Optimal. Leaf size=72 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {b x^3-a}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \sqrt [3]{b}} \]
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Rubi [A] time = 0.19, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {2140, 203} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {b x^3-a}}\right )}{\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
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 \tan ^{-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 = 0.43, size = 447, normalized size = 6.21 \begin {gather*} \frac {2 \sqrt {\frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (4 \sqrt {3} \sqrt [3]{a} \sqrt {-\frac {2 i \sqrt [3]{a}+\left (\sqrt {3}+i\right ) \sqrt [3]{b} x}{\left (\sqrt {3}-3 i\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 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {-\frac {i \left (\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )+\sqrt {\frac {\left (\sqrt {3}-i\right ) \sqrt [3]{a}+\left (\sqrt {3}+i\right ) \sqrt [3]{b} x}{\left (\sqrt {3}-3 i\right ) \sqrt [3]{a}}} \left (\left (-3+(2+i) \sqrt {3}\right ) \sqrt [3]{a}+\left ((1+2 i) \sqrt {3}-3 i\right ) \sqrt [3]{b} x\right ) F\left (\sin ^{-1}\left (\sqrt {-\frac {i \left (\left (1-i \sqrt {3}\right ) \sqrt [3]{b} x+2 \sqrt [3]{a}\right )}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )\right )}{\left ((1+2 i) \sqrt {3}-3 i\right ) \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {b x^3-a}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 12.45, size = 116, normalized size = 1.61 \begin {gather*} \frac {2 \sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\frac {\sqrt {1+\frac {2}{\sqrt {3}}} b^{2/3} x^2}{\sqrt [6]{a}}+\sqrt {1+\frac {2}{\sqrt {3}}} \sqrt [6]{a} \sqrt [3]{b} x+\sqrt {1+\frac {2}{\sqrt {3}}} \sqrt {a}}{\sqrt {b x^3-a}}\right )}{\sqrt [6]{a} \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 4.25, size = 1245, normalized size = 17.29
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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