Optimal. Leaf size=354 \[ \frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-a^2 x^2}\right )}{2^{2/3} a b^{2/3}}-\frac {\log \left (b^{2/3} \left (b^2-a^2 x^2\right )^{2/3}\right )}{2\ 2^{2/3} a b^{2/3}}-\frac {\log \left (2 \sqrt [3]{b} \sqrt [3]{b^2-a^2 x^2}+2^{2/3} a x+2^{2/3} b\right )}{2^{2/3} a b^{2/3}}+\frac {\log \left (-2^{2/3} a \sqrt [3]{b} x \sqrt [3]{b^2-a^2 x^2}-2^{2/3} b^{4/3} \sqrt [3]{b^2-a^2 x^2}+2 b^{2/3} \left (b^2-a^2 x^2\right )^{2/3}+\sqrt [3]{2} a^2 x^2+2 \sqrt [3]{2} a b x+\sqrt [3]{2} b^2\right )}{2\ 2^{2/3} a b^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{b^2-a^2 x^2}}{\sqrt [3]{b} \sqrt [3]{b^2-a^2 x^2}-2^{2/3} a x-2^{2/3} b}\right )}{2^{2/3} a b^{2/3}} \]
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Rubi [A] time = 0.08, antiderivative size = 229, normalized size of antiderivative = 0.65, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1009, 1008} \begin {gather*} \frac {\sqrt [3]{1-\frac {a^2 x^2}{b^2}} \log \left (a^2 x^2+3 b^2\right )}{2\ 2^{2/3} a \sqrt [3]{b^2-a^2 x^2}}-\frac {3 \sqrt [3]{1-\frac {a^2 x^2}{b^2}} \log \left (\left (\frac {a x}{b}+1\right )^{2/3}+\sqrt [3]{2} \sqrt [3]{1-\frac {a x}{b}}\right )}{2\ 2^{2/3} a \sqrt [3]{b^2-a^2 x^2}}+\frac {\sqrt {3} \sqrt [3]{1-\frac {a^2 x^2}{b^2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \left (\frac {a x}{b}+1\right )^{2/3}}{\sqrt {3} \sqrt [3]{1-\frac {a x}{b}}}\right )}{2^{2/3} a \sqrt [3]{b^2-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1008
Rule 1009
Rubi steps
\begin {align*} \int \frac {-3 b+a x}{\sqrt [3]{b^2-a^2 x^2} \left (3 b^2+a^2 x^2\right )} \, dx &=\frac {\sqrt [3]{1-\frac {a^2 x^2}{b^2}} \int \frac {-3 b+a x}{\left (3 b^2+a^2 x^2\right ) \sqrt [3]{1-\frac {a^2 x^2}{b^2}}} \, dx}{\sqrt [3]{b^2-a^2 x^2}}\\ &=\frac {\sqrt {3} \sqrt [3]{1-\frac {a^2 x^2}{b^2}} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} \left (1+\frac {a x}{b}\right )^{2/3}}{\sqrt {3} \sqrt [3]{1-\frac {a x}{b}}}\right )}{2^{2/3} a \sqrt [3]{b^2-a^2 x^2}}+\frac {\sqrt [3]{1-\frac {a^2 x^2}{b^2}} \log \left (3 b^2+a^2 x^2\right )}{2\ 2^{2/3} a \sqrt [3]{b^2-a^2 x^2}}-\frac {3 \sqrt [3]{1-\frac {a^2 x^2}{b^2}} \log \left (\sqrt [3]{2} \sqrt [3]{1-\frac {a x}{b}}+\left (1+\frac {a x}{b}\right )^{2/3}\right )}{2\ 2^{2/3} a \sqrt [3]{b^2-a^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.47, size = 250, normalized size = 0.71 \begin {gather*} \frac {x \left (a x \sqrt [3]{1-\frac {a^2 x^2}{b^2}} F_1\left (1;\frac {1}{3},1;2;\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{3 b^2}\right )-\frac {162 b^5 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{3 b^2}\right )}{\left (a^2 x^2+3 b^2\right ) \left (9 b^2 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{3 b^2}\right )+2 a^2 x^2 \left (F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{3 b^2}\right )-F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{3 b^2}\right )\right )\right )}\right )}{6 b^2 \sqrt [3]{b^2-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.59, size = 354, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{b^2-a^2 x^2}}{-2^{2/3} b-2^{2/3} a x+\sqrt [3]{b} \sqrt [3]{b^2-a^2 x^2}}\right )}{2^{2/3} a b^{2/3}}+\frac {\log \left (\sqrt [3]{b} \sqrt [3]{b^2-a^2 x^2}\right )}{2^{2/3} a b^{2/3}}-\frac {\log \left (b^{2/3} \left (b^2-a^2 x^2\right )^{2/3}\right )}{2\ 2^{2/3} a b^{2/3}}-\frac {\log \left (2^{2/3} b+2^{2/3} a x+2 \sqrt [3]{b} \sqrt [3]{b^2-a^2 x^2}\right )}{2^{2/3} a b^{2/3}}+\frac {\log \left (\sqrt [3]{2} b^2+2 \sqrt [3]{2} a b x+\sqrt [3]{2} a^2 x^2-2^{2/3} b^{4/3} \sqrt [3]{b^2-a^2 x^2}-2^{2/3} a \sqrt [3]{b} x \sqrt [3]{b^2-a^2 x^2}+2 b^{2/3} \left (b^2-a^2 x^2\right )^{2/3}\right )}{2\ 2^{2/3} a b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {a x - 3 \, b}{{\left (a^{2} x^{2} + 3 \, b^{2}\right )} {\left (-a^{2} x^{2} + b^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {a x -3 b}{\left (-a^{2} x^{2}+b^{2}\right )^{\frac {1}{3}} \left (a^{2} x^{2}+3 b^{2}\right )}\, dx\]
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 \begin {gather*} \int \frac {a x - 3 \, b}{{\left (a^{2} x^{2} + 3 \, b^{2}\right )} {\left (-a^{2} x^{2} + b^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}
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 \begin {gather*} \int -\frac {3\,b-a\,x}{{\left (b^2-a^2\,x^2\right )}^{1/3}\,\left (a^2\,x^2+3\,b^2\right )} \,d x \end {gather*}
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 \begin {gather*} \int \frac {a x - 3 b}{\sqrt [3]{- \left (a x - b\right ) \left (a x + b\right )} \left (a^{2} x^{2} + 3 b^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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