3.1133 \(\int \frac{(A+B x) (d+e x)^5}{(a+b x)^3} \, dx\)

Optimal. Leaf size=230 \[ \frac{e^4 (a+b x)^3 (-6 a B e+A b e+5 b B d)}{3 b^7}+\frac{5 e^3 (a+b x)^2 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}+\frac{10 e^2 x (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^6}-\frac{(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{b^7 (a+b x)}-\frac{(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}+\frac{5 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^7}+\frac{B e^5 (a+b x)^4}{4 b^7} \]

[Out]

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

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Rubi [A]  time = 0.346598, antiderivative size = 230, 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, Rules used = {77} \[ \frac{e^4 (a+b x)^3 (-6 a B e+A b e+5 b B d)}{3 b^7}+\frac{5 e^3 (a+b x)^2 (b d-a e) (-3 a B e+A b e+2 b B d)}{2 b^7}+\frac{10 e^2 x (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^6}-\frac{(b d-a e)^4 (-6 a B e+5 A b e+b B d)}{b^7 (a+b x)}-\frac{(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}+\frac{5 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)}{b^7}+\frac{B e^5 (a+b x)^4}{4 b^7} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^5)/(a + b*x)^3,x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^5}{(a+b x)^3} \, dx &=\int \left (\frac{10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e)}{b^6}+\frac{(A b-a B) (b d-a e)^5}{b^6 (a+b x)^3}+\frac{(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^6 (a+b x)^2}+\frac{5 e (b d-a e)^3 (b B d+2 A b e-3 a B e)}{b^6 (a+b x)}+\frac{5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)}{b^6}+\frac{e^4 (5 b B d+A b e-6 a B e) (a+b x)^2}{b^6}+\frac{B e^5 (a+b x)^3}{b^6}\right ) \, dx\\ &=\frac{10 e^2 (b d-a e)^2 (b B d+A b e-2 a B e) x}{b^6}-\frac{(A b-a B) (b d-a e)^5}{2 b^7 (a+b x)^2}-\frac{(b d-a e)^4 (b B d+5 A b e-6 a B e)}{b^7 (a+b x)}+\frac{5 e^3 (b d-a e) (2 b B d+A b e-3 a B e) (a+b x)^2}{2 b^7}+\frac{e^4 (5 b B d+A b e-6 a B e) (a+b x)^3}{3 b^7}+\frac{B e^5 (a+b x)^4}{4 b^7}+\frac{5 e (b d-a e)^3 (b B d+2 A b e-3 a B e) \log (a+b x)}{b^7}\\ \end{align*}

Mathematica [A]  time = 0.140605, size = 254, normalized size = 1.1 \[ \frac{12 b e^2 x \left (6 a^2 b e^2 (A e+5 B d)-10 a^3 B e^3-15 a b^2 d e (A e+2 B d)+10 b^3 d^2 (A e+B d)\right )+6 b^2 e^3 x^2 \left (6 a^2 B e^2-3 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+4 b^3 e^4 x^3 (-3 a B e+A b e+5 b B d)-\frac{12 (b d-a e)^4 (-6 a B e+5 A b e+b B d)}{a+b x}-\frac{6 (A b-a B) (b d-a e)^5}{(a+b x)^2}+60 e (b d-a e)^3 \log (a+b x) (-3 a B e+2 A b e+b B d)+3 b^4 B e^5 x^4}{12 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x)^3,x]

[Out]

(12*b*e^2*(-10*a^3*B*e^3 + 10*b^3*d^2*(B*d + A*e) - 15*a*b^2*d*e*(2*B*d + A*e) + 6*a^2*b*e^2*(5*B*d + A*e))*x
+ 6*b^2*e^3*(6*a^2*B*e^2 + 5*b^2*d*(2*B*d + A*e) - 3*a*b*e*(5*B*d + A*e))*x^2 + 4*b^3*e^4*(5*b*B*d + A*b*e - 3
*a*B*e)*x^3 + 3*b^4*B*e^5*x^4 - (6*(A*b - a*B)*(b*d - a*e)^5)/(a + b*x)^2 - (12*(b*d - a*e)^4*(b*B*d + 5*A*b*e
- 6*a*B*e))/(a + b*x) + 60*e*(b*d - a*e)^3*(b*B*d + 2*A*b*e - 3*a*B*e)*Log[a + b*x])/(12*b^7)

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Maple [B]  time = 0.015, size = 833, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5/(b*x+a)^3,x)

[Out]

-30/b^4*e^2*ln(b*x+a)*B*a*d^3+20/b^5/(b*x+a)*A*a^3*d*e^4-30/b^4/(b*x+a)*A*a^2*d^2*e^3+20/b^3/(b*x+a)*A*a*d^3*e
^2-25/b^6/(b*x+a)*B*a^4*d*e^4+40/b^5/(b*x+a)*B*a^3*d^2*e^3-30/b^4/(b*x+a)*B*a^2*d^3*e^2+10/b^3/(b*x+a)*B*a*d^4
*e-5/2/b^5/(b*x+a)^2*A*a^4*d*e^4+5/b^4/(b*x+a)^2*A*a^3*d^2*e^3-5/b^3/(b*x+a)^2*A*a^2*d^3*e^2+5/2/b^2/(b*x+a)^2
*A*a*d^4*e+5/2/b^6/(b*x+a)^2*B*a^5*d*e^4-5/b^5/(b*x+a)^2*B*a^4*d^2*e^3+5/b^4/(b*x+a)^2*B*a^3*d^3*e^2-5/2/b^3/(
b*x+a)^2*B*a^2*d^4*e-15/2*e^4/b^4*B*x^2*a*d-15*e^4/b^4*A*a*d*x+30*e^4/b^5*B*a^2*d*x-30*e^3/b^4*B*a*d^2*x+30/b^
5*e^4*ln(b*x+a)*A*a^2*d-30/b^4*e^3*ln(b*x+a)*A*a*d^2-50/b^6*e^4*ln(b*x+a)*B*a^3*d+60/b^5*e^3*ln(b*x+a)*B*a^2*d
^2+3*e^5/b^5*B*x^2*a^2+5*e^3/b^3*B*x^2*d^2+6*e^5/b^5*A*a^2*x+10*e^3/b^3*A*d^2*x-10*e^5/b^6*a^3*B*x+10*e^2/b^3*
B*d^3*x-3/2*e^5/b^4*A*x^2*a+5/2*e^4/b^3*A*x^2*d+1/4*e^5/b^3*B*x^4+1/3*e^5/b^3*A*x^3-1/b^2/(b*x+a)*B*d^5-1/2/b/
(b*x+a)^2*A*d^5-10/b^6*e^5*ln(b*x+a)*A*a^3+10/b^3*e^2*ln(b*x+a)*A*d^3+15/b^7*e^5*ln(b*x+a)*B*a^4+5/b^3*e*ln(b*
x+a)*B*d^4-5/b^6/(b*x+a)*A*a^4*e^5-5/b^2/(b*x+a)*A*d^4*e+6/b^7/(b*x+a)*B*a^5*e^5+1/2/b^6/(b*x+a)^2*A*a^5*e^5-1
/2/b^7/(b*x+a)^2*B*a^6*e^5+1/2/b^2/(b*x+a)^2*B*a*d^5-e^5/b^4*B*x^3*a+5/3*e^4/b^3*B*x^3*d

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Maxima [B]  time = 1.27091, size = 799, normalized size = 3.47 \begin{align*} -\frac{{\left (B a b^{5} + A b^{6}\right )} d^{5} - 5 \,{\left (3 \, B a^{2} b^{4} - A a b^{5}\right )} d^{4} e + 10 \,{\left (5 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{3} e^{2} - 10 \,{\left (7 \, B a^{4} b^{2} - 5 \, A a^{3} b^{3}\right )} d^{2} e^{3} + 5 \,{\left (9 \, B a^{5} b - 7 \, A a^{4} b^{2}\right )} d e^{4} -{\left (11 \, B a^{6} - 9 \, A a^{5} b\right )} e^{5} + 2 \,{\left (B b^{6} d^{5} - 5 \,{\left (2 \, B a b^{5} - A b^{6}\right )} d^{4} e + 10 \,{\left (3 \, B a^{2} b^{4} - 2 \, A a b^{5}\right )} d^{3} e^{2} - 10 \,{\left (4 \, B a^{3} b^{3} - 3 \, A a^{2} b^{4}\right )} d^{2} e^{3} + 5 \,{\left (5 \, B a^{4} b^{2} - 4 \, A a^{3} b^{3}\right )} d e^{4} -{\left (6 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} e^{5}\right )} x}{2 \,{\left (b^{9} x^{2} + 2 \, a b^{8} x + a^{2} b^{7}\right )}} + \frac{3 \, B b^{3} e^{5} x^{4} + 4 \,{\left (5 \, B b^{3} d e^{4} -{\left (3 \, B a b^{2} - A b^{3}\right )} e^{5}\right )} x^{3} + 6 \,{\left (10 \, B b^{3} d^{2} e^{3} - 5 \,{\left (3 \, B a b^{2} - A b^{3}\right )} d e^{4} + 3 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} e^{5}\right )} x^{2} + 12 \,{\left (10 \, B b^{3} d^{3} e^{2} - 10 \,{\left (3 \, B a b^{2} - A b^{3}\right )} d^{2} e^{3} + 15 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} d e^{4} - 2 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} e^{5}\right )} x}{12 \, b^{6}} + \frac{5 \,{\left (B b^{4} d^{4} e - 2 \,{\left (3 \, B a b^{3} - A b^{4}\right )} d^{3} e^{2} + 6 \,{\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} d^{2} e^{3} - 2 \,{\left (5 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} e^{5}\right )} \log \left (b x + a\right )}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*((B*a*b^5 + A*b^6)*d^5 - 5*(3*B*a^2*b^4 - A*a*b^5)*d^4*e + 10*(5*B*a^3*b^3 - 3*A*a^2*b^4)*d^3*e^2 - 10*(7
*B*a^4*b^2 - 5*A*a^3*b^3)*d^2*e^3 + 5*(9*B*a^5*b - 7*A*a^4*b^2)*d*e^4 - (11*B*a^6 - 9*A*a^5*b)*e^5 + 2*(B*b^6*
d^5 - 5*(2*B*a*b^5 - A*b^6)*d^4*e + 10*(3*B*a^2*b^4 - 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 - 3*A*a^2*b^4)*d^2*
e^3 + 5*(5*B*a^4*b^2 - 4*A*a^3*b^3)*d*e^4 - (6*B*a^5*b - 5*A*a^4*b^2)*e^5)*x)/(b^9*x^2 + 2*a*b^8*x + a^2*b^7)
+ 1/12*(3*B*b^3*e^5*x^4 + 4*(5*B*b^3*d*e^4 - (3*B*a*b^2 - A*b^3)*e^5)*x^3 + 6*(10*B*b^3*d^2*e^3 - 5*(3*B*a*b^2
- A*b^3)*d*e^4 + 3*(2*B*a^2*b - A*a*b^2)*e^5)*x^2 + 12*(10*B*b^3*d^3*e^2 - 10*(3*B*a*b^2 - A*b^3)*d^2*e^3 + 1
5*(2*B*a^2*b - A*a*b^2)*d*e^4 - 2*(5*B*a^3 - 3*A*a^2*b)*e^5)*x)/b^6 + 5*(B*b^4*d^4*e - 2*(3*B*a*b^3 - A*b^4)*d
^3*e^2 + 6*(2*B*a^2*b^2 - A*a*b^3)*d^2*e^3 - 2*(5*B*a^3*b - 3*A*a^2*b^2)*d*e^4 + (3*B*a^4 - 2*A*a^3*b)*e^5)*lo
g(b*x + a)/b^7

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Fricas [B]  time = 1.60385, size = 1855, normalized size = 8.07 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(b*x+a)^3,x, algorithm="fricas")

[Out]

1/12*(3*B*b^6*e^5*x^6 - 6*(B*a*b^5 + A*b^6)*d^5 + 30*(3*B*a^2*b^4 - A*a*b^5)*d^4*e - 60*(5*B*a^3*b^3 - 3*A*a^2
*b^4)*d^3*e^2 + 60*(7*B*a^4*b^2 - 5*A*a^3*b^3)*d^2*e^3 - 30*(9*B*a^5*b - 7*A*a^4*b^2)*d*e^4 + 6*(11*B*a^6 - 9*
A*a^5*b)*e^5 + 2*(10*B*b^6*d*e^4 - (3*B*a*b^5 - 2*A*b^6)*e^5)*x^5 + 5*(12*B*b^6*d^2*e^3 - 2*(5*B*a*b^5 - 3*A*b
^6)*d*e^4 + (3*B*a^2*b^4 - 2*A*a*b^5)*e^5)*x^4 + 20*(6*B*b^6*d^3*e^2 - 6*(2*B*a*b^5 - A*b^6)*d^2*e^3 + 2*(5*B*
a^2*b^4 - 3*A*a*b^5)*d*e^4 - (3*B*a^3*b^3 - 2*A*a^2*b^4)*e^5)*x^3 + 6*(40*B*a*b^5*d^3*e^2 - 10*(11*B*a^2*b^4 -
4*A*a*b^5)*d^2*e^3 + 5*(21*B*a^3*b^3 - 11*A*a^2*b^4)*d*e^4 - (34*B*a^4*b^2 - 21*A*a^3*b^3)*e^5)*x^2 - 12*(B*b
^6*d^5 - 5*(2*B*a*b^5 - A*b^6)*d^4*e + 20*(B*a^2*b^4 - A*a*b^5)*d^3*e^2 - 10*(B*a^3*b^3 - 2*A*a^2*b^4)*d^2*e^3
- 5*(B*a^4*b^2 + A*a^3*b^3)*d*e^4 + (4*B*a^5*b - A*a^4*b^2)*e^5)*x + 60*(B*a^2*b^4*d^4*e - 2*(3*B*a^3*b^3 - A
*a^2*b^4)*d^3*e^2 + 6*(2*B*a^4*b^2 - A*a^3*b^3)*d^2*e^3 - 2*(5*B*a^5*b - 3*A*a^4*b^2)*d*e^4 + (3*B*a^6 - 2*A*a
^5*b)*e^5 + (B*b^6*d^4*e - 2*(3*B*a*b^5 - A*b^6)*d^3*e^2 + 6*(2*B*a^2*b^4 - A*a*b^5)*d^2*e^3 - 2*(5*B*a^3*b^3
- 3*A*a^2*b^4)*d*e^4 + (3*B*a^4*b^2 - 2*A*a^3*b^3)*e^5)*x^2 + 2*(B*a*b^5*d^4*e - 2*(3*B*a^2*b^4 - A*a*b^5)*d^3
*e^2 + 6*(2*B*a^3*b^3 - A*a^2*b^4)*d^2*e^3 - 2*(5*B*a^4*b^2 - 3*A*a^3*b^3)*d*e^4 + (3*B*a^5*b - 2*A*a^4*b^2)*e
^5)*x)*log(b*x + a))/(b^9*x^2 + 2*a*b^8*x + a^2*b^7)

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Sympy [B]  time = 13.6227, size = 602, normalized size = 2.62 \begin{align*} \frac{B e^{5} x^{4}}{4 b^{3}} + \frac{- 9 A a^{5} b e^{5} + 35 A a^{4} b^{2} d e^{4} - 50 A a^{3} b^{3} d^{2} e^{3} + 30 A a^{2} b^{4} d^{3} e^{2} - 5 A a b^{5} d^{4} e - A b^{6} d^{5} + 11 B a^{6} e^{5} - 45 B a^{5} b d e^{4} + 70 B a^{4} b^{2} d^{2} e^{3} - 50 B a^{3} b^{3} d^{3} e^{2} + 15 B a^{2} b^{4} d^{4} e - B a b^{5} d^{5} + x \left (- 10 A a^{4} b^{2} e^{5} + 40 A a^{3} b^{3} d e^{4} - 60 A a^{2} b^{4} d^{2} e^{3} + 40 A a b^{5} d^{3} e^{2} - 10 A b^{6} d^{4} e + 12 B a^{5} b e^{5} - 50 B a^{4} b^{2} d e^{4} + 80 B a^{3} b^{3} d^{2} e^{3} - 60 B a^{2} b^{4} d^{3} e^{2} + 20 B a b^{5} d^{4} e - 2 B b^{6} d^{5}\right )}{2 a^{2} b^{7} + 4 a b^{8} x + 2 b^{9} x^{2}} - \frac{x^{3} \left (- A b e^{5} + 3 B a e^{5} - 5 B b d e^{4}\right )}{3 b^{4}} + \frac{x^{2} \left (- 3 A a b e^{5} + 5 A b^{2} d e^{4} + 6 B a^{2} e^{5} - 15 B a b d e^{4} + 10 B b^{2} d^{2} e^{3}\right )}{2 b^{5}} - \frac{x \left (- 6 A a^{2} b e^{5} + 15 A a b^{2} d e^{4} - 10 A b^{3} d^{2} e^{3} + 10 B a^{3} e^{5} - 30 B a^{2} b d e^{4} + 30 B a b^{2} d^{2} e^{3} - 10 B b^{3} d^{3} e^{2}\right )}{b^{6}} + \frac{5 e \left (a e - b d\right )^{3} \left (- 2 A b e + 3 B a e - B b d\right ) \log{\left (a + b x \right )}}{b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5/(b*x+a)**3,x)

[Out]

B*e**5*x**4/(4*b**3) + (-9*A*a**5*b*e**5 + 35*A*a**4*b**2*d*e**4 - 50*A*a**3*b**3*d**2*e**3 + 30*A*a**2*b**4*d
**3*e**2 - 5*A*a*b**5*d**4*e - A*b**6*d**5 + 11*B*a**6*e**5 - 45*B*a**5*b*d*e**4 + 70*B*a**4*b**2*d**2*e**3 -
50*B*a**3*b**3*d**3*e**2 + 15*B*a**2*b**4*d**4*e - B*a*b**5*d**5 + x*(-10*A*a**4*b**2*e**5 + 40*A*a**3*b**3*d*
e**4 - 60*A*a**2*b**4*d**2*e**3 + 40*A*a*b**5*d**3*e**2 - 10*A*b**6*d**4*e + 12*B*a**5*b*e**5 - 50*B*a**4*b**2
*d*e**4 + 80*B*a**3*b**3*d**2*e**3 - 60*B*a**2*b**4*d**3*e**2 + 20*B*a*b**5*d**4*e - 2*B*b**6*d**5))/(2*a**2*b
**7 + 4*a*b**8*x + 2*b**9*x**2) - x**3*(-A*b*e**5 + 3*B*a*e**5 - 5*B*b*d*e**4)/(3*b**4) + x**2*(-3*A*a*b*e**5
+ 5*A*b**2*d*e**4 + 6*B*a**2*e**5 - 15*B*a*b*d*e**4 + 10*B*b**2*d**2*e**3)/(2*b**5) - x*(-6*A*a**2*b*e**5 + 15
*A*a*b**2*d*e**4 - 10*A*b**3*d**2*e**3 + 10*B*a**3*e**5 - 30*B*a**2*b*d*e**4 + 30*B*a*b**2*d**2*e**3 - 10*B*b*
*3*d**3*e**2)/b**6 + 5*e*(a*e - b*d)**3*(-2*A*b*e + 3*B*a*e - B*b*d)*log(a + b*x)/b**7

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Giac [B]  time = 1.66586, size = 807, normalized size = 3.51 \begin{align*} \frac{5 \,{\left (B b^{4} d^{4} e - 6 \, B a b^{3} d^{3} e^{2} + 2 \, A b^{4} d^{3} e^{2} + 12 \, B a^{2} b^{2} d^{2} e^{3} - 6 \, A a b^{3} d^{2} e^{3} - 10 \, B a^{3} b d e^{4} + 6 \, A a^{2} b^{2} d e^{4} + 3 \, B a^{4} e^{5} - 2 \, A a^{3} b e^{5}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{7}} - \frac{B a b^{5} d^{5} + A b^{6} d^{5} - 15 \, B a^{2} b^{4} d^{4} e + 5 \, A a b^{5} d^{4} e + 50 \, B a^{3} b^{3} d^{3} e^{2} - 30 \, A a^{2} b^{4} d^{3} e^{2} - 70 \, B a^{4} b^{2} d^{2} e^{3} + 50 \, A a^{3} b^{3} d^{2} e^{3} + 45 \, B a^{5} b d e^{4} - 35 \, A a^{4} b^{2} d e^{4} - 11 \, B a^{6} e^{5} + 9 \, A a^{5} b e^{5} + 2 \,{\left (B b^{6} d^{5} - 10 \, B a b^{5} d^{4} e + 5 \, A b^{6} d^{4} e + 30 \, B a^{2} b^{4} d^{3} e^{2} - 20 \, A a b^{5} d^{3} e^{2} - 40 \, B a^{3} b^{3} d^{2} e^{3} + 30 \, A a^{2} b^{4} d^{2} e^{3} + 25 \, B a^{4} b^{2} d e^{4} - 20 \, A a^{3} b^{3} d e^{4} - 6 \, B a^{5} b e^{5} + 5 \, A a^{4} b^{2} e^{5}\right )} x}{2 \,{\left (b x + a\right )}^{2} b^{7}} + \frac{3 \, B b^{9} x^{4} e^{5} + 20 \, B b^{9} d x^{3} e^{4} + 60 \, B b^{9} d^{2} x^{2} e^{3} + 120 \, B b^{9} d^{3} x e^{2} - 12 \, B a b^{8} x^{3} e^{5} + 4 \, A b^{9} x^{3} e^{5} - 90 \, B a b^{8} d x^{2} e^{4} + 30 \, A b^{9} d x^{2} e^{4} - 360 \, B a b^{8} d^{2} x e^{3} + 120 \, A b^{9} d^{2} x e^{3} + 36 \, B a^{2} b^{7} x^{2} e^{5} - 18 \, A a b^{8} x^{2} e^{5} + 360 \, B a^{2} b^{7} d x e^{4} - 180 \, A a b^{8} d x e^{4} - 120 \, B a^{3} b^{6} x e^{5} + 72 \, A a^{2} b^{7} x e^{5}}{12 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(b*x+a)^3,x, algorithm="giac")

[Out]

5*(B*b^4*d^4*e - 6*B*a*b^3*d^3*e^2 + 2*A*b^4*d^3*e^2 + 12*B*a^2*b^2*d^2*e^3 - 6*A*a*b^3*d^2*e^3 - 10*B*a^3*b*d
*e^4 + 6*A*a^2*b^2*d*e^4 + 3*B*a^4*e^5 - 2*A*a^3*b*e^5)*log(abs(b*x + a))/b^7 - 1/2*(B*a*b^5*d^5 + A*b^6*d^5 -
15*B*a^2*b^4*d^4*e + 5*A*a*b^5*d^4*e + 50*B*a^3*b^3*d^3*e^2 - 30*A*a^2*b^4*d^3*e^2 - 70*B*a^4*b^2*d^2*e^3 + 5
0*A*a^3*b^3*d^2*e^3 + 45*B*a^5*b*d*e^4 - 35*A*a^4*b^2*d*e^4 - 11*B*a^6*e^5 + 9*A*a^5*b*e^5 + 2*(B*b^6*d^5 - 10
*B*a*b^5*d^4*e + 5*A*b^6*d^4*e + 30*B*a^2*b^4*d^3*e^2 - 20*A*a*b^5*d^3*e^2 - 40*B*a^3*b^3*d^2*e^3 + 30*A*a^2*b
^4*d^2*e^3 + 25*B*a^4*b^2*d*e^4 - 20*A*a^3*b^3*d*e^4 - 6*B*a^5*b*e^5 + 5*A*a^4*b^2*e^5)*x)/((b*x + a)^2*b^7) +
1/12*(3*B*b^9*x^4*e^5 + 20*B*b^9*d*x^3*e^4 + 60*B*b^9*d^2*x^2*e^3 + 120*B*b^9*d^3*x*e^2 - 12*B*a*b^8*x^3*e^5
+ 4*A*b^9*x^3*e^5 - 90*B*a*b^8*d*x^2*e^4 + 30*A*b^9*d*x^2*e^4 - 360*B*a*b^8*d^2*x*e^3 + 120*A*b^9*d^2*x*e^3 +
36*B*a^2*b^7*x^2*e^5 - 18*A*a*b^8*x^2*e^5 + 360*B*a^2*b^7*d*x*e^4 - 180*A*a*b^8*d*x*e^4 - 120*B*a^3*b^6*x*e^5
+ 72*A*a^2*b^7*x*e^5)/b^12