\(\int \frac {1}{x^5 \sqrt [3]{1+x^6}} \, dx\) [104]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {1}{x^5 \sqrt [3]{1+x^6}} \, dx=-\frac {\left (1+x^6\right )^{2/3}}{4 x^4} \]

[Out]

-1/4*(x^6+1)^(2/3)/x^4

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \int \frac {1}{x^5 \sqrt [3]{1+x^6}} \, dx=-\frac {\left (x^6+1\right )^{2/3}}{4 x^4} \]

[In]

Int[1/(x^5*(1 + x^6)^(1/3)),x]

[Out]

-1/4*(1 + x^6)^(2/3)/x^4

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (1+x^6\right )^{2/3}}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \sqrt [3]{1+x^6}} \, dx=-\frac {\left (1+x^6\right )^{2/3}}{4 x^4} \]

[In]

Integrate[1/(x^5*(1 + x^6)^(1/3)),x]

[Out]

-1/4*(1 + x^6)^(2/3)/x^4

Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
trager	\(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{4 x^{4}}\)	\(13\)
meijerg	\(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{4 x^{4}}\)	\(13\)
risch	\(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{4 x^{4}}\)	\(13\)
pseudoelliptic	\(-\frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{4 x^{4}}\)	\(13\)
gosper	\(-\frac {\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}{4 x^{4} \left (x^{6}+1\right )^{\frac {1}{3}}}\)	\(28\)

[In]

int(1/x^5/(x^6+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

-1/4*(x^6+1)^(2/3)/x^4

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^5 \sqrt [3]{1+x^6}} \, dx=-\frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{4 \, x^{4}} \]

[In]

integrate(1/x^5/(x^6+1)^(1/3),x, algorithm="fricas")

[Out]

-1/4*(x^6 + 1)^(2/3)/x^4

Sympy [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {1}{x^5 \sqrt [3]{1+x^6}} \, dx=\frac {\left (1 + \frac {1}{x^{6}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {2}{3}\right )}{6 \Gamma \left (\frac {1}{3}\right )} \]

[In]

integrate(1/x**5/(x**6+1)**(1/3),x)

[Out]

(1 + x**(-6))**(2/3)*gamma(-2/3)/(6*gamma(1/3))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^5 \sqrt [3]{1+x^6}} \, dx=-\frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}}}{4 \, x^{4}} \]

[In]

integrate(1/x^5/(x^6+1)^(1/3),x, algorithm="maxima")

[Out]

-1/4*(x^6 + 1)^(2/3)/x^4

Giac [F]

\[ \int \frac {1}{x^5 \sqrt [3]{1+x^6}} \, dx=\int { \frac {1}{{\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{5}} \,d x } \]

[In]

integrate(1/x^5/(x^6+1)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((x^6 + 1)^(1/3)*x^5), x)

Mupad [B] (verification not implemented)

Time = 4.95 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^5 \sqrt [3]{1+x^6}} \, dx=-\frac {{\left (x^6+1\right )}^{2/3}}{4\,x^4} \]

[In]

int(1/(x^5*(x^6 + 1)^(1/3)),x)

[Out]

-(x^6 + 1)^(2/3)/(4*x^4)