3.1693 \(\int \frac{(A+B x) (d+e x)^4}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=187 \[ \frac{e^3 (a+b x)^3 (-5 a B e+A b e+4 b B d)}{3 b^6}+\frac{e^2 (a+b x)^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac{(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac{(b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{b^6}+\frac{2 e x (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5}+\frac{B e^4 (a+b x)^4}{4 b^6} \]

[Out]

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

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Rubi [A]  time = 0.551284, antiderivative size = 187, 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 \[ \frac{e^3 (a+b x)^3 (-5 a B e+A b e+4 b B d)}{3 b^6}+\frac{e^2 (a+b x)^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac{(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac{(b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{b^6}+\frac{2 e x (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5}+\frac{B e^4 (a+b x)^4}{4 b^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B e^{4} \left (a + b x\right )^{4}}{4 b^{6}} + \frac{4 e \left (a e - b d\right )^{2} \left (3 A b e - 5 B a e + 2 B b d\right ) \int \frac{1}{2}\, dx}{b^{5}} + \frac{e^{3} \left (a + b x\right )^{3} \left (A b e - 5 B a e + 4 B b d\right )}{3 b^{6}} - \frac{e^{2} \left (a + b x\right )^{2} \left (a e - b d\right ) \left (2 A b e - 5 B a e + 3 B b d\right )}{b^{6}} - \frac{\left (a e - b d\right )^{3} \left (4 A b e - 5 B a e + B b d\right ) \log{\left (a + b x \right )}}{b^{6}} - \frac{\left (A b - B a\right ) \left (a e - b d\right )^{4}}{b^{6} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.335124, size = 365, normalized size = 1.95 \[ \frac{-4 A b \left (3 a^4 e^4-3 a^3 b e^3 (4 d+3 e x)+6 a^2 b^2 e^2 \left (3 d^2+4 d e x-e^2 x^2\right )+2 a b^3 e \left (-6 d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )+b^4 \left (3 d^4-18 d^2 e^2 x^2-6 d e^3 x^3-e^4 x^4\right )\right )+B \left (12 a^5 e^4-48 a^4 b e^3 (d+e x)+6 a^3 b^2 e^2 \left (12 d^2+24 d e x-5 e^2 x^2\right )+2 a^2 b^3 e \left (-24 d^3-72 d^2 e x+48 d e^2 x^2+5 e^3 x^3\right )+a b^4 \left (12 d^4+48 d^3 e x-108 d^2 e^2 x^2-32 d e^3 x^3-5 e^4 x^4\right )+b^5 e x^2 \left (48 d^3+36 d^2 e x+16 d e^2 x^2+3 e^3 x^3\right )\right )+12 (a+b x) (b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{12 b^6 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(d + e*x)^4)/(a^2 + 2*a*b*x + b^2*x^2),x]

[Out]

(B*(12*a^5*e^4 - 48*a^4*b*e^3*(d + e*x) + 6*a^3*b^2*e^2*(12*d^2 + 24*d*e*x - 5*e
^2*x^2) + b^5*e*x^2*(48*d^3 + 36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3) + 2*a^2*b^3
*e*(-24*d^3 - 72*d^2*e*x + 48*d*e^2*x^2 + 5*e^3*x^3) + a*b^4*(12*d^4 + 48*d^3*e*
x - 108*d^2*e^2*x^2 - 32*d*e^3*x^3 - 5*e^4*x^4)) - 4*A*b*(3*a^4*e^4 - 3*a^3*b*e^
3*(4*d + 3*e*x) + 6*a^2*b^2*e^2*(3*d^2 + 4*d*e*x - e^2*x^2) + 2*a*b^3*e*(-6*d^3
- 9*d^2*e*x + 9*d*e^2*x^2 + e^3*x^3) + b^4*(3*d^4 - 18*d^2*e^2*x^2 - 6*d*e^3*x^3
- e^4*x^4)) + 12*(b*d - a*e)^3*(b*B*d + 4*A*b*e - 5*a*B*e)*(a + b*x)*Log[a + b*
x])/(12*b^6*(a + b*x))

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Maple [B]  time = 0.017, size = 564, normalized size = 3. \[{\frac{B{e}^{4}{x}^{4}}{4\,{b}^{2}}}+{\frac{A{e}^{4}{x}^{3}}{3\,{b}^{2}}}-{\frac{A{a}^{4}{e}^{4}}{{b}^{5} \left ( bx+a \right ) }}-{\frac{2\,B{e}^{4}{x}^{3}a}{3\,{b}^{3}}}+3\,{\frac{{e}^{2}B{x}^{2}{d}^{2}}{{b}^{2}}}+3\,{\frac{A{e}^{4}{a}^{2}x}{{b}^{4}}}+6\,{\frac{A{e}^{2}{d}^{2}x}{{b}^{2}}}-4\,{\frac{B{e}^{4}{a}^{3}x}{{b}^{5}}}+4\,{\frac{Be{d}^{3}x}{{b}^{2}}}-4\,{\frac{\ln \left ( bx+a \right ) A{a}^{3}{e}^{4}}{{b}^{5}}}+4\,{\frac{\ln \left ( bx+a \right ) A{d}^{3}e}{{b}^{2}}}+2\,{\frac{A{e}^{3}{x}^{2}d}{{b}^{2}}}+{\frac{Ba{d}^{4}}{{b}^{2} \left ( bx+a \right ) }}+{\frac{B{a}^{5}{e}^{4}}{{b}^{6} \left ( bx+a \right ) }}-6\,{\frac{A{a}^{2}{d}^{2}{e}^{2}}{{b}^{3} \left ( bx+a \right ) }}+4\,{\frac{aA{d}^{3}e}{{b}^{2} \left ( bx+a \right ) }}-4\,{\frac{{e}^{3}B{x}^{2}ad}{{b}^{3}}}-4\,{\frac{B{a}^{4}d{e}^{3}}{{b}^{5} \left ( bx+a \right ) }}+6\,{\frac{B{a}^{3}{d}^{2}{e}^{2}}{{b}^{4} \left ( bx+a \right ) }}-4\,{\frac{{a}^{2}B{d}^{3}e}{{b}^{3} \left ( bx+a \right ) }}-8\,{\frac{aA{e}^{3}dx}{{b}^{3}}}+12\,{\frac{Bd{a}^{2}{e}^{3}x}{{b}^{4}}}-12\,{\frac{aB{e}^{2}{d}^{2}x}{{b}^{3}}}+12\,{\frac{\ln \left ( bx+a \right ) Ad{a}^{2}{e}^{3}}{{b}^{4}}}-12\,{\frac{\ln \left ( bx+a \right ) A{d}^{2}a{e}^{2}}{{b}^{3}}}-16\,{\frac{\ln \left ( bx+a \right ) Bd{a}^{3}{e}^{3}}{{b}^{5}}}+18\,{\frac{\ln \left ( bx+a \right ) B{d}^{2}{a}^{2}{e}^{2}}{{b}^{4}}}-8\,{\frac{\ln \left ( bx+a \right ) B{d}^{3}ae}{{b}^{3}}}+4\,{\frac{A{a}^{3}d{e}^{3}}{{b}^{4} \left ( bx+a \right ) }}+{\frac{3\,B{e}^{4}{x}^{2}{a}^{2}}{2\,{b}^{4}}}+5\,{\frac{\ln \left ( bx+a \right ) B{e}^{4}{a}^{4}}{{b}^{6}}}+{\frac{4\,{e}^{3}B{x}^{3}d}{3\,{b}^{2}}}-{\frac{A{e}^{4}{x}^{2}a}{{b}^{3}}}-{\frac{{d}^{4}A}{b \left ( bx+a \right ) }}+{\frac{\ln \left ( bx+a \right ) B{d}^{4}}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^4/(b^2*x^2+2*a*b*x+a^2),x)

[Out]

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

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Maxima [A]  time = 0.698461, size = 555, normalized size = 2.97 \[ \frac{{\left (B a b^{4} - A b^{5}\right )} d^{4} - 4 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} +{\left (B a^{5} - A a^{4} b\right )} e^{4}}{b^{7} x + a b^{6}} + \frac{3 \, B b^{3} e^{4} x^{4} + 4 \,{\left (4 \, B b^{3} d e^{3} -{\left (2 \, B a b^{2} - A b^{3}\right )} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B b^{3} d^{2} e^{2} - 4 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d e^{3} +{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} e^{4}\right )} x^{2} + 12 \,{\left (4 \, B b^{3} d^{3} e - 6 \,{\left (2 \, B a b^{2} - A b^{3}\right )} d^{2} e^{2} + 4 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} d e^{3} -{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} e^{4}\right )} x}{12 \, b^{5}} + \frac{{\left (B b^{4} d^{4} - 4 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{3} +{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} e^{4}\right )} \log \left (b x + a\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.273538, size = 824, normalized size = 4.41 \[ \frac{3 \, B b^{5} e^{4} x^{5} + 12 \,{\left (B a b^{4} - A b^{5}\right )} d^{4} - 48 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 72 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{2} - 48 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{3} + 12 \,{\left (B a^{5} - A a^{4} b\right )} e^{4} +{\left (16 \, B b^{5} d e^{3} -{\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} e^{4}\right )} x^{4} + 2 \,{\left (18 \, B b^{5} d^{2} e^{2} - 4 \,{\left (4 \, B a b^{4} - 3 \, A b^{5}\right )} d e^{3} +{\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} e^{4}\right )} x^{3} + 6 \,{\left (8 \, B b^{5} d^{3} e - 6 \,{\left (3 \, B a b^{4} - 2 \, A b^{5}\right )} d^{2} e^{2} + 4 \,{\left (4 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} d e^{3} -{\left (5 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} e^{4}\right )} x^{2} + 12 \,{\left (4 \, B a b^{4} d^{3} e - 6 \,{\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{2} + 4 \,{\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d e^{3} -{\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} e^{4}\right )} x + 12 \,{\left (B a b^{4} d^{4} - 4 \,{\left (2 \, B a^{2} b^{3} - A a b^{4}\right )} d^{3} e + 6 \,{\left (3 \, B a^{3} b^{2} - 2 \, A a^{2} b^{3}\right )} d^{2} e^{2} - 4 \,{\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} d e^{3} +{\left (5 \, B a^{5} - 4 \, A a^{4} b\right )} e^{4} +{\left (B b^{5} d^{4} - 4 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d^{3} e + 6 \,{\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{2} e^{2} - 4 \,{\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d e^{3} +{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} e^{4}\right )} x\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")

[Out]

1/12*(3*B*b^5*e^4*x^5 + 12*(B*a*b^4 - A*b^5)*d^4 - 48*(B*a^2*b^3 - A*a*b^4)*d^3*
e + 72*(B*a^3*b^2 - A*a^2*b^3)*d^2*e^2 - 48*(B*a^4*b - A*a^3*b^2)*d*e^3 + 12*(B*
a^5 - A*a^4*b)*e^4 + (16*B*b^5*d*e^3 - (5*B*a*b^4 - 4*A*b^5)*e^4)*x^4 + 2*(18*B*
b^5*d^2*e^2 - 4*(4*B*a*b^4 - 3*A*b^5)*d*e^3 + (5*B*a^2*b^3 - 4*A*a*b^4)*e^4)*x^3
+ 6*(8*B*b^5*d^3*e - 6*(3*B*a*b^4 - 2*A*b^5)*d^2*e^2 + 4*(4*B*a^2*b^3 - 3*A*a*b
^4)*d*e^3 - (5*B*a^3*b^2 - 4*A*a^2*b^3)*e^4)*x^2 + 12*(4*B*a*b^4*d^3*e - 6*(2*B*
a^2*b^3 - A*a*b^4)*d^2*e^2 + 4*(3*B*a^3*b^2 - 2*A*a^2*b^3)*d*e^3 - (4*B*a^4*b -
3*A*a^3*b^2)*e^4)*x + 12*(B*a*b^4*d^4 - 4*(2*B*a^2*b^3 - A*a*b^4)*d^3*e + 6*(3*B
*a^3*b^2 - 2*A*a^2*b^3)*d^2*e^2 - 4*(4*B*a^4*b - 3*A*a^3*b^2)*d*e^3 + (5*B*a^5 -
4*A*a^4*b)*e^4 + (B*b^5*d^4 - 4*(2*B*a*b^4 - A*b^5)*d^3*e + 6*(3*B*a^2*b^3 - 2*
A*a*b^4)*d^2*e^2 - 4*(4*B*a^3*b^2 - 3*A*a^2*b^3)*d*e^3 + (5*B*a^4*b - 4*A*a^3*b^
2)*e^4)*x)*log(b*x + a))/(b^7*x + a*b^6)

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Sympy [A]  time = 10.0411, size = 384, normalized size = 2.05 \[ \frac{B e^{4} x^{4}}{4 b^{2}} + \frac{- A a^{4} b e^{4} + 4 A a^{3} b^{2} d e^{3} - 6 A a^{2} b^{3} d^{2} e^{2} + 4 A a b^{4} d^{3} e - A b^{5} d^{4} + B a^{5} e^{4} - 4 B a^{4} b d e^{3} + 6 B a^{3} b^{2} d^{2} e^{2} - 4 B a^{2} b^{3} d^{3} e + B a b^{4} d^{4}}{a b^{6} + b^{7} x} - \frac{x^{3} \left (- A b e^{4} + 2 B a e^{4} - 4 B b d e^{3}\right )}{3 b^{3}} + \frac{x^{2} \left (- 2 A a b e^{4} + 4 A b^{2} d e^{3} + 3 B a^{2} e^{4} - 8 B a b d e^{3} + 6 B b^{2} d^{2} e^{2}\right )}{2 b^{4}} - \frac{x \left (- 3 A a^{2} b e^{4} + 8 A a b^{2} d e^{3} - 6 A b^{3} d^{2} e^{2} + 4 B a^{3} e^{4} - 12 B a^{2} b d e^{3} + 12 B a b^{2} d^{2} e^{2} - 4 B b^{3} d^{3} e\right )}{b^{5}} + \frac{\left (a e - b d\right )^{3} \left (- 4 A b e + 5 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**4/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.344218, size = 585, normalized size = 3.13 \[ \frac{{\left (B b^{4} d^{4} - 8 \, B a b^{3} d^{3} e + 4 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} - 12 \, A a b^{3} d^{2} e^{2} - 16 \, B a^{3} b d e^{3} + 12 \, A a^{2} b^{2} d e^{3} + 5 \, B a^{4} e^{4} - 4 \, A a^{3} b e^{4}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6}} + \frac{B a b^{4} d^{4} - A b^{5} d^{4} - 4 \, B a^{2} b^{3} d^{3} e + 4 \, A a b^{4} d^{3} e + 6 \, B a^{3} b^{2} d^{2} e^{2} - 6 \, A a^{2} b^{3} d^{2} e^{2} - 4 \, B a^{4} b d e^{3} + 4 \, A a^{3} b^{2} d e^{3} + B a^{5} e^{4} - A a^{4} b e^{4}}{{\left (b x + a\right )} b^{6}} + \frac{3 \, B b^{6} x^{4} e^{4} + 16 \, B b^{6} d x^{3} e^{3} + 36 \, B b^{6} d^{2} x^{2} e^{2} + 48 \, B b^{6} d^{3} x e - 8 \, B a b^{5} x^{3} e^{4} + 4 \, A b^{6} x^{3} e^{4} - 48 \, B a b^{5} d x^{2} e^{3} + 24 \, A b^{6} d x^{2} e^{3} - 144 \, B a b^{5} d^{2} x e^{2} + 72 \, A b^{6} d^{2} x e^{2} + 18 \, B a^{2} b^{4} x^{2} e^{4} - 12 \, A a b^{5} x^{2} e^{4} + 144 \, B a^{2} b^{4} d x e^{3} - 96 \, A a b^{5} d x e^{3} - 48 \, B a^{3} b^{3} x e^{4} + 36 \, A a^{2} b^{4} x e^{4}}{12 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(e*x + d)^4/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")

[Out]

(B*b^4*d^4 - 8*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 - 16*B*a^3*b*d*e^3 + 12*A*a^2*b^2*d*e^3 + 5*B*a^4*e^4 - 4*A*a^3*b*e^4)*
ln(abs(b*x + a))/b^6 + (B*a*b^4*d^4 - A*b^5*d^4 - 4*B*a^2*b^3*d^3*e + 4*A*a*b^4*
d^3*e + 6*B*a^3*b^2*d^2*e^2 - 6*A*a^2*b^3*d^2*e^2 - 4*B*a^4*b*d*e^3 + 4*A*a^3*b^
2*d*e^3 + B*a^5*e^4 - A*a^4*b*e^4)/((b*x + a)*b^6) + 1/12*(3*B*b^6*x^4*e^4 + 16*
B*b^6*d*x^3*e^3 + 36*B*b^6*d^2*x^2*e^2 + 48*B*b^6*d^3*x*e - 8*B*a*b^5*x^3*e^4 +
4*A*b^6*x^3*e^4 - 48*B*a*b^5*d*x^2*e^3 + 24*A*b^6*d*x^2*e^3 - 144*B*a*b^5*d^2*x*
e^2 + 72*A*b^6*d^2*x*e^2 + 18*B*a^2*b^4*x^2*e^4 - 12*A*a*b^5*x^2*e^4 + 144*B*a^2
*b^4*d*x*e^3 - 96*A*a*b^5*d*x*e^3 - 48*B*a^3*b^3*x*e^4 + 36*A*a^2*b^4*x*e^4)/b^8