Optimal. Leaf size=44 \[ \frac{3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}}+\frac{9 x}{2 \sqrt [3]{a-b x^2}} \]
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Rubi [A] time = 0.0172769, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {413, 383} \[ \frac{3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}}+\frac{9 x}{2 \sqrt [3]{a-b x^2}} \]
Antiderivative was successfully verified.
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Rule 413
Rule 383
Rubi steps
\begin{align*} \int \frac{\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{7/3}} \, dx &=\frac{3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}}-\frac{3 \int \frac{-12 a^2 b+4 a b^2 x^2}{\left (a-b x^2\right )^{4/3}} \, dx}{8 a b}\\ &=\frac{9 x}{2 \sqrt [3]{a-b x^2}}+\frac{3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}}\\ \end{align*}
Mathematica [A] time = 5.02633, size = 24, normalized size = 0.55 \[ \frac{9 a x-3 b x^3}{\left (a-b x^2\right )^{4/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 24, normalized size = 0.6 \begin{align*} 3\,{\frac{x \left ( -b{x}^{2}+3\,a \right ) }{ \left ( -b{x}^{2}+a \right ) ^{4/3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14021, size = 45, normalized size = 1.02 \begin{align*} \frac{3 \,{\left (b x^{3} - 3 \, a x\right )}}{{\left (b x^{2} - a\right )}{\left (-b x^{2} + a\right )}^{\frac{1}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77402, size = 90, normalized size = 2.05 \begin{align*} -\frac{3 \,{\left (b x^{3} - 3 \, a x\right )}{\left (-b x^{2} + a\right )}^{\frac{2}{3}}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 a + b x^{2}\right )^{2}}{\left (a - b x^{2}\right )^{\frac{7}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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