\(\int \frac {\sqrt [3]{-1+x^6} (1+x^6)}{x^5} \, dx\) [108]

Optimal result

Integrand size = 18, antiderivative size = 16 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {\left (-1+x^6\right )^{4/3}}{4 x^4} \]

[Out]

1/4*(x^6-1)^(4/3)/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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460} \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {\left (x^6-1\right )^{4/3}}{4 x^4} \]

[In]

Int[((-1 + x^6)^(1/3)*(1 + x^6))/x^5,x]

[Out]

(-1 + x^6)^(4/3)/(4*x^4)

Rule 460

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

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {\left (-1+x^6\right )^{4/3}}{4 x^4} \]

[In]

Integrate[((-1 + x^6)^(1/3)*(1 + x^6))/x^5,x]

[Out]

(-1 + x^6)^(4/3)/(4*x^4)

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
trager	\(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{4 x^{4}}\)	\(13\)
pseudoelliptic	\(\frac {\left (x^{6}-1\right )^{\frac {4}{3}}}{4 x^{4}}\)	\(13\)
risch	\(\frac {x^{12}-2 x^{6}+1}{4 \left (x^{6}-1\right )^{\frac {2}{3}} x^{4}}\)	\(23\)
gosper	\(\frac {\left (x^{6}-1\right )^{\frac {1}{3}} \left (x -1\right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}{4 x^{4}}\)	\(33\)
meijerg	\(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], x^{6}\right )}{4 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{4}}\)	\(66\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {{\left (x^{6} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \]

[In]

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

[Out]

1/4*(x^6 - 1)^(4/3)/x^4

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.50 (sec) , antiderivative size = 71, normalized size of antiderivative = 4.44 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {x^{2} e^{\frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} - \frac {e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{3} \\ \frac {1}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{4} \Gamma \left (\frac {1}{3}\right )} \]

[In]

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

[Out]

x**2*exp(I*pi/3)*gamma(1/3)*hyper((-1/3, 1/3), (4/3,), x**6)/(6*gamma(4/3)) - exp(-2*I*pi/3)*gamma(-2/3)*hyper
((-2/3, -1/3), (1/3,), x**6)/(6*x**4*gamma(1/3))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (12) = 24\).

Time = 0.33 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.38 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {{\left (x^{6} - 1\right )} {\left (x^{2} + x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^5} \, dx=\frac {{\left (x^6-1\right )}^{4/3}}{4\,x^4} \]

[In]

int(((x^6 - 1)^(1/3)*(x^6 + 1))/x^5,x)

[Out]

(x^6 - 1)^(4/3)/(4*x^4)