Optimal. Leaf size=23 \[ -\frac {4 \left (x^5+x^3\right )^{3/4}}{x^2 \left (x^2+1\right )} \]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2056, 449} \begin {gather*} -\frac {4 x}{\sqrt [4]{x^5+x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 449
Rule 2056
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [4]{x^3+x^5}} \, dx &=\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] time = 0.03, size = 14, normalized size = 0.61 \begin {gather*} -\frac {4 x}{\sqrt [4]{x^5+x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.14, size = 23, normalized size = 1.00 \begin {gather*} -\frac {4 \left (x^3+x^5\right )^{3/4}}{x^2 \left (1+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 20, normalized size = 0.87 \begin {gather*} -\frac {4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 13, normalized size = 0.57
method | result | size |
gosper | \(-\frac {4 x}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\) | \(13\) |
risch | \(-\frac {4 x}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(15\) |
trager | \(-\frac {4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (x^{2}+1\right )}\) | \(22\) |
meijerg | \(-4 \hypergeom \left (\left [\frac {1}{8}, \frac {5}{4}\right ], \left [\frac {9}{8}\right ], -x^{2}\right ) x^{\frac {1}{4}}+\frac {4 \hypergeom \left (\left [\frac {9}{8}, \frac {5}{4}\right ], \left [\frac {17}{8}\right ], -x^{2}\right ) x^{\frac {9}{4}}}{9}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.15, size = 21, normalized size = 0.91 \begin {gather*} -\frac {4\,{\left (x^5+x^3\right )}^{3/4}}{x^2\,\left (x^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________