3.1.83 \(\int \frac {(-3+x^4) \sqrt [3]{1+x^4}}{x^5} \, dx\)

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

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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}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-3 + x^4)*(1 + x^4)^(1/3))/x^5,x]

[Out]

(3*(1 + x^4)^(4/3))/(4*x^4)

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 {\left (-3+x^4\right ) \sqrt [3]{1+x^4}}{x^5} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {(-3+x) \sqrt [3]{1+x}}{x^2} \, dx,x,x^4\right )\\ &=\frac {3 \left (1+x^4\right )^{4/3}}{4 x^4}\\ \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}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-3 + x^4)*(1 + x^4)^(1/3))/x^5,x]

[Out]

(3*(1 + x^4)^(4/3))/(4*x^4)

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IntegrateAlgebraic [A]  time = 0.03, size = 16, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^4\right )^{4/3}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-3 + x^4)*(1 + x^4)^(1/3))/x^5,x]

[Out]

(3*(1 + x^4)^(4/3))/(4*x^4)

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fricas [A]  time = 0.47, size = 12, normalized size = 0.75 \begin {gather*} \frac {3 \, {\left (x^{4} + 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3)*(x^4+1)^(1/3)/x^5,x, algorithm="fricas")

[Out]

3/4*(x^4 + 1)^(4/3)/x^4

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giac [A]  time = 0.27, size = 22, normalized size = 1.38 \begin {gather*} \frac {3}{4} \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {3 \, {\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^4-3)*(x^4+1)^(1/3)/x^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.10, size = 13, normalized size = 0.81

method	result	size
gosper	\(\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{4 x^{4}}\)	\(13\)
trager	\(\frac {3 \left (x^{4}+1\right )^{\frac {4}{3}}}{4 x^{4}}\)	\(13\)
risch	\(\frac {\frac {3}{2} x^{4}+\frac {3}{4}+\frac {3}{4} x^{8}}{\left (x^{4}+1\right )^{\frac {2}{3}} x^{4}}\)	\(23\)
meijerg	\(\frac {\frac {3 \Gamma \left (\frac {2}{3}\right )}{x^{4}}-\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )+\frac {\hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 3\right ], -x^{4}\right ) \Gamma \left (\frac {2}{3}\right ) x^{4}}{3}}{4 \Gamma \left (\frac {2}{3}\right )}-\frac {-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+4 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )-\hypergeom \left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -x^{4}\right ) \Gamma \left (\frac {2}{3}\right ) x^{4}}{12 \Gamma \left (\frac {2}{3}\right )}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-3)*(x^4+1)^(1/3)/x^5,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.46, size = 22, normalized size = 1.38 \begin {gather*} \frac {3}{4} \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + \frac {3 \, {\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^4-3)*(x^4+1)^(1/3)/x^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.10, size = 12, normalized size = 0.75 \begin {gather*} \frac {3\,{\left (x^4+1\right )}^{4/3}}{4\,x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*(x^4 + 1)^(4/3))/(4*x^4)

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sympy [C]  time = 178.66, size = 71, normalized size = 4.44 \begin {gather*} - \frac {x^{\frac {4}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 \Gamma \left (\frac {2}{3}\right )} + \frac {3 \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x^{\frac {8}{3}} \Gamma \left (\frac {5}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-3)*(x**4+1)**(1/3)/x**5,x)

[Out]

-x**(4/3)*gamma(-1/3)*hyper((-1/3, -1/3), (2/3,), exp_polar(I*pi)/x**4)/(4*gamma(2/3)) + 3*gamma(2/3)*hyper((-
1/3, 2/3), (5/3,), exp_polar(I*pi)/x**4)/(4*x**(8/3)*gamma(5/3))

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