Optimal. Leaf size=284 \[ -\frac {16 \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),4 \sqrt {3}-7\right )}{9 \sqrt [4]{3} x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {8 b \sqrt [3]{a+b x^2}}{9 x}-\frac {\left (a+b x^2\right )^{4/3}}{3 x^3} \]
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Rubi [A] time = 0.15, antiderivative size = 284, 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 = {277, 236, 219} \[ -\frac {16 \sqrt {2-\sqrt {3}} b \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{9 \sqrt [4]{3} x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {\left (a+b x^2\right )^{4/3}}{3 x^3}-\frac {8 b \sqrt [3]{a+b x^2}}{9 x} \]
Antiderivative was successfully verified.
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Rule 219
Rule 236
Rule 277
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{4/3}}{x^4} \, dx &=-\frac {\left (a+b x^2\right )^{4/3}}{3 x^3}+\frac {1}{9} (8 b) \int \frac {\sqrt [3]{a+b x^2}}{x^2} \, dx\\ &=-\frac {8 b \sqrt [3]{a+b x^2}}{9 x}-\frac {\left (a+b x^2\right )^{4/3}}{3 x^3}+\frac {1}{27} \left (16 b^2\right ) \int \frac {1}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=-\frac {8 b \sqrt [3]{a+b x^2}}{9 x}-\frac {\left (a+b x^2\right )^{4/3}}{3 x^3}+\frac {\left (8 b \sqrt {b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{9 x}\\ &=-\frac {8 b \sqrt [3]{a+b x^2}}{9 x}-\frac {\left (a+b x^2\right )^{4/3}}{3 x^3}-\frac {16 \sqrt {2-\sqrt {3}} b \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 )}{9 \sqrt [4]{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 = 0.01, size = 52, normalized size = 0.18 \[ -\frac {a \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac {3}{2},-\frac {4}{3};-\frac {1}{2};-\frac {b x^2}{a}\right )}{3 x^3 \sqrt [3]{\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {4}{3}}}{x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {4}{3}}}{x^{4}}\,{d x} \]
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 \[ \int \frac {\left (b \,x^{2}+a \right )^{\frac {4}{3}}}{x^{4}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {4}{3}}}{x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^{4/3}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.09, size = 34, normalized size = 0.12 \[ - \frac {a^{\frac {4}{3}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {4}{3} \\ - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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