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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 = 0.01, size = 36, normalized size = 1.00 \[ \frac {1}{4} \left (x^4-8 x^3+54 x^2-432 x+\frac {972}{x+3}-2079\right )+405 \log (x+3) \]

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]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.00, size = 39, normalized size = 1.08 \[ \frac {x^5-5 x^4+30 x^3-270 x^2-1296 x+972}{4 (x+3)}+405 \log (x+3) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^5/(3 + x)^2,x]

[Out]

(972 - 1296*x - 270*x^2 + 30*x^3 - 5*x^4 + x^5)/(4*(3 + x)) + 405*Log[3 + x]

________________________________________________________________________________________

fricas [A]  time = 1.28, size = 39, normalized size = 1.08 \[ \frac {x^{5} - 5 \, x^{4} + 30 \, x^{3} - 270 \, x^{2} + 1620 \, {\left (x + 3\right )} \log \left (x + 3\right ) - 1296 \, x + 972}{4 \, {\left (x + 3\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.88, size = 45, normalized size = 1.25 \[ -\frac {1}{4} \, {\left (x + 3\right )}^{4} {\left (\frac {20}{x + 3} - \frac {180}{{\left (x + 3\right )}^{2}} + \frac {1080}{{\left (x + 3\right )}^{3}} - 1\right )} + \frac {243}{x + 3} + 405 \, \log \left ({\left | x + 3 \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 33, normalized size = 0.92

method	result	size
default	\(-108 x +\frac {27 x^{2}}{2}-2 x^{3}+\frac {x^{4}}{4}+\frac {243}{3+x}+405 \ln \left (3+x \right )\)	\(33\)
risch	\(-108 x +\frac {27 x^{2}}{2}-2 x^{3}+\frac {x^{4}}{4}+\frac {243}{3+x}+405 \ln \left (3+x \right )\)	\(33\)
norman	\(\frac {-\frac {135}{2} x^{2}+\frac {15}{2} x^{3}-\frac {5}{4} x^{4}+\frac {1}{4} x^{5}+1215}{3+x}+405 \ln \left (3+x \right )\)	\(36\)
meijerg	\(-\frac {9 x \left (-\frac {1}{27} x^{4}+\frac {5}{27} x^{3}-\frac {10}{9} x^{2}+10 x +60\right )}{4 \left (1+\frac {x}{3}\right )}+405 \ln \left (1+\frac {x}{3}\right )\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(3+x)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.51, size = 32, normalized size = 0.89 \[ \frac {1}{4} \, x^{4} - 2 \, x^{3} + \frac {27}{2} \, x^{2} - 108 \, x + \frac {243}{x + 3} + 405 \, \log \left (x + 3\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.00, size = 32, normalized size = 0.89 \[ 405\,\ln \left (x+3\right )-108\,x+\frac {243}{x+3}+\frac {27\,x^2}{2}-2\,x^3+\frac {x^4}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(x + 3)^2,x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.09, size = 31, normalized size = 0.86 \[ \frac {x^{4}}{4} - 2 x^{3} + \frac {27 x^{2}}{2} - 108 x + 405 \log {\left (x + 3 \right )} + \frac {243}{x + 3} \]

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)

________________________________________________________________________________________