Optimal. Leaf size=484 \[ \frac{2 \sqrt{2} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}+\frac{2 x \left (a+b x^3\right )}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{a x^2+b x^5}} \]
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Rubi [A] time = 0.189697, antiderivative size = 484, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2032, 303, 218, 1877} \[ \frac{2 \sqrt{2} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{b} x+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}+\frac{2 x \left (a+b x^3\right )}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{a x^2+b x^5}} \]
Antiderivative was successfully verified.
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Rule 2032
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{a x^2+b x^5}} \, dx &=\frac{\left (x \sqrt{a+b x^3}\right ) \int \frac{x}{\sqrt{a+b x^3}} \, dx}{\sqrt{a x^2+b x^5}}\\ &=\frac{\left (x \sqrt{a+b x^3}\right ) \int \frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt{a+b x^3}} \, dx}{\sqrt [3]{b} \sqrt{a x^2+b x^5}}+\frac{\left (\sqrt{2 \left (2-\sqrt{3}\right )} \sqrt [3]{a} x \sqrt{a+b x^3}\right ) \int \frac{1}{\sqrt{a+b x^3}} \, dx}{\sqrt [3]{b} \sqrt{a x^2+b x^5}}\\ &=\frac{2 x \left (a+b x^3\right )}{b^{2/3} \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{a x^2+b x^5}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}+\frac{2 \sqrt{2} \sqrt [3]{a} x \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt{\frac{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt{a x^2+b x^5}}\\ \end{align*}
Mathematica [C] time = 0.0110842, size = 55, normalized size = 0.11 \[ \frac{x^3 \sqrt{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 \sqrt{x^2 \left (a+b x^3\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 394, normalized size = 0.8 \begin{align*}{\frac{-{\frac{i}{6}}x\sqrt{3}}{{b}^{2}} \left ( -{b}^{2}a \right ) ^{{\frac{2}{3}}}\sqrt{{-i\sqrt{3} \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}-2\,bx-\sqrt [3]{-{b}^{2}a} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}\sqrt{-2\,{\frac{-bx+\sqrt [3]{-{b}^{2}a}}{\sqrt [3]{-{b}^{2}a} \left ( i\sqrt{3}-3 \right ) }}}\sqrt{{-i\sqrt{3} \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}+2\,bx+\sqrt [3]{-{b}^{2}a} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}} \left ( i{\it EllipticE} \left ({\frac{\sqrt{2}\sqrt{3}}{6}\sqrt{{-i\sqrt{3} \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}-2\,bx-\sqrt [3]{-{b}^{2}a} \right ){\frac{1}{\sqrt [3]{-{b}^{2}a}}}}}},\sqrt{2}\sqrt{{\frac{i\sqrt{3}}{i\sqrt{3}-3}}} \right ) \sqrt{3}-3\,{\it EllipticE} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{{\frac{-i \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}-2\,bx-\sqrt [3]{-{b}^{2}a} \right ) \sqrt{3}}{\sqrt [3]{-{b}^{2}a}}}},\sqrt{2}\sqrt{{\frac{i\sqrt{3}}{i\sqrt{3}-3}}} \right ) +2\,{\it EllipticF} \left ( 1/6\,\sqrt{3}\sqrt{2}\sqrt{{\frac{-i \left ( i\sqrt{3}\sqrt [3]{-{b}^{2}a}-2\,bx-\sqrt [3]{-{b}^{2}a} \right ) \sqrt{3}}{\sqrt [3]{-{b}^{2}a}}}},\sqrt{2}\sqrt{{\frac{i\sqrt{3}}{i\sqrt{3}-3}}} \right ) \right ){\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}} \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{x^{2}}{\sqrt{b x^{5} + a x^{2}}}\,{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{\sqrt{b x^{5} + a x^{2}}}{b x^{3} + a}, x\right ) \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{x^{2}}{\sqrt{x^{2} \left (a + b x^{3}\right )}}\, 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{x^{2}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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