Optimal. Leaf size=30 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x^5+x}}{x^4+1}\right )}{\sqrt {a}} \]
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Rubi [F] time = 1.22, 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 {-1+3 x^4}{\left (1-a x+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 {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx &=\frac {\left (\sqrt {x} \sqrt {1+x^4}\right ) \int \frac {-1+3 x^4}{\sqrt {x} \sqrt {1+x^4} \left (1-a x+x^4\right )} \, dx}{\sqrt {x+x^5}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {-1+3 x^8}{\sqrt {1+x^8} \left (1-a x^2+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}{\sqrt {1+x^8}}-\frac {4-3 a x^2}{\sqrt {1+x^8} \left (1-a x^2+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 {4-3 a x^2}{\sqrt {1+x^8} \left (1-a x^2+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 {1}{\sqrt {1+x^8}} \, 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 {4}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}}+\frac {3 a x^2}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}+\frac {\left (3 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{\sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ &=\frac {3 x^2 \sqrt {\frac {(1+x)^2}{x}} \sqrt {-\frac {1+x^4}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {-\frac {\sqrt {2}-2 x+\sqrt {2} x^2}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2+\sqrt {2}} (1+x) \sqrt {x+x^5}}-\frac {3 \sqrt {-\frac {(1-x)^2}{x}} x^2 \sqrt {-\frac {1+x^4}{x^2}} F\left (\sin ^{-1}\left (\frac {1}{2} \sqrt {\frac {\sqrt {2}+2 x+\sqrt {2} x^2}{x}}\right )|-2 \left (1-\sqrt {2}\right )\right )}{\sqrt {2+\sqrt {2}} (1-x) \sqrt {x+x^5}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}-\frac {\left (6 a \sqrt {x} \sqrt {1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+a x^2-x^8\right ) \sqrt {1+x^8}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^5}}\\ \end {align*}
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Mathematica [F] time = 0.47, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+3 x^4}{\left (1-a x+x^4\right ) \sqrt {x+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.51, size = 30, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x+x^5}}{1+x^4}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 127, normalized size = 4.23 \begin {gather*} \left [\frac {\log \left (\frac {x^{8} + 6 \, a x^{5} + a^{2} x^{2} + 2 \, x^{4} - 4 \, \sqrt {x^{5} + x} {\left (x^{4} + a x + 1\right )} \sqrt {a} + 6 \, a x + 1}{x^{8} - 2 \, a x^{5} + a^{2} x^{2} + 2 \, x^{4} - 2 \, a x + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {x^{5} + x} {\left (x^{4} + a x + 1\right )} \sqrt {-a}}{2 \, {\left (a x^{5} + a x\right )}}\right )}{a}\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^{4} - 1}{\sqrt {x^{5} + x} {\left (x^{4} - a x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {3 x^{4}-1}{\left (x^{4}-a x +1\right ) \sqrt {x^{5}+x}}\, dx\]
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^{4} - 1}{\sqrt {x^{5} + x} {\left (x^{4} - a x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 37, normalized size = 1.23 \begin {gather*} \frac {\ln \left (\frac {a\,x-2\,\sqrt {a}\,\sqrt {x^5+x}+x^4+1}{x^4-a\,x+1}\right )}{\sqrt {a}} \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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