\(\int \frac {-1+x^2}{(1+x^2) \sqrt [4]{x^3+x^5}} \, dx\) [232]

Optimal result

Integrand size = 24, antiderivative size = 23 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [4]{x^3+x^5}} \, dx=-\frac {4 \left (x^3+x^5\right )^{3/4}}{x^2 \left (1+x^2\right )} \]

[Out]

-4*(x^5+x^3)^(3/4)/x^2/(x^2+1)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2081, 460} \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [4]{x^3+x^5}} \, dx=-\frac {4 x}{\sqrt [4]{x^5+x^3}} \]

[In]

Int[(-1 + x^2)/((1 + x^2)*(x^3 + x^5)^(1/4)),x]

[Out]

(-4*x)/(x^3 + x^5)^(1/4)

Rule 460

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]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \int \frac {-1+x^2}{x^{3/4} \left (1+x^2\right )^{5/4}} \, dx}{\sqrt [4]{x^3+x^5}} \\ & = -\frac {4 x}{\sqrt [4]{x^3+x^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [4]{x^3+x^5}} \, dx=-\frac {4 x}{\sqrt [4]{x^3+x^5}} \]

[In]

Integrate[(-1 + x^2)/((1 + x^2)*(x^3 + x^5)^(1/4)),x]

[Out]

(-4*x)/(x^3 + x^5)^(1/4)

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57

method	result	size
gosper	\(-\frac {4 x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\)	\(13\)
risch	\(-\frac {4 x}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\)	\(15\)
pseudoelliptic	\(-\frac {4 x}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\)	\(15\)
trager	\(-\frac {4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (x^{2}+1\right )}\)	\(22\)
meijerg	\(-4 x^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{8}, \frac {5}{4}\right ], \left [\frac {9}{8}\right ], -x^{2}\right )+\frac {4 x^{\frac {9}{4}} \operatorname {hypergeom}\left (\left [\frac {9}{8}, \frac {5}{4}\right ], \left [\frac {17}{8}\right ], -x^{2}\right )}{9}\)	\(34\)

[In]

int((x^2-1)/(x^2+1)/(x^5+x^3)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-4*x/(x^5+x^3)^(1/4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [4]{x^3+x^5}} \, dx=-\frac {4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{2}} \]

[In]

integrate((x^2-1)/(x^2+1)/(x^5+x^3)^(1/4),x, algorithm="fricas")

[Out]

-4*(x^5 + x^3)^(3/4)/(x^4 + x^2)

Sympy [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [4]{x^3+x^5}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\sqrt [4]{x^{3} \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \]

[In]

integrate((x**2-1)/(x**2+1)/(x**5+x**3)**(1/4),x)

[Out]

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

Maxima [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [4]{x^3+x^5}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}} \,d x } \]

[In]

integrate((x^2-1)/(x^2+1)/(x^5+x^3)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^2 - 1)/((x^5 + x^3)^(1/4)*(x^2 + 1)), x)

Giac [F]

\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [4]{x^3+x^5}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}} \,d x } \]

[In]

integrate((x^2-1)/(x^2+1)/(x^5+x^3)^(1/4),x, algorithm="giac")

[Out]

integrate((x^2 - 1)/((x^5 + x^3)^(1/4)*(x^2 + 1)), x)

Mupad [B] (verification not implemented)

Time = 5.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [4]{x^3+x^5}} \, dx=-\frac {4\,{\left (x^5+x^3\right )}^{3/4}}{x^2\,\left (x^2+1\right )} \]

[In]

int((x^2 - 1)/((x^3 + x^5)^(1/4)*(x^2 + 1)),x)

[Out]

-(4*(x^3 + x^5)^(3/4))/(x^2*(x^2 + 1))