\(\int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx\) [139]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2059, 777} \[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\frac {3 \left (x^4+x^2\right )^{2/3}}{2 x^2} \]

[In]

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

[Out]

(3*(x^2 + x^4)^(2/3))/(2*x^2)

Rule 777

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

Rule 2059

Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n
, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x]
/; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] &&  !IntegerQ[p] && NeQ[k, j] && IntegerQ[Simplify[j/n]] && Integ
erQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\frac {3 \left (1+x^2\right )}{2 \sqrt [3]{x^2+x^4}} \]

[In]

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

[Out]

(3*(1 + x^2))/(2*(x^2 + x^4)^(1/3))

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
trager	\(\frac {3 \left (x^{4}+x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(15\)
gosper	\(\frac {\frac {3 x^{2}}{2}+\frac {3}{2}}{\left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\)	\(17\)
pseudoelliptic	\(\frac {3 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(17\)
risch	\(\frac {\frac {3 x^{2}}{2}+\frac {3}{2}}{\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}\)	\(19\)
meijerg	\(\frac {3 \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{2}\right )}{2 x^{\frac {2}{3}}}+\frac {3 x^{\frac {4}{3}} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right )}{4}\)	\(34\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\frac {3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]

[In]

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

[Out]

3/2*(x^4 + x^2)^(2/3)/x^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx=\frac {3\,{\left (x^4+x^2\right )}^{2/3}}{2\,x^2} \]

[In]

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

[Out]

(3*(x^2 + x^4)^(2/3))/(2*x^2)