3.1683 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=193 \[ -\frac{b^3 (d+e x)^2 (-4 a B e-A b e+5 b B d)}{2 e^6}+\frac{2 b^2 x (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^5}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6 (d+e x)}+\frac{(b d-a e)^4 (B d-A e)}{2 e^6 (d+e x)^2}-\frac{2 b (b d-a e)^2 \log (d+e x) (-2 a B e-3 A b e+5 b B d)}{e^6}+\frac{b^4 B (d+e x)^3}{3 e^6} \]

[Out]

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

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Rubi [A]  time = 0.595004, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{b^3 (d+e x)^2 (-4 a B e-A b e+5 b B d)}{2 e^6}+\frac{2 b^2 x (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^5}-\frac{(b d-a e)^3 (-a B e-4 A b e+5 b B d)}{e^6 (d+e x)}+\frac{(b d-a e)^4 (B d-A e)}{2 e^6 (d+e x)^2}-\frac{2 b (b d-a e)^2 \log (d+e x) (-2 a B e-3 A b e+5 b B d)}{e^6}+\frac{b^4 B (d+e x)^3}{3 e^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 113.022, size = 192, normalized size = 0.99 \[ \frac{B b^{4} \left (d + e x\right )^{3}}{3 e^{6}} + \frac{b^{3} \left (d + e x\right )^{2} \left (A b e + 4 B a e - 5 B b d\right )}{2 e^{6}} + \frac{2 b^{2} x \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{e^{5}} + \frac{2 b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{\left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{e^{6} \left (d + e x\right )} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{4}}{2 e^{6} \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b**4*(d + e*x)**3/(3*e**6) + b**3*(d + e*x)**2*(A*b*e + 4*B*a*e - 5*B*b*d)/(2*
e**6) + 2*b**2*x*(a*e - b*d)*(2*A*b*e + 3*B*a*e - 5*B*b*d)/e**5 + 2*b*(a*e - b*d
)**2*(3*A*b*e + 2*B*a*e - 5*B*b*d)*log(d + e*x)/e**6 - (a*e - b*d)**3*(4*A*b*e +
B*a*e - 5*B*b*d)/(e**6*(d + e*x)) - (A*e - B*d)*(a*e - b*d)**4/(2*e**6*(d + e*x
)**2)

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Mathematica [A]  time = 0.188304, size = 187, normalized size = 0.97 \[ \frac{-6 b^2 e x \left (-6 a^2 B e^2-4 a b e (A e-3 B d)+3 b^2 d (A e-2 B d)\right )+3 b^3 e^2 x^2 (4 a B e+A b e-3 b B d)-\frac{6 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{d+e x}+\frac{3 (b d-a e)^4 (B d-A e)}{(d+e x)^2}-12 b (b d-a e)^2 \log (d+e x) (-2 a B e-3 A b e+5 b B d)+2 b^4 B e^3 x^3}{6 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(-6*b^2*e*(-6*a^2*B*e^2 - 4*a*b*e*(-3*B*d + A*e) + 3*b^2*d*(-2*B*d + A*e))*x + 3
*b^3*e^2*(-3*b*B*d + A*b*e + 4*a*B*e)*x^2 + 2*b^4*B*e^3*x^3 + (3*(b*d - a*e)^4*(
B*d - A*e))/(d + e*x)^2 - (6*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e))/(d + e*x
) - 12*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*Log[d + e*x])/(6*e^6)

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Maple [B]  time = 0.017, size = 601, normalized size = 3.1 \[ 4\,{\frac{b\ln \left ( ex+d \right ) B{a}^{3}}{{e}^{3}}}-10\,{\frac{{b}^{4}\ln \left ( ex+d \right ) B{d}^{3}}{{e}^{6}}}+2\,{\frac{B{x}^{2}{b}^{3}a}{{e}^{3}}}-{\frac{3\,{b}^{4}B{x}^{2}d}{2\,{e}^{4}}}+6\,{\frac{{b}^{2}\ln \left ( ex+d \right ) A{a}^{2}}{{e}^{3}}}-{\frac{A{d}^{4}{b}^{4}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{Bd{a}^{4}}{2\,{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{B{b}^{4}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{b}^{4}\ln \left ( ex+d \right ) A{d}^{2}}{{e}^{5}}}+4\,{\frac{Aa{b}^{3}x}{{e}^{3}}}-3\,{\frac{Ad{b}^{4}x}{{e}^{4}}}+6\,{\frac{{b}^{2}B{a}^{2}x}{{e}^{3}}}+6\,{\frac{B{b}^{4}{d}^{2}x}{{e}^{5}}}+{\frac{A{b}^{4}{x}^{2}}{2\,{e}^{3}}}-{\frac{B{a}^{4}}{{e}^{2} \left ( ex+d \right ) }}-5\,{\frac{B{b}^{4}{d}^{4}}{{e}^{6} \left ( ex+d \right ) }}-2\,{\frac{B{d}^{2}{a}^{3}b}{{e}^{3} \left ( ex+d \right ) ^{2}}}+3\,{\frac{{b}^{2}B{a}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{A{d}^{3}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{Ad{a}^{3}b}{{e}^{2} \left ( ex+d \right ) ^{2}}}-3\,{\frac{A{d}^{2}{a}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) ^{2}}}-12\,{\frac{Ba{b}^{3}dx}{{e}^{4}}}-18\,{\frac{{b}^{2}B{a}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) }}+16\,{\frac{Ba{b}^{3}{d}^{3}}{{e}^{5} \left ( ex+d \right ) }}-12\,{\frac{{b}^{3}\ln \left ( ex+d \right ) Aad}{{e}^{4}}}-18\,{\frac{{b}^{2}\ln \left ( ex+d \right ) B{a}^{2}d}{{e}^{4}}}+24\,{\frac{{b}^{3}\ln \left ( ex+d \right ) Ba{d}^{2}}{{e}^{5}}}+12\,{\frac{Ad{a}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}-12\,{\frac{A{d}^{2}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+8\,{\frac{{a}^{3}bBd}{{e}^{3} \left ( ex+d \right ) }}-4\,{\frac{A{a}^{3}b}{{e}^{2} \left ( ex+d \right ) }}+4\,{\frac{A{d}^{3}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-2\,{\frac{Ba{b}^{3}{d}^{4}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{A{a}^{4}}{2\,e \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{4}B{x}^{3}}{3\,{e}^{3}}} \]

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)^2/(e*x+d)^3,x)

[Out]

4*b/e^3*ln(e*x+d)*B*a^3-10*b^4/e^6*ln(e*x+d)*B*d^3+2*b^3/e^3*B*x^2*a-3/2*b^4/e^4
*B*x^2*d+6*b^2/e^3*ln(e*x+d)*A*a^2-1/2/e^5/(e*x+d)^2*A*b^4*d^4+1/2/e^2/(e*x+d)^2
*B*d*a^4+1/2/e^6/(e*x+d)^2*B*b^4*d^5+6*b^4/e^5*ln(e*x+d)*A*d^2+4*b^3/e^3*A*a*x-3
*b^4/e^4*A*d*x+6*b^2/e^3*a^2*B*x+6*b^4/e^5*B*d^2*x+1/2*b^4/e^3*A*x^2-1/e^2/(e*x+
d)*B*a^4-5/e^6/(e*x+d)*B*b^4*d^4-2/e^3/(e*x+d)^2*B*d^2*a^3*b+3/e^4/(e*x+d)^2*B*a
^2*b^2*d^3+2/e^4/(e*x+d)^2*A*a*b^3*d^3+2/e^2/(e*x+d)^2*A*d*a^3*b-3/e^3/(e*x+d)^2
*A*d^2*a^2*b^2-12*b^3/e^4*B*a*d*x-18/e^4/(e*x+d)*B*a^2*b^2*d^2+16/e^5/(e*x+d)*B*
a*b^3*d^3-12*b^3/e^4*ln(e*x+d)*A*a*d-18*b^2/e^4*ln(e*x+d)*B*a^2*d+24*b^3/e^5*ln(
e*x+d)*B*a*d^2+12/e^3/(e*x+d)*A*a^2*b^2*d-12/e^4/(e*x+d)*A*a*b^3*d^2+8/e^3/(e*x+
d)*B*a^3*b*d-4/e^2/(e*x+d)*A*a^3*b+4/e^5/(e*x+d)*A*b^4*d^3-2/e^5/(e*x+d)^2*B*a*b
^3*d^4-1/2/e/(e*x+d)^2*A*a^4+1/3*b^4/e^3*B*x^3

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Maxima [A]  time = 0.701758, size = 566, normalized size = 2.93 \[ -\frac{9 \, B b^{4} d^{5} + A a^{4} e^{5} - 7 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 10 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 6 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 2 \,{\left (5 \, B b^{4} d^{4} e - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{2 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac{2 \, B b^{4} e^{2} x^{3} - 3 \,{\left (3 \, B b^{4} d e -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{2}\right )} x^{2} + 6 \,{\left (6 \, B b^{4} d^{2} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} x}{6 \, e^{5}} - \frac{2 \,{\left (5 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(9*B*b^4*d^5 + A*a^4*e^5 - 7*(4*B*a*b^3 + A*b^4)*d^4*e + 10*(3*B*a^2*b^2 +
2*A*a*b^3)*d^3*e^2 - 6*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*A*a^3*b)*d
*e^4 + 2*(5*B*b^4*d^4*e - 4*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 6*(3*B*a^2*b^2 + 2*A*a
*b^3)*d^2*e^3 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + (B*a^4 + 4*A*a^3*b)*e^5)*x)/
(e^8*x^2 + 2*d*e^7*x + d^2*e^6) + 1/6*(2*B*b^4*e^2*x^3 - 3*(3*B*b^4*d*e - (4*B*a
*b^3 + A*b^4)*e^2)*x^2 + 6*(6*B*b^4*d^2 - 3*(4*B*a*b^3 + A*b^4)*d*e + 2*(3*B*a^2
*b^2 + 2*A*a*b^3)*e^2)*x)/e^5 - 2*(5*B*b^4*d^3 - 3*(4*B*a*b^3 + A*b^4)*d^2*e + 3
*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 - (2*B*a^3*b + 3*A*a^2*b^2)*e^3)*log(e*x + d)/e
^6

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Fricas [A]  time = 0.292719, size = 880, normalized size = 4.56 \[ \frac{2 \, B b^{4} e^{5} x^{5} - 27 \, B b^{4} d^{5} - 3 \, A a^{4} e^{5} + 21 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 30 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 18 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} -{\left (5 \, B b^{4} d e^{4} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 4 \,{\left (5 \, B b^{4} d^{2} e^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 3 \,{\left (21 \, B b^{4} d^{3} e^{2} - 11 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 8 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4}\right )} x^{2} + 6 \,{\left (B b^{4} d^{4} e +{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} - 4 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} -{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x - 12 \,{\left (5 \, B b^{4} d^{5} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (5 \, B b^{4} d^{3} e^{2} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 2 \,{\left (5 \, B b^{4} d^{4} e - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} -{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*B*b^4*e^5*x^5 - 27*B*b^4*d^5 - 3*A*a^4*e^5 + 21*(4*B*a*b^3 + A*b^4)*d^4*e
- 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 18*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 -
3*(B*a^4 + 4*A*a^3*b)*d*e^4 - (5*B*b^4*d*e^4 - 3*(4*B*a*b^3 + A*b^4)*e^5)*x^4 +
4*(5*B*b^4*d^2*e^3 - 3*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*
e^5)*x^3 + 3*(21*B*b^4*d^3*e^2 - 11*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 8*(3*B*a^2*b^2
+ 2*A*a*b^3)*d*e^4)*x^2 + 6*(B*b^4*d^4*e + (4*B*a*b^3 + A*b^4)*d^3*e^2 - 4*(3*B
*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 - (B*a^4 + 4*A
*a^3*b)*e^5)*x - 12*(5*B*b^4*d^5 - 3*(4*B*a*b^3 + A*b^4)*d^4*e + 3*(3*B*a^2*b^2
+ 2*A*a*b^3)*d^3*e^2 - (2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (5*B*b^4*d^3*e^2 - 3*
(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 - (2*B*a^3*b + 3
*A*a^2*b^2)*e^5)*x^2 + 2*(5*B*b^4*d^4*e - 3*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B
*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 - (2*B*a^3*b + 3*A*a^2*b^2)*d*e^4)*x)*log(e*x + d)
)/(e^8*x^2 + 2*d*e^7*x + d^2*e^6)

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Sympy [A]  time = 33.0395, size = 435, normalized size = 2.25 \[ \frac{B b^{4} x^{3}}{3 e^{3}} + \frac{2 b \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{A a^{4} e^{5} + 4 A a^{3} b d e^{4} - 18 A a^{2} b^{2} d^{2} e^{3} + 20 A a b^{3} d^{3} e^{2} - 7 A b^{4} d^{4} e + B a^{4} d e^{4} - 12 B a^{3} b d^{2} e^{3} + 30 B a^{2} b^{2} d^{3} e^{2} - 28 B a b^{3} d^{4} e + 9 B b^{4} d^{5} + x \left (8 A a^{3} b e^{5} - 24 A a^{2} b^{2} d e^{4} + 24 A a b^{3} d^{2} e^{3} - 8 A b^{4} d^{3} e^{2} + 2 B a^{4} e^{5} - 16 B a^{3} b d e^{4} + 36 B a^{2} b^{2} d^{2} e^{3} - 32 B a b^{3} d^{3} e^{2} + 10 B b^{4} d^{4} e\right )}{2 d^{2} e^{6} + 4 d e^{7} x + 2 e^{8} x^{2}} + \frac{x^{2} \left (A b^{4} e + 4 B a b^{3} e - 3 B b^{4} d\right )}{2 e^{4}} + \frac{x \left (4 A a b^{3} e^{2} - 3 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 12 B a b^{3} d e + 6 B b^{4} d^{2}\right )}{e^{5}} \]

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)**2/(e*x+d)**3,x)

[Out]

B*b**4*x**3/(3*e**3) + 2*b*(a*e - b*d)**2*(3*A*b*e + 2*B*a*e - 5*B*b*d)*log(d +
e*x)/e**6 - (A*a**4*e**5 + 4*A*a**3*b*d*e**4 - 18*A*a**2*b**2*d**2*e**3 + 20*A*a
*b**3*d**3*e**2 - 7*A*b**4*d**4*e + B*a**4*d*e**4 - 12*B*a**3*b*d**2*e**3 + 30*B
*a**2*b**2*d**3*e**2 - 28*B*a*b**3*d**4*e + 9*B*b**4*d**5 + x*(8*A*a**3*b*e**5 -
24*A*a**2*b**2*d*e**4 + 24*A*a*b**3*d**2*e**3 - 8*A*b**4*d**3*e**2 + 2*B*a**4*e
**5 - 16*B*a**3*b*d*e**4 + 36*B*a**2*b**2*d**2*e**3 - 32*B*a*b**3*d**3*e**2 + 10
*B*b**4*d**4*e))/(2*d**2*e**6 + 4*d*e**7*x + 2*e**8*x**2) + x**2*(A*b**4*e + 4*B
*a*b**3*e - 3*B*b**4*d)/(2*e**4) + x*(4*A*a*b**3*e**2 - 3*A*b**4*d*e + 6*B*a**2*
b**2*e**2 - 12*B*a*b**3*d*e + 6*B*b**4*d**2)/e**5

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GIAC/XCAS [A]  time = 0.303304, size = 564, normalized size = 2.92 \[ -2 \,{\left (5 \, B b^{4} d^{3} - 12 \, B a b^{3} d^{2} e - 3 \, A b^{4} d^{2} e + 9 \, B a^{2} b^{2} d e^{2} + 6 \, A a b^{3} d e^{2} - 2 \, B a^{3} b e^{3} - 3 \, A a^{2} b^{2} e^{3}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{6} \,{\left (2 \, B b^{4} x^{3} e^{6} - 9 \, B b^{4} d x^{2} e^{5} + 36 \, B b^{4} d^{2} x e^{4} + 12 \, B a b^{3} x^{2} e^{6} + 3 \, A b^{4} x^{2} e^{6} - 72 \, B a b^{3} d x e^{5} - 18 \, A b^{4} d x e^{5} + 36 \, B a^{2} b^{2} x e^{6} + 24 \, A a b^{3} x e^{6}\right )} e^{\left (-9\right )} - \frac{{\left (9 \, B b^{4} d^{5} - 28 \, B a b^{3} d^{4} e - 7 \, A b^{4} d^{4} e + 30 \, B a^{2} b^{2} d^{3} e^{2} + 20 \, A a b^{3} d^{3} e^{2} - 12 \, B a^{3} b d^{2} e^{3} - 18 \, A a^{2} b^{2} d^{2} e^{3} + B a^{4} d e^{4} + 4 \, A a^{3} b d e^{4} + A a^{4} e^{5} + 2 \,{\left (5 \, B b^{4} d^{4} e - 16 \, B a b^{3} d^{3} e^{2} - 4 \, A b^{4} d^{3} e^{2} + 18 \, B a^{2} b^{2} d^{2} e^{3} + 12 \, A a b^{3} d^{2} e^{3} - 8 \, B a^{3} b d e^{4} - 12 \, A a^{2} b^{2} d e^{4} + B a^{4} e^{5} + 4 \, A a^{3} b e^{5}\right )} x\right )} e^{\left (-6\right )}}{2 \,{\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(5*B*b^4*d^3 - 12*B*a*b^3*d^2*e - 3*A*b^4*d^2*e + 9*B*a^2*b^2*d*e^2 + 6*A*a*b
^3*d*e^2 - 2*B*a^3*b*e^3 - 3*A*a^2*b^2*e^3)*e^(-6)*ln(abs(x*e + d)) + 1/6*(2*B*b
^4*x^3*e^6 - 9*B*b^4*d*x^2*e^5 + 36*B*b^4*d^2*x*e^4 + 12*B*a*b^3*x^2*e^6 + 3*A*b
^4*x^2*e^6 - 72*B*a*b^3*d*x*e^5 - 18*A*b^4*d*x*e^5 + 36*B*a^2*b^2*x*e^6 + 24*A*a
*b^3*x*e^6)*e^(-9) - 1/2*(9*B*b^4*d^5 - 28*B*a*b^3*d^4*e - 7*A*b^4*d^4*e + 30*B*
a^2*b^2*d^3*e^2 + 20*A*a*b^3*d^3*e^2 - 12*B*a^3*b*d^2*e^3 - 18*A*a^2*b^2*d^2*e^3
+ B*a^4*d*e^4 + 4*A*a^3*b*d*e^4 + A*a^4*e^5 + 2*(5*B*b^4*d^4*e - 16*B*a*b^3*d^3
*e^2 - 4*A*b^4*d^3*e^2 + 18*B*a^2*b^2*d^2*e^3 + 12*A*a*b^3*d^2*e^3 - 8*B*a^3*b*d
*e^4 - 12*A*a^2*b^2*d*e^4 + B*a^4*e^5 + 4*A*a^3*b*e^5)*x)*e^(-6)/(x*e + d)^2