Optimal. Leaf size=419 \[ \frac {8\ 3^{3/4} b^2 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} \operatorname {EllipticF}\left (\cos ^{-1}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{55 a c^{17/3} \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}-\frac {24 b \sqrt [3]{a+b x^2}}{55 c^3 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{11 c (c x)^{11/3}} \]
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Rubi [A] time = 0.70, antiderivative size = 419, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {277, 329, 241, 225} \[ \frac {8\ 3^{3/4} b^2 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{55 a c^{17/3} \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}-\frac {24 b \sqrt [3]{a+b x^2}}{55 c^3 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{11 c (c x)^{11/3}} \]
Antiderivative was successfully verified.
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Rule 225
Rule 241
Rule 277
Rule 329
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{4/3}}{(c x)^{14/3}} \, dx &=-\frac {3 \left (a+b x^2\right )^{4/3}}{11 c (c x)^{11/3}}+\frac {(8 b) \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{8/3}} \, dx}{11 c^2}\\ &=-\frac {24 b \sqrt [3]{a+b x^2}}{55 c^3 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{11 c (c x)^{11/3}}+\frac {\left (16 b^2\right ) \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )^{2/3}} \, dx}{55 c^4}\\ &=-\frac {24 b \sqrt [3]{a+b x^2}}{55 c^3 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{11 c (c x)^{11/3}}+\frac {\left (48 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+\frac {b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{55 c^5}\\ &=-\frac {24 b \sqrt [3]{a+b x^2}}{55 c^3 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{11 c (c x)^{11/3}}+\frac {\left (48 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {b x^6}{c^2}}} \, dx,x,\frac {\sqrt [3]{c x}}{\sqrt [6]{a+b x^2}}\right )}{55 c^5 \sqrt {\frac {a}{a+b x^2}} \sqrt {a+b x^2}}\\ &=-\frac {24 b \sqrt [3]{a+b x^2}}{55 c^3 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{11 c (c x)^{11/3}}+\frac {8\ 3^{3/4} b^2 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {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}}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{55 a c^{17/3} \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 57, normalized size = 0.14 \[ -\frac {3 a x \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac {11}{6},-\frac {4}{3};-\frac {5}{6};-\frac {b x^2}{a}\right )}{11 (c x)^{14/3} \sqrt [3]{\frac {b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {4}{3}} \left (c x\right )^{\frac {1}{3}}}{c^{5} x^{5}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {4}{3}}}{\left (c x\right )^{\frac {14}{3}}}\,{d x} \]
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 \[ \int \frac {\left (b \,x^{2}+a \right )^{\frac {4}{3}}}{\left (c x \right )^{\frac {14}{3}}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {4}{3}}}{\left (c x\right )^{\frac {14}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+a\right )}^{4/3}}{{\left (c\,x\right )}^{14/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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