3.1122 \(\int \frac{(A+B x) (d+e x)^5}{(a+b x)^2} \, dx\)

Optimal. Leaf size=227 \[ \frac{5 e^2 (a+b x)^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7}+\frac{5 e^3 (a+b x)^3 (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac{e^4 (a+b x)^4 (-6 a B e+A b e+5 b B d)}{4 b^7}-\frac{(A b-a B) (b d-a e)^5}{b^7 (a+b x)}+\frac{5 e x (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{b^6}+\frac{(b d-a e)^4 \log (a+b x) (-6 a B e+5 A b e+b B d)}{b^7}+\frac{B e^5 (a+b x)^5}{5 b^7} \]

[Out]

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

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Rubi [A]  time = 0.389174, antiderivative size = 227, 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} \[ \frac{5 e^2 (a+b x)^2 (b d-a e)^2 (-2 a B e+A b e+b B d)}{b^7}+\frac{5 e^3 (a+b x)^3 (b d-a e) (-3 a B e+A b e+2 b B d)}{3 b^7}+\frac{e^4 (a+b x)^4 (-6 a B e+A b e+5 b B d)}{4 b^7}-\frac{(A b-a B) (b d-a e)^5}{b^7 (a+b x)}+\frac{5 e x (b d-a e)^3 (-3 a B e+2 A b e+b B d)}{b^6}+\frac{(b d-a e)^4 \log (a+b x) (-6 a B e+5 A b e+b B d)}{b^7}+\frac{B e^5 (a+b x)^5}{5 b^7} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^5)/(a + b*x)^2,x]

[Out]

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

Mathematica [B]  time = 0.245818, size = 500, normalized size = 2.2 \[ \frac{-5 A b \left (-10 a^2 b^3 e^2 \left (-24 d^2 e x-12 d^3+12 d e^2 x^2+e^3 x^3\right )+30 a^3 b^2 e^3 \left (-4 d^2-6 d e x+e^2 x^2\right )+12 a^4 b e^4 (5 d+4 e x)-12 a^5 e^5+5 a b^4 e \left (36 d^2 e^2 x^2-24 d^3 e x-12 d^4+8 d e^3 x^3+e^4 x^4\right )+b^5 \left (-120 d^3 e^2 x^2-60 d^2 e^3 x^3+12 d^5-20 d e^4 x^4-3 e^5 x^5\right )\right )+B \left (60 a^4 b^2 e^3 \left (-10 d^2-20 d e x+3 e^2 x^2\right )+30 a^3 b^3 e^2 \left (60 d^2 e x+20 d^3-25 d e^2 x^2-2 e^3 x^3\right )+10 a^2 b^4 e \left (120 d^2 e^2 x^2-120 d^3 e x-30 d^4+25 d e^3 x^3+3 e^4 x^4\right )+300 a^5 b e^4 (d+e x)-60 a^6 e^5+a b^5 \left (-900 d^3 e^2 x^2-400 d^2 e^3 x^3+300 d^4 e x+60 d^5-125 d e^4 x^4-18 e^5 x^5\right )+b^6 e x^2 \left (200 d^2 e^2 x^2+300 d^3 e x+300 d^4+75 d e^3 x^3+12 e^4 x^4\right )\right )+60 (a+b x) (b d-a e)^4 \log (a+b x) (-6 a B e+5 A b e+b B d)}{60 b^7 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^5)/(a + b*x)^2,x]

[Out]

(B*(-60*a^6*e^5 + 300*a^5*b*e^4*(d + e*x) + 60*a^4*b^2*e^3*(-10*d^2 - 20*d*e*x + 3*e^2*x^2) + 30*a^3*b^3*e^2*(
20*d^3 + 60*d^2*e*x - 25*d*e^2*x^2 - 2*e^3*x^3) + 10*a^2*b^4*e*(-30*d^4 - 120*d^3*e*x + 120*d^2*e^2*x^2 + 25*d
*e^3*x^3 + 3*e^4*x^4) + b^6*e*x^2*(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) + a*b^
5*(60*d^5 + 300*d^4*e*x - 900*d^3*e^2*x^2 - 400*d^2*e^3*x^3 - 125*d*e^4*x^4 - 18*e^5*x^5)) - 5*A*b*(-12*a^5*e^
5 + 12*a^4*b*e^4*(5*d + 4*e*x) + 30*a^3*b^2*e^3*(-4*d^2 - 6*d*e*x + e^2*x^2) - 10*a^2*b^3*e^2*(-12*d^3 - 24*d^
2*e*x + 12*d*e^2*x^2 + e^3*x^3) + 5*a*b^4*e*(-12*d^4 - 24*d^3*e*x + 36*d^2*e^2*x^2 + 8*d*e^3*x^3 + e^4*x^4) +
b^5*(12*d^5 - 120*d^3*e^2*x^2 - 60*d^2*e^3*x^3 - 20*d*e^4*x^4 - 3*e^5*x^5)) + 60*(b*d - a*e)^4*(b*B*d + 5*A*b*
e - 6*a*B*e)*(a + b*x)*Log[a + b*x])/(60*b^7*(a + b*x))

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Maple [B]  time = 0.012, size = 787, normalized size = 3.5 \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)^2,x)

[Out]

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

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Maxima [B]  time = 1.27104, size = 782, normalized size = 3.44 \begin{align*} \frac{{\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}}{b^{8} x + a b^{7}} + \frac{12 \, B b^{4} e^{5} x^{5} + 15 \,{\left (5 \, B b^{4} d e^{4} -{\left (2 \, B a b^{3} - A b^{4}\right )} e^{5}\right )} x^{4} + 20 \,{\left (10 \, B b^{4} d^{2} e^{3} - 5 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d e^{4} +{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 30 \,{\left (10 \, B b^{4} d^{3} e^{2} - 10 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{2} e^{3} + 5 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d e^{4} -{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 60 \,{\left (5 \, B b^{4} d^{4} e - 10 \,{\left (2 \, B a b^{3} - A b^{4}\right )} d^{3} e^{2} + 10 \,{\left (3 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} d^{2} e^{3} - 5 \,{\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} d e^{4} +{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} e^{5}\right )} x}{60 \, b^{6}} + \frac{{\left (B b^{5} d^{5} - 5 \,{\left (2 \, B a b^{4} - A b^{5}\right )} d^{4} e + 10 \,{\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4}\right )} d^{3} e^{2} - 10 \,{\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} d^{2} e^{3} + 5 \,{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} d e^{4} -{\left (6 \, B a^{5} - 5 \, A a^{4} 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)^2,x, algorithm="maxima")

[Out]

((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)/(b^8*x + a*b^7) + 1/60*(12*B*b^4*e
^5*x^5 + 15*(5*B*b^4*d*e^4 - (2*B*a*b^3 - A*b^4)*e^5)*x^4 + 20*(10*B*b^4*d^2*e^3 - 5*(2*B*a*b^3 - A*b^4)*d*e^4
+ (3*B*a^2*b^2 - 2*A*a*b^3)*e^5)*x^3 + 30*(10*B*b^4*d^3*e^2 - 10*(2*B*a*b^3 - A*b^4)*d^2*e^3 + 5*(3*B*a^2*b^2
- 2*A*a*b^3)*d*e^4 - (4*B*a^3*b - 3*A*a^2*b^2)*e^5)*x^2 + 60*(5*B*b^4*d^4*e - 10*(2*B*a*b^3 - A*b^4)*d^3*e^2
+ 10*(3*B*a^2*b^2 - 2*A*a*b^3)*d^2*e^3 - 5*(4*B*a^3*b - 3*A*a^2*b^2)*d*e^4 + (5*B*a^4 - 4*A*a^3*b)*e^5)*x)/b^6
+ (B*b^5*d^5 - 5*(2*B*a*b^4 - A*b^5)*d^4*e + 10*(3*B*a^2*b^3 - 2*A*a*b^4)*d^3*e^2 - 10*(4*B*a^3*b^2 - 3*A*a^2
*b^3)*d^2*e^3 + 5*(5*B*a^4*b - 4*A*a^3*b^2)*d*e^4 - (6*B*a^5 - 5*A*a^4*b)*e^5)*log(b*x + a)/b^7

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Fricas [B]  time = 1.59897, size = 1713, normalized size = 7.55 \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)^2,x, algorithm="fricas")

[Out]

1/60*(12*B*b^6*e^5*x^6 + 60*(B*a*b^5 - A*b^6)*d^5 - 300*(B*a^2*b^4 - A*a*b^5)*d^4*e + 600*(B*a^3*b^3 - A*a^2*b
^4)*d^3*e^2 - 600*(B*a^4*b^2 - A*a^3*b^3)*d^2*e^3 + 300*(B*a^5*b - A*a^4*b^2)*d*e^4 - 60*(B*a^6 - A*a^5*b)*e^5
+ 3*(25*B*b^6*d*e^4 - (6*B*a*b^5 - 5*A*b^6)*e^5)*x^5 + 5*(40*B*b^6*d^2*e^3 - 5*(5*B*a*b^5 - 4*A*b^6)*d*e^4 +
(6*B*a^2*b^4 - 5*A*a*b^5)*e^5)*x^4 + 10*(30*B*b^6*d^3*e^2 - 10*(4*B*a*b^5 - 3*A*b^6)*d^2*e^3 + 5*(5*B*a^2*b^4
- 4*A*a*b^5)*d*e^4 - (6*B*a^3*b^3 - 5*A*a^2*b^4)*e^5)*x^3 + 30*(10*B*b^6*d^4*e - 10*(3*B*a*b^5 - 2*A*b^6)*d^3*
e^2 + 10*(4*B*a^2*b^4 - 3*A*a*b^5)*d^2*e^3 - 5*(5*B*a^3*b^3 - 4*A*a^2*b^4)*d*e^4 + (6*B*a^4*b^2 - 5*A*a^3*b^3)
*e^5)*x^2 + 60*(5*B*a*b^5*d^4*e - 10*(2*B*a^2*b^4 - A*a*b^5)*d^3*e^2 + 10*(3*B*a^3*b^3 - 2*A*a^2*b^4)*d^2*e^3
- 5*(4*B*a^4*b^2 - 3*A*a^3*b^3)*d*e^4 + (5*B*a^5*b - 4*A*a^4*b^2)*e^5)*x + 60*(B*a*b^5*d^5 - 5*(2*B*a^2*b^4 -
A*a*b^5)*d^4*e + 10*(3*B*a^3*b^3 - 2*A*a^2*b^4)*d^3*e^2 - 10*(4*B*a^4*b^2 - 3*A*a^3*b^3)*d^2*e^3 + 5*(5*B*a^5*
b - 4*A*a^4*b^2)*d*e^4 - (6*B*a^6 - 5*A*a^5*b)*e^5 + (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)*log(b*x + a))/(b^8*x + a*b^7)

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Sympy [B]  time = 3.37277, size = 552, normalized size = 2.43 \begin{align*} \frac{B e^{5} x^{5}}{5 b^{2}} - \frac{- A a^{5} b e^{5} + 5 A a^{4} b^{2} d e^{4} - 10 A a^{3} b^{3} d^{2} e^{3} + 10 A a^{2} b^{4} d^{3} e^{2} - 5 A a b^{5} d^{4} e + A b^{6} d^{5} + B a^{6} e^{5} - 5 B a^{5} b d e^{4} + 10 B a^{4} b^{2} d^{2} e^{3} - 10 B a^{3} b^{3} d^{3} e^{2} + 5 B a^{2} b^{4} d^{4} e - B a b^{5} d^{5}}{a b^{7} + b^{8} x} - \frac{x^{4} \left (- A b e^{5} + 2 B a e^{5} - 5 B b d e^{4}\right )}{4 b^{3}} + \frac{x^{3} \left (- 2 A a b e^{5} + 5 A b^{2} d e^{4} + 3 B a^{2} e^{5} - 10 B a b d e^{4} + 10 B b^{2} d^{2} e^{3}\right )}{3 b^{4}} - \frac{x^{2} \left (- 3 A a^{2} b e^{5} + 10 A a b^{2} d e^{4} - 10 A b^{3} d^{2} e^{3} + 4 B a^{3} e^{5} - 15 B a^{2} b d e^{4} + 20 B a b^{2} d^{2} e^{3} - 10 B b^{3} d^{3} e^{2}\right )}{2 b^{5}} + \frac{x \left (- 4 A a^{3} b e^{5} + 15 A a^{2} b^{2} d e^{4} - 20 A a b^{3} d^{2} e^{3} + 10 A b^{4} d^{3} e^{2} + 5 B a^{4} e^{5} - 20 B a^{3} b d e^{4} + 30 B a^{2} b^{2} d^{2} e^{3} - 20 B a b^{3} d^{3} e^{2} + 5 B b^{4} d^{4} e\right )}{b^{6}} - \frac{\left (a e - b d\right )^{4} \left (- 5 A b e + 6 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)**2,x)

[Out]

B*e**5*x**5/(5*b**2) - (-A*a**5*b*e**5 + 5*A*a**4*b**2*d*e**4 - 10*A*a**3*b**3*d**2*e**3 + 10*A*a**2*b**4*d**3
*e**2 - 5*A*a*b**5*d**4*e + A*b**6*d**5 + B*a**6*e**5 - 5*B*a**5*b*d*e**4 + 10*B*a**4*b**2*d**2*e**3 - 10*B*a*
*3*b**3*d**3*e**2 + 5*B*a**2*b**4*d**4*e - B*a*b**5*d**5)/(a*b**7 + b**8*x) - x**4*(-A*b*e**5 + 2*B*a*e**5 - 5
*B*b*d*e**4)/(4*b**3) + x**3*(-2*A*a*b*e**5 + 5*A*b**2*d*e**4 + 3*B*a**2*e**5 - 10*B*a*b*d*e**4 + 10*B*b**2*d*
*2*e**3)/(3*b**4) - x**2*(-3*A*a**2*b*e**5 + 10*A*a*b**2*d*e**4 - 10*A*b**3*d**2*e**3 + 4*B*a**3*e**5 - 15*B*a
**2*b*d*e**4 + 20*B*a*b**2*d**2*e**3 - 10*B*b**3*d**3*e**2)/(2*b**5) + x*(-4*A*a**3*b*e**5 + 15*A*a**2*b**2*d*
e**4 - 20*A*a*b**3*d**2*e**3 + 10*A*b**4*d**3*e**2 + 5*B*a**4*e**5 - 20*B*a**3*b*d*e**4 + 30*B*a**2*b**2*d**2*
e**3 - 20*B*a*b**3*d**3*e**2 + 5*B*b**4*d**4*e)/b**6 - (a*e - b*d)**4*(-5*A*b*e + 6*B*a*e - B*b*d)*log(a + b*x
)/b**7

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Giac [B]  time = 2.59964, size = 953, normalized size = 4.2 \begin{align*} \frac{{\left (b x + a\right )}^{5}{\left (12 \, B e^{5} + \frac{15 \,{\left (5 \, B b^{2} d e^{4} - 6 \, B a b e^{5} + A b^{2} e^{5}\right )}}{{\left (b x + a\right )} b} + \frac{100 \,{\left (2 \, B b^{4} d^{2} e^{3} - 5 \, B a b^{3} d e^{4} + A b^{4} d e^{4} + 3 \, B a^{2} b^{2} e^{5} - A a b^{3} e^{5}\right )}}{{\left (b x + a\right )}^{2} b^{2}} + \frac{300 \,{\left (B b^{6} d^{3} e^{2} - 4 \, B a b^{5} d^{2} e^{3} + A b^{6} d^{2} e^{3} + 5 \, B a^{2} b^{4} d e^{4} - 2 \, A a b^{5} d e^{4} - 2 \, B a^{3} b^{3} e^{5} + A a^{2} b^{4} e^{5}\right )}}{{\left (b x + a\right )}^{3} b^{3}} + \frac{300 \,{\left (B b^{8} d^{4} e - 6 \, B a b^{7} d^{3} e^{2} + 2 \, A b^{8} d^{3} e^{2} + 12 \, B a^{2} b^{6} d^{2} e^{3} - 6 \, A a b^{7} d^{2} e^{3} - 10 \, B a^{3} b^{5} d e^{4} + 6 \, A a^{2} b^{6} d e^{4} + 3 \, B a^{4} b^{4} e^{5} - 2 \, A a^{3} b^{5} e^{5}\right )}}{{\left (b x + a\right )}^{4} b^{4}}\right )}}{60 \, b^{7}} - \frac{{\left (B b^{5} d^{5} - 10 \, B a b^{4} d^{4} e + 5 \, A b^{5} d^{4} e + 30 \, B a^{2} b^{3} d^{3} e^{2} - 20 \, A a b^{4} d^{3} e^{2} - 40 \, B a^{3} b^{2} d^{2} e^{3} + 30 \, A a^{2} b^{3} d^{2} e^{3} + 25 \, B a^{4} b d e^{4} - 20 \, A a^{3} b^{2} d e^{4} - 6 \, B a^{5} e^{5} + 5 \, A a^{4} b e^{5}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{7}} + \frac{\frac{B a b^{10} d^{5}}{b x + a} - \frac{A b^{11} d^{5}}{b x + a} - \frac{5 \, B a^{2} b^{9} d^{4} e}{b x + a} + \frac{5 \, A a b^{10} d^{4} e}{b x + a} + \frac{10 \, B a^{3} b^{8} d^{3} e^{2}}{b x + a} - \frac{10 \, A a^{2} b^{9} d^{3} e^{2}}{b x + a} - \frac{10 \, B a^{4} b^{7} d^{2} e^{3}}{b x + a} + \frac{10 \, A a^{3} b^{8} d^{2} e^{3}}{b x + a} + \frac{5 \, B a^{5} b^{6} d e^{4}}{b x + a} - \frac{5 \, A a^{4} b^{7} d e^{4}}{b x + a} - \frac{B a^{6} b^{5} e^{5}}{b x + a} + \frac{A a^{5} b^{6} e^{5}}{b x + a}}{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)^2,x, algorithm="giac")

[Out]

1/60*(b*x + a)^5*(12*B*e^5 + 15*(5*B*b^2*d*e^4 - 6*B*a*b*e^5 + A*b^2*e^5)/((b*x + a)*b) + 100*(2*B*b^4*d^2*e^3
- 5*B*a*b^3*d*e^4 + A*b^4*d*e^4 + 3*B*a^2*b^2*e^5 - A*a*b^3*e^5)/((b*x + a)^2*b^2) + 300*(B*b^6*d^3*e^2 - 4*B
*a*b^5*d^2*e^3 + A*b^6*d^2*e^3 + 5*B*a^2*b^4*d*e^4 - 2*A*a*b^5*d*e^4 - 2*B*a^3*b^3*e^5 + A*a^2*b^4*e^5)/((b*x
+ a)^3*b^3) + 300*(B*b^8*d^4*e - 6*B*a*b^7*d^3*e^2 + 2*A*b^8*d^3*e^2 + 12*B*a^2*b^6*d^2*e^3 - 6*A*a*b^7*d^2*e^
3 - 10*B*a^3*b^5*d*e^4 + 6*A*a^2*b^6*d*e^4 + 3*B*a^4*b^4*e^5 - 2*A*a^3*b^5*e^5)/((b*x + a)^4*b^4))/b^7 - (B*b^
5*d^5 - 10*B*a*b^4*d^4*e + 5*A*b^5*d^4*e + 30*B*a^2*b^3*d^3*e^2 - 20*A*a*b^4*d^3*e^2 - 40*B*a^3*b^2*d^2*e^3 +
30*A*a^2*b^3*d^2*e^3 + 25*B*a^4*b*d*e^4 - 20*A*a^3*b^2*d*e^4 - 6*B*a^5*e^5 + 5*A*a^4*b*e^5)*log(abs(b*x + a)/(
(b*x + a)^2*abs(b)))/b^7 + (B*a*b^10*d^5/(b*x + a) - A*b^11*d^5/(b*x + a) - 5*B*a^2*b^9*d^4*e/(b*x + a) + 5*A*
a*b^10*d^4*e/(b*x + a) + 10*B*a^3*b^8*d^3*e^2/(b*x + a) - 10*A*a^2*b^9*d^3*e^2/(b*x + a) - 10*B*a^4*b^7*d^2*e^
3/(b*x + a) + 10*A*a^3*b^8*d^2*e^3/(b*x + a) + 5*B*a^5*b^6*d*e^4/(b*x + a) - 5*A*a^4*b^7*d*e^4/(b*x + a) - B*a
^6*b^5*e^5/(b*x + a) + A*a^5*b^6*e^5/(b*x + a))/b^12