Optimal. Leaf size=637 \[ \frac {28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac {14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {306}{475} a x \left (a-b x^2\right )^{8/3}-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac {114048 a^4 x}{8645 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {57024 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {38016 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
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Rubi [A]
time = 0.44, antiderivative size = 637, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {427, 396, 201,
241, 310, 225, 1893} \begin {gather*} \frac {38016 \sqrt {2} 3^{3/4} a^{13/3} \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]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8645 b 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 {57024 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \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]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8645 b 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 {114048 a^4 x}{8645 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac {14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {306}{475} a x \left (a-b x^2\right )^{8/3}-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 225
Rule 241
Rule 310
Rule 396
Rule 427
Rule 1893
Rubi steps
\begin {align*} \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2 \, dx &=-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac {3 \int \left (a-b x^2\right )^{5/3} \left (-78 a^2 b-34 a b^2 x^2\right ) \, dx}{25 b}\\ &=-\frac {306}{475} a x \left (a-b x^2\right )^{8/3}-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )+\frac {1}{475} \left (4752 a^2\right ) \int \left (a-b x^2\right )^{5/3} \, dx\\ &=\frac {14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {306}{475} a x \left (a-b x^2\right )^{8/3}-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )+\frac {\left (9504 a^3\right ) \int \left (a-b x^2\right )^{2/3} \, dx}{1235}\\ &=\frac {28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac {14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {306}{475} a x \left (a-b x^2\right )^{8/3}-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )+\frac {\left (38016 a^4\right ) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{8645}\\ &=\frac {28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac {14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {306}{475} a x \left (a-b x^2\right )^{8/3}-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac {\left (57024 a^4 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{8645 b x}\\ &=\frac {28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac {14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {306}{475} a x \left (a-b x^2\right )^{8/3}-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )+\frac {\left (57024 a^4 \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 )}{8645 b x}-\frac {\left (57024 \sqrt {2 \left (2+\sqrt {3}\right )} a^{13/3} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{8645 b x}\\ &=\frac {28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac {14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {306}{475} a x \left (a-b x^2\right )^{8/3}-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac {114048 a^4 x}{8645 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {57024 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \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 )}{8645 b 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 {38016 \sqrt {2} 3^{3/4} a^{13/3} \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 )}{8645 b 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 = 10.63, size = 173, normalized size = 0.27 \begin {gather*} \frac {x \left (a-b x^2\right )^{2/3} \left (21 a \left (45 a^2+10 a b x^2+b^2 x^4\right ) \Gamma \left (-\frac {5}{3}\right ) \, _2F_1\left (-\frac {5}{3},\frac {1}{2};\frac {7}{2};\frac {b x^2}{a}\right )+8 b x^2 \left (18 a^2+9 a b x^2+b^2 x^4\right ) \Gamma \left (-\frac {2}{3}\right ) \, _2F_1\left (-\frac {2}{3},\frac {3}{2};\frac {9}{2};\frac {b x^2}{a}\right )+4 b \left (3 a x+b x^3\right )^2 \Gamma \left (-\frac {2}{3}\right ) \, _3F_2\left (-\frac {2}{3},\frac {3}{2},2;1,\frac {9}{2};\frac {b x^2}{a}\right )\right )}{105 \left (1-\frac {b x^2}{a}\right )^{2/3} \Gamma \left (-\frac {5}{3}\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (-b \,x^{2}+a \right )^{\frac {5}{3}} \left (b \,x^{2}+3 a \right )^{2}\, dx\]
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 = 2.59, size = 131, normalized size = 0.21 \begin {gather*} 9 a^{\frac {11}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} - a^{\frac {8}{3}} b x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} - a^{\frac {5}{3}} b^{2} 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^{2 i \pi }}{a}} \right )} - \frac {a^{\frac {2}{3}} b^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7} \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 {\left (a-b\,x^2\right )}^{5/3}\,{\left (b\,x^2+3\,a\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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