3.102 \(\int \frac{x^5}{(3+x)^2} \, dx\)

Optimal. Leaf size=36 \[ \frac{x^4}{4}-2 x^3+\frac{27 x^2}{2}-108 x+\frac{243}{x+3}+405 \log (x+3) \]

[Out]

-108*x + (27*x^2)/2 - 2*x^3 + x^4/4 + 243/(3 + x) + 405*Log[3 + x]

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Rubi [A]  time = 0.0166641, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{x^4}{4}-2 x^3+\frac{27 x^2}{2}-108 x+\frac{243}{x+3}+405 \log (x+3) \]

Antiderivative was successfully verified.

[In]

Int[x^5/(3 + x)^2,x]

[Out]

-108*x + (27*x^2)/2 - 2*x^3 + x^4/4 + 243/(3 + x) + 405*Log[3 + x]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^5}{(3+x)^2} \, dx &=\int \left (-108+27 x-6 x^2+x^3-\frac{243}{(3+x)^2}+\frac{405}{3+x}\right ) \, dx\\ &=-108 x+\frac{27 x^2}{2}-2 x^3+\frac{x^4}{4}+\frac{243}{3+x}+405 \log (3+x)\\ \end{align*}

Mathematica [A]  time = 0.0142851, size = 36, normalized size = 1. \[ \frac{1}{4} \left (x^4-8 x^3+54 x^2-432 x+\frac{972}{x+3}-2079\right )+405 \log (x+3) \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(3 + x)^2,x]

[Out]

(-2079 - 432*x + 54*x^2 - 8*x^3 + x^4 + 972/(3 + x))/4 + 405*Log[3 + x]

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Maple [A]  time = 0.005, size = 33, normalized size = 0.9 \begin{align*} -108\,x+{\frac{27\,{x}^{2}}{2}}-2\,{x}^{3}+{\frac{{x}^{4}}{4}}+243\, \left ( 3+x \right ) ^{-1}+405\,\ln \left ( 3+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(3+x)^2,x)

[Out]

-108*x+27/2*x^2-2*x^3+1/4*x^4+243/(3+x)+405*ln(3+x)

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Maxima [A]  time = 0.921378, size = 43, normalized size = 1.19 \begin{align*} \frac{1}{4} \, x^{4} - 2 \, x^{3} + \frac{27}{2} \, x^{2} - 108 \, x + \frac{243}{x + 3} + 405 \, \log \left (x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(3+x)^2,x, algorithm="maxima")

[Out]

1/4*x^4 - 2*x^3 + 27/2*x^2 - 108*x + 243/(x + 3) + 405*log(x + 3)

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Fricas [A]  time = 2.10633, size = 117, normalized size = 3.25 \begin{align*} \frac{x^{5} - 5 \, x^{4} + 30 \, x^{3} - 270 \, x^{2} + 1620 \,{\left (x + 3\right )} \log \left (x + 3\right ) - 1296 \, x + 972}{4 \,{\left (x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(3+x)^2,x, algorithm="fricas")

[Out]

1/4*(x^5 - 5*x^4 + 30*x^3 - 270*x^2 + 1620*(x + 3)*log(x + 3) - 1296*x + 972)/(x + 3)

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Sympy [A]  time = 0.079906, size = 31, normalized size = 0.86 \begin{align*} \frac{x^{4}}{4} - 2 x^{3} + \frac{27 x^{2}}{2} - 108 x + 405 \log{\left (x + 3 \right )} + \frac{243}{x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(3+x)**2,x)

[Out]

x**4/4 - 2*x**3 + 27*x**2/2 - 108*x + 405*log(x + 3) + 243/(x + 3)

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Giac [A]  time = 1.05035, size = 61, normalized size = 1.69 \begin{align*} -\frac{1}{4} \,{\left (x + 3\right )}^{4}{\left (\frac{20}{x + 3} - \frac{180}{{\left (x + 3\right )}^{2}} + \frac{1080}{{\left (x + 3\right )}^{3}} - 1\right )} + \frac{243}{x + 3} + 405 \, \log \left ({\left | x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(3+x)^2,x, algorithm="giac")

[Out]

-1/4*(x + 3)^4*(20/(x + 3) - 180/(x + 3)^2 + 1080/(x + 3)^3 - 1) + 243/(x + 3) + 405*log(abs(x + 3))