Optimal. Leaf size=153 \[ -\frac {a \sqrt [3]{b} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{c^{5/3}}-\frac {2 a \sqrt [3]{b} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{5/3}}+\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}} \]
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Rubi [A] time = 0.29, antiderivative size = 233, normalized size of antiderivative = 1.52, number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {277, 279, 329, 275, 331, 292, 31, 634, 617, 204, 628} \[ \frac {a \sqrt [3]{b} \log \left (\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{3 c^{5/3}}+\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {2 a \sqrt [3]{b} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 c^{5/3}}-\frac {2 a \sqrt [3]{b} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt {3} c^{2/3}}\right )}{\sqrt {3} c^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 277
Rule 279
Rule 292
Rule 329
Rule 331
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{4/3}}{(c x)^{5/3}} \, dx &=-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}+\frac {(4 b) \int \sqrt [3]{c x} \sqrt [3]{a+b x^2} \, dx}{c^2}\\ &=\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}+\frac {(4 a b) \int \frac {\sqrt [3]{c x}}{\left (a+b x^2\right )^{2/3}} \, dx}{3 c^2}\\ &=\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}+\frac {(4 a b) \operatorname {Subst}\left (\int \frac {x^3}{\left (a+\frac {b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{c^3}\\ &=\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}+\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x}{\left (a+\frac {b x^3}{c^2}\right )^{2/3}} \, dx,x,(c x)^{2/3}\right )}{c^3}\\ &=\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}+\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x}{1-\frac {b x^3}{c^2}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{c^3}\\ &=\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}+\frac {\left (2 a b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\frac {\sqrt [3]{b} x}{c^{2/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 c^{7/3}}-\frac {\left (2 a b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt [3]{b} x}{c^{2/3}}}{1+\frac {\sqrt [3]{b} x}{c^{2/3}}+\frac {b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 c^{7/3}}\\ &=\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}-\frac {2 a \sqrt [3]{b} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 c^{5/3}}-\frac {\left (a b^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\frac {\sqrt [3]{b} x}{c^{2/3}}+\frac {b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{c^{7/3}}+\frac {\left (a \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt [3]{b}}{c^{2/3}}+\frac {2 b^{2/3} x}{c^{4/3}}}{1+\frac {\sqrt [3]{b} x}{c^{2/3}}+\frac {b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 c^{5/3}}\\ &=\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}-\frac {2 a \sqrt [3]{b} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 c^{5/3}}+\frac {a \sqrt [3]{b} \log \left (c^{4/3}+\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 c^{5/3}}+\frac {\left (2 a \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}\right )}{c^{5/3}}\\ &=\frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}-\frac {2 a \sqrt [3]{b} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}}{\sqrt {3}}\right )}{\sqrt {3} c^{5/3}}-\frac {2 a \sqrt [3]{b} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 c^{5/3}}+\frac {a \sqrt [3]{b} \log \left (c^{4/3}+\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 c^{5/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 57, normalized size = 0.37 \[ -\frac {3 a x \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac {4}{3},-\frac {1}{3};\frac {2}{3};-\frac {b x^2}{a}\right )}{2 (c x)^{5/3} \sqrt [3]{\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {4}{3}}}{\left (c x\right )^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{\frac {4}{3}}}{\left (c x \right )^{\frac {5}{3}}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {4}{3}}}{\left (c x\right )^{\frac {5}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^{4/3}}{{\left (c\,x\right )}^{5/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.23, size = 49, normalized size = 0.32 \[ \frac {a^{\frac {4}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac {5}{3}} x^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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