Optimal. Leaf size=601 \[ -\frac {108 a^2 x \left (a+b x^2\right )^{2/3}}{1729 b^2}+\frac {12 a x^3 \left (a+b x^2\right )^{2/3}}{247 b}+\frac {3}{19} x^5 \left (a+b x^2\right )^{2/3}-\frac {324 a^3 x}{1729 b^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}+\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{1729 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {108 \sqrt {2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{1729 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \]
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Rubi [A]
time = 0.33, antiderivative size = 601, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {285, 327, 241,
310, 225, 1893} \begin {gather*} -\frac {108 \sqrt {2} 3^{3/4} a^{10/3} \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 )}{1729 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 {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{10/3} \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}} E\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 )}{1729 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 {324 a^3 x}{1729 b^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {108 a^2 x \left (a+b x^2\right )^{2/3}}{1729 b^2}+\frac {3}{19} x^5 \left (a+b x^2\right )^{2/3}+\frac {12 a x^3 \left (a+b x^2\right )^{2/3}}{247 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 225
Rule 241
Rule 285
Rule 310
Rule 327
Rule 1893
Rubi steps
\begin {align*} \int x^4 \left (a+b x^2\right )^{2/3} \, dx &=\frac {3}{19} x^5 \left (a+b x^2\right )^{2/3}+\frac {1}{19} (4 a) \int \frac {x^4}{\sqrt [3]{a+b x^2}} \, dx\\ &=\frac {12 a x^3 \left (a+b x^2\right )^{2/3}}{247 b}+\frac {3}{19} x^5 \left (a+b x^2\right )^{2/3}-\frac {\left (36 a^2\right ) \int \frac {x^2}{\sqrt [3]{a+b x^2}} \, dx}{247 b}\\ &=-\frac {108 a^2 x \left (a+b x^2\right )^{2/3}}{1729 b^2}+\frac {12 a x^3 \left (a+b x^2\right )^{2/3}}{247 b}+\frac {3}{19} x^5 \left (a+b x^2\right )^{2/3}+\frac {\left (108 a^3\right ) \int \frac {1}{\sqrt [3]{a+b x^2}} \, dx}{1729 b^2}\\ &=-\frac {108 a^2 x \left (a+b x^2\right )^{2/3}}{1729 b^2}+\frac {12 a x^3 \left (a+b x^2\right )^{2/3}}{247 b}+\frac {3}{19} x^5 \left (a+b x^2\right )^{2/3}+\frac {\left (162 a^3 \sqrt {b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{1729 b^3 x}\\ &=-\frac {108 a^2 x \left (a+b x^2\right )^{2/3}}{1729 b^2}+\frac {12 a x^3 \left (a+b x^2\right )^{2/3}}{247 b}+\frac {3}{19} x^5 \left (a+b x^2\right )^{2/3}-\frac {\left (162 a^3 \sqrt {b x^2}\right ) \text {Subst}\left (\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{1729 b^3 x}+\frac {\left (162 \sqrt {2 \left (2+\sqrt {3}\right )} a^{10/3} \sqrt {b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{1729 b^3 x}\\ &=-\frac {108 a^2 x \left (a+b x^2\right )^{2/3}}{1729 b^2}+\frac {12 a x^3 \left (a+b x^2\right )^{2/3}}{247 b}+\frac {3}{19} x^5 \left (a+b x^2\right )^{2/3}-\frac {324 a^3 x}{1729 b^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}+\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{10/3} \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}} E\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 )}{1729 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 {108 \sqrt {2} 3^{3/4} a^{10/3} \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 )}{1729 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 = 6.87, size = 94, normalized size = 0.16 \begin {gather*} \frac {3 x \left (a+b x^2\right )^{2/3} \left (\left (1+\frac {b x^2}{a}\right )^{2/3} \left (-9 a^2+4 a b x^2+13 b^2 x^4\right )+9 a^2 \, _2F_1\left (-\frac {2}{3},\frac {1}{2};\frac {3}{2};-\frac {b x^2}{a}\right )\right )}{247 b^2 \left (1+\frac {b x^2}{a}\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int x^{4} \left (b \,x^{2}+a \right )^{\frac {2}{3}}\, dx\]
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 \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 = 0.48, size = 29, normalized size = 0.05 \begin {gather*} \frac {a^{\frac {2}{3}} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{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 )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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