3.2.9 \(\int \frac {2+x^6}{x^4 \sqrt [4]{-1+x^6}} \, dx\)

Optimal. Leaf size=16 \[ \frac {2 \left (x^6-1\right )^{3/4}}{3 x^3} \]

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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 )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1 + x^6)^(3/4))/(3*x^3)

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 {2+x^6}{x^4 \sqrt [4]{-1+x^6}} \, dx &=\frac {2 \left (-1+x^6\right )^{3/4}}{3 x^3}\\ \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 )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1 + x^6)^(3/4))/(3*x^3)

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IntegrateAlgebraic [A]  time = 0.14, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \left (-1+x^6\right )^{3/4}}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-1 + x^6)^(3/4))/(3*x^3)

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fricas [A]  time = 0.45, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (x^{6} - 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(x^6 - 1)^(3/4)/x^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 13, normalized size = 0.81

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.55, size = 38, normalized size = 2.38 \begin {gather*} \frac {2 \, {\left (x^{6} - 1\right )}}{3 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.15, size = 12, normalized size = 0.75 \begin {gather*} \frac {2\,{\left (x^6-1\right )}^{3/4}}{3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(x^6 - 1)^(3/4))/(3*x^3)

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sympy [C]  time = 2.62, size = 48, normalized size = 3.00 \begin {gather*} \frac {x^{3} e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {x^{6}} \right )}}{3} + \frac {2 e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{6}} \right )}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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