Optimal. Leaf size=592 \[ -\frac {108 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt {3}-7\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {324 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}+\frac {45}{7} x \left (a-b x^2\right )^{2/3} \]
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Rubi [A] time = 0.37, antiderivative size = 592, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {413, 388, 235, 304, 219, 1879} \[ -\frac {108 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {324 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}+\frac {45}{7} x \left (a-b x^2\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 219
Rule 235
Rule 304
Rule 388
Rule 413
Rule 1879
Rubi steps
\begin {align*} \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx &=\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-\frac {3 \int \frac {6 a^2 b+10 a b^2 x^2}{\sqrt [3]{a-b x^2}} \, dx}{2 a b}\\ &=\frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-\frac {1}{7} (108 a) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx\\ &=\frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}+\frac {\left (162 a \sqrt {-b x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x}\\ &=\frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-\frac {\left (162 a \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 )}{7 b x}+\frac {\left (162 \sqrt {2 \left (2+\sqrt {3}\right )} a^{4/3} \sqrt {-b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x}\\ &=\frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}+\frac {324 a x}{7 \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^{4/3} \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 )}{7 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 {108 \sqrt {2} 3^{3/4} a^{4/3} \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 )}{7 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 = 5.05, size = 62, normalized size = 0.10 \[ -\frac {3 x \left (36 a \sqrt [3]{1-\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {b x^2}{a}\right )-57 a+b x^2\right )}{7 \sqrt [3]{a-b x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} x^{4} + 6 \, a b x^{2} + 9 \, a^{2}\right )} {\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.32, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+3 a \right )^{2}}{\left (-b \,x^{2}+a \right )^{\frac {4}{3}}}\, dx \]
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 \[ \int \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^2+3\,a\right )}^2}{{\left (a-b\,x^2\right )}^{4/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 a + b x^{2}\right )^{2}}{\left (a - b x^{2}\right )^{\frac {4}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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