Optimal. Leaf size=131 \[ \frac {a c^{7/3} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{2 b^{5/3}}+\frac {a c^{7/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}+1}{\sqrt {3}}\right )}{\sqrt {3} b^{5/3}}+\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b} \]
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Rubi [A] time = 0.26, antiderivative size = 209, normalized size of antiderivative = 1.60, number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {321, 329, 275, 331, 292, 31, 634, 617, 204, 628} \[ \frac {a c^{7/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}-\frac {a c^{7/3} \log \left (\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}\right )}{6 b^{5/3}}+\frac {a c^{7/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{2/3}}{\sqrt {3} c^{2/3}}\right )}{\sqrt {3} b^{5/3}}+\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 321
Rule 329
Rule 331
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {(c x)^{7/3}}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {\left (2 a c^2\right ) \int \frac {\sqrt [3]{c x}}{\left (a+b x^2\right )^{2/3}} \, dx}{3 b}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {(2 a c) \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 )}{b}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {(a c) \operatorname {Subst}\left (\int \frac {x}{\left (a+\frac {b x^3}{c^2}\right )^{2/3}} \, dx,x,(c x)^{2/3}\right )}{b}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {(a c) \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 )}{b}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}-\frac {\left (a c^{5/3}\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 b^{4/3}}+\frac {\left (a c^{5/3}\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 b^{4/3}}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}+\frac {a c^{7/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}+\frac {\left (a c^{5/3}\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 )}{2 b^{4/3}}-\frac {\left (a c^{7/3}\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 )}{6 b^{5/3}}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}+\frac {a c^{7/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}-\frac {a c^{7/3} \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 )}{6 b^{5/3}}-\frac {\left (a c^{7/3}\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 )}{b^{5/3}}\\ &=\frac {c (c x)^{4/3} \sqrt [3]{a+b x^2}}{2 b}+\frac {a c^{7/3} \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} b^{5/3}}+\frac {a c^{7/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{3 b^{5/3}}-\frac {a c^{7/3} \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 )}{6 b^{5/3}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 58, normalized size = 0.44 \[ \frac {c (c x)^{4/3} \left (-a \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {b x^2}{b x^2+a}\right )+a+b x^2\right )}{2 b \left (a+b x^2\right )^{2/3}} \]
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 \[ \int \frac {\left (c x\right )^{\frac {7}{3}}}{{\left (b x^{2} + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{\frac {7}{3}}}{\left (b \,x^{2}+a \right )^{\frac {2}{3}}}\, dx \]
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 \[ \int \frac {\left (c x\right )^{\frac {7}{3}}}{{\left (b x^{2} + a\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,x\right )}^{7/3}}{{\left (b\,x^2+a\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 30.01, size = 44, normalized size = 0.34 \[ \frac {c^{\frac {7}{3}} x^{\frac {10}{3}} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {2}{3}} \Gamma \left (\frac {8}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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