Optimal. Leaf size=25 \[ \frac {2}{77} \left (x^2+1\right )^{3/4} \left (7 x^4+3 x^2-4\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.08, 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}{11} \left (x^2+1\right )^{11/4}-\frac {2}{7} \left (x^2+1\right )^{7/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \left (1+x^2\right )^{3/4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (1+x)^{3/4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-(1+x)^{3/4}+(1+x)^{7/4}\right ) \, dx,x,x^2\right )\\ &=-\frac {2}{7} \left (1+x^2\right )^{7/4}+\frac {2}{11} \left (1+x^2\right )^{11/4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {2}{77} \left (x^2+1\right )^{7/4} \left (7 x^2-4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 20, normalized size = 0.80 \begin {gather*} \frac {2}{77} \left (x^2+1\right )^{7/4} \left (7 x^2-4\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 21, normalized size = 0.84 \begin {gather*} \frac {2}{77} \, {\left (7 \, x^{4} + 3 \, x^{2} - 4\right )} {\left (x^{2} + 1\right )}^{\frac {3}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 19, normalized size = 0.76 \begin {gather*} \frac {2}{11} \, {\left (x^{2} + 1\right )}^{\frac {11}{4}} - \frac {2}{7} \, {\left (x^{2} + 1\right )}^{\frac {7}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.68 \begin {gather*} \frac {2 \left (x^{2}+1\right )^{\frac {7}{4}} \left (7 x^{2}-4\right )}{77} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 19, normalized size = 0.76 \begin {gather*} \frac {2}{11} \, {\left (x^{2} + 1\right )}^{\frac {11}{4}} - \frac {2}{7} \, {\left (x^{2} + 1\right )}^{\frac {7}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 20, normalized size = 0.80 \begin {gather*} {\left (x^2+1\right )}^{3/4}\,\left (\frac {2\,x^4}{11}+\frac {6\,x^2}{77}-\frac {8}{77}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.80, size = 41, normalized size = 1.64 \begin {gather*} \frac {2 x^{4} \left (x^{2} + 1\right )^{\frac {3}{4}}}{11} + \frac {6 x^{2} \left (x^{2} + 1\right )^{\frac {3}{4}}}{77} - \frac {8 \left (x^{2} + 1\right )^{\frac {3}{4}}}{77} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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