3.51 \(\int \frac{x^5 (d+e x)}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=90 \[ \frac{b^3 (c d-b e)}{c^5 (b+c x)}+\frac{b^2 (3 c d-4 b e) \log (b+c x)}{c^5}+\frac{x^2 (c d-2 b e)}{2 c^3}-\frac{b x (2 c d-3 b e)}{c^4}+\frac{e x^3}{3 c^2} \]

[Out]

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

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Rubi [A]  time = 0.0943529, antiderivative size = 90, 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 = {765} \[ \frac{b^3 (c d-b e)}{c^5 (b+c x)}+\frac{b^2 (3 c d-4 b e) \log (b+c x)}{c^5}+\frac{x^2 (c d-2 b e)}{2 c^3}-\frac{b x (2 c d-3 b e)}{c^4}+\frac{e x^3}{3 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0552571, size = 87, normalized size = 0.97 \[ \frac{\frac{6 b^3 (c d-b e)}{b+c x}+6 b^2 (3 c d-4 b e) \log (b+c x)+3 c^2 x^2 (c d-2 b e)+6 b c x (3 b e-2 c d)+2 c^3 e x^3}{6 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 109, normalized size = 1.2 \begin{align*}{\frac{e{x}^{3}}{3\,{c}^{2}}}-{\frac{b{x}^{2}e}{{c}^{3}}}+{\frac{d{x}^{2}}{2\,{c}^{2}}}+3\,{\frac{{b}^{2}ex}{{c}^{4}}}-2\,{\frac{bdx}{{c}^{3}}}-{\frac{{b}^{4}e}{{c}^{5} \left ( cx+b \right ) }}+{\frac{{b}^{3}d}{{c}^{4} \left ( cx+b \right ) }}-4\,{\frac{{b}^{3}\ln \left ( cx+b \right ) e}{{c}^{5}}}+3\,{\frac{{b}^{2}\ln \left ( cx+b \right ) d}{{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x+d)/(c*x^2+b*x)^2,x)

[Out]

1/3*e*x^3/c^2-1/c^3*x^2*b*e+1/2*d*x^2/c^2+3/c^4*b^2*e*x-2/c^3*b*d*x-b^4/c^5/(c*x+b)*e+b^3/c^4/(c*x+b)*d-4*b^3/
c^5*ln(c*x+b)*e+3*b^2/c^4*ln(c*x+b)*d

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Maxima [A]  time = 1.10355, size = 132, normalized size = 1.47 \begin{align*} \frac{b^{3} c d - b^{4} e}{c^{6} x + b c^{5}} + \frac{2 \, c^{2} e x^{3} + 3 \,{\left (c^{2} d - 2 \, b c e\right )} x^{2} - 6 \,{\left (2 \, b c d - 3 \, b^{2} e\right )} x}{6 \, c^{4}} + \frac{{\left (3 \, b^{2} c d - 4 \, b^{3} e\right )} \log \left (c x + b\right )}{c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^3*c*d - b^4*e)/(c^6*x + b*c^5) + 1/6*(2*c^2*e*x^3 + 3*(c^2*d - 2*b*c*e)*x^2 - 6*(2*b*c*d - 3*b^2*e)*x)/c^4
+ (3*b^2*c*d - 4*b^3*e)*log(c*x + b)/c^5

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Fricas [A]  time = 1.7408, size = 297, normalized size = 3.3 \begin{align*} \frac{2 \, c^{4} e x^{4} + 6 \, b^{3} c d - 6 \, b^{4} e +{\left (3 \, c^{4} d - 4 \, b c^{3} e\right )} x^{3} - 3 \,{\left (3 \, b c^{3} d - 4 \, b^{2} c^{2} e\right )} x^{2} - 6 \,{\left (2 \, b^{2} c^{2} d - 3 \, b^{3} c e\right )} x + 6 \,{\left (3 \, b^{3} c d - 4 \, b^{4} e +{\left (3 \, b^{2} c^{2} d - 4 \, b^{3} c e\right )} x\right )} \log \left (c x + b\right )}{6 \,{\left (c^{6} x + b c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

1/6*(2*c^4*e*x^4 + 6*b^3*c*d - 6*b^4*e + (3*c^4*d - 4*b*c^3*e)*x^3 - 3*(3*b*c^3*d - 4*b^2*c^2*e)*x^2 - 6*(2*b^
2*c^2*d - 3*b^3*c*e)*x + 6*(3*b^3*c*d - 4*b^4*e + (3*b^2*c^2*d - 4*b^3*c*e)*x)*log(c*x + b))/(c^6*x + b*c^5)

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Sympy [A]  time = 0.752955, size = 90, normalized size = 1. \begin{align*} - \frac{b^{2} \left (4 b e - 3 c d\right ) \log{\left (b + c x \right )}}{c^{5}} - \frac{b^{4} e - b^{3} c d}{b c^{5} + c^{6} x} + \frac{e x^{3}}{3 c^{2}} - \frac{x^{2} \left (2 b e - c d\right )}{2 c^{3}} + \frac{x \left (3 b^{2} e - 2 b c d\right )}{c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**2*(4*b*e - 3*c*d)*log(b + c*x)/c**5 - (b**4*e - b**3*c*d)/(b*c**5 + c**6*x) + e*x**3/(3*c**2) - x**2*(2*b*
e - c*d)/(2*c**3) + x*(3*b**2*e - 2*b*c*d)/c**4

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Giac [A]  time = 1.22645, size = 144, normalized size = 1.6 \begin{align*} \frac{{\left (3 \, b^{2} c d - 4 \, b^{3} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{5}} + \frac{2 \, c^{4} x^{3} e + 3 \, c^{4} d x^{2} - 6 \, b c^{3} x^{2} e - 12 \, b c^{3} d x + 18 \, b^{2} c^{2} x e}{6 \, c^{6}} + \frac{b^{3} c d - b^{4} e}{{\left (c x + b\right )} c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*b^2*c*d - 4*b^3*e)*log(abs(c*x + b))/c^5 + 1/6*(2*c^4*x^3*e + 3*c^4*d*x^2 - 6*b*c^3*x^2*e - 12*b*c^3*d*x +
18*b^2*c^2*x*e)/c^6 + (b^3*c*d - b^4*e)/((c*x + b)*c^5)