3.1106 \(\int \frac{(A+B x) (d+e x)^4}{(a+b x)^2} \, dx\)

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 = 0.546661, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{e^3 (a+b x)^3 (-5 a B e+A b e+4 b B d)}{3 b^6}+\frac{e^2 (a+b x)^2 (b d-a e) (-5 a B e+2 A b e+3 b B d)}{b^6}-\frac{(A b-a B) (b d-a e)^4}{b^6 (a+b x)}+\frac{(b d-a e)^3 \log (a+b x) (-5 a B e+4 A b e+b B d)}{b^6}+\frac{2 e x (b d-a e)^2 (-5 a B e+3 A b e+2 b B d)}{b^5}+\frac{B e^4 (a+b x)^4}{4 b^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(d + e*x)^4)/(a + b*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 [A]  time = 71.7643, size = 189, normalized size = 1.01 \[ \frac{B e^{4} \left (a + b x\right )^{4}}{4 b^{6}} + \frac{2 e x \left (a e - b d\right )^{2} \left (3 A b e - 5 B a e + 2 B b d\right )}{b^{5}} + \frac{e^{3} \left (a + b x\right )^{3} \left (A b e - 5 B a e + 4 B b d\right )}{3 b^{6}} - \frac{e^{2} \left (a + b x\right )^{2} \left (a e - b d\right ) \left (2 A b e - 5 B a e + 3 B b d\right )}{b^{6}} - \frac{\left (a e - b d\right )^{3} \left (4 A b e - 5 B a e + B b d\right ) \log{\left (a + b x \right )}}{b^{6}} - \frac{\left (A b - B a\right ) \left (a e - b d\right )^{4}}{b^{6} \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*e**4*(a + b*x)**4/(4*b**6) + 2*e*x*(a*e - b*d)**2*(3*A*b*e - 5*B*a*e + 2*B*b*d
)/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 = 0.374233, 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 + b*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. \[ 5\,{\frac{\ln \left ( bx+a \right ) B{a}^{4}{e}^{4}}{{b}^{6}}}-{\frac{A{a}^{4}{e}^{4}}{ \left ( bx+a \right ){b}^{5}}}+{\frac{B{a}^{5}{e}^{4}}{ \left ( bx+a \right ){b}^{6}}}+{\frac{Ba{d}^{4}}{ \left ( bx+a \right ){b}^{2}}}+2\,{\frac{{e}^{3}A{x}^{2}d}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) B{d}^{4}}{{b}^{2}}}-{\frac{A{d}^{4}}{b \left ( bx+a \right ) }}+{\frac{B{e}^{4}{x}^{4}}{4\,{b}^{2}}}+{\frac{{e}^{4}A{x}^{3}}{3\,{b}^{2}}}+4\,{\frac{{a}^{3}Ad{e}^{3}}{ \left ( bx+a \right ){b}^{4}}}-6\,{\frac{{a}^{2}A{d}^{2}{e}^{2}}{ \left ( bx+a \right ){b}^{3}}}+4\,{\frac{Aa{d}^{3}e}{ \left ( bx+a \right ){b}^{2}}}+12\,{\frac{\ln \left ( bx+a \right ) A{a}^{2}d{e}^{3}}{{b}^{4}}}-12\,{\frac{\ln \left ( bx+a \right ) Aa{d}^{2}{e}^{2}}{{b}^{3}}}-16\,{\frac{\ln \left ( bx+a \right ) B{a}^{3}d{e}^{3}}{{b}^{5}}}+18\,{\frac{\ln \left ( bx+a \right ) B{a}^{2}{d}^{2}{e}^{2}}{{b}^{4}}}-8\,{\frac{\ln \left ( bx+a \right ) Ba{d}^{3}e}{{b}^{3}}}-12\,{\frac{{e}^{2}Ba{d}^{2}x}{{b}^{3}}}+12\,{\frac{{e}^{3}B{a}^{2}dx}{{b}^{4}}}-4\,{\frac{{e}^{3}B{x}^{2}ad}{{b}^{3}}}-4\,{\frac{B{a}^{4}d{e}^{3}}{ \left ( bx+a \right ){b}^{5}}}+6\,{\frac{B{a}^{3}{d}^{2}{e}^{2}}{ \left ( bx+a \right ){b}^{4}}}-4\,{\frac{B{a}^{2}{d}^{3}e}{ \left ( bx+a \right ){b}^{3}}}-8\,{\frac{aA{e}^{3}dx}{{b}^{3}}}+{\frac{3\,B{e}^{4}{x}^{2}{a}^{2}}{2\,{b}^{4}}}+3\,{\frac{{e}^{2}B{x}^{2}{d}^{2}}{{b}^{2}}}+3\,{\frac{{a}^{2}A{e}^{4}x}{{b}^{4}}}+6\,{\frac{{e}^{2}A{d}^{2}x}{{b}^{2}}}-4\,{\frac{B{a}^{3}{e}^{4}x}{{b}^{5}}}+4\,{\frac{eB{d}^{3}x}{{b}^{2}}}-{\frac{{e}^{4}A{x}^{2}a}{{b}^{3}}}-{\frac{2\,B{e}^{4}{x}^{3}a}{3\,{b}^{3}}}+{\frac{4\,{e}^{3}B{x}^{3}d}{3\,{b}^{2}}}-4\,{\frac{\ln \left ( bx+a \right ) A{a}^{3}{e}^{4}}{{b}^{5}}}+4\,{\frac{\ln \left ( bx+a \right ) A{d}^{3}e}{{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37238, 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*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 = 0.214714, 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*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 = 10.0499, 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*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 = 0.232748, size = 705, normalized size = 3.77 \[ \frac{{\left (b x + a\right )}^{4}{\left (3 \, B e^{4} + \frac{4 \,{\left (4 \, B b^{2} d e^{3} - 5 \, B a b e^{4} + A b^{2} e^{4}\right )}}{{\left (b x + a\right )} b} + \frac{12 \,{\left (3 \, B b^{4} d^{2} e^{2} - 8 \, B a b^{3} d e^{3} + 2 \, A b^{4} d e^{3} + 5 \, B a^{2} b^{2} e^{4} - 2 \, A a b^{3} e^{4}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{24 \,{\left (2 \, B b^{6} d^{3} e - 9 \, B a b^{5} d^{2} e^{2} + 3 \, A b^{6} d^{2} e^{2} + 12 \, B a^{2} b^{4} d e^{3} - 6 \, A a b^{5} d e^{3} - 5 \, B a^{3} b^{3} e^{4} + 3 \, A a^{2} b^{4} e^{4}\right )}}{{\left (b x + a\right )}^{3} b^{3}}\right )}}{12 \, b^{6}} - \frac{{\left (B b^{4} d^{4} - 8 \, B a b^{3} d^{3} e + 4 \, A b^{4} d^{3} e + 18 \, B a^{2} b^{2} d^{2} e^{2} - 12 \, A a b^{3} d^{2} e^{2} - 16 \, B a^{3} b d e^{3} + 12 \, A a^{2} b^{2} d e^{3} + 5 \, B a^{4} e^{4} - 4 \, A a^{3} b e^{4}\right )}{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{6}} + \frac{\frac{B a b^{8} d^{4}}{b x + a} - \frac{A b^{9} d^{4}}{b x + a} - \frac{4 \, B a^{2} b^{7} d^{3} e}{b x + a} + \frac{4 \, A a b^{8} d^{3} e}{b x + a} + \frac{6 \, B a^{3} b^{6} d^{2} e^{2}}{b x + a} - \frac{6 \, A a^{2} b^{7} d^{2} e^{2}}{b x + a} - \frac{4 \, B a^{4} b^{5} d e^{3}}{b x + a} + \frac{4 \, A a^{3} b^{6} d e^{3}}{b x + a} + \frac{B a^{5} b^{4} e^{4}}{b x + a} - \frac{A a^{4} b^{5} e^{4}}{b x + a}}{b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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