\(\int \frac {(-1+x^2) \sqrt [3]{x^2+x^4}}{x^3} \, dx\) [142]

Optimal result

Integrand size = 20, antiderivative size = 18 \[ \int \frac {\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}}{x^3} \, dx=\frac {3 \left (x^2+x^4\right )^{4/3}}{4 x^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {1604} \[ \int \frac {\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}}{x^3} \, dx=\frac {3 \left (x^4+x^2\right )^{4/3}}{4 x^4} \]

[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 1604

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*} \text {integral}& = \frac {3 \left (x^2+x^4\right )^{4/3}}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}}{x^3} \, dx=\frac {3 \left (1+x^2\right ) \sqrt [3]{x^2+x^4}}{4 x^2} \]

[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)

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 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	\(\frac {3 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\)	\(20\)
pseudoelliptic	\(\frac {3 \left (x^{2}+1\right ) \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}{4 x^{2}}\)	\(22\)
meijerg	\(\frac {3 \operatorname {hypergeom}\left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], -x^{2}\right )}{4 x^{\frac {4}{3}}}+\frac {3 x^{\frac {2}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right )}{2}\)	\(34\)
risch	\(\frac {3 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} \left (x^{4}+2 x^{2}+1\right )}{4 x^{2} \left (x^{2}+1\right )}\)	\(34\)

[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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}}{x^3} \, dx=\frac {3 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{4 \, x^{2}} \]

[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

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [B] (verification not implemented)

Time = 5.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {\left (-1+x^2\right ) \sqrt [3]{x^2+x^4}}{x^3} \, dx=\frac {3\,{\left (x^4+x^2\right )}^{1/3}\,\left (x^2+1\right )}{4\,x^2} \]

[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)