Optimal. Leaf size=807 \[ \frac {x}{12 a^3 \sqrt [3]{a-b x^2}}+\frac {x}{24 a^2 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}+\frac {x}{12 a^3 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{48\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{16\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8\ 3^{3/4} a^{8/3} b \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{8/3} b \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]
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Rubi [A] time = 0.53, antiderivative size = 807, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {x}{12 a^3 \sqrt [3]{a-b x^2}}+\frac {x}{24 a^2 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}+\frac {x}{12 a^3 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{48\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{16\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{8\ 3^{3/4} a^{8/3} b \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{8/3} b \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]
Antiderivative was successfully verified.
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Rule 219
Rule 235
Rule 304
Rule 393
Rule 414
Rule 527
Rule 530
Rule 1879
Rubi steps
\begin {align*} \int \frac {1}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^2} \, dx &=\frac {x}{24 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {\int \frac {-7 a b+\frac {5 b^2 x^2}{3}}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )} \, dx}{24 a^2 b}\\ &=\frac {x}{12 a^3 \sqrt [3]{a-b x^2}}+\frac {x}{24 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {\int \frac {-\frac {8}{3} a^2 b^2+\frac {16}{9} a b^3 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{64 a^4 b^2}\\ &=\frac {x}{12 a^3 \sqrt [3]{a-b x^2}}+\frac {x}{24 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {\int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{36 a^3}+\frac {\int \frac {1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{8 a^2}\\ &=\frac {x}{12 a^3 \sqrt [3]{a-b x^2}}+\frac {x}{24 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {\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 )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{48\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\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 )}{16\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\sqrt {-b x^2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{24 a^3 b x}\\ &=\frac {x}{12 a^3 \sqrt [3]{a-b x^2}}+\frac {x}{24 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {\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 )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{48\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\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 )}{16\ 2^{2/3} a^{17/6} \sqrt {b}}-\frac {\sqrt {-b x^2} \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 )}{24 a^3 b x}+\frac {\left (\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 )}{12 a^{8/3} b x}\\ &=\frac {x}{12 a^3 \sqrt [3]{a-b x^2}}+\frac {x}{24 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}+\frac {x}{12 a^3 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {\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 )}{16\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{48\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\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 )}{16\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\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 )}{8\ 3^{3/4} a^{8/3} 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 {\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 )}{6 \sqrt {2} \sqrt [4]{3} a^{8/3} 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*}
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Mathematica [C] time = 0.18, size = 236, normalized size = 0.29 \[ \frac {x \left (\frac {27 a \left (\frac {9 a^2 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{2 b x^2 \left (F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )-F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )+9 a F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}+7 a+2 b x^2\right )}{3 a+b x^2}-2 b x^2 \sqrt [3]{1-\frac {b x^2}{a}} F_1\left (\frac {3}{2};\frac {1}{3},1;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )}{648 a^4 \sqrt [3]{a-b x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {1}{{\left (b x^{2} + 3 \, a\right )}^{2} {\left (-b x^{2} + a\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {4}{3}} \left (b \,x^{2}+3 a \right )^{2}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{2} + 3 \, a\right )}^{2} {\left (-b x^{2} + a\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a-b\,x^2\right )}^{4/3}\,{\left (b\,x^2+3\,a\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a - b x^{2}\right )^{\frac {4}{3}} \left (3 a + b x^{2}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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