Optimal. Leaf size=80 \[ \frac{9 a^2 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac{3 \left (a+b x^2\right )^{16/3}}{32 b^4}-\frac{9 a \left (a+b x^2\right )^{13/3}}{26 b^4} \]
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Rubi [A] time = 0.048907, antiderivative size = 80, normalized size of antiderivative = 1., 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} \[ \frac{9 a^2 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{3 a^3 \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac{3 \left (a+b x^2\right )^{16/3}}{32 b^4}-\frac{9 a \left (a+b x^2\right )^{13/3}}{26 b^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^7 \left (a+b x^2\right )^{4/3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^3 (a+b x)^{4/3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3 (a+b x)^{4/3}}{b^3}+\frac{3 a^2 (a+b x)^{7/3}}{b^3}-\frac{3 a (a+b x)^{10/3}}{b^3}+\frac{(a+b x)^{13/3}}{b^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a^3 \left (a+b x^2\right )^{7/3}}{14 b^4}+\frac{9 a^2 \left (a+b x^2\right )^{10/3}}{20 b^4}-\frac{9 a \left (a+b x^2\right )^{13/3}}{26 b^4}+\frac{3 \left (a+b x^2\right )^{16/3}}{32 b^4}\\ \end{align*}
Mathematica [A] time = 0.026091, size = 50, normalized size = 0.62 \[ \frac{3 \left (a+b x^2\right )^{7/3} \left (189 a^2 b x^2-81 a^3-315 a b^2 x^4+455 b^3 x^6\right )}{14560 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 47, normalized size = 0.6 \begin{align*} -{\frac{-1365\,{b}^{3}{x}^{6}+945\,a{b}^{2}{x}^{4}-567\,{a}^{2}b{x}^{2}+243\,{a}^{3}}{14560\,{b}^{4}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.85292, size = 86, normalized size = 1.08 \begin{align*} \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{16}{3}}}{32 \, b^{4}} - \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} a}{26 \, b^{4}} + \frac{9 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a^{2}}{20 \, b^{4}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{3}}{14 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74078, size = 162, normalized size = 2.02 \begin{align*} \frac{3 \,{\left (455 \, b^{5} x^{10} + 595 \, a b^{4} x^{8} + 14 \, a^{2} b^{3} x^{6} - 18 \, a^{3} b^{2} x^{4} + 27 \, a^{4} b x^{2} - 81 \, a^{5}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{14560 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.78559, size = 136, normalized size = 1.7 \begin{align*} \begin{cases} - \frac{243 a^{5} \sqrt [3]{a + b x^{2}}}{14560 b^{4}} + \frac{81 a^{4} x^{2} \sqrt [3]{a + b x^{2}}}{14560 b^{3}} - \frac{27 a^{3} x^{4} \sqrt [3]{a + b x^{2}}}{7280 b^{2}} + \frac{3 a^{2} x^{6} \sqrt [3]{a + b x^{2}}}{1040 b} + \frac{51 a x^{8} \sqrt [3]{a + b x^{2}}}{416} + \frac{3 b x^{10} \sqrt [3]{a + b x^{2}}}{32} & \text{for}\: b \neq 0 \\\frac{a^{\frac{4}{3}} x^{8}}{8} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.53501, size = 181, normalized size = 2.26 \begin{align*} \frac{3 \,{\left (\frac{4 \,{\left (140 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} - 546 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a + 780 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{2} - 455 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{3}\right )} a}{b^{3}} + \frac{455 \,{\left (b x^{2} + a\right )}^{\frac{16}{3}} - 2240 \,{\left (b x^{2} + a\right )}^{\frac{13}{3}} a + 4368 \,{\left (b x^{2} + a\right )}^{\frac{10}{3}} a^{2} - 4160 \,{\left (b x^{2} + a\right )}^{\frac{7}{3}} a^{3} + 1820 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} a^{4}}{b^{3}}\right )}}{14560 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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