3.1094 \(\int \frac{(A+B x) (d+e x)^5}{a+b x} \, dx\)

Optimal. Leaf size=187 \[ \frac{(A b-a B) (b d-a e)^5 \log (a+b x)}{b^7}+\frac{e x (A b-a B) (b d-a e)^4}{b^6}+\frac{(d+e x)^2 (A b-a B) (b d-a e)^3}{2 b^5}+\frac{(d+e x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac{(d+e x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac{(d+e x)^5 (A b-a B)}{5 b^2}+\frac{B (d+e x)^6}{6 b e} \]

[Out]

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

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Rubi [A]  time = 0.269328, 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 \[ \frac{(A b-a B) (b d-a e)^5 \log (a+b x)}{b^7}+\frac{e x (A b-a B) (b d-a e)^4}{b^6}+\frac{(d+e x)^2 (A b-a B) (b d-a e)^3}{2 b^5}+\frac{(d+e x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac{(d+e x)^4 (A b-a B) (b d-a e)}{4 b^3}+\frac{(d+e x)^5 (A b-a B)}{5 b^2}+\frac{B (d+e x)^6}{6 b e} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.357371, size = 368, normalized size = 1.97 \[ \frac{b x \left (-60 a^5 B e^5+30 a^4 b e^4 (2 A e+10 B d+B e x)-10 a^3 b^2 e^3 \left (3 A e (10 d+e x)+B \left (60 d^2+15 d e x+2 e^2 x^2\right )\right )+5 a^2 b^3 e^2 \left (2 A e \left (60 d^2+15 d e x+2 e^2 x^2\right )+B \left (120 d^3+60 d^2 e x+20 d e^2 x^2+3 e^3 x^3\right )\right )-a b^4 e \left (5 A e \left (120 d^3+60 d^2 e x+20 d e^2 x^2+3 e^3 x^3\right )+B \left (300 d^4+300 d^3 e x+200 d^2 e^2 x^2+75 d e^3 x^3+12 e^4 x^4\right )\right )+b^5 \left (A e \left (300 d^4+300 d^3 e x+200 d^2 e^2 x^2+75 d e^3 x^3+12 e^4 x^4\right )+10 B \left (6 d^5+15 d^4 e x+20 d^3 e^2 x^2+15 d^2 e^3 x^3+6 d e^4 x^4+e^5 x^5\right )\right )\right )+60 (A b-a B) (b d-a e)^5 \log (a+b x)}{60 b^7} \]

Antiderivative was successfully verified.

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

[Out]

(b*x*(-60*a^5*B*e^5 + 30*a^4*b*e^4*(10*B*d + 2*A*e + B*e*x) - 10*a^3*b^2*e^3*(3*
A*e*(10*d + e*x) + B*(60*d^2 + 15*d*e*x + 2*e^2*x^2)) + 5*a^2*b^3*e^2*(2*A*e*(60
*d^2 + 15*d*e*x + 2*e^2*x^2) + B*(120*d^3 + 60*d^2*e*x + 20*d*e^2*x^2 + 3*e^3*x^
3)) - a*b^4*e*(5*A*e*(120*d^3 + 60*d^2*e*x + 20*d*e^2*x^2 + 3*e^3*x^3) + B*(300*
d^4 + 300*d^3*e*x + 200*d^2*e^2*x^2 + 75*d*e^3*x^3 + 12*e^4*x^4)) + b^5*(A*e*(30
0*d^4 + 300*d^3*e*x + 200*d^2*e^2*x^2 + 75*d*e^3*x^3 + 12*e^4*x^4) + 10*B*(6*d^5
+ 15*d^4*e*x + 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 + 6*d*e^4*x^4 + e^5*x^5))) + 60*
(A*b - a*B)*(b*d - a*e)^5*Log[a + b*x])/(60*b^7)

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Maple [B]  time = 0.01, size = 737, normalized size = 3.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36747, size = 760, normalized size = 4.06 \[ \frac{10 \, B b^{5} e^{5} x^{6} + 12 \,{\left (5 \, B b^{5} d e^{4} -{\left (B a b^{4} - A b^{5}\right )} e^{5}\right )} x^{5} + 15 \,{\left (10 \, B b^{5} d^{2} e^{3} - 5 \,{\left (B a b^{4} - A b^{5}\right )} d e^{4} +{\left (B a^{2} b^{3} - A a b^{4}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B b^{5} d^{3} e^{2} - 10 \,{\left (B a b^{4} - A b^{5}\right )} d^{2} e^{3} + 5 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d e^{4} -{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} e^{5}\right )} x^{3} + 30 \,{\left (5 \, B b^{5} d^{4} e - 10 \,{\left (B a b^{4} - A b^{5}\right )} d^{3} e^{2} + 10 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{2} e^{3} - 5 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d e^{4} +{\left (B a^{4} b - A a^{3} b^{2}\right )} e^{5}\right )} x^{2} + 60 \,{\left (B b^{5} d^{5} - 5 \,{\left (B a b^{4} - A b^{5}\right )} d^{4} e + 10 \,{\left (B a^{2} b^{3} - A a b^{4}\right )} d^{3} e^{2} - 10 \,{\left (B a^{3} b^{2} - A a^{2} b^{3}\right )} d^{2} e^{3} + 5 \,{\left (B a^{4} b - A a^{3} b^{2}\right )} d e^{4} -{\left (B a^{5} - A a^{4} b\right )} e^{5}\right )} x}{60 \, b^{6}} - \frac{{\left ({\left (B a b^{5} - A b^{6}\right )} d^{5} - 5 \,{\left (B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \,{\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \,{\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \,{\left (B a^{5} b - A a^{4} b^{2}\right )} d e^{4} -{\left (B a^{6} - A a^{5} b\right )} e^{5}\right )} \log \left (b x + a\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*B*b^5*e^5*x^6 + 12*(5*B*b^5*d*e^4 - (B*a*b^4 - A*b^5)*e^5)*x^5 + 15*(10
*B*b^5*d^2*e^3 - 5*(B*a*b^4 - A*b^5)*d*e^4 + (B*a^2*b^3 - A*a*b^4)*e^5)*x^4 + 20
*(10*B*b^5*d^3*e^2 - 10*(B*a*b^4 - A*b^5)*d^2*e^3 + 5*(B*a^2*b^3 - A*a*b^4)*d*e^
4 - (B*a^3*b^2 - A*a^2*b^3)*e^5)*x^3 + 30*(5*B*b^5*d^4*e - 10*(B*a*b^4 - A*b^5)*
d^3*e^2 + 10*(B*a^2*b^3 - A*a*b^4)*d^2*e^3 - 5*(B*a^3*b^2 - A*a^2*b^3)*d*e^4 + (
B*a^4*b - A*a^3*b^2)*e^5)*x^2 + 60*(B*b^5*d^5 - 5*(B*a*b^4 - A*b^5)*d^4*e + 10*(
B*a^2*b^3 - A*a*b^4)*d^3*e^2 - 10*(B*a^3*b^2 - A*a^2*b^3)*d^2*e^3 + 5*(B*a^4*b -
A*a^3*b^2)*d*e^4 - (B*a^5 - A*a^4*b)*e^5)*x)/b^6 - ((B*a*b^5 - A*b^6)*d^5 - 5*(
B*a^2*b^4 - A*a*b^5)*d^4*e + 10*(B*a^3*b^3 - A*a^2*b^4)*d^3*e^2 - 10*(B*a^4*b^2
- A*a^3*b^3)*d^2*e^3 + 5*(B*a^5*b - A*a^4*b^2)*d*e^4 - (B*a^6 - A*a^5*b)*e^5)*lo
g(b*x + a)/b^7

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Fricas [A]  time = 0.208087, size = 764, normalized size = 4.09 \[ \frac{10 \, B b^{6} e^{5} x^{6} + 12 \,{\left (5 \, B b^{6} d e^{4} -{\left (B a b^{5} - A b^{6}\right )} e^{5}\right )} x^{5} + 15 \,{\left (10 \, B b^{6} d^{2} e^{3} - 5 \,{\left (B a b^{5} - A b^{6}\right )} d e^{4} +{\left (B a^{2} b^{4} - A a b^{5}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B b^{6} d^{3} e^{2} - 10 \,{\left (B a b^{5} - A b^{6}\right )} d^{2} e^{3} + 5 \,{\left (B a^{2} b^{4} - A a b^{5}\right )} d e^{4} -{\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} e^{5}\right )} x^{3} + 30 \,{\left (5 \, B b^{6} d^{4} e - 10 \,{\left (B a b^{5} - A b^{6}\right )} d^{3} e^{2} + 10 \,{\left (B a^{2} b^{4} - A a b^{5}\right )} d^{2} e^{3} - 5 \,{\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d e^{4} +{\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} e^{5}\right )} x^{2} + 60 \,{\left (B b^{6} d^{5} - 5 \,{\left (B a b^{5} - A b^{6}\right )} d^{4} e + 10 \,{\left (B a^{2} b^{4} - A a b^{5}\right )} d^{3} e^{2} - 10 \,{\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \,{\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d e^{4} -{\left (B a^{5} b - A a^{4} b^{2}\right )} e^{5}\right )} x - 60 \,{\left ({\left (B a b^{5} - A b^{6}\right )} d^{5} - 5 \,{\left (B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \,{\left (B a^{3} b^{3} - A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \,{\left (B a^{4} b^{2} - A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \,{\left (B a^{5} b - A a^{4} b^{2}\right )} d e^{4} -{\left (B a^{6} - A a^{5} b\right )} e^{5}\right )} \log \left (b x + a\right )}{60 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*B*b^6*e^5*x^6 + 12*(5*B*b^6*d*e^4 - (B*a*b^5 - A*b^6)*e^5)*x^5 + 15*(10
*B*b^6*d^2*e^3 - 5*(B*a*b^5 - A*b^6)*d*e^4 + (B*a^2*b^4 - A*a*b^5)*e^5)*x^4 + 20
*(10*B*b^6*d^3*e^2 - 10*(B*a*b^5 - A*b^6)*d^2*e^3 + 5*(B*a^2*b^4 - A*a*b^5)*d*e^
4 - (B*a^3*b^3 - A*a^2*b^4)*e^5)*x^3 + 30*(5*B*b^6*d^4*e - 10*(B*a*b^5 - A*b^6)*
d^3*e^2 + 10*(B*a^2*b^4 - A*a*b^5)*d^2*e^3 - 5*(B*a^3*b^3 - A*a^2*b^4)*d*e^4 + (
B*a^4*b^2 - A*a^3*b^3)*e^5)*x^2 + 60*(B*b^6*d^5 - 5*(B*a*b^5 - A*b^6)*d^4*e + 10
*(B*a^2*b^4 - A*a*b^5)*d^3*e^2 - 10*(B*a^3*b^3 - A*a^2*b^4)*d^2*e^3 + 5*(B*a^4*b
^2 - A*a^3*b^3)*d*e^4 - (B*a^5*b - A*a^4*b^2)*e^5)*x - 60*((B*a*b^5 - A*b^6)*d^5
- 5*(B*a^2*b^4 - A*a*b^5)*d^4*e + 10*(B*a^3*b^3 - A*a^2*b^4)*d^3*e^2 - 10*(B*a^
4*b^2 - A*a^3*b^3)*d^2*e^3 + 5*(B*a^5*b - A*a^4*b^2)*d*e^4 - (B*a^6 - A*a^5*b)*e
^5)*log(b*x + a))/b^7

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Sympy [A]  time = 6.48152, size = 510, normalized size = 2.73 \[ \frac{B e^{5} x^{6}}{6 b} - \frac{x^{5} \left (- A b e^{5} + B a e^{5} - 5 B b d e^{4}\right )}{5 b^{2}} + \frac{x^{4} \left (- A a b e^{5} + 5 A b^{2} d e^{4} + B a^{2} e^{5} - 5 B a b d e^{4} + 10 B b^{2} d^{2} e^{3}\right )}{4 b^{3}} - \frac{x^{3} \left (- A a^{2} b e^{5} + 5 A a b^{2} d e^{4} - 10 A b^{3} d^{2} e^{3} + B a^{3} e^{5} - 5 B a^{2} b d e^{4} + 10 B a b^{2} d^{2} e^{3} - 10 B b^{3} d^{3} e^{2}\right )}{3 b^{4}} + \frac{x^{2} \left (- A a^{3} b e^{5} + 5 A a^{2} b^{2} d e^{4} - 10 A a b^{3} d^{2} e^{3} + 10 A b^{4} d^{3} e^{2} + B a^{4} e^{5} - 5 B a^{3} b d e^{4} + 10 B a^{2} b^{2} d^{2} e^{3} - 10 B a b^{3} d^{3} e^{2} + 5 B b^{4} d^{4} e\right )}{2 b^{5}} - \frac{x \left (- A a^{4} b e^{5} + 5 A a^{3} b^{2} d e^{4} - 10 A a^{2} b^{3} d^{2} e^{3} + 10 A a b^{4} d^{3} e^{2} - 5 A b^{5} d^{4} e + B a^{5} e^{5} - 5 B a^{4} b d e^{4} + 10 B a^{3} b^{2} d^{2} e^{3} - 10 B a^{2} b^{3} d^{3} e^{2} + 5 B a b^{4} d^{4} e - B b^{5} d^{5}\right )}{b^{6}} + \frac{\left (- A b + B a\right ) \left (a e - b d\right )^{5} \log{\left (a + b x \right )}}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223438, size = 857, normalized size = 4.58 \[ \frac{10 \, B b^{5} x^{6} e^{5} + 60 \, B b^{5} d x^{5} e^{4} + 150 \, B b^{5} d^{2} x^{4} e^{3} + 200 \, B b^{5} d^{3} x^{3} e^{2} + 150 \, B b^{5} d^{4} x^{2} e + 60 \, B b^{5} d^{5} x - 12 \, B a b^{4} x^{5} e^{5} + 12 \, A b^{5} x^{5} e^{5} - 75 \, B a b^{4} d x^{4} e^{4} + 75 \, A b^{5} d x^{4} e^{4} - 200 \, B a b^{4} d^{2} x^{3} e^{3} + 200 \, A b^{5} d^{2} x^{3} e^{3} - 300 \, B a b^{4} d^{3} x^{2} e^{2} + 300 \, A b^{5} d^{3} x^{2} e^{2} - 300 \, B a b^{4} d^{4} x e + 300 \, A b^{5} d^{4} x e + 15 \, B a^{2} b^{3} x^{4} e^{5} - 15 \, A a b^{4} x^{4} e^{5} + 100 \, B a^{2} b^{3} d x^{3} e^{4} - 100 \, A a b^{4} d x^{3} e^{4} + 300 \, B a^{2} b^{3} d^{2} x^{2} e^{3} - 300 \, A a b^{4} d^{2} x^{2} e^{3} + 600 \, B a^{2} b^{3} d^{3} x e^{2} - 600 \, A a b^{4} d^{3} x e^{2} - 20 \, B a^{3} b^{2} x^{3} e^{5} + 20 \, A a^{2} b^{3} x^{3} e^{5} - 150 \, B a^{3} b^{2} d x^{2} e^{4} + 150 \, A a^{2} b^{3} d x^{2} e^{4} - 600 \, B a^{3} b^{2} d^{2} x e^{3} + 600 \, A a^{2} b^{3} d^{2} x e^{3} + 30 \, B a^{4} b x^{2} e^{5} - 30 \, A a^{3} b^{2} x^{2} e^{5} + 300 \, B a^{4} b d x e^{4} - 300 \, A a^{3} b^{2} d x e^{4} - 60 \, B a^{5} x e^{5} + 60 \, A a^{4} b x e^{5}}{60 \, b^{6}} - \frac{{\left (B a b^{5} d^{5} - A b^{6} d^{5} - 5 \, B a^{2} b^{4} d^{4} e + 5 \, A a b^{5} d^{4} e + 10 \, B a^{3} b^{3} d^{3} e^{2} - 10 \, A a^{2} b^{4} d^{3} e^{2} - 10 \, B a^{4} b^{2} d^{2} e^{3} + 10 \, A a^{3} b^{3} d^{2} e^{3} + 5 \, B a^{5} b d e^{4} - 5 \, A a^{4} b^{2} d e^{4} - B a^{6} e^{5} + A a^{5} b e^{5}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*B*b^5*x^6*e^5 + 60*B*b^5*d*x^5*e^4 + 150*B*b^5*d^2*x^4*e^3 + 200*B*b^5*
d^3*x^3*e^2 + 150*B*b^5*d^4*x^2*e + 60*B*b^5*d^5*x - 12*B*a*b^4*x^5*e^5 + 12*A*b
^5*x^5*e^5 - 75*B*a*b^4*d*x^4*e^4 + 75*A*b^5*d*x^4*e^4 - 200*B*a*b^4*d^2*x^3*e^3
+ 200*A*b^5*d^2*x^3*e^3 - 300*B*a*b^4*d^3*x^2*e^2 + 300*A*b^5*d^3*x^2*e^2 - 300
*B*a*b^4*d^4*x*e + 300*A*b^5*d^4*x*e + 15*B*a^2*b^3*x^4*e^5 - 15*A*a*b^4*x^4*e^5
+ 100*B*a^2*b^3*d*x^3*e^4 - 100*A*a*b^4*d*x^3*e^4 + 300*B*a^2*b^3*d^2*x^2*e^3 -
300*A*a*b^4*d^2*x^2*e^3 + 600*B*a^2*b^3*d^3*x*e^2 - 600*A*a*b^4*d^3*x*e^2 - 20*
B*a^3*b^2*x^3*e^5 + 20*A*a^2*b^3*x^3*e^5 - 150*B*a^3*b^2*d*x^2*e^4 + 150*A*a^2*b
^3*d*x^2*e^4 - 600*B*a^3*b^2*d^2*x*e^3 + 600*A*a^2*b^3*d^2*x*e^3 + 30*B*a^4*b*x^
2*e^5 - 30*A*a^3*b^2*x^2*e^5 + 300*B*a^4*b*d*x*e^4 - 300*A*a^3*b^2*d*x*e^4 - 60*
B*a^5*x*e^5 + 60*A*a^4*b*x*e^5)/b^6 - (B*a*b^5*d^5 - A*b^6*d^5 - 5*B*a^2*b^4*d^4
*e + 5*A*a*b^5*d^4*e + 10*B*a^3*b^3*d^3*e^2 - 10*A*a^2*b^4*d^3*e^2 - 10*B*a^4*b^
2*d^2*e^3 + 10*A*a^3*b^3*d^2*e^3 + 5*B*a^5*b*d*e^4 - 5*A*a^4*b^2*d*e^4 - B*a^6*e
^5 + A*a^5*b*e^5)*ln(abs(b*x + a))/b^7