3.1154 \(\int \frac{(A+B x) (d+e x)^5}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=257 \[ -\frac{(c d-b e)^3 \log (b+c x) \left (-b^2 c e (4 B d-A e)-3 b c^2 d (B d-A e)+6 A c^3 d^2-3 b^3 B e^2\right )}{b^5 c^4}+\frac{d^3 \log (x) \left (5 b^2 e (2 A e+B d)-3 b c d (5 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac{(c d-b e)^4 \left (-2 A b c e-3 A c^2 d+3 b^2 B e+2 b B c d\right )}{b^4 c^4 (b+c x)}-\frac{(b B-A c) (c d-b e)^5}{2 b^3 c^4 (b+c x)^2}-\frac{d^4 (5 A b e-3 A c d+b B d)}{b^4 x}-\frac{A d^5}{2 b^3 x^2}+\frac{B e^5 x}{c^3} \]

[Out]

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

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Rubi [A]  time = 0.427852, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ -\frac{(c d-b e)^3 \log (b+c x) \left (-b^2 c e (4 B d-A e)-3 b c^2 d (B d-A e)+6 A c^3 d^2-3 b^3 B e^2\right )}{b^5 c^4}+\frac{d^3 \log (x) \left (5 b^2 e (2 A e+B d)-3 b c d (5 A e+B d)+6 A c^2 d^2\right )}{b^5}-\frac{(c d-b e)^4 \left (-2 A b c e-3 A c^2 d+3 b^2 B e+2 b B c d\right )}{b^4 c^4 (b+c x)}-\frac{(b B-A c) (c d-b e)^5}{2 b^3 c^4 (b+c x)^2}-\frac{d^4 (5 A b e-3 A c d+b B d)}{b^4 x}-\frac{A d^5}{2 b^3 x^2}+\frac{B e^5 x}{c^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^5)/(b*x + c*x^2)^3,x]

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.145606, size = 254, normalized size = 0.99 \[ -\frac{(c d-b e)^3 \log (b+c x) \left (b^2 c e (A e-4 B d)+3 b c^2 d (A e-B d)+6 A c^3 d^2-3 b^3 B e^2\right )}{b^5 c^4}+\frac{d^3 \log (x) \left (5 b^2 e (2 A e+B d)-3 b c d (5 A e+B d)+6 A c^2 d^2\right )}{b^5}+\frac{(c d-b e)^4 \left (2 A b c e+3 A c^2 d-3 b^2 B e-2 b B c d\right )}{b^4 c^4 (b+c x)}+\frac{(b B-A c) (b e-c d)^5}{2 b^3 c^4 (b+c x)^2}-\frac{d^4 (5 A b e-3 A c d+b B d)}{b^4 x}-\frac{A d^5}{2 b^3 x^2}+\frac{B e^5 x}{c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^5)/(b*x + c*x^2)^3,x]

[Out]

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

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Maple [B]  time = 0.021, size = 661, normalized size = 2.6 \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/(c*x^2+b*x)^3,x)

[Out]

-1/2*A*d^5/b^3/x^2+15*c/b^4*ln(c*x+b)*A*d^4*e-15*d^4/b^4*ln(x)*A*c*e-10*c/b^3/(c*x+b)*A*d^4*e+10/c^3*b/(c*x+b)
*B*d*e^4-5/2*c/b^2/(c*x+b)^2*A*d^4*e-5/2/c^3*b^2/(c*x+b)^2*B*d*e^4+5/c^2*b/(c*x+b)^2*B*d^2*e^3+5/2/c^2*b/(c*x+
b)^2*A*d*e^4+B*e^5*x/c^3+1/c^3*ln(c*x+b)*A*e^5-d^5/b^3/x*B-6*c^2/b^5*ln(c*x+b)*A*d^5-3/c^4*b*ln(c*x+b)*B*e^5+5
/c^3*ln(c*x+b)*B*d*e^4-5/b^3*ln(c*x+b)*B*d^4*e+3*c/b^4*ln(c*x+b)*B*d^5+2/c^3*b/(c*x+b)*A*e^5+3*c^2/b^4/(c*x+b)
*A*d^5-3/c^4*b^2/(c*x+b)*B*e^5-2*c/b^3/(c*x+b)*B*d^5-1/2/c^3*b^2/(c*x+b)^2*A*e^5+1/2*c^2/b^3/(c*x+b)^2*A*d^5+1
/2/c^4*b^3/(c*x+b)^2*B*e^5-1/2*c/b^2/(c*x+b)^2*B*d^5+10*d^3/b^3*ln(x)*A*e^2+6*d^5/b^5*ln(x)*A*c^2+5*d^4/b^3*ln
(x)*B*e-3*d^5/b^4*ln(x)*B*c-5*d^4/b^3/x*A*e+3*d^5/b^4/x*A*c-5/c/(c*x+b)^2*B*d^3*e^2-5/c/(c*x+b)^2*A*d^2*e^3-10
/c^2/(c*x+b)*B*d^2*e^3+5/b^2/(c*x+b)*B*d^4*e-5/c^2/(c*x+b)*A*d*e^4+10/b^2/(c*x+b)*A*d^3*e^2-10/b^3*ln(c*x+b)*A
*d^3*e^2+5/2/b/(c*x+b)^2*B*d^4*e+5/b/(c*x+b)^2*A*d^3*e^2

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Maxima [B]  time = 1.16519, size = 693, normalized size = 2.7 \begin{align*} \frac{B e^{5} x}{c^{3}} - \frac{A b^{3} c^{4} d^{5} - 2 \,{\left (10 \, A b^{2} c^{5} d^{3} e^{2} - 10 \, B b^{4} c^{3} d^{2} e^{3} - 3 \,{\left (B b c^{6} - 2 \, A c^{7}\right )} d^{5} + 5 \,{\left (B b^{2} c^{5} - 3 \, A b c^{6}\right )} d^{4} e + 5 \,{\left (2 \, B b^{5} c^{2} - A b^{4} c^{3}\right )} d e^{4} -{\left (3 \, B b^{6} c - 2 \, A b^{5} c^{2}\right )} e^{5}\right )} x^{3} +{\left (9 \,{\left (B b^{2} c^{5} - 2 \, A b c^{6}\right )} d^{5} - 15 \,{\left (B b^{3} c^{4} - 3 \, A b^{2} c^{5}\right )} d^{4} e + 10 \,{\left (B b^{4} c^{3} - 3 \, A b^{3} c^{4}\right )} d^{3} e^{2} + 10 \,{\left (B b^{5} c^{2} + A b^{4} c^{3}\right )} d^{2} e^{3} - 5 \,{\left (3 \, B b^{6} c - A b^{5} c^{2}\right )} d e^{4} +{\left (5 \, B b^{7} - 3 \, A b^{6} c\right )} e^{5}\right )} x^{2} + 2 \,{\left (5 \, A b^{3} c^{4} d^{4} e +{\left (B b^{3} c^{4} - 2 \, A b^{2} c^{5}\right )} d^{5}\right )} x}{2 \,{\left (b^{4} c^{6} x^{4} + 2 \, b^{5} c^{5} x^{3} + b^{6} c^{4} x^{2}\right )}} + \frac{{\left (10 \, A b^{2} d^{3} e^{2} - 3 \,{\left (B b c - 2 \, A c^{2}\right )} d^{5} + 5 \,{\left (B b^{2} - 3 \, A b c\right )} d^{4} e\right )} \log \left (x\right )}{b^{5}} - \frac{{\left (10 \, A b^{2} c^{4} d^{3} e^{2} - 5 \, B b^{5} c d e^{4} - 3 \,{\left (B b c^{5} - 2 \, A c^{6}\right )} d^{5} + 5 \,{\left (B b^{2} c^{4} - 3 \, A b c^{5}\right )} d^{4} e +{\left (3 \, B b^{6} - A b^{5} c\right )} e^{5}\right )} \log \left (c x + b\right )}{b^{5} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

B*e^5*x/c^3 - 1/2*(A*b^3*c^4*d^5 - 2*(10*A*b^2*c^5*d^3*e^2 - 10*B*b^4*c^3*d^2*e^3 - 3*(B*b*c^6 - 2*A*c^7)*d^5
+ 5*(B*b^2*c^5 - 3*A*b*c^6)*d^4*e + 5*(2*B*b^5*c^2 - A*b^4*c^3)*d*e^4 - (3*B*b^6*c - 2*A*b^5*c^2)*e^5)*x^3 + (
9*(B*b^2*c^5 - 2*A*b*c^6)*d^5 - 15*(B*b^3*c^4 - 3*A*b^2*c^5)*d^4*e + 10*(B*b^4*c^3 - 3*A*b^3*c^4)*d^3*e^2 + 10
*(B*b^5*c^2 + A*b^4*c^3)*d^2*e^3 - 5*(3*B*b^6*c - A*b^5*c^2)*d*e^4 + (5*B*b^7 - 3*A*b^6*c)*e^5)*x^2 + 2*(5*A*b
^3*c^4*d^4*e + (B*b^3*c^4 - 2*A*b^2*c^5)*d^5)*x)/(b^4*c^6*x^4 + 2*b^5*c^5*x^3 + b^6*c^4*x^2) + (10*A*b^2*d^3*e
^2 - 3*(B*b*c - 2*A*c^2)*d^5 + 5*(B*b^2 - 3*A*b*c)*d^4*e)*log(x)/b^5 - (10*A*b^2*c^4*d^3*e^2 - 5*B*b^5*c*d*e^4
- 3*(B*b*c^5 - 2*A*c^6)*d^5 + 5*(B*b^2*c^4 - 3*A*b*c^5)*d^4*e + (3*B*b^6 - A*b^5*c)*e^5)*log(c*x + b)/(b^5*c^
4)

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Fricas [B]  time = 2.62662, size = 1785, normalized size = 6.95 \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/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

1/2*(2*B*b^5*c^3*e^5*x^5 + 4*B*b^6*c^2*e^5*x^4 - A*b^4*c^4*d^5 + 2*(10*A*b^3*c^5*d^3*e^2 - 10*B*b^5*c^3*d^2*e^
3 - 3*(B*b^2*c^6 - 2*A*b*c^7)*d^5 + 5*(B*b^3*c^5 - 3*A*b^2*c^6)*d^4*e + 5*(2*B*b^6*c^2 - A*b^5*c^3)*d*e^4 - 2*
(B*b^7*c - A*b^6*c^2)*e^5)*x^3 - (9*(B*b^3*c^5 - 2*A*b^2*c^6)*d^5 - 15*(B*b^4*c^4 - 3*A*b^3*c^5)*d^4*e + 10*(B
*b^5*c^3 - 3*A*b^4*c^4)*d^3*e^2 + 10*(B*b^6*c^2 + A*b^5*c^3)*d^2*e^3 - 5*(3*B*b^7*c - A*b^6*c^2)*d*e^4 + (5*B*
b^8 - 3*A*b^7*c)*e^5)*x^2 - 2*(5*A*b^4*c^4*d^4*e + (B*b^4*c^4 - 2*A*b^3*c^5)*d^5)*x - 2*((10*A*b^2*c^6*d^3*e^2
- 5*B*b^5*c^3*d*e^4 - 3*(B*b*c^7 - 2*A*c^8)*d^5 + 5*(B*b^2*c^6 - 3*A*b*c^7)*d^4*e + (3*B*b^6*c^2 - A*b^5*c^3)
*e^5)*x^4 + 2*(10*A*b^3*c^5*d^3*e^2 - 5*B*b^6*c^2*d*e^4 - 3*(B*b^2*c^6 - 2*A*b*c^7)*d^5 + 5*(B*b^3*c^5 - 3*A*b
^2*c^6)*d^4*e + (3*B*b^7*c - A*b^6*c^2)*e^5)*x^3 + (10*A*b^4*c^4*d^3*e^2 - 5*B*b^7*c*d*e^4 - 3*(B*b^3*c^5 - 2*
A*b^2*c^6)*d^5 + 5*(B*b^4*c^4 - 3*A*b^3*c^5)*d^4*e + (3*B*b^8 - A*b^7*c)*e^5)*x^2)*log(c*x + b) + 2*((10*A*b^2
*c^6*d^3*e^2 - 3*(B*b*c^7 - 2*A*c^8)*d^5 + 5*(B*b^2*c^6 - 3*A*b*c^7)*d^4*e)*x^4 + 2*(10*A*b^3*c^5*d^3*e^2 - 3*
(B*b^2*c^6 - 2*A*b*c^7)*d^5 + 5*(B*b^3*c^5 - 3*A*b^2*c^6)*d^4*e)*x^3 + (10*A*b^4*c^4*d^3*e^2 - 3*(B*b^3*c^5 -
2*A*b^2*c^6)*d^5 + 5*(B*b^4*c^4 - 3*A*b^3*c^5)*d^4*e)*x^2)*log(x))/(b^5*c^6*x^4 + 2*b^6*c^5*x^3 + b^7*c^4*x^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.27685, size = 690, normalized size = 2.68 \begin{align*} \frac{B x e^{5}}{c^{3}} - \frac{{\left (3 \, B b c d^{5} - 6 \, A c^{2} d^{5} - 5 \, B b^{2} d^{4} e + 15 \, A b c d^{4} e - 10 \, A b^{2} d^{3} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} + \frac{{\left (3 \, B b c^{5} d^{5} - 6 \, A c^{6} d^{5} - 5 \, B b^{2} c^{4} d^{4} e + 15 \, A b c^{5} d^{4} e - 10 \, A b^{2} c^{4} d^{3} e^{2} + 5 \, B b^{5} c d e^{4} - 3 \, B b^{6} e^{5} + A b^{5} c e^{5}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c^{4}} - \frac{A b^{3} c^{4} d^{5} + 2 \,{\left (3 \, B b c^{6} d^{5} - 6 \, A c^{7} d^{5} - 5 \, B b^{2} c^{5} d^{4} e + 15 \, A b c^{6} d^{4} e - 10 \, A b^{2} c^{5} d^{3} e^{2} + 10 \, B b^{4} c^{3} d^{2} e^{3} - 10 \, B b^{5} c^{2} d e^{4} + 5 \, A b^{4} c^{3} d e^{4} + 3 \, B b^{6} c e^{5} - 2 \, A b^{5} c^{2} e^{5}\right )} x^{3} +{\left (9 \, B b^{2} c^{5} d^{5} - 18 \, A b c^{6} d^{5} - 15 \, B b^{3} c^{4} d^{4} e + 45 \, A b^{2} c^{5} d^{4} e + 10 \, B b^{4} c^{3} d^{3} e^{2} - 30 \, A b^{3} c^{4} d^{3} e^{2} + 10 \, B b^{5} c^{2} d^{2} e^{3} + 10 \, A b^{4} c^{3} d^{2} e^{3} - 15 \, B b^{6} c d e^{4} + 5 \, A b^{5} c^{2} d e^{4} + 5 \, B b^{7} e^{5} - 3 \, A b^{6} c e^{5}\right )} x^{2} + 2 \,{\left (B b^{3} c^{4} d^{5} - 2 \, A b^{2} c^{5} d^{5} + 5 \, A b^{3} c^{4} d^{4} e\right )} x}{2 \,{\left (c x + b\right )}^{2} b^{4} c^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

B*x*e^5/c^3 - (3*B*b*c*d^5 - 6*A*c^2*d^5 - 5*B*b^2*d^4*e + 15*A*b*c*d^4*e - 10*A*b^2*d^3*e^2)*log(abs(x))/b^5
+ (3*B*b*c^5*d^5 - 6*A*c^6*d^5 - 5*B*b^2*c^4*d^4*e + 15*A*b*c^5*d^4*e - 10*A*b^2*c^4*d^3*e^2 + 5*B*b^5*c*d*e^4
- 3*B*b^6*e^5 + A*b^5*c*e^5)*log(abs(c*x + b))/(b^5*c^4) - 1/2*(A*b^3*c^4*d^5 + 2*(3*B*b*c^6*d^5 - 6*A*c^7*d^
5 - 5*B*b^2*c^5*d^4*e + 15*A*b*c^6*d^4*e - 10*A*b^2*c^5*d^3*e^2 + 10*B*b^4*c^3*d^2*e^3 - 10*B*b^5*c^2*d*e^4 +
5*A*b^4*c^3*d*e^4 + 3*B*b^6*c*e^5 - 2*A*b^5*c^2*e^5)*x^3 + (9*B*b^2*c^5*d^5 - 18*A*b*c^6*d^5 - 15*B*b^3*c^4*d^
4*e + 45*A*b^2*c^5*d^4*e + 10*B*b^4*c^3*d^3*e^2 - 30*A*b^3*c^4*d^3*e^2 + 10*B*b^5*c^2*d^2*e^3 + 10*A*b^4*c^3*d
^2*e^3 - 15*B*b^6*c*d*e^4 + 5*A*b^5*c^2*d*e^4 + 5*B*b^7*e^5 - 3*A*b^6*c*e^5)*x^2 + 2*(B*b^3*c^4*d^5 - 2*A*b^2*
c^5*d^5 + 5*A*b^3*c^4*d^4*e)*x)/((c*x + b)^2*b^4*c^4*x^2)