3.1.88 \(\int \frac {\sqrt [3]{1+x^4} (3+x^4)}{x^9} \, dx\)

Optimal. Leaf size=16 \[ -\frac {3 \left (x^4+1\right )^{4/3}}{8 x^8} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {446, 74} \begin {gather*} -\frac {3 \left (x^4+1\right )^{4/3}}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x^4)^(1/3)*(3 + x^4))/x^9,x]

[Out]

(-3*(1 + x^4)^(4/3))/(8*x^8)

Rule 74

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{1+x^4} \left (3+x^4\right )}{x^9} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x} (3+x)}{x^3} \, dx,x,x^4\right )\\ &=-\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8}\\ \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 )^{4/3}}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^4)^(1/3)*(3 + x^4))/x^9,x]

[Out]

(-3*(1 + x^4)^(4/3))/(8*x^8)

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IntegrateAlgebraic [A]  time = 0.04, size = 16, normalized size = 1.00 \begin {gather*} -\frac {3 \left (1+x^4\right )^{4/3}}{8 x^8} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^4)^(1/3)*(3 + x^4))/x^9,x]

[Out]

(-3*(1 + x^4)^(4/3))/(8*x^8)

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fricas [A]  time = 0.44, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(1/3)*(x^4+3)/x^9,x, algorithm="fricas")

[Out]

-3/8*(x^4 + 1)^(4/3)/x^8

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giac [A]  time = 0.55, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}}}{8 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(1/3)*(x^4+3)/x^9,x, algorithm="giac")

[Out]

-3/8*(x^4 + 1)^(4/3)/x^8

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

method	result	size
gosper	\(-\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{8 x^{8}}\)	\(13\)
trager	\(-\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{8 x^{8}}\)	\(13\)
risch	\(-\frac {3 \left (x^{8}+2 x^{4}+1\right )}{8 \left (x^{4}+1\right )^{\frac {2}{3}} x^{8}}\)	\(23\)
meijerg	\(-\frac {\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{4}}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {\hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{4}\right ) \Gamma \left (\frac {2}{3}\right ) x^{4}}{3}}{12 \Gamma \left (\frac {2}{3}\right )}-\frac {\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{8}}+\frac {\Gamma \left (\frac {2}{3}\right )}{x^{4}}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{3}-\frac {5 \hypergeom \left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], -x^{4}\right ) \Gamma \left (\frac {2}{3}\right ) x^{4}}{27}}{4 \Gamma \left (\frac {2}{3}\right )}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+1)^(1/3)*(x^4+3)/x^9,x,method=_RETURNVERBOSE)

[Out]

-3/8*(x^4+1)^(4/3)/x^8

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maxima [B]  time = 0.59, size = 50, normalized size = 3.12 \begin {gather*} \frac {{\left (x^{4} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{8 \, {\left (2 \, x^{4} - {\left (x^{4} + 1\right )}^{2} + 1\right )}} - \frac {{\left (x^{4} + 1\right )}^{\frac {1}{3}}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+1)^(1/3)*(x^4+3)/x^9,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.23, size = 12, normalized size = 0.75 \begin {gather*} -\frac {3\,{\left (x^4+1\right )}^{4/3}}{8\,x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 1)^(1/3)*(x^4 + 3))/x^9,x)

[Out]

-(3*(x^4 + 1)^(4/3))/(8*x^8)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+1)**(1/3)*(x**4+3)/x**9,x)

[Out]

Timed out

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