Optimal. Leaf size=28 \[ \frac {4 \left (3 x^4-7 x^3+3\right ) \left (x^5+x\right )^{3/4}}{21 x^6} \]
________________________________________________________________________________________
Rubi [A] time = 0.30, antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 16, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2052, 2025, 2032, 364} \begin {gather*} \frac {4 \left (x^5+x\right )^{3/4}}{7 x^6}-\frac {4 \left (x^5+x\right )^{3/4}}{3 x^3}+\frac {4 \left (x^5+x\right )^{3/4}}{7 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 364
Rule 2025
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx &=\int \left (-\frac {3}{x^6 \sqrt [4]{x+x^5}}+\frac {3}{x^3 \sqrt [4]{x+x^5}}-\frac {2}{x^2 \sqrt [4]{x+x^5}}-\frac {x}{\sqrt [4]{x+x^5}}+\frac {x^2}{\sqrt [4]{x+x^5}}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x^2 \sqrt [4]{x+x^5}} \, dx\right )-3 \int \frac {1}{x^6 \sqrt [4]{x+x^5}} \, dx+3 \int \frac {1}{x^3 \sqrt [4]{x+x^5}} \, dx-\int \frac {x}{\sqrt [4]{x+x^5}} \, dx+\int \frac {x^2}{\sqrt [4]{x+x^5}} \, dx\\ &=\frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^5\right )^{3/4}}{3 x^3}+\frac {8 \left (x+x^5\right )^{3/4}}{5 x^2}+\frac {9}{7} \int \frac {1}{x^2 \sqrt [4]{x+x^5}} \, dx-\frac {14}{5} \int \frac {x^2}{\sqrt [4]{x+x^5}} \, dx-\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{1+x^4}} \, dx}{\sqrt [4]{x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^4}} \, dx}{\sqrt [4]{x+x^5}}+\int \frac {x}{\sqrt [4]{x+x^5}} \, dx\\ &=\frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^5\right )^{3/4}}{7 x^2}-\frac {4 x^2 \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{4},\frac {7}{16};\frac {23}{16};-x^4\right )}{7 \sqrt [4]{x+x^5}}+\frac {4 x^3 \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{4},\frac {11}{16};\frac {27}{16};-x^4\right )}{11 \sqrt [4]{x+x^5}}+\frac {9}{5} \int \frac {x^2}{\sqrt [4]{x+x^5}} \, dx+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{1+x^4}} \, dx}{\sqrt [4]{x+x^5}}-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^4}} \, dx}{5 \sqrt [4]{x+x^5}}\\ &=\frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^5\right )^{3/4}}{7 x^2}-\frac {36 x^3 \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{4},\frac {11}{16};\frac {27}{16};-x^4\right )}{55 \sqrt [4]{x+x^5}}+\frac {\left (9 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^4}} \, dx}{5 \sqrt [4]{x+x^5}}\\ &=\frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^5\right )^{3/4}}{7 x^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.10, size = 123, normalized size = 4.39 \begin {gather*} \frac {4 \sqrt [4]{x^4+1} \left (165 \, _2F_1\left (-\frac {21}{16},\frac {1}{4};-\frac {5}{16};-x^4\right )+x^3 \left (-165 x^4 \, _2F_1\left (\frac {1}{4},\frac {7}{16};\frac {23}{16};-x^4\right )+462 x \, _2F_1\left (-\frac {5}{16},\frac {1}{4};\frac {11}{16};-x^4\right )-385 \, _2F_1\left (-\frac {9}{16},\frac {1}{4};\frac {7}{16};-x^4\right )+105 x^5 \, _2F_1\left (\frac {1}{4},\frac {11}{16};\frac {27}{16};-x^4\right )\right )\right )}{1155 x^5 \sqrt [4]{x^5+x}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.59, size = 28, normalized size = 1.00 \begin {gather*} \frac {4 \left (3-7 x^3+3 x^4\right ) \left (x+x^5\right )^{3/4}}{21 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.48, size = 24, normalized size = 0.86 \begin {gather*} \frac {4 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} {\left (3 \, x^{4} - 7 \, x^{3} + 3\right )}}{21 \, x^{6}} \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^{4} - x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.10, size = 25, normalized size = 0.89
method | result | size |
trager | \(\frac {4 \left (3 x^{4}-7 x^{3}+3\right ) \left (x^{5}+x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(25\) |
gosper | \(\frac {4 \left (x^{4}+1\right ) \left (3 x^{4}-7 x^{3}+3\right )}{21 x^{5} \left (x^{5}+x \right )^{\frac {1}{4}}}\) | \(30\) |
risch | \(\frac {-\frac {4}{3} x^{7}-\frac {4}{3} x^{3}+\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}}{x^{5} \left (x \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\) | \(37\) |
meijerg | \(\frac {4 \hypergeom \left (\left [-\frac {21}{16}, \frac {1}{4}\right ], \left [-\frac {5}{16}\right ], -x^{4}\right )}{7 x^{\frac {21}{4}}}+\frac {8 \hypergeom \left (\left [-\frac {5}{16}, \frac {1}{4}\right ], \left [\frac {11}{16}\right ], -x^{4}\right )}{5 x^{\frac {5}{4}}}-\frac {4 \hypergeom \left (\left [-\frac {9}{16}, \frac {1}{4}\right ], \left [\frac {7}{16}\right ], -x^{4}\right )}{3 x^{\frac {9}{4}}}+\frac {4 \hypergeom \left (\left [\frac {1}{4}, \frac {11}{16}\right ], \left [\frac {27}{16}\right ], -x^{4}\right ) x^{\frac {11}{4}}}{11}-\frac {4 \hypergeom \left (\left [\frac {1}{4}, \frac {7}{16}\right ], \left [\frac {23}{16}\right ], -x^{4}\right ) x^{\frac {7}{4}}}{7}\) | \(82\) |
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^{4} - x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.29, size = 39, normalized size = 1.39 \begin {gather*} \frac {12\,{\left (x^5+x\right )}^{3/4}-28\,x^3\,{\left (x^5+x\right )}^{3/4}+12\,x^4\,{\left (x^5+x\right )}^{3/4}}{21\,x^6} \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^{4} - 3\right ) \left (x^{4} - x^{3} + 1\right )}{x^{6} \sqrt [4]{x \left (x^{4} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________