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

Optimal. Leaf size=14 \[ -\frac {4 \sqrt [4]{x^3-1}}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 14, 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 {4 \sqrt [4]{x^3-1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(-1 + x^3)^(1/4))/x

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

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{x^3-1}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(-1 + x^3)^(1/4))/x

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IntegrateAlgebraic [A]  time = 2.10, size = 14, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{-1+x^3}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(-1 + x^3)^(1/4))/x

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fricas [A]  time = 0.46, size = 12, normalized size = 0.86 \begin {gather*} -\frac {4 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(x^3 - 1)^(1/4)/x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 13, normalized size = 0.93

method	result	size
trager	\(-\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}\)	\(13\)
risch	\(-\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}\)	\(13\)
gosper	\(-\frac {4 \left (-1+x \right ) \left (x^{2}+x +1\right )}{x \left (x^{3}-1\right )^{\frac {3}{4}}}\)	\(22\)
meijerg	\(\frac {\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {2}{3}, \frac {3}{4}\right ], \left [\frac {5}{3}\right ], x^{3}\right ) x^{2}}{2 \mathrm {signum}\left (x^{3}-1\right )^{\frac {3}{4}}}+\frac {4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {1}{3}, \frac {3}{4}\right ], \left [\frac {2}{3}\right ], x^{3}\right )}{\mathrm {signum}\left (x^{3}-1\right )^{\frac {3}{4}} x}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 18, normalized size = 1.29 \begin {gather*} -\frac {4 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.12, size = 12, normalized size = 0.86 \begin {gather*} -\frac {4\,{\left (x^3-1\right )}^{1/4}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*(x^3 - 1)^(1/4))/x

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sympy [C]  time = 1.69, size = 68, normalized size = 4.86 \begin {gather*} \frac {x^{2} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {3}{4} \\ \frac {5}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {4 e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{4} \\ \frac {2}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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