Optimal. Leaf size=335 \[ -\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}-\frac {432\ 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{21505 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \]
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Rubi [A]
time = 0.16, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {285, 327, 242,
225} \begin {gather*} -\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}-\frac {432\ 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \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 (\text {ArcSin}\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 )}{21505 b^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 {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 225
Rule 242
Rule 285
Rule 327
Rubi steps
\begin {align*} \int x^4 \left (a+b x^2\right )^{4/3} \, dx &=\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {1}{23} (8 a) \int x^4 \sqrt [3]{a+b x^2} \, dx\\ &=\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {1}{391} \left (16 a^2\right ) \int \frac {x^4}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}-\frac {\left (144 a^3\right ) \int \frac {x^2}{\left (a+b x^2\right )^{2/3}} \, dx}{4301 b}\\ &=-\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {\left (432 a^4\right ) \int \frac {1}{\left (a+b x^2\right )^{2/3}} \, dx}{21505 b^2}\\ &=-\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}+\frac {\left (648 a^4 \sqrt {b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{21505 b^3 x}\\ &=-\frac {432 a^3 x \sqrt [3]{a+b x^2}}{21505 b^2}+\frac {48 a^2 x^3 \sqrt [3]{a+b x^2}}{4301 b}+\frac {24}{391} a x^5 \sqrt [3]{a+b x^2}+\frac {3}{23} x^5 \left (a+b x^2\right )^{4/3}-\frac {432\ 3^{3/4} \sqrt {2-\sqrt {3}} a^4 \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 )}{21505 b^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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 7.71, size = 79, normalized size = 0.24 \begin {gather*} \frac {3 x \sqrt [3]{a+b x^2} \left (-\left (\left (9 a-17 b x^2\right ) \left (a+b x^2\right )^2\right )+\frac {9 a^3 \, _2F_1\left (-\frac {4}{3},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )}{\sqrt [3]{1+\frac {b x^2}{a}}}\right )}{391 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{4} \left (b \,x^{2}+a \right )^{\frac {4}{3}}\, dx\]
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]
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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Sympy [A]
time = 0.59, size = 29, normalized size = 0.09 \begin {gather*} \frac {a^{\frac {4}{3}} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} \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 = -0.00 \begin {gather*} \int x^4\,{\left (b\,x^2+a\right )}^{4/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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