Optimal. Leaf size=59 \[ \frac {3 a^2 \left (a+b x^2\right )^{7/3}}{14 b^3}+\frac {3 \left (a+b x^2\right )^{13/3}}{26 b^3}-\frac {3 a \left (a+b x^2\right )^{10/3}}{10 b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {3 a^2 \left (a+b x^2\right )^{7/3}}{14 b^3}+\frac {3 \left (a+b x^2\right )^{13/3}}{26 b^3}-\frac {3 a \left (a+b x^2\right )^{10/3}}{10 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^5 \left (a+b x^2\right )^{4/3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x^2 (a+b x)^{4/3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2 (a+b x)^{4/3}}{b^2}-\frac {2 a (a+b x)^{7/3}}{b^2}+\frac {(a+b x)^{10/3}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac {3 a^2 \left (a+b x^2\right )^{7/3}}{14 b^3}-\frac {3 a \left (a+b x^2\right )^{10/3}}{10 b^3}+\frac {3 \left (a+b x^2\right )^{13/3}}{26 b^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.66 \[ \frac {3 \left (a+b x^2\right )^{7/3} \left (9 a^2-21 a b x^2+35 b^2 x^4\right )}{910 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 57, normalized size = 0.97 \[ \frac {3 \, {\left (35 \, b^{4} x^{8} + 49 \, a b^{3} x^{6} + 2 \, a^{2} b^{2} x^{4} - 3 \, a^{3} b x^{2} + 9 \, a^{4}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{910 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.65, size = 43, normalized size = 0.73 \[ \frac {3 \, {\left (35 \, {\left (b x^{2} + a\right )}^{\frac {13}{3}} - 91 \, {\left (b x^{2} + a\right )}^{\frac {10}{3}} a + 65 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} a^{2}\right )}}{910 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 36, normalized size = 0.61 \[ \frac {3 \left (b \,x^{2}+a \right )^{\frac {7}{3}} \left (35 b^{2} x^{4}-21 a b \,x^{2}+9 a^{2}\right )}{910 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 47, normalized size = 0.80 \[ \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {13}{3}}}{26 \, b^{3}} - \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {10}{3}} a}{10 \, b^{3}} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {7}{3}} a^{2}}{14 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.10, size = 53, normalized size = 0.90 \[ {\left (b\,x^2+a\right )}^{1/3}\,\left (\frac {21\,a\,x^6}{130}+\frac {3\,b\,x^8}{26}+\frac {27\,a^4}{910\,b^3}+\frac {3\,a^2\,x^4}{455\,b}-\frac {9\,a^3\,x^2}{910\,b^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.98, size = 112, normalized size = 1.90 \[ \begin {cases} \frac {27 a^{4} \sqrt [3]{a + b x^{2}}}{910 b^{3}} - \frac {9 a^{3} x^{2} \sqrt [3]{a + b x^{2}}}{910 b^{2}} + \frac {3 a^{2} x^{4} \sqrt [3]{a + b x^{2}}}{455 b} + \frac {21 a x^{6} \sqrt [3]{a + b x^{2}}}{130} + \frac {3 b x^{8} \sqrt [3]{a + b x^{2}}}{26} & \text {for}\: b \neq 0 \\\frac {a^{\frac {4}{3}} x^{6}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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