3.176 \(\int \frac{x^{14}}{(3+2 x^5)^3} \, dx\)

Optimal. Leaf size=39 \[ \frac{3}{20 \left (2 x^5+3\right )}-\frac{9}{80 \left (2 x^5+3\right )^2}+\frac{1}{40} \log \left (2 x^5+3\right ) \]

[Out]

-9/(80*(3 + 2*x^5)^2) + 3/(20*(3 + 2*x^5)) + Log[3 + 2*x^5]/40

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Rubi [A]  time = 0.0249247, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3}{20 \left (2 x^5+3\right )}-\frac{9}{80 \left (2 x^5+3\right )^2}+\frac{1}{40} \log \left (2 x^5+3\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^14/(3 + 2*x^5)^3,x]

[Out]

-9/(80*(3 + 2*x^5)^2) + 3/(20*(3 + 2*x^5)) + Log[3 + 2*x^5]/40

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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^{14}}{\left (3+2 x^5\right )^3} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x^2}{(3+2 x)^3} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{9}{4 (3+2 x)^3}-\frac{3}{2 (3+2 x)^2}+\frac{1}{4 (3+2 x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{9}{80 \left (3+2 x^5\right )^2}+\frac{3}{20 \left (3+2 x^5\right )}+\frac{1}{40} \log \left (3+2 x^5\right )\\ \end{align*}

Mathematica [A]  time = 0.0134275, size = 33, normalized size = 0.85 \[ \frac{1}{80} \left (\frac{3 \left (8 x^5+9\right )}{\left (2 x^5+3\right )^2}+2 \log \left (2 x^5+3\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^14/(3 + 2*x^5)^3,x]

[Out]

((3*(9 + 8*x^5))/(3 + 2*x^5)^2 + 2*Log[3 + 2*x^5])/80

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Maple [A]  time = 0.009, size = 34, normalized size = 0.9 \begin{align*} -{\frac{9}{80\, \left ( 2\,{x}^{5}+3 \right ) ^{2}}}+{\frac{3}{40\,{x}^{5}+60}}+{\frac{\ln \left ( 2\,{x}^{5}+3 \right ) }{40}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(2*x^5+3)^3,x)

[Out]

-9/80/(2*x^5+3)^2+3/20/(2*x^5+3)+1/40*ln(2*x^5+3)

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Maxima [A]  time = 0.923762, size = 46, normalized size = 1.18 \begin{align*} \frac{3 \,{\left (8 \, x^{5} + 9\right )}}{80 \,{\left (4 \, x^{10} + 12 \, x^{5} + 9\right )}} + \frac{1}{40} \, \log \left (2 \, x^{5} + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(2*x^5+3)^3,x, algorithm="maxima")

[Out]

3/80*(8*x^5 + 9)/(4*x^10 + 12*x^5 + 9) + 1/40*log(2*x^5 + 3)

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Fricas [A]  time = 1.9503, size = 112, normalized size = 2.87 \begin{align*} \frac{24 \, x^{5} + 2 \,{\left (4 \, x^{10} + 12 \, x^{5} + 9\right )} \log \left (2 \, x^{5} + 3\right ) + 27}{80 \,{\left (4 \, x^{10} + 12 \, x^{5} + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(2*x^5+3)^3,x, algorithm="fricas")

[Out]

1/80*(24*x^5 + 2*(4*x^10 + 12*x^5 + 9)*log(2*x^5 + 3) + 27)/(4*x^10 + 12*x^5 + 9)

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Sympy [A]  time = 0.171112, size = 27, normalized size = 0.69 \begin{align*} \frac{24 x^{5} + 27}{320 x^{10} + 960 x^{5} + 720} + \frac{\log{\left (2 x^{5} + 3 \right )}}{40} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**14/(2*x**5+3)**3,x)

[Out]

(24*x**5 + 27)/(320*x**10 + 960*x**5 + 720) + log(2*x**5 + 3)/40

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Giac [A]  time = 1.10811, size = 41, normalized size = 1.05 \begin{align*} -\frac{3 \,{\left (x^{10} + x^{5}\right )}}{20 \,{\left (2 \, x^{5} + 3\right )}^{2}} + \frac{1}{40} \, \log \left ({\left | 2 \, x^{5} + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^14/(2*x^5+3)^3,x, algorithm="giac")

[Out]

-3/20*(x^10 + x^5)/(2*x^5 + 3)^2 + 1/40*log(abs(2*x^5 + 3))