Optimal. Leaf size=827 \[ \frac{79 x}{768 a^4 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (b x^2+3 a\right )}+\frac{79 x}{768 a^4 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{79 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{512\ 3^{3/4} a^{11/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac{79 \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt{3}\right )}{384 \sqrt{2} \sqrt [4]{3} a^{11/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]
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Rubi [A] time = 0.63431, antiderivative size = 827, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {414, 527, 530, 235, 304, 219, 1879, 393} \[ \frac{79 x}{768 a^4 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (b x^2+3 a\right )}+\frac{79 x}{768 a^4 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{79 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{512\ 3^{3/4} a^{11/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac{79 \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{384 \sqrt{2} \sqrt [4]{3} a^{11/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]
Antiderivative was successfully verified.
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Rule 414
Rule 527
Rule 530
Rule 235
Rule 304
Rule 219
Rule 1879
Rule 393
Rubi steps
\begin{align*} \int \frac{1}{\left (a-b x^2\right )^{7/3} \left (3 a+b x^2\right )^2} \, dx &=\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )}-\frac{\int \frac{-7 a b+\frac{11 b^2 x^2}{3}}{\left (a-b x^2\right )^{7/3} \left (3 a+b x^2\right )} \, dx}{24 a^2 b}\\ &=\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )}-\frac{\int \frac{-\frac{194}{3} a^2 b^2-\frac{50}{9} a b^3 x^2}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )} \, dx}{256 a^4 b^2}\\ &=\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{79 x}{768 a^4 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )}-\frac{3 \int \frac{\frac{344 a^3 b^3}{9}+\frac{632}{27} a^2 b^4 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{2048 a^6 b^3}\\ &=\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{79 x}{768 a^4 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )}-\frac{79 \int \frac{1}{\sqrt [3]{a-b x^2}} \, dx}{2304 a^4}+\frac{3 \int \frac{1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{64 a^3}\\ &=\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{79 x}{768 a^4 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{\left (79 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1536 a^4 b x}\\ &=\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{79 x}{768 a^4 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}-\frac{\left (79 \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 )}{1536 a^4 b x}+\frac{\left (79 \sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{768 a^{11/3} b x}\\ &=\frac{5 x}{384 a^3 \left (a-b x^2\right )^{4/3}}+\frac{79 x}{768 a^4 \sqrt [3]{a-b x^2}}+\frac{x}{24 a^2 \left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )}+\frac{79 x}{768 a^4 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{128\ 2^{2/3} a^{23/6} \sqrt{b}}+\frac{79 \sqrt{2+\sqrt{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 )}{512\ 3^{3/4} a^{11/3} 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{79 \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 )}{384 \sqrt{2} \sqrt [4]{3} a^{11/3} 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*}
Mathematica [C] time = 0.241185, size = 259, normalized size = 0.31 \[ \frac{x \left (\frac{27 a \left (\frac{299 a^2-148 a b x^2-79 b^2 x^4}{a-b x^2}-\frac{387 a^2 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}\right )}{3 a+b x^2}-79 b x^2 \sqrt [3]{1-\frac{b x^2}{a}} F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )}{20736 a^5 \sqrt [3]{a-b x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( b{x}^{2}+3\,a \right ) ^{2}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{7}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + 3 \, a\right )}^{2}{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + 3 \, a\right )}^{2}{\left (-b x^{2} + a\right )}^{\frac{7}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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