Optimal. Leaf size=659 \[ \frac {1264896 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt {3}-7\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {1897344 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {3794688 a^4 x}{8645 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \]
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Rubi [A] time = 0.53, antiderivative size = 659, 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 = {416, 528, 388, 235, 304, 219, 1879} \[ -\frac {3794688 a^4 x}{8645 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}+\frac {1264896 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {1897344 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8645 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \]
Antiderivative was successfully verified.
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Rule 219
Rule 235
Rule 304
Rule 388
Rule 416
Rule 528
Rule 1879
Rubi steps
\begin {align*} \int \frac {\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx &=-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac {3 \int \frac {\left (3 a+b x^2\right )^2 \left (-78 a^2 b-50 a b^2 x^2\right )}{\sqrt [3]{a-b x^2}} \, dx}{25 b}\\ &=-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3+\frac {9 \int \frac {\left (3 a+b x^2\right ) \left (1632 a^3 b^2+1344 a^2 b^3 x^2\right )}{\sqrt [3]{a-b x^2}} \, dx}{475 b^2}\\ &=-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac {27 \int \frac {-25248 a^4 b^3-19168 a^3 b^4 x^2}{\sqrt [3]{a-b x^2}} \, dx}{6175 b^3}\\ &=-\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3+\frac {\left (1264896 a^4\right ) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{8645}\\ &=-\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac {\left (1897344 a^4 \sqrt {-b x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{8645 b x}\\ &=-\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3+\frac {\left (1897344 a^4 \sqrt {-b x^2}\right ) \operatorname {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 (1897344 \sqrt {2 \left (2+\sqrt {3}\right )} a^{13/3} \sqrt {-b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{8645 b x}\\ &=-\frac {1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac {36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac {18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac {3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac {3794688 a^4 x}{8645 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {1897344 \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 {1264896 \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] time = 5.06, size = 98, normalized size = 0.15 \[ \frac {3 x \left (2108160 a^4 \sqrt [3]{1-\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {b x^2}{a}\right )-941085 a^4+727830 a^3 b x^2+184044 a^2 b^2 x^4+27482 a b^3 x^6+1729 b^4 x^8\right )}{43225 \sqrt [3]{a-b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (b^{4} x^{8} + 12 \, a b^{3} x^{6} + 54 \, a^{2} b^{2} x^{4} + 108 \, a^{3} b x^{2} + 81 \, a^{4}\right )} {\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{b x^{2} - a}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+3 a \right )^{4}}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+3\,a\right )}^4}{{\left (a-b\,x^2\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.40, size = 165, normalized size = 0.25 \[ 81 a^{\frac {11}{3}} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + 36 a^{\frac {8}{3}} b x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {54 a^{\frac {5}{3}} b^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac {12 a^{\frac {2}{3}} b^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7} + \frac {b^{4} x^{9} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {9}{2} \\ \frac {11}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{9 \sqrt [3]{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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