\(\int \frac {1+x^2}{(-1+x^2) \sqrt [4]{-x^3+x^5}} \, dx\) [270]

Optimal result

Integrand size = 26, antiderivative size = 25 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [4]{-x^3+x^5}} \, dx=-\frac {4 \left (-x^3+x^5\right )^{3/4}}{x^2 \left (-1+x^2\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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 = 16, normalized size of antiderivative = 0.64 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [4]{-x^3+x^5}} \, dx=-\frac {4 x}{\sqrt [4]{x^3 \left (-1+x^2\right )}} \]

[In]

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

[Out]

(-4*x)/(x^3*(-1 + x^2))^(1/4)

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60

method	result	size
gosper	\(-\frac {4 x}{\left (x^{5}-x^{3}\right )^{\frac {1}{4}}}\)	\(15\)
risch	\(-\frac {4 x}{\left (x^{3} \left (x^{2}-1\right )\right )^{\frac {1}{4}}}\)	\(15\)
pseudoelliptic	\(-\frac {4 x}{\left (x^{5}-x^{3}\right )^{\frac {1}{4}}}\)	\(15\)
trager	\(-\frac {4 \left (x^{5}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (x^{2}-1\right )}\)	\(24\)
meijerg	\(-\frac {4 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}} x^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [\frac {1}{8}, \frac {5}{4}\right ], \left [\frac {9}{8}\right ], x^{2}\right )}{\operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}}}-\frac {4 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{4}} x^{\frac {9}{4}} \operatorname {hypergeom}\left (\left [\frac {9}{8}, \frac {5}{4}\right ], \left [\frac {17}{8}\right ], x^{2}\right )}{9 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{4}}}\)	\(66\)

[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 = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [4]{-x^3+x^5}} \, dx=-\frac {4 \, {\left (x^{5} - x^{3}\right )}^{\frac {3}{4}}}{x^{4} - x^{2}} \]

[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]

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

[In]

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

[Out]

Integral((x**2 + 1)/((x**3*(x - 1)*(x + 1))**(1/4)*(x - 1)*(x + 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 = 5.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [4]{-x^3+x^5}} \, dx=-\frac {4\,{\left (x^5-x^3\right )}^{3/4}}{x^2\,\left (x^2-1\right )} \]

[In]

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

[Out]

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