Optimal. Leaf size=18 \[ \frac {2}{5} \sqrt [4]{x^2-1} \left (x^2+4\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.39, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {2}{5} \left (x^2-1\right )^{5/4}+2 \sqrt [4]{x^2-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\left (-1+x^2\right )^{3/4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(-1+x)^{3/4}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{(-1+x)^{3/4}}+\sqrt [4]{-1+x}\right ) \, dx,x,x^2\right )\\ &=2 \sqrt [4]{-1+x^2}+\frac {2}{5} \left (-1+x^2\right )^{5/4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {2}{5} \sqrt [4]{x^2-1} \left (x^2+4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {2}{5} \sqrt [4]{x^2-1} \left (x^2+4\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 14, normalized size = 0.78 \begin {gather*} \frac {2}{5} \, {\left (x^{2} + 4\right )} {\left (x^{2} - 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 19, normalized size = 1.06 \begin {gather*} \frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 1.17 \begin {gather*} \frac {2 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+4\right )}{5 \left (x^{2}-1\right )^{\frac {3}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 19, normalized size = 1.06 \begin {gather*} \frac {2}{5} \, {\left (x^{2} - 1\right )}^{\frac {5}{4}} + 2 \, {\left (x^{2} - 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 14, normalized size = 0.78 \begin {gather*} \frac {2\,{\left (x^2-1\right )}^{1/4}\,\left (x^2+4\right )}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.83, size = 75, normalized size = 4.17 \begin {gather*} \begin {cases} \frac {2 x^{2} \sqrt [4]{x^{2} - 1}}{5} + \frac {8 \sqrt [4]{x^{2} - 1}}{5} & \text {for}\: \left |{x^{2}}\right | > 1 \\- \frac {2 x^{2} \sqrt [4]{1 - x^{2}} e^{- \frac {3 i \pi }{4}}}{5} - \frac {8 \sqrt [4]{1 - x^{2}} e^{- \frac {3 i \pi }{4}}}{5} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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