3.2.21 \(\int \frac {3+x}{(-1+x)^2 \sqrt [3]{-1+x^2}} \, dx\)

Optimal. Leaf size=18 \[ -\frac {3 \left (x^2-1\right )^{2/3}}{2 (x-1)^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.11, 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 = {787} \begin {gather*} -\frac {3 \left (x^2-1\right )^{2/3}}{2 (1-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-1 + x^2)^(2/3))/(2*(1 - x)^2)

Rule 787

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(g*(d + e*x)^m
*(a + c*x^2)^(p + 1))/(c*(m + 2*p + 2)), x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&
EqQ[m*(d*g + e*f) + 2*e*f*(p + 1), 0]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 21, normalized size = 1.17 \begin {gather*} -\frac {3 (x+1)}{2 (x-1) \sqrt [3]{x^2-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.06, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 \left (-1+x^2\right )^{2/3}}{2 (-1+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-1 + x^2)^(2/3))/(2*(-1 + x)^2)

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fricas [A]  time = 0.46, size = 19, normalized size = 1.06 \begin {gather*} -\frac {3 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(x^2 - 1)^(2/3)/(x^2 - 2*x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.09, size = 15, normalized size = 0.83

method	result	size
trager	\(-\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{2 \left (-1+x \right )^{2}}\)	\(15\)
gosper	\(-\frac {3 \left (1+x \right )}{2 \left (-1+x \right ) \left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(18\)
risch	\(-\frac {3 \left (1+x \right )}{2 \left (-1+x \right ) \left (x^{2}-1\right )^{\frac {1}{3}}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 12, normalized size = 0.67 \begin {gather*} -\frac {3 \, {\left (x + 1\right )}^{\frac {2}{3}}}{2 \, {\left (x - 1\right )}^{\frac {4}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/2*(x + 1)^(2/3)/(x - 1)^(4/3)

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mupad [B]  time = 0.14, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3\,{\left (x^2-1\right )}^{2/3}}{2\,{\left (x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(3*(x^2 - 1)^(2/3))/(2*(x - 1)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x + 3)/(((x - 1)*(x + 1))**(1/3)*(x - 1)**2), x)

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