Optimal. Leaf size=18 \[ \frac {4 \left (x^5+x^3\right )^{9/4}}{9 x^9} \]
________________________________________________________________________________________
Rubi [B] time = 0.18, antiderivative size = 53, normalized size of antiderivative = 2.94, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2052, 2004, 2032, 364, 2020, 2025} \begin {gather*} \frac {4}{9} \sqrt [4]{x^5+x^3} x+\frac {8 \sqrt [4]{x^5+x^3}}{9 x}+\frac {4 \sqrt [4]{x^5+x^3}}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 2004
Rule 2020
Rule 2025
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^5}}{x^4} \, dx &=\int \left (\sqrt [4]{x^3+x^5}-\frac {\sqrt [4]{x^3+x^5}}{x^4}\right ) \, dx\\ &=\int \sqrt [4]{x^3+x^5} \, dx-\int \frac {\sqrt [4]{x^3+x^5}}{x^4} \, dx\\ &=\frac {4 \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {4}{9} x \sqrt [4]{x^3+x^5}-\frac {2}{9} \int \frac {x}{\left (x^3+x^5\right )^{3/4}} \, dx+\frac {2}{9} \int \frac {x^3}{\left (x^3+x^5\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {8 \sqrt [4]{x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{x^3+x^5}-\frac {2}{9} \int \frac {x^3}{\left (x^3+x^5\right )^{3/4}} \, dx+\frac {\left (2 x^{9/4} \left (1+x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (1+x^2\right )^{3/4}} \, dx}{9 \left (x^3+x^5\right )^{3/4}}\\ &=\frac {4 \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {8 \sqrt [4]{x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{x^3+x^5}+\frac {8 x^4 \left (1+x^2\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^2\right )}{63 \left (x^3+x^5\right )^{3/4}}-\frac {\left (2 x^{9/4} \left (1+x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (1+x^2\right )^{3/4}} \, dx}{9 \left (x^3+x^5\right )^{3/4}}\\ &=\frac {4 \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {8 \sqrt [4]{x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{x^3+x^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 65, normalized size = 3.61 \begin {gather*} \frac {4 \sqrt [4]{x^5+x^3} \left (7 \, _2F_1\left (-\frac {9}{8},-\frac {1}{4};-\frac {1}{8};-x^2\right )+9 x^4 \, _2F_1\left (-\frac {1}{4},\frac {7}{8};\frac {15}{8};-x^2\right )\right )}{63 x^3 \sqrt [4]{x^2+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.09, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^3+x^5\right )^{9/4}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 24, normalized size = 1.33 \begin {gather*} \frac {4 \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{2} + 1\right )}}{9 \, x^{3}} \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 {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 22, normalized size = 1.22
method | result | size |
gosper | \(\frac {4 \left (x^{2}+1\right )^{2} \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(22\) |
trager | \(\frac {4 \left (x^{4}+2 x^{2}+1\right ) \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}\) | \(25\) |
meijerg | \(\frac {4 \hypergeom \left (\left [-\frac {9}{8}, -\frac {1}{4}\right ], \left [-\frac {1}{8}\right ], -x^{2}\right )}{9 x^{\frac {9}{4}}}+\frac {4 \hypergeom \left (\left [-\frac {1}{4}, \frac {7}{8}\right ], \left [\frac {15}{8}\right ], -x^{2}\right ) x^{\frac {7}{4}}}{7}\) | \(34\) |
risch | \(\frac {4 \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \left (x^{6}+3 x^{4}+3 x^{2}+1\right )}{9 x^{3} \left (x^{2}+1\right )}\) | \(39\) |
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 {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.27, size = 41, normalized size = 2.28 \begin {gather*} \frac {4\,x\,{\left (x^5+x^3\right )}^{1/4}}{9}+\frac {8\,{\left (x^5+x^3\right )}^{1/4}}{9\,x}+\frac {4\,{\left (x^5+x^3\right )}^{1/4}}{9\,x^3} \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 {\sqrt [4]{x^{3} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________