Optimal. Leaf size=72 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \]
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Rubi [A] time = 0.17, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2140, 206} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2140
Rubi steps
\begin {align*} \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (\left (1+\sqrt {3}\right ) \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}{1-\left (3+2 \sqrt {3}\right ) a x^2} \, dx,x,\frac {1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {-a-b x^3}}\right )}{\sqrt [3]{b}}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {-a-b x^3}}\right )}{\sqrt {3+2 \sqrt {3}} \sqrt [6]{a} \sqrt [3]{b}}\\ \end {align*}
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Mathematica [C] time = 0.49, size = 323, normalized size = 4.49 \begin {gather*} \frac {2 \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (\frac {4 \sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} \sqrt {\frac {b^{2/3} x^2}{a^{2/3}}-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac {2 i \sqrt {3}}{3+(2+i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\left (3+(2+i) \sqrt {3}\right ) \sqrt [3]{b}}-\frac {\left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a}}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}}\right )}{\sqrt {-a-b x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 12.47, size = 118, normalized size = 1.64 \begin {gather*} \frac {2 \sqrt {\frac {1}{3} \left (2 \sqrt {3}-3\right )} \tanh ^{-1}\left (\frac {\frac {\sqrt {\frac {2}{\sqrt {3}}-1} b^{2/3} x^2}{\sqrt [6]{a}}-\sqrt {\frac {2}{\sqrt {3}}-1} \sqrt [6]{a} \sqrt [3]{b} x+\sqrt {\frac {2}{\sqrt {3}}-1} \sqrt {a}}{\sqrt {-a-b x^3}}\right )}{\sqrt [6]{a} \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 4.43, size = 1299, normalized size = 18.04
result too large to display
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b^{\frac {1}{3}} x +\left (1-\sqrt {3}\right ) a^{\frac {1}{3}}}{\left (b^{\frac {1}{3}} x +\left (1+\sqrt {3}\right ) a^{\frac {1}{3}}\right ) \sqrt {-b \,x^{3}-a}}\, dx \end {gather*}
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 {b^{\frac {1}{3}} x - a^{\frac {1}{3}} {\left (\sqrt {3} - 1\right )}}{\sqrt {-b x^{3} - a} {\left (b^{\frac {1}{3}} x + a^{\frac {1}{3}} {\left (\sqrt {3} + 1\right )}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \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 {- \sqrt {3} \sqrt [3]{a} + \sqrt [3]{a} + \sqrt [3]{b} x}{\sqrt {- a - b x^{3}} \left (\sqrt [3]{a} + \sqrt {3} \sqrt [3]{a} + \sqrt [3]{b} x\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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