Optimal. Leaf size=597 \[ \frac{1080 \sqrt{2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt{3}-7\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{3240 a^2 x}{91 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{1620 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{198}{91} a x \left (a-b x^2\right )^{2/3}-\frac{3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \]
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Rubi [A] time = 0.380998, antiderivative size = 597, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {416, 388, 235, 304, 219, 1879} \[ -\frac{3240 a^2 x}{91 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{1080 \sqrt{2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{1620 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{198}{91} a x \left (a-b x^2\right )^{2/3}-\frac{3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \]
Antiderivative was successfully verified.
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Rule 416
Rule 388
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{\left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}} \, dx &=-\frac{3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3 \int \frac{-42 a^2 b-22 a b^2 x^2}{\sqrt [3]{a-b x^2}} \, dx}{13 b}\\ &=-\frac{198}{91} a x \left (a-b x^2\right )^{2/3}-\frac{3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )+\frac{1}{91} \left (1080 a^2\right ) \int \frac{1}{\sqrt [3]{a-b x^2}} \, dx\\ &=-\frac{198}{91} a x \left (a-b x^2\right )^{2/3}-\frac{3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{\left (1620 a^2 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{91 b x}\\ &=-\frac{198}{91} a x \left (a-b x^2\right )^{2/3}-\frac{3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )+\frac{\left (1620 a^2 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{91 b x}-\frac{\left (1620 \sqrt{2 \left (2+\sqrt{3}\right )} a^{7/3} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{91 b x}\\ &=-\frac{198}{91} a x \left (a-b x^2\right )^{2/3}-\frac{3}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3240 a^2 x}{91 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{1620 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{1080 \sqrt{2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{91 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 4.43264, size = 158, normalized size = 0.26 \[ \frac{x \sqrt [3]{1-\frac{b x^2}{a}} \left (4 b \left (3 a x+b x^3\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{4}{3},\frac{3}{2},2\right \},\left \{1,\frac{9}{2}\right \},\frac{b x^2}{a}\right )+8 b x^2 \left (18 a^2+9 a b x^2+b^2 x^4\right ) \, _2F_1\left (\frac{4}{3},\frac{3}{2};\frac{9}{2};\frac{b x^2}{a}\right )+63 a \left (45 a^2+10 a b x^2+b^2 x^4\right ) \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{7}{2};\frac{b x^2}{a}\right )\right )}{315 a \sqrt [3]{a-b x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+3\,a \right ) ^{2}{\frac{1}{\sqrt [3]{-b{x}^{2}+a}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (b^{2} x^{4} + 6 \, a b x^{2} + 9 \, a^{2}\right )}{\left (-b x^{2} + a\right )}^{\frac{2}{3}}}{b x^{2} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.54655, size = 94, normalized size = 0.16 \begin{align*} 9 a^{\frac{5}{3}} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + 2 a^{\frac{2}{3}} b x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + \frac{b^{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5 \sqrt [3]{a}} \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{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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