3.1682 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^2} \, dx\)

Optimal. Leaf size=188 \[ -\frac{b^3 (d+e x)^3 (-4 a B e-A b e+5 b B d)}{3 e^6}+\frac{b^2 (d+e x)^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac{(b d-a e)^4 (B d-A e)}{e^6 (d+e x)}+\frac{(b d-a e)^3 \log (d+e x) (-a B e-4 A b e+5 b B d)}{e^6}-\frac{2 b x (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^5}+\frac{b^4 B (d+e x)^4}{4 e^6} \]

[Out]

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

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Rubi [A]  time = 0.69816, antiderivative size = 188, 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 \[ -\frac{b^3 (d+e x)^3 (-4 a B e-A b e+5 b B d)}{3 e^6}+\frac{b^2 (d+e x)^2 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{e^6}+\frac{(b d-a e)^4 (B d-A e)}{e^6 (d+e x)}+\frac{(b d-a e)^3 \log (d+e x) (-a B e-4 A b e+5 b B d)}{e^6}-\frac{2 b x (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{e^5}+\frac{b^4 B (d+e x)^4}{4 e^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.270232, size = 354, normalized size = 1.88 \[ \frac{12 a^4 e^4 (B d-A e)+48 a^3 b e^3 \left (A d e+B \left (-d^2+d e x+e^2 x^2\right )\right )+36 a^2 b^2 e^2 \left (2 A e \left (-d^2+d e x+e^2 x^2\right )+B \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )\right )+8 a b^3 e \left (3 A e \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+2 B \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )\right )+12 (d+e x) (b d-a e)^3 \log (d+e x) (-a B e-4 A b e+5 b B d)+b^4 \left (4 A e \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+B \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )\right )}{12 e^6 (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

(12*a^4*e^4*(B*d - A*e) + 48*a^3*b*e^3*(A*d*e + B*(-d^2 + d*e*x + e^2*x^2)) + 36
*a^2*b^2*e^2*(2*A*e*(-d^2 + d*e*x + e^2*x^2) + B*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^
2 + e^3*x^3)) + 8*a*b^3*e*(3*A*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2
*B*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + b^4*(4*A*e*(-
3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + B*(12*d^5 - 48*d^4*
e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x^5)) + 12*(b*d - a*
e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x)*Log[d + e*x])/(12*e^6*(d + e*x))

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Maple [B]  time = 0.016, size = 564, normalized size = 3. \[ -4\,{\frac{{a}^{3}bB{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{{b}^{2}B{a}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-8\,{\frac{\ln \left ( ex+d \right ) B{a}^{3}bd}{{e}^{3}}}+18\,{\frac{\ln \left ( ex+d \right ) B{a}^{2}{b}^{2}{d}^{2}}{{e}^{4}}}-16\,{\frac{\ln \left ( ex+d \right ) Ba{b}^{3}{d}^{3}}{{e}^{5}}}+4\,{\frac{Ad{a}^{3}b}{{e}^{2} \left ( ex+d \right ) }}-6\,{\frac{A{d}^{2}{a}^{2}{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+4\,{\frac{A{d}^{3}a{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{A{b}^{4}{x}^{2}d}{{e}^{3}}}+3\,{\frac{A{d}^{2}{b}^{4}x}{{e}^{4}}}+{\frac{A{b}^{4}{x}^{3}}{3\,{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) B{a}^{4}}{{e}^{2}}}-{\frac{A{a}^{4}}{e \left ( ex+d \right ) }}+{\frac{{b}^{4}B{x}^{4}}{4\,{e}^{2}}}+4\,{\frac{{a}^{3}bBx}{{e}^{2}}}-4\,{\frac{B{b}^{4}{d}^{3}x}{{e}^{5}}}+4\,{\frac{\ln \left ( ex+d \right ) A{a}^{3}b}{{e}^{2}}}-{\frac{2\,{b}^{4}B{x}^{3}d}{3\,{e}^{3}}}+6\,{\frac{A{a}^{2}{b}^{2}x}{{e}^{2}}}+3\,{\frac{{b}^{2}B{x}^{2}{a}^{2}}{{e}^{2}}}+{\frac{3\,{b}^{4}B{x}^{2}{d}^{2}}{2\,{e}^{4}}}+{\frac{4\,B{b}^{3}{x}^{3}a}{3\,{e}^{2}}}-4\,{\frac{\ln \left ( ex+d \right ) A{b}^{4}{d}^{3}}{{e}^{5}}}+5\,{\frac{\ln \left ( ex+d \right ) B{b}^{4}{d}^{4}}{{e}^{6}}}-{\frac{A{d}^{4}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{Bd{a}^{4}}{{e}^{2} \left ( ex+d \right ) }}+{\frac{B{b}^{4}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}+2\,{\frac{A{b}^{3}{x}^{2}a}{{e}^{2}}}-12\,{\frac{\ln \left ( ex+d \right ) A{a}^{2}{b}^{2}d}{{e}^{3}}}+12\,{\frac{\ln \left ( ex+d \right ) Aa{b}^{3}{d}^{2}}{{e}^{4}}}-4\,{\frac{Ba{b}^{3}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}-4\,{\frac{B{x}^{2}{b}^{3}ad}{{e}^{3}}}-8\,{\frac{Ada{b}^{3}x}{{e}^{3}}}-12\,{\frac{{b}^{2}B{a}^{2}dx}{{e}^{3}}}+12\,{\frac{Ba{b}^{3}{d}^{2}x}{{e}^{4}}} \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 0.697615, size = 554, normalized size = 2.95 \[ \frac{B b^{4} d^{5} - A a^{4} e^{5} -{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}}{e^{7} x + d e^{6}} + \frac{3 \, B b^{4} e^{3} x^{4} - 4 \,{\left (2 \, B b^{4} d e^{2} -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{3}\right )} x^{3} + 6 \,{\left (3 \, B b^{4} d^{2} e - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{2} + 2 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{3}\right )} x^{2} - 12 \,{\left (4 \, B b^{4} d^{3} - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 4 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} x}{12 \, e^{5}} + \frac{{\left (5 \, B b^{4} d^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \]

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)^2,x, algorithm="maxima")

[Out]

(B*b^4*d^5 - A*a^4*e^5 - (4*B*a*b^3 + A*b^4)*d^4*e + 2*(3*B*a^2*b^2 + 2*A*a*b^3)
*d^3*e^2 - 2*(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)/(e^7
*x + d*e^6) + 1/12*(3*B*b^4*e^3*x^4 - 4*(2*B*b^4*d*e^2 - (4*B*a*b^3 + A*b^4)*e^3
)*x^3 + 6*(3*B*b^4*d^2*e - 2*(4*B*a*b^3 + A*b^4)*d*e^2 + 2*(3*B*a^2*b^2 + 2*A*a*
b^3)*e^3)*x^2 - 12*(4*B*b^4*d^3 - 3*(4*B*a*b^3 + A*b^4)*d^2*e + 4*(3*B*a^2*b^2 +
2*A*a*b^3)*d*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^3)*x)/e^5 + (5*B*b^4*d^4 - 4*(
4*B*a*b^3 + A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^2 - 4*(2*B*a^3*b +
3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^3*b)*e^4)*log(e*x + d)/e^6

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Fricas [A]  time = 0.274688, size = 807, normalized size = 4.29 \[ \frac{3 \, B b^{4} e^{5} x^{5} + 12 \, B b^{4} d^{5} - 12 \, A a^{4} e^{5} - 12 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 24 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 24 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 12 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} -{\left (5 \, B b^{4} d e^{4} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 2 \,{\left (5 \, B b^{4} d^{2} e^{3} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} - 6 \,{\left (5 \, B b^{4} d^{3} e^{2} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} - 12 \,{\left (4 \, B b^{4} d^{4} e - 3 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 4 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4}\right )} x + 12 \,{\left (5 \, B b^{4} d^{5} - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} +{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} +{\left (5 \, B b^{4} d^{4} e - 4 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{7} x + d e^{6}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/12*(3*B*b^4*e^5*x^5 + 12*B*b^4*d^5 - 12*A*a^4*e^5 - 12*(4*B*a*b^3 + A*b^4)*d^4
*e + 24*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 24*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3
+ 12*(B*a^4 + 4*A*a^3*b)*d*e^4 - (5*B*b^4*d*e^4 - 4*(4*B*a*b^3 + A*b^4)*e^5)*x^
4 + 2*(5*B*b^4*d^2*e^3 - 4*(4*B*a*b^3 + A*b^4)*d*e^4 + 6*(3*B*a^2*b^2 + 2*A*a*b^
3)*e^5)*x^3 - 6*(5*B*b^4*d^3*e^2 - 4*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 6*(3*B*a^2*b^
2 + 2*A*a*b^3)*d*e^4 - 4*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 - 12*(4*B*b^4*d^4*e
- 3*(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 - 2*(2*B*a
^3*b + 3*A*a^2*b^2)*d*e^4)*x + 12*(5*B*b^4*d^5 - 4*(4*B*a*b^3 + A*b^4)*d^4*e + 6
*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 4*(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 + (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)*log(e*x + d))/(e^7*x + d*e^6)

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Sympy [A]  time = 10.0247, size = 384, normalized size = 2.04 \[ \frac{B b^{4} x^{4}}{4 e^{2}} + \frac{- A a^{4} e^{5} + 4 A a^{3} b d e^{4} - 6 A a^{2} b^{2} d^{2} e^{3} + 4 A a b^{3} d^{3} e^{2} - A b^{4} d^{4} e + B a^{4} d e^{4} - 4 B a^{3} b d^{2} e^{3} + 6 B a^{2} b^{2} d^{3} e^{2} - 4 B a b^{3} d^{4} e + B b^{4} d^{5}}{d e^{6} + e^{7} x} + \frac{x^{3} \left (A b^{4} e + 4 B a b^{3} e - 2 B b^{4} d\right )}{3 e^{3}} + \frac{x^{2} \left (4 A a b^{3} e^{2} - 2 A b^{4} d e + 6 B a^{2} b^{2} e^{2} - 8 B a b^{3} d e + 3 B b^{4} d^{2}\right )}{2 e^{4}} + \frac{x \left (6 A a^{2} b^{2} e^{3} - 8 A a b^{3} d e^{2} + 3 A b^{4} d^{2} e + 4 B a^{3} b e^{3} - 12 B a^{2} b^{2} d e^{2} + 12 B a b^{3} d^{2} e - 4 B b^{4} d^{3}\right )}{e^{5}} + \frac{\left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right ) \log{\left (d + e x \right )}}{e^{6}} \]

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)**2,x)

[Out]

B*b**4*x**4/(4*e**2) + (-A*a**4*e**5 + 4*A*a**3*b*d*e**4 - 6*A*a**2*b**2*d**2*e*
*3 + 4*A*a*b**3*d**3*e**2 - A*b**4*d**4*e + B*a**4*d*e**4 - 4*B*a**3*b*d**2*e**3
+ 6*B*a**2*b**2*d**3*e**2 - 4*B*a*b**3*d**4*e + B*b**4*d**5)/(d*e**6 + e**7*x)
+ x**3*(A*b**4*e + 4*B*a*b**3*e - 2*B*b**4*d)/(3*e**3) + x**2*(4*A*a*b**3*e**2 -
2*A*b**4*d*e + 6*B*a**2*b**2*e**2 - 8*B*a*b**3*d*e + 3*B*b**4*d**2)/(2*e**4) +
x*(6*A*a**2*b**2*e**3 - 8*A*a*b**3*d*e**2 + 3*A*b**4*d**2*e + 4*B*a**3*b*e**3 -
12*B*a**2*b**2*d*e**2 + 12*B*a*b**3*d**2*e - 4*B*b**4*d**3)/e**5 + (a*e - b*d)**
3*(4*A*b*e + B*a*e - 5*B*b*d)*log(d + e*x)/e**6

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GIAC/XCAS [A]  time = 0.298667, size = 710, normalized size = 3.78 \[ \frac{1}{12} \,{\left (3 \, B b^{4} - \frac{4 \,{\left (5 \, B b^{4} d e - 4 \, B a b^{3} e^{2} - A b^{4} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{12 \,{\left (5 \, B b^{4} d^{2} e^{2} - 8 \, B a b^{3} d e^{3} - 2 \, A b^{4} d e^{3} + 3 \, B a^{2} b^{2} e^{4} + 2 \, A a b^{3} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{24 \,{\left (5 \, B b^{4} d^{3} e^{3} - 12 \, B a b^{3} d^{2} e^{4} - 3 \, A b^{4} d^{2} e^{4} + 9 \, B a^{2} b^{2} d e^{5} + 6 \, A a b^{3} d e^{5} - 2 \, B a^{3} b e^{6} - 3 \, A a^{2} b^{2} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )}{\left (x e + d\right )}^{4} e^{\left (-6\right )} -{\left (5 \, B b^{4} d^{4} - 16 \, B a b^{3} d^{3} e - 4 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} + 12 \, A a b^{3} d^{2} e^{2} - 8 \, B a^{3} b d e^{3} - 12 \, A a^{2} b^{2} d e^{3} + B a^{4} e^{4} + 4 \, A a^{3} b e^{4}\right )} e^{\left (-6\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{B b^{4} d^{5} e^{4}}{x e + d} - \frac{4 \, B a b^{3} d^{4} e^{5}}{x e + d} - \frac{A b^{4} d^{4} e^{5}}{x e + d} + \frac{6 \, B a^{2} b^{2} d^{3} e^{6}}{x e + d} + \frac{4 \, A a b^{3} d^{3} e^{6}}{x e + d} - \frac{4 \, B a^{3} b d^{2} e^{7}}{x e + d} - \frac{6 \, A a^{2} b^{2} d^{2} e^{7}}{x e + d} + \frac{B a^{4} d e^{8}}{x e + d} + \frac{4 \, A a^{3} b d e^{8}}{x e + d} - \frac{A a^{4} e^{9}}{x e + d}\right )} e^{\left (-10\right )} \]

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)^2,x, algorithm="giac")

[Out]

1/12*(3*B*b^4 - 4*(5*B*b^4*d*e - 4*B*a*b^3*e^2 - A*b^4*e^2)*e^(-1)/(x*e + d) + 1
2*(5*B*b^4*d^2*e^2 - 8*B*a*b^3*d*e^3 - 2*A*b^4*d*e^3 + 3*B*a^2*b^2*e^4 + 2*A*a*b
^3*e^4)*e^(-2)/(x*e + d)^2 - 24*(5*B*b^4*d^3*e^3 - 12*B*a*b^3*d^2*e^4 - 3*A*b^4*
d^2*e^4 + 9*B*a^2*b^2*d*e^5 + 6*A*a*b^3*d*e^5 - 2*B*a^3*b*e^6 - 3*A*a^2*b^2*e^6)
*e^(-3)/(x*e + d)^3)*(x*e + d)^4*e^(-6) - (5*B*b^4*d^4 - 16*B*a*b^3*d^3*e - 4*A*
b^4*d^3*e + 18*B*a^2*b^2*d^2*e^2 + 12*A*a*b^3*d^2*e^2 - 8*B*a^3*b*d*e^3 - 12*A*a
^2*b^2*d*e^3 + B*a^4*e^4 + 4*A*a^3*b*e^4)*e^(-6)*ln(abs(x*e + d)*e^(-1)/(x*e + d
)^2) + (B*b^4*d^5*e^4/(x*e + d) - 4*B*a*b^3*d^4*e^5/(x*e + d) - A*b^4*d^4*e^5/(x
*e + d) + 6*B*a^2*b^2*d^3*e^6/(x*e + d) + 4*A*a*b^3*d^3*e^6/(x*e + d) - 4*B*a^3*
b*d^2*e^7/(x*e + d) - 6*A*a^2*b^2*d^2*e^7/(x*e + d) + B*a^4*d*e^8/(x*e + d) + 4*
A*a^3*b*d*e^8/(x*e + d) - A*a^4*e^9/(x*e + d))*e^(-10)