3.2.42 \(\int \frac {(-1+x^2) \sqrt [3]{x^2+x^4}}{x^3} \, dx\)

Optimal. Leaf size=18 \[ \frac {3 \left (x^4+x^2\right )^{4/3}}{4 x^4} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1590} \begin {gather*} \frac {3 \left (x^4+x^2\right )^{4/3}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(x^2 + x^4)^(4/3))/(4*x^4)

Rule 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 23, normalized size = 1.28 \begin {gather*} \frac {3 \left (x^2+1\right ) \sqrt [3]{x^4+x^2}}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.07, size = 18, normalized size = 1.00 \begin {gather*} \frac {3 \left (x^2+x^4\right )^{4/3}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(x^2 + x^4)^(4/3))/(4*x^4)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 19, normalized size = 1.06 \begin {gather*} \frac {3 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{4 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*(x^4 + x^2)^(1/3)*(x^2 + 1)/x^2

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.08, size = 20, normalized size = 1.11

method	result	size
gosper	\(\frac {3 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\)	\(20\)
trager	\(\frac {3 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\)	\(20\)
meijerg	\(\frac {3 \hypergeom \left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], -x^{2}\right )}{4 x^{\frac {4}{3}}}+\frac {3 \hypergeom \left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right ) x^{\frac {2}{3}}}{2}\)	\(34\)
risch	\(\frac {3 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} \left (x^{4}+2 x^{2}+1\right )}{4 x^{2} \left (x^{2}+1\right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.21, size = 19, normalized size = 1.06 \begin {gather*} \frac {3\,{\left (x^4+x^2\right )}^{1/3}\,\left (x^2+1\right )}{4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________