Optimal. Leaf size=47 \[ \frac {2 \sqrt {x^5-x}}{x^4-1}-2 \sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x^5-x}}\right ) \]
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Rubi [F] time = 2.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x+3 x^5}{\left (-1+x^4\right ) \left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {x+3 x^5}{\left (-1+x^4\right ) \left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx &=\int \frac {x \left (1+3 x^4\right )}{\left (-1+x^4\right ) \left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^4}\right ) \int \frac {\sqrt {x} \left (1+3 x^4\right )}{\left (-1+x^4\right )^{3/2} \left (-a-x+a x^4\right )} \, dx}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1+3 x^8\right )}{\left (-1+x^8\right )^{3/2} \left (-a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {3 x^2}{a \left (-1+x^8\right )^{3/2}}+\frac {x^2 \left (4 a+3 x^2\right )}{a \left (-1+x^8\right )^{3/2} \left (-a-x^2+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (4 a+3 x^2\right )}{\left (-1+x^8\right )^{3/2} \left (-a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^8\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^5}}\\ &=-\frac {3 x^2}{2 a \sqrt {-x+x^5}}-\frac {\left (3 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^8}} \, dx,x,\sqrt {x}\right )}{2 a \sqrt {-x+x^5}}+\frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {4 a x^2}{\left (-1+x^8\right )^{3/2} \left (-a-x^2+a x^8\right )}+\frac {3 x^4}{\left (-1+x^8\right )^{3/2} \left (-a-x^2+a x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^5}}\\ &=-\frac {3 x^2}{2 a \sqrt {-x+x^5}}+\frac {\left (8 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^8\right )^{3/2} \left (-a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^8\right )^{3/2} \left (-a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^5}}-\frac {\left (3 (-1)^{3/4} \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt [4]{-1} x^2}{\sqrt {-1+x^8}} \, dx,x,\sqrt {x}\right )}{4 a \sqrt {-x+x^5}}+\frac {\left (3 (-1)^{3/4} \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt [4]{-1} x^2}{\sqrt {-1+x^8}} \, dx,x,\sqrt {x}\right )}{4 a \sqrt {-x+x^5}}\\ &=-\frac {3 x^2}{2 a \sqrt {-x+x^5}}+\frac {3 x^2 \sqrt {-\frac {(-1)^{3/4} \left (1+\sqrt [4]{-1} x\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {(-1)^{3/4} \left (\sqrt {2}-2 \sqrt [4]{-1} x+i \sqrt {2} x^2\right )}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{4 \sqrt {2+\sqrt {2}} a \left (1+\sqrt [4]{-1} x\right ) \sqrt {-x+x^5}}+\frac {3 x^2 \sqrt {\frac {(-1)^{3/4} \left (1-\sqrt [4]{-1} x\right )^2}{x}} \sqrt {\frac {i \left (1-x^4\right )}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {(-1)^{3/4} \left (\sqrt {2}+2 \sqrt [4]{-1} x+i \sqrt {2} x^2\right )}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{4 \sqrt {2+\sqrt {2}} a \left (1-\sqrt [4]{-1} x\right ) \sqrt {-x+x^5}}+\frac {\left (8 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^8\right )^{3/2} \left (-a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}+\frac {\left (6 \sqrt {x} \sqrt {-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^8\right )^{3/2} \left (-a-x^2+a x^8\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {-x+x^5}}\\ \end {align*}
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Mathematica [F] time = 1.49, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x+3 x^5}{\left (-1+x^4\right ) \left (-a-x+a x^4\right ) \sqrt {-x+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 2.33, size = 47, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {x^5-x}}{x^4-1}-2 \sqrt {a} \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x^5-x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 193, normalized size = 4.11 \begin {gather*} \left [\frac {{\left (x^{4} - 1\right )} \sqrt {a} \log \left (\frac {a^{2} x^{8} - 2 \, a^{2} x^{4} + 6 \, a x^{5} - 4 \, {\left (a x^{4} - a + x\right )} \sqrt {x^{5} - x} \sqrt {a} + a^{2} - 6 \, a x + x^{2}}{a^{2} x^{8} - 2 \, a^{2} x^{4} - 2 \, a x^{5} + a^{2} + 2 \, a x + x^{2}}\right ) + 4 \, \sqrt {x^{5} - x}}{2 \, {\left (x^{4} - 1\right )}}, \frac {{\left (x^{4} - 1\right )} \sqrt {-a} \arctan \left (\frac {{\left (a x^{4} - a + x\right )} \sqrt {x^{5} - x} \sqrt {-a}}{2 \, {\left (a x^{5} - a x\right )}}\right ) + 2 \, \sqrt {x^{5} - x}}{x^{4} - 1}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + x}{{\left (a x^{4} - a - x\right )} \sqrt {x^{5} - x} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{5}+x}{\left (x^{4}-1\right ) \left (a \,x^{4}-a -x \right ) \sqrt {x^{5}-x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + x}{{\left (a x^{4} - a - x\right )} \sqrt {x^{5} - x} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 61, normalized size = 1.30 \begin {gather*} \frac {2\,\sqrt {x^5-x}}{x^4-1}+\sqrt {a}\,\ln \left (\frac {a-x+2\,\sqrt {a}\,\sqrt {x^5-x}-a\,x^4}{-a\,x^4+x+a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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