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

Optimal. Leaf size=36 \[ -\frac{c d-b e}{2 c^2 (b+c x)^2}-\frac{e}{c^2 (b+c x)} \]

[Out]

-(c*d - b*e)/(2*c^2*(b + c*x)^2) - e/(c^2*(b + c*x))

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Rubi [A]  time = 0.0292562, antiderivative size = 36, 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} \[ -\frac{c d-b e}{2 c^2 (b+c x)^2}-\frac{e}{c^2 (b+c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(c*d - b*e)/(2*c^2*(b + c*x)^2) - e/(c^2*(b + c*x))

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

Mathematica [A]  time = 0.0087728, size = 26, normalized size = 0.72 \[ -\frac{b e+c (d+2 e x)}{2 c^2 (b+c x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*e + c*(d + 2*e*x))/(2*c^2*(b + c*x)^2)

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Maple [A]  time = 0.006, size = 35, normalized size = 1. \begin{align*} -{\frac{e}{{c}^{2} \left ( cx+b \right ) }}-{\frac{-be+cd}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08501, size = 51, normalized size = 1.42 \begin{align*} -\frac{2 \, c e x + c d + b e}{2 \,{\left (c^{4} x^{2} + 2 \, b c^{3} x + b^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.67242, size = 81, normalized size = 2.25 \begin{align*} -\frac{2 \, c e x + c d + b e}{2 \,{\left (c^{4} x^{2} + 2 \, b c^{3} x + b^{2} c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.678642, size = 39, normalized size = 1.08 \begin{align*} - \frac{b e + c d + 2 c e x}{2 b^{2} c^{2} + 4 b c^{3} x + 2 c^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15295, size = 35, normalized size = 0.97 \begin{align*} -\frac{2 \, c x e + c d + b e}{2 \,{\left (c x + b\right )}^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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