Optimal. Leaf size=153 \[ \frac {2 b (c x)^{4/3} \sqrt [3]{a+b x^2}}{c^3}-\frac {3 \left (a+b x^2\right )^{4/3}}{2 c (c x)^{2/3}}-\frac {2 a \sqrt [3]{b} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}}{\sqrt {3}}\right )}{\sqrt {3} c^{5/3}}-\frac {a \sqrt [3]{b} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{c^{5/3}} \]
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Rubi [A]
time = 0.10, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {283, 285, 335,
281, 337} \begin {gather*} -\frac {2 a \sqrt [3]{b} \text {ArcTan}\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 {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 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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 281
Rule 283
Rule 285
Rule 335
Rule 337
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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 0.74, size = 203, normalized size = 1.33 \begin {gather*} \frac {x \left (-9 a \sqrt [3]{a+b x^2}+3 b x^2 \sqrt [3]{a+b x^2}-4 \sqrt {3} a \sqrt [3]{b} x^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x^{2/3}}{\sqrt [3]{b} x^{2/3}+2 \sqrt [3]{a+b x^2}}\right )-4 a \sqrt [3]{b} x^{2/3} \log \left (-\sqrt [3]{b} x^{2/3}+\sqrt [3]{a+b x^2}\right )+2 a \sqrt [3]{b} x^{2/3} \log \left (b^{2/3} x^{4/3}+\sqrt [3]{b} x^{2/3} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )\right )}{6 (c x)^{5/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.03, size = 49, normalized size = 0.32 \begin {gather*} \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 )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{4/3}}{{\left (c\,x\right )}^{5/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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