3.2.10 \(\int \frac {(-1+x^6)^{3/4} (2+x^6)}{x^8} \, dx\)

Optimal. Leaf size=16 \[ \frac {2 \left (x^6-1\right )^{7/4}}{7 x^7} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {449} \begin {gather*} \frac {2 \left (x^6-1\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^6)^(3/4)*(2 + x^6))/x^8,x]

[Out]

(2*(-1 + x^6)^(7/4))/(7*x^7)

Rule 449

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*} \int \frac {\left (-1+x^6\right )^{3/4} \left (2+x^6\right )}{x^8} \, dx &=\frac {2 \left (-1+x^6\right )^{7/4}}{7 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^6-1\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^6)^(3/4)*(2 + x^6))/x^8,x]

[Out]

(2*(-1 + x^6)^(7/4))/(7*x^7)

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IntegrateAlgebraic [A]  time = 0.15, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \left (-1+x^6\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^6)^(3/4)*(2 + x^6))/x^8,x]

[Out]

(2*(-1 + x^6)^(7/4))/(7*x^7)

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fricas [A]  time = 0.48, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (x^{6} - 1\right )}^{\frac {7}{4}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(3/4)*(x^6+2)/x^8,x, algorithm="fricas")

[Out]

2/7*(x^6 - 1)^(7/4)/x^7

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 2\right )} {\left (x^{6} - 1\right )}^{\frac {3}{4}}}{x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(3/4)*(x^6+2)/x^8,x, algorithm="giac")

[Out]

integrate((x^6 + 2)*(x^6 - 1)^(3/4)/x^8, x)

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maple [A]  time = 0.11, size = 13, normalized size = 0.81

method	result	size
trager	\(\frac {2 \left (x^{6}-1\right )^{\frac {7}{4}}}{7 x^{7}}\)	\(13\)
risch	\(\frac {\frac {2}{7} x^{12}-\frac {4}{7} x^{6}+\frac {2}{7}}{x^{7} \left (x^{6}-1\right )^{\frac {1}{4}}}\)	\(23\)
gosper	\(\frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{6}-1\right )^{\frac {3}{4}}}{7 x^{7}}\)	\(33\)
meijerg	\(-\frac {\mathrm {signum}\left (x^{6}-1\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {3}{4}, -\frac {1}{6}\right ], \left [\frac {5}{6}\right ], x^{6}\right )}{\left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {3}{4}} x}-\frac {2 \mathrm {signum}\left (x^{6}-1\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {7}{6}, -\frac {3}{4}\right ], \left [-\frac {1}{6}\right ], x^{6}\right )}{7 \left (-\mathrm {signum}\left (x^{6}-1\right )\right )^{\frac {3}{4}} x^{7}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-1)^(3/4)*(x^6+2)/x^8,x,method=_RETURNVERBOSE)

[Out]

2/7*(x^6-1)^(7/4)/x^7

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maxima [B]  time = 0.58, size = 38, normalized size = 2.38 \begin {gather*} \frac {2 \, {\left (x^{6} - 1\right )} {\left (x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x^{2} - x + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}}}{7 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-1)^(3/4)*(x^6+2)/x^8,x, algorithm="maxima")

[Out]

2/7*(x^6 - 1)*(x^2 + x + 1)^(3/4)*(x^2 - x + 1)^(3/4)*(x + 1)^(3/4)*(x - 1)^(3/4)/x^7

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mupad [B]  time = 0.22, size = 27, normalized size = 1.69 \begin {gather*} -\frac {2\,{\left (x^6-1\right )}^{3/4}-2\,x^6\,{\left (x^6-1\right )}^{3/4}}{7\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 - 1)^(3/4)*(x^6 + 2))/x^8,x)

[Out]

-(2*(x^6 - 1)^(3/4) - 2*x^6*(x^6 - 1)^(3/4))/(7*x^7)

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sympy [C]  time = 3.11, size = 76, normalized size = 4.75 \begin {gather*} - \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {1}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{6} \\ \frac {5}{6} \end {matrix}\middle | {x^{6}} \right )}}{6 x \Gamma \left (\frac {5}{6}\right )} - \frac {e^{- \frac {i \pi }{4}} \Gamma \left (- \frac {7}{6}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{6}, - \frac {3}{4} \\ - \frac {1}{6} \end {matrix}\middle | {x^{6}} \right )}}{3 x^{7} \Gamma \left (- \frac {1}{6}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-1)**(3/4)*(x**6+2)/x**8,x)

[Out]

-exp(-I*pi/4)*gamma(-1/6)*hyper((-3/4, -1/6), (5/6,), x**6)/(6*x*gamma(5/6)) - exp(-I*pi/4)*gamma(-7/6)*hyper(
(-7/6, -3/4), (-1/6,), x**6)/(3*x**7*gamma(-1/6))

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