}+30\,{\frac{d{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-45\,{\frac{{d}^{2}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+30\,{\frac{{d}^{3}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{6}{b}^{6}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{b}^{2}{a}^{4}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{d}^{4}{b}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{b}^{5}xa}{{e}^{5}}}-5\,{\frac{{b}^{6}xd}{{e}^{6}}}-2\,{\frac{{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{3}}}+2\,{\frac{{d}^{5}{b}^{6}}{{e}^{7} \left ( ex+d \right ) ^{3}}}+15\,{\frac{{b}^{4}\ln \left ( ex+d \right ){a}^{2}}{{e}^{5}}}+15\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{2}}{{e}^{7}}}-20\,{\frac{{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+20\,{\frac{{d}^{3}{b}^{6}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{15\,{d}^{2}{b}^{2}{a}^{4}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+10\,{\frac{d{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-20\,{\frac{{d}^{2}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^5,x)

[Out]

-1/4/e/(e*x+d)^4*a^6-30*b^5/e^6*ln(e*x+d)*d*a+3/2/e^2/(e*x+d)^4*d*a^5*b+20*b^4/e
^5/(e*x+d)^3*a^2*d^3+60*b^4/e^5/(e*x+d)*a^2*d-60*b^5/e^6/(e*x+d)*a*d^2-10*b^5/e^
6/(e*x+d)^3*a*d^4+5/e^4/(e*x+d)^4*d^3*a^3*b^3-15/4/e^5/(e*x+d)^4*d^4*a^2*b^4+1/2
*b^6*x^2/e^5+3/2/e^6/(e*x+d)^4*d^5*a*b^5+30*b^3/e^4/(e*x+d)^2*a^3*d-45*b^4/e^5/(
e*x+d)^2*d^2*a^2+30*b^5/e^6/(e*x+d)^2*d^3*a-1/4/e^7/(e*x+d)^4*d^6*b^6-15/2*b^2/e
^3/(e*x+d)^2*a^4-15/2*b^6/e^7/(e*x+d)^2*d^4+6*b^5/e^5*x*a-5*b^6/e^6*x*d-2*b/e^2/
(e*x+d)^3*a^5+2*b^6/e^7/(e*x+d)^3*d^5+15*b^4/e^5*ln(e*x+d)*a^2+15*b^6/e^7*ln(e*x
+d)*d^2-20*b^3/e^4/(e*x+d)*a^3+20*b^6/e^7/(e*x+d)*d^3-15/4/e^3/(e*x+d)^4*d^2*b^2
*a^4+10*b^2/e^3/(e*x+d)^3*a^4*d-20*b^3/e^4/(e*x+d)^3*a^3*d^2

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Maxima [A]  time = 0.707129, size = 522, normalized size = 3.37 \[ \frac{57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} + 80 \,{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 30 \,{\left (7 \, b^{6} d^{4} e^{2} - 20 \, a b^{5} d^{3} e^{3} + 18 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (47 \, b^{6} d^{5} e - 130 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{b^{6} e x^{2} - 2 \,{\left (5 \, b^{6} d - 6 \, a b^{5} e\right )} x}{2 \, e^{6}} + \frac{15 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^5,x, algorithm="maxima")

[Out]

1/4*(57*b^6*d^6 - 154*a*b^5*d^5*e + 125*a^2*b^4*d^4*e^2 - 20*a^3*b^3*d^3*e^3 - 5
*a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 - a^6*e^6 + 80*(b^6*d^3*e^3 - 3*a*b^5*d^2*e^4 +
3*a^2*b^4*d*e^5 - a^3*b^3*e^6)*x^3 + 30*(7*b^6*d^4*e^2 - 20*a*b^5*d^3*e^3 + 18*
a^2*b^4*d^2*e^4 - 4*a^3*b^3*d*e^5 - a^4*b^2*e^6)*x^2 + 4*(47*b^6*d^5*e - 130*a*b
^5*d^4*e^2 + 110*a^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 5*a^4*b^2*d*e^5 - 2*a^5*
b*e^6)*x)/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7) + 1/
2*(b^6*e*x^2 - 2*(5*b^6*d - 6*a*b^5*e)*x)/e^6 + 15*(b^6*d^2 - 2*a*b^5*d*e + a^2*
b^4*e^2)*log(e*x + d)/e^7

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Fricas [A]  time = 0.203162, size = 771, normalized size = 4.97 \[ \frac{2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \,{\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \,{\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \,{\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \,{\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} +{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \,{\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \,{\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^5,x, algorithm="fricas")

[Out]

1/4*(2*b^6*e^6*x^6 + 57*b^6*d^6 - 154*a*b^5*d^5*e + 125*a^2*b^4*d^4*e^2 - 20*a^3
*b^3*d^3*e^3 - 5*a^4*b^2*d^2*e^4 - 2*a^5*b*d*e^5 - a^6*e^6 - 12*(b^6*d*e^5 - 2*a
*b^5*e^6)*x^5 - 4*(17*b^6*d^2*e^4 - 24*a*b^5*d*e^5)*x^4 - 16*(2*b^6*d^3*e^3 + 6*
a*b^5*d^2*e^4 - 15*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 6*(22*b^6*d^4*e^2 - 84*a
*b^5*d^3*e^3 + 90*a^2*b^4*d^2*e^4 - 20*a^3*b^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 4*(4
2*b^6*d^5*e - 124*a*b^5*d^4*e^2 + 110*a^2*b^4*d^3*e^3 - 20*a^3*b^3*d^2*e^4 - 5*a
^4*b^2*d*e^5 - 2*a^5*b*e^6)*x + 60*(b^6*d^6 - 2*a*b^5*d^5*e + a^2*b^4*d^4*e^2 +
(b^6*d^2*e^4 - 2*a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 4*(b^6*d^3*e^3 - 2*a*b^5*d^2*e
^4 + a^2*b^4*d*e^5)*x^3 + 6*(b^6*d^4*e^2 - 2*a*b^5*d^3*e^3 + a^2*b^4*d^2*e^4)*x^
2 + 4*(b^6*d^5*e - 2*a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3)*x)*log(e*x + d))/(e^11*x^4
+ 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)

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Sympy [A]  time = 37.1173, size = 393, normalized size = 2.54 \[ \frac{b^{6} x^{2}}{2 e^{5}} + \frac{15 b^{4} \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} + 2 a^{5} b d e^{5} + 5 a^{4} b^{2} d^{2} e^{4} + 20 a^{3} b^{3} d^{3} e^{3} - 125 a^{2} b^{4} d^{4} e^{2} + 154 a b^{5} d^{5} e - 57 b^{6} d^{6} + x^{3} \left (80 a^{3} b^{3} e^{6} - 240 a^{2} b^{4} d e^{5} + 240 a b^{5} d^{2} e^{4} - 80 b^{6} d^{3} e^{3}\right ) + x^{2} \left (30 a^{4} b^{2} e^{6} + 120 a^{3} b^{3} d e^{5} - 540 a^{2} b^{4} d^{2} e^{4} + 600 a b^{5} d^{3} e^{3} - 210 b^{6} d^{4} e^{2}\right ) + x \left (8 a^{5} b e^{6} + 20 a^{4} b^{2} d e^{5} + 80 a^{3} b^{3} d^{2} e^{4} - 440 a^{2} b^{4} d^{3} e^{3} + 520 a b^{5} d^{4} e^{2} - 188 b^{6} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} + \frac{x \left (6 a b^{5} e - 5 b^{6} d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**5,x)

[Out]

b**6*x**2/(2*e**5) + 15*b**4*(a*e - b*d)**2*log(d + e*x)/e**7 - (a**6*e**6 + 2*a
**5*b*d*e**5 + 5*a**4*b**2*d**2*e**4 + 20*a**3*b**3*d**3*e**3 - 125*a**2*b**4*d*
*4*e**2 + 154*a*b**5*d**5*e - 57*b**6*d**6 + x**3*(80*a**3*b**3*e**6 - 240*a**2*
b**4*d*e**5 + 240*a*b**5*d**2*e**4 - 80*b**6*d**3*e**3) + x**2*(30*a**4*b**2*e**
6 + 120*a**3*b**3*d*e**5 - 540*a**2*b**4*d**2*e**4 + 600*a*b**5*d**3*e**3 - 210*
b**6*d**4*e**2) + x*(8*a**5*b*e**6 + 20*a**4*b**2*d*e**5 + 80*a**3*b**3*d**2*e**
4 - 440*a**2*b**4*d**3*e**3 + 520*a*b**5*d**4*e**2 - 188*b**6*d**5*e))/(4*d**4*e
**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + x*(
6*a*b**5*e - 5*b**6*d)/e**6

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GIAC/XCAS [A]  time = 0.217456, size = 694, normalized size = 4.48 \[ \frac{1}{2} \,{\left (b^{6} - \frac{12 \,{\left (b^{6} d e - a b^{5} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-7\right )} - 15 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} e^{\left (-7\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{4} \,{\left (\frac{80 \, b^{6} d^{3} e^{29}}{x e + d} - \frac{30 \, b^{6} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac{8 \, b^{6} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac{b^{6} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} - \frac{240 \, a b^{5} d^{2} e^{30}}{x e + d} + \frac{120 \, a b^{5} d^{3} e^{30}}{{\left (x e + d\right )}^{2}} - \frac{40 \, a b^{5} d^{4} e^{30}}{{\left (x e + d\right )}^{3}} + \frac{6 \, a b^{5} d^{5} e^{30}}{{\left (x e + d\right )}^{4}} + \frac{240 \, a^{2} b^{4} d e^{31}}{x e + d} - \frac{180 \, a^{2} b^{4} d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac{80 \, a^{2} b^{4} d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac{15 \, a^{2} b^{4} d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac{80 \, a^{3} b^{3} e^{32}}{x e + d} + \frac{120 \, a^{3} b^{3} d e^{32}}{{\left (x e + d\right )}^{2}} - \frac{80 \, a^{3} b^{3} d^{2} e^{32}}{{\left (x e + d\right )}^{3}} + \frac{20 \, a^{3} b^{3} d^{3} e^{32}}{{\left (x e + d\right )}^{4}} - \frac{30 \, a^{4} b^{2} e^{33}}{{\left (x e + d\right )}^{2}} + \frac{40 \, a^{4} b^{2} d e^{33}}{{\left (x e + d\right )}^{3}} - \frac{15 \, a^{4} b^{2} d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac{8 \, a^{5} b e^{34}}{{\left (x e + d\right )}^{3}} + \frac{6 \, a^{5} b d e^{34}}{{\left (x e + d\right )}^{4}} - \frac{a^{6} e^{35}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^5,x, algorithm="giac")

[Out]

1/2*(b^6 - 12*(b^6*d*e - a*b^5*e^2)*e^(-1)/(x*e + d))*(x*e + d)^2*e^(-7) - 15*(b
^6*d^2 - 2*a*b^5*d*e + a^2*b^4*e^2)*e^(-7)*ln(abs(x*e + d)*e^(-1)/(x*e + d)^2) +
1/4*(80*b^6*d^3*e^29/(x*e + d) - 30*b^6*d^4*e^29/(x*e + d)^2 + 8*b^6*d^5*e^29/(
x*e + d)^3 - b^6*d^6*e^29/(x*e + d)^4 - 240*a*b^5*d^2*e^30/(x*e + d) + 120*a*b^5
*d^3*e^30/(x*e + d)^2 - 40*a*b^5*d^4*e^30/(x*e + d)^3 + 6*a*b^5*d^5*e^30/(x*e +
d)^4 + 240*a^2*b^4*d*e^31/(x*e + d) - 180*a^2*b^4*d^2*e^31/(x*e + d)^2 + 80*a^2*
b^4*d^3*e^31/(x*e + d)^3 - 15*a^2*b^4*d^4*e^31/(x*e + d)^4 - 80*a^3*b^3*e^32/(x*
e + d) + 120*a^3*b^3*d*e^32/(x*e + d)^2 - 80*a^3*b^3*d^2*e^32/(x*e + d)^3 + 20*a
^3*b^3*d^3*e^32/(x*e + d)^4 - 30*a^4*b^2*e^33/(x*e + d)^2 + 40*a^4*b^2*d*e^33/(x
*e + d)^3 - 15*a^4*b^2*d^2*e^33/(x*e + d)^4 - 8*a^5*b*e^34/(x*e + d)^3 + 6*a^5*b
*d*e^34/(x*e + d)^4 - a^6*e^35/(x*e + d)^4)*e^(-36)