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

Optimal. Leaf size=71 \[ -\frac{b^2 (c d-b e)}{2 c^4 (b+c x)^2}+\frac{b (2 c d-3 b e)}{c^4 (b+c x)}+\frac{(c d-3 b e) \log (b+c x)}{c^4}+\frac{e x}{c^3} \]

[Out]

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

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Rubi [A]  time = 0.0640509, antiderivative size = 71, 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} \[ -\frac{b^2 (c d-b e)}{2 c^4 (b+c x)^2}+\frac{b (2 c d-3 b e)}{c^4 (b+c x)}+\frac{(c d-3 b e) \log (b+c x)}{c^4}+\frac{e x}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0272333, size = 75, normalized size = 1.06 \[ \frac{2 b c d-3 b^2 e}{c^4 (b+c x)}+\frac{b^3 e-b^2 c d}{2 c^4 (b+c x)^2}+\frac{(c d-3 b e) \log (b+c x)}{c^4}+\frac{e x}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 94, normalized size = 1.3 \begin{align*}{\frac{ex}{{c}^{3}}}-3\,{\frac{{b}^{2}e}{{c}^{4} \left ( cx+b \right ) }}+2\,{\frac{bd}{{c}^{3} \left ( cx+b \right ) }}-3\,{\frac{\ln \left ( cx+b \right ) be}{{c}^{4}}}+{\frac{\ln \left ( cx+b \right ) d}{{c}^{3}}}+{\frac{{b}^{3}e}{2\,{c}^{4} \left ( cx+b \right ) ^{2}}}-{\frac{{b}^{2}d}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(e*x+d)/(c*x^2+b*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.08401, size = 112, normalized size = 1.58 \begin{align*} \frac{3 \, b^{2} c d - 5 \, b^{3} e + 2 \,{\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x}{2 \,{\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\right )}} + \frac{e x}{c^{3}} + \frac{{\left (c d - 3 \, b e\right )} \log \left (c x + b\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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73074, size = 278, normalized size = 3.92 \begin{align*} \frac{2 \, c^{3} e x^{3} + 4 \, b c^{2} e x^{2} + 3 \, b^{2} c d - 5 \, b^{3} e + 4 \,{\left (b c^{2} d - b^{2} c e\right )} x + 2 \,{\left (b^{2} c d - 3 \, b^{3} e +{\left (c^{3} d - 3 \, b c^{2} e\right )} x^{2} + 2 \,{\left (b c^{2} d - 3 \, b^{2} c e\right )} x\right )} \log \left (c x + b\right )}{2 \,{\left (c^{6} x^{2} + 2 \, b c^{5} x + b^{2} c^{4}\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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.939123, size = 83, normalized size = 1.17 \begin{align*} - \frac{5 b^{3} e - 3 b^{2} c d + x \left (6 b^{2} c e - 4 b c^{2} d\right )}{2 b^{2} c^{4} + 4 b c^{5} x + 2 c^{6} x^{2}} + \frac{e x}{c^{3}} - \frac{\left (3 b e - c d\right ) \log{\left (b + c x \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)**3,x)

[Out]

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

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Giac [A]  time = 1.15848, size = 100, normalized size = 1.41 \begin{align*} \frac{x e}{c^{3}} + \frac{{\left (c d - 3 \, b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{4}} + \frac{3 \, b^{2} c d - 5 \, b^{3} e + 2 \,{\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x}{2 \,{\left (c x + b\right )}^{2} 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)^3,x, algorithm="giac")

[Out]

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