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

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

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 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])

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]]

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 = 0.01, 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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.02, size = 33, normalized size = 0.85 \[ \frac {3 \left (8 x^5+9\right )}{80 \left (2 x^5+3\right )^2}+\frac {1}{40} \log \left (2 x^5+3\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^14/(3 + 2*x^5)^3,x]

[Out]

(3*(9 + 8*x^5))/(80*(3 + 2*x^5)^2) + Log[3 + 2*x^5]/40

________________________________________________________________________________________

fricas [A]  time = 1.32, size = 45, normalized size = 1.15 \[ \frac {24 \, x^{5} + 2 \, {\left (4 \, x^{10} + 12 \, x^{5} + 9\right )} \log \left (2 \, x^{5} + 3\right ) + 27}{80 \, {\left (4 \, x^{10} + 12 \, x^{5} + 9\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.97, size = 30, normalized size = 0.77 \[ -\frac {3 \, {\left (x^{10} + x^{5}\right )}}{20 \, {\left (2 \, x^{5} + 3\right )}^{2}} + \frac {1}{40} \, \log \left ({\left | 2 \, x^{5} + 3 \right |}\right ) \]

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))

________________________________________________________________________________________

maple [A]  time = 0.26, size = 29, normalized size = 0.74

method	result	size
norman	\(\frac {\frac {3 x^{5}}{10}+\frac {27}{80}}{\left (2 x^{5}+3\right )^{2}}+\frac {\ln \left (2 x^{5}+3\right )}{40}\)	\(29\)
risch	\(\frac {\frac {3 x^{5}}{10}+\frac {27}{80}}{\left (2 x^{5}+3\right )^{2}}+\frac {\ln \left (2 x^{5}+3\right )}{40}\)	\(30\)
meijerg	\(-\frac {x^{5} \left (6 x^{5}+6\right )}{360 \left (1+\frac {2 x^{5}}{3}\right )^{2}}+\frac {\ln \left (1+\frac {2 x^{5}}{3}\right )}{40}\)	\(33\)
default	\(-\frac {9}{80 \left (2 x^{5}+3\right )^{2}}+\frac {3}{20 \left (2 x^{5}+3\right )}+\frac {\ln \left (2 x^{5}+3\right )}{40}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3/10*x^5+27/80)/(2*x^5+3)^2+1/40*ln(2*x^5+3)

________________________________________________________________________________________

maxima [A]  time = 0.60, size = 34, normalized size = 0.87 \[ \frac {3 \, {\left (8 \, x^{5} + 9\right )}}{80 \, {\left (4 \, x^{10} + 12 \, x^{5} + 9\right )}} + \frac {1}{40} \, \log \left (2 \, x^{5} + 3\right ) \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 29, normalized size = 0.74 \[ \frac {\ln \left (x^5+\frac {3}{2}\right )}{40}+\frac {\frac {3\,x^5}{40}+\frac {27}{320}}{x^{10}+3\,x^5+\frac {9}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^14/(2*x^5 + 3)^3,x)

[Out]

log(x^5 + 3/2)/40 + ((3*x^5)/40 + 27/320)/(3*x^5 + x^10 + 9/4)

________________________________________________________________________________________

sympy [A]  time = 0.18, size = 27, normalized size = 0.69 \[ \frac {24 x^{5} + 27}{320 x^{10} + 960 x^{5} + 720} + \frac {\log {\left (2 x^{5} + 3 \right )}}{40} \]

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

________________________________________________________________________________________