Optimal. Leaf size=298 \[ \frac {3 x \left (a+b x^2\right )}{5 b \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}+\frac {3\ 3^{3/4} \sqrt {2-\sqrt {3}} a^2 \left (\frac {b x^2}{a}+1\right )^{2/3} \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{5 b^2 x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}} \]
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Rubi [A] time = 0.23, antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1113, 321, 236, 219} \[ \frac {3 x \left (a+b x^2\right )}{5 b \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}+\frac {3\ 3^{3/4} \sqrt {2-\sqrt {3}} a^2 \left (\frac {b x^2}{a}+1\right )^{2/3} \left (1-\sqrt [3]{\frac {b x^2}{a}+1}\right ) \sqrt {\frac {\left (\frac {b x^2}{a}+1\right )^{2/3}+\sqrt [3]{\frac {b x^2}{a}+1}+1}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {-\sqrt [3]{\frac {b x^2}{a}+1}+\sqrt {3}+1}{-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{5 b^2 x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \sqrt {-\frac {1-\sqrt [3]{\frac {b x^2}{a}+1}}{\left (-\sqrt [3]{\frac {b x^2}{a}+1}-\sqrt {3}+1\right )^2}}} \]
Antiderivative was successfully verified.
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Rule 219
Rule 236
Rule 321
Rule 1113
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt [3]{a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac {\left (1+\frac {b x^2}{a}\right )^{2/3} \int \frac {x^2}{\left (1+\frac {b x^2}{a}\right )^{2/3}} \, dx}{\sqrt [3]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {3 x \left (a+b x^2\right )}{5 b \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}-\frac {\left (3 a \left (1+\frac {b x^2}{a}\right )^{2/3}\right ) \int \frac {1}{\left (1+\frac {b x^2}{a}\right )^{2/3}} \, dx}{5 b \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {3 x \left (a+b x^2\right )}{5 b \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}-\frac {\left (9 a^2 \sqrt {\frac {b x^2}{a}} \left (1+\frac {b x^2}{a}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+\frac {b x^2}{a}}\right )}{10 b^2 x \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {3 x \left (a+b x^2\right )}{5 b \sqrt [3]{a^2+2 a b x^2+b^2 x^4}}+\frac {3\ 3^{3/4} \sqrt {2-\sqrt {3}} a^2 \left (1+\frac {b x^2}{a}\right )^{2/3} \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 )}{5 b^2 x \sqrt [3]{a^2+2 a b x^2+b^2 x^4} \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*}
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Mathematica [C] time = 0.03, size = 64, normalized size = 0.21 \[ \frac {3 x \left (-a \left (\frac {b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};-\frac {b x^2}{a}\right )+a+b x^2\right )}{5 b \sqrt [3]{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{3}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {1}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\sqrt [3]{\left (a + b x^{2}\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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