Optimal. Leaf size=293 \[ -\frac {27\ 3^{3/4} \sqrt {2-\sqrt {3}} a^2 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),4 \sqrt {3}-7\right )}{55 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {27 a x \sqrt [3]{a+b x^2}}{55 b^2}+\frac {3 x^3 \sqrt [3]{a+b x^2}}{11 b} \]
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Rubi [A] time = 0.16, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {321, 236, 219} \[ -\frac {27\ 3^{3/4} \sqrt {2-\sqrt {3}} a^2 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{55 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {27 a x \sqrt [3]{a+b x^2}}{55 b^2}+\frac {3 x^3 \sqrt [3]{a+b x^2}}{11 b} \]
Antiderivative was successfully verified.
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Rule 219
Rule 236
Rule 321
Rubi steps
\begin {align*} \int \frac {x^4}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac {3 x^3 \sqrt [3]{a+b x^2}}{11 b}-\frac {(9 a) \int \frac {x^2}{\left (a+b x^2\right )^{2/3}} \, dx}{11 b}\\ &=-\frac {27 a x \sqrt [3]{a+b x^2}}{55 b^2}+\frac {3 x^3 \sqrt [3]{a+b x^2}}{11 b}+\frac {\left (27 a^2\right ) \int \frac {1}{\left (a+b x^2\right )^{2/3}} \, dx}{55 b^2}\\ &=-\frac {27 a x \sqrt [3]{a+b x^2}}{55 b^2}+\frac {3 x^3 \sqrt [3]{a+b x^2}}{11 b}+\frac {\left (81 a^2 \sqrt {b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{110 b^3 x}\\ &=-\frac {27 a x \sqrt [3]{a+b x^2}}{55 b^2}+\frac {3 x^3 \sqrt [3]{a+b x^2}}{11 b}-\frac {27\ 3^{3/4} \sqrt {2-\sqrt {3}} a^2 \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 )}{55 b^3 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*}
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Mathematica [C] time = 0.03, size = 79, normalized size = 0.27 \[ \frac {3 \left (9 a^2 x \left (\frac {b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {3}{2};-\frac {b x^2}{a}\right )-9 a^2 x-4 a b x^3+5 b^2 x^5\right )}{55 b^2 \left (a+b x^2\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{4}}{{\left (b x^{2} + a\right )}^{\frac {2}{3}}}, x\right ) \]
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 \[ \int \frac {x^{4}}{{\left (b x^{2} + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (b \,x^{2}+a \right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {x^{4}}{{\left (b x^{2} + a\right )}^{\frac {2}{3}}}\,{d x} \]
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 \[ \int \frac {x^4}{{\left (b\,x^2+a\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 27, normalized size = 0.09 \[ \frac {x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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