Optimal. Leaf size=118 \[ \frac {\log \left (2^{2/3} \sqrt [3]{x^5+x}-2 x\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^5+x}+x}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2^{2/3} \sqrt [3]{x^5+x} x+\sqrt [3]{2} \left (x^5+x\right )^{2/3}+2 x^2\right )}{4 \sqrt [3]{2}} \]
________________________________________________________________________________________
Rubi [C] time = 0.60, antiderivative size = 123, normalized size of antiderivative = 1.04, number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2056, 6715, 6725, 245, 1438, 429, 465, 510} \begin {gather*} -\frac {3 \sqrt [3]{x^4+1} x F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^4,-x^4\right )}{\sqrt [3]{x^5+x}}-\frac {3 \sqrt [3]{x^4+1} x^3 F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};x^4,-x^4\right )}{4 \sqrt [3]{x^5+x}}+\frac {3 \sqrt [3]{x^4+1} x \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x^5+x}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 245
Rule 429
Rule 465
Rule 510
Rule 1438
Rule 2056
Rule 6715
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x+x^5}} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \int \frac {1+x^2}{\sqrt [3]{x} \left (-1+x^2\right ) \sqrt [3]{1+x^4}} \, dx}{\sqrt [3]{x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt [3]{1+x^6}}+\frac {2}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}}+\frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}}\right ) \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x+x^5}}\\ &=-\frac {3 x \sqrt [3]{1+x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^4,-x^4\right )}{\sqrt [3]{x+x^5}}+\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}+\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\left (-1+x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{4/3}\right )}{2 \sqrt [3]{x+x^5}}\\ &=-\frac {3 x \sqrt [3]{1+x^4} F_1\left (\frac {1}{6};1,\frac {1}{3};\frac {7}{6};x^4,-x^4\right )}{\sqrt [3]{x+x^5}}-\frac {3 x^3 \sqrt [3]{1+x^4} F_1\left (\frac {2}{3};1,\frac {1}{3};\frac {5}{3};x^4,-x^4\right )}{4 \sqrt [3]{x+x^5}}+\frac {3 x \sqrt [3]{1+x^4} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};-x^4\right )}{2 \sqrt [3]{x+x^5}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt [3]{x+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.39, size = 118, normalized size = 1.00 \begin {gather*} \frac {\log \left (2^{2/3} \sqrt [3]{x^5+x}-2 x\right )}{2 \sqrt [3]{2}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^5+x}+x}\right )}{2 \sqrt [3]{2}}-\frac {\log \left (2^{2/3} \sqrt [3]{x^5+x} x+\sqrt [3]{2} \left (x^5+x\right )^{2/3}+2 x^2\right )}{4 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 4.14, size = 294, normalized size = 2.49 \begin {gather*} -\frac {1}{12} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{12} - 24 \, x^{10} - 57 \, x^{8} - 56 \, x^{6} - 57 \, x^{4} - 24 \, x^{2} + 1\right )} + 24 \, \sqrt {2} {\left (x^{9} - x^{7} - x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} - 12 \cdot 2^{\frac {1}{6}} {\left (x^{8} + 14 \, x^{6} + 6 \, x^{4} + 14 \, x^{2} + 1\right )} {\left (x^{5} + x\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (x^{12} + 48 \, x^{10} + 15 \, x^{8} + 88 \, x^{6} + 15 \, x^{4} + 48 \, x^{2} + 1\right )}}\right ) - \frac {1}{24} \cdot 2^{\frac {2}{3}} \log \left (\frac {2^{\frac {2}{3}} {\left (x^{8} + 14 \, x^{6} + 6 \, x^{4} + 14 \, x^{2} + 1\right )} + 12 \cdot 2^{\frac {1}{3}} {\left (x^{5} + x^{3} + x\right )} {\left (x^{5} + x\right )}^{\frac {1}{3}} + 6 \, {\left (x^{5} + x\right )}^{\frac {2}{3}} {\left (x^{4} + 4 \, x^{2} + 1\right )}}{x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1}\right ) + \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{5} + x\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (x^{4} - 2 \, x^{2} + 1\right )} - 6 \, {\left (x^{5} + x\right )}^{\frac {1}{3}} x}{x^{4} - 2 \, x^{2} + 1}\right ) \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\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 20.51, size = 951, normalized size = 8.06
result too large to display
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\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2+1}{\left (x^2-1\right )\,{\left (x^5+x\right )}^{1/3}} \,d x \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 {x^{2} + 1}{\sqrt [3]{x \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________