Optimal. Leaf size=618 \[ -\frac{3\ 3^{3/4} a^2 \left (\frac{b x^2}{a}+1\right )^{4/3} \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{\sqrt{2} b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}}-\frac{9 a x \left (\frac{b x^2}{a}+1\right )^{4/3}}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )}-\frac{3 x \left (a+b x^2\right )}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac{9 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^2 \left (\frac{b x^2}{a}+1\right )^{4/3} \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}} \]
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Rubi [A] time = 0.411956, antiderivative size = 618, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1113, 288, 235, 304, 219, 1879} \[ -\frac{9 a x \left (\frac{b x^2}{a}+1\right )^{4/3}}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )}-\frac{3 x \left (a+b x^2\right )}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac{3\ 3^{3/4} a^2 \left (\frac{b x^2}{a}+1\right )^{4/3} \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt{2} b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}}+\frac{9 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^2 \left (\frac{b x^2}{a}+1\right )^{4/3} \left (1-\sqrt [3]{\frac{b x^2}{a}+1}\right ) \sqrt{\frac{\left (\frac{b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac{b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-\sqrt [3]{\frac{b x^2}{a}+1}+\sqrt{3}+1}{-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{4 b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{\frac{b x^2}{a}+1}}{\left (-\sqrt [3]{\frac{b x^2}{a}+1}-\sqrt{3}+1\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 1113
Rule 288
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}} \, dx &=\frac{\left (1+\frac{b x^2}{a}\right )^{4/3} \int \frac{x^2}{\left (1+\frac{b x^2}{a}\right )^{4/3}} \, dx}{\left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=-\frac{3 x \left (a+b x^2\right )}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac{\left (3 a \left (1+\frac{b x^2}{a}\right )^{4/3}\right ) \int \frac{1}{\sqrt [3]{1+\frac{b x^2}{a}}} \, dx}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=-\frac{3 x \left (a+b x^2\right )}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac{\left (9 a^2 \sqrt{\frac{b x^2}{a}} \left (1+\frac{b x^2}{a}\right )^{4/3}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1+\frac{b x^2}{a}}\right )}{4 b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=-\frac{3 x \left (a+b x^2\right )}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac{\left (9 a^2 \sqrt{\frac{b x^2}{a}} \left (1+\frac{b x^2}{a}\right )^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1+\sqrt{3}-x}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1+\frac{b x^2}{a}}\right )}{4 b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}+\frac{\left (9 \sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} a^2 \sqrt{\frac{b x^2}{a}} \left (1+\frac{b x^2}{a}\right )^{4/3}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x^3}} \, dx,x,\sqrt [3]{1+\frac{b x^2}{a}}\right )}{2 b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}\\ &=-\frac{3 x \left (a+b x^2\right )}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3}}-\frac{9 a x \left (1+\frac{b x^2}{a}\right )^{4/3}}{2 b \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )}+\frac{9 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^2 \left (1+\frac{b x^2}{a}\right )^{4/3} \left (1-\sqrt [3]{1+\frac{b x^2}{a}}\right ) \sqrt{\frac{1+\sqrt [3]{1+\frac{b x^2}{a}}+\left (1+\frac{b x^2}{a}\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )^2}} E\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}}{1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}}\right )|-7+4 \sqrt{3}\right )}{4 b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{1+\frac{b x^2}{a}}}{\left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )^2}}}-\frac{3\ 3^{3/4} a^2 \left (1+\frac{b x^2}{a}\right )^{4/3} \left (1-\sqrt [3]{1+\frac{b x^2}{a}}\right ) \sqrt{\frac{1+\sqrt [3]{1+\frac{b x^2}{a}}+\left (1+\frac{b x^2}{a}\right )^{2/3}}{\left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )^2}} F\left (\sin ^{-1}\left (\frac{1+\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}}{1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}}\right )|-7+4 \sqrt{3}\right )}{\sqrt{2} b^2 x \left (a^2+2 a b x^2+b^2 x^4\right )^{2/3} \sqrt{-\frac{1-\sqrt [3]{1+\frac{b x^2}{a}}}{\left (1-\sqrt{3}-\sqrt [3]{1+\frac{b x^2}{a}}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0331036, size = 64, normalized size = 0.1 \[ \frac{3 x \left (a+b x^2\right ) \left (\sqrt [3]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-1\right )}{2 b \left (\left (a+b x^2\right )^2\right )^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.204, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{-{\frac{2}{3}}}}\, 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{x^{2}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{2}{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{x^{2}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{2}{3}}}, 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}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{2}{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{x^{2}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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