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

Optimal. Leaf size=16 \[ \frac {3 \left (x^4-1\right )^{5/3}}{5 x^5} \]

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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 {3 \left (x^4-1\right )^{5/3}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-1 + x^4)^(5/3))/(5*x^5)

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^4\right )^{2/3} \left (3+x^4\right )}{x^6} \, dx &=\frac {3 \left (-1+x^4\right )^{5/3}}{5 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {3 \left (x^4-1\right )^{5/3}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-1 + x^4)^(5/3))/(5*x^5)

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IntegrateAlgebraic [A]  time = 0.08, size = 16, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+x^4\right )^{5/3}}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(-1 + x^4)^(5/3))/(5*x^5)

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fricas [A]  time = 0.45, size = 12, normalized size = 0.75 \begin {gather*} \frac {3 \, {\left (x^{4} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(x^4 - 1)^(5/3)/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

method	result	size
trager	\(\frac {3 \left (x^{4}-1\right )^{\frac {5}{3}}}{5 x^{5}}\)	\(13\)
risch	\(\frac {\frac {3}{5} x^{8}-\frac {6}{5} x^{4}+\frac {3}{5}}{x^{5} \left (x^{4}-1\right )^{\frac {1}{3}}}\)	\(23\)
gosper	\(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}}{5 x^{5}}\)	\(24\)
meijerg	\(-\frac {\mathrm {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], x^{4}\right )}{\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {2}{3}} x}-\frac {3 \mathrm {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {5}{4}, -\frac {2}{3}\right ], \left [-\frac {1}{4}\right ], x^{4}\right )}{5 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {2}{3}} x^{5}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(x^4-1)^(5/3)/x^5

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maxima [B]  time = 0.60, size = 27, normalized size = 1.69 \begin {gather*} \frac {3 \, {\left (x^{4} - 1\right )} {\left (x^{2} + 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {2}{3}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*(x^4 - 1)*(x^2 + 1)^(2/3)*(x + 1)^(2/3)*(x - 1)^(2/3)/x^5

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mupad [B]  time = 0.19, size = 27, normalized size = 1.69 \begin {gather*} -\frac {3\,{\left (x^4-1\right )}^{2/3}-3\,x^4\,{\left (x^4-1\right )}^{2/3}}{5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*(x^4 - 1)^(2/3) - 3*x^4*(x^4 - 1)^(2/3))/(5*x^5)

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sympy [C]  time = 2.40, size = 78, normalized size = 4.88 \begin {gather*} - \frac {e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {2}{3} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4}} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(-I*pi/3)*gamma(-1/4)*hyper((-2/3, -1/4), (3/4,), x**4)/(4*x*gamma(3/4)) - 3*exp(-I*pi/3)*gamma(-5/4)*hype
r((-5/4, -2/3), (-1/4,), x**4)/(4*x**5*gamma(-1/4))

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