Integrand size = 24, antiderivative size = 653 \[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=-\frac {3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac {81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac {9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac {36936 a^2 x}{91 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {18468 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {12312 \sqrt {2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
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Time = 0.37 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {424, 540, 542, 396, 241, 310, 225, 1893} \[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\frac {12312 \sqrt {2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {18468 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {36936 a^2 x}{91 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac {9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}-\frac {81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac {3240}{91} a x \left (a-b x^2\right )^{2/3} \]
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Rule 225
Rule 241
Rule 310
Rule 396
Rule 424
Rule 540
Rule 542
Rule 1893
Rubi steps \begin{align*} \text {integral}& = \frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac {3 \int \frac {\left (3 a+b x^2\right )^2 \left (-12 a^2 b+20 a b^2 x^2\right )}{\left (a-b x^2\right )^{4/3}} \, dx}{8 a b} \\ & = -\frac {9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac {9 \int \frac {\left (3 a+b x^2\right ) \left (-48 a^3 b^2-48 a^2 b^3 x^2\right )}{\sqrt [3]{a-b x^2}} \, dx}{16 a^2 b^2} \\ & = -\frac {81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac {9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}+\frac {27 \int \frac {768 a^4 b^3+640 a^3 b^4 x^2}{\sqrt [3]{a-b x^2}} \, dx}{208 a^2 b^3} \\ & = -\frac {3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac {81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac {9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}+\frac {1}{91} \left (12312 a^2\right ) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx \\ & = -\frac {3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac {81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac {9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac {\left (18468 a^2 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{91 b x} \\ & = -\frac {3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac {81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac {9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}+\frac {\left (18468 a^2 \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 )}{91 b x}-\frac {\left (18468 \left (1+\sqrt {3}\right ) a^{7/3} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{91 b x} \\ & = -\frac {3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac {81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac {9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac {36936 a^2 x}{91 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {18468 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{7/3} \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 )}{91 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 {12312 \sqrt {2} 3^{3/4} a^{7/3} \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 )}{91 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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 15.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.15 \[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=-\frac {3 \left (1647 a^3 x-4743 a^2 b x^3+177 a b^2 x^5+7 b^3 x^7-4104 a^2 x \left (a-b x^2\right ) \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )\right )}{91 \left (a-b x^2\right )^{4/3}} \]
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\[\int \frac {\left (b \,x^{2}+3 a \right )^{4}}{\left (-b \,x^{2}+a \right )^{\frac {7}{3}}}d x\]
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\[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \]
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\[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int \frac {\left (3 a + b x^{2}\right )^{4}}{\left (a - b x^{2}\right )^{\frac {7}{3}}}\, dx \]
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\[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \]
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\[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int \frac {{\left (b\,x^2+3\,a\right )}^4}{{\left (a-b\,x^2\right )}^{7/3}} \,d x \]
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