Optimal. Leaf size=52 \[ -\frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
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Rubi [A] time = 0.14, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2138, 206} \[ -\frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2138
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx &=-\frac {\left (2 \sqrt [3]{a}\right ) \operatorname {Subst}\left (\int \frac {1}{9-a x^2} \, dx,x,\frac {\left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2}{\sqrt {a-b x^3}}\right )}{\sqrt [3]{b}}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 1.02 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2}{3 \sqrt {a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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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(-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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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {-b^{\frac {1}{3}} x +a^{\frac {1}{3}}}{\left (b^{\frac {1}{3}} x +2 a^{\frac {1}{3}}\right ) \sqrt {-b \,x^{3}+a}}\, dx \]
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 \[ -\int \frac {b^{\frac {1}{3}} x - a^{\frac {1}{3}}}{\sqrt {-b x^{3} + a} {\left (b^{\frac {1}{3}} x + 2 \, a^{\frac {1}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 67, normalized size = 1.29 \[ \frac {\ln \left (\frac {\left (\sqrt {a-b\,x^3}-\sqrt {a}\right )\,{\left (\sqrt {a-b\,x^3}+\sqrt {a}+2\,a^{1/6}\,b^{1/3}\,x\right )}^3}{x^3\,{\left (b^{1/3}\,x+2\,a^{1/3}\right )}^3}\right )}{3\,a^{1/6}\,b^{1/3}} \]
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 \[ - \int \left (- \frac {\sqrt [3]{a}}{2 \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\right )\, dx - \int \frac {\sqrt [3]{b} x}{2 \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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