Optimal. Leaf size=278 \[ \frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}} \]
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Rubi [A] time = 0.41, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2141, 218, 2140, 206} \[ \frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {a+b x^3}}\right )}{3^{3/4} \sqrt [6]{a} b^{2/3}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 218
Rule 2140
Rule 2141
Rubi steps
\begin {align*} \int \frac {x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx &=\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \left (-22 a b+\left (1-\sqrt {3}\right )^3 a b\right )+6 a b^{4/3} x}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a+b x^3}} \, dx}{6 \left (3+\sqrt {3}\right ) a b^{4/3}}+\frac {\left (2+\sqrt {3}\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{\left (3+\sqrt {3}\right ) \sqrt [3]{b}}\\ &=\frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\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 )}{\left (3+\sqrt {3}\right ) b^{2/3}}\\ &=-\frac {\sqrt {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 )}{3^{3/4} \sqrt [6]{a} b^{2/3}}+\frac {2 \sqrt {\frac {7}{6}+\frac {2}{\sqrt {3}}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [C] time = 1.25, size = 427, normalized size = 1.54 \[ -\frac {4 \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \left (i \left (\sqrt {3}-1\right ) \sqrt [3]{a} \sqrt {\frac {\left (\sqrt {3}+i\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (\sqrt {3}-3 i\right ) \sqrt [3]{a}}} \sqrt {\frac {b^{2/3} x^2}{a^{2/3}}-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}+1} \Pi \left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {\left (i+\sqrt {3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )-\frac {i \sqrt [4]{3} \left (\left (\sqrt {3}+(-2-i)\right ) \sqrt [3]{a}+\left ((1+2 i)-i \sqrt {3}\right ) \sqrt [3]{b} x\right ) \sqrt {-\frac {2 i \sqrt [3]{b} x}{\sqrt [3]{a}}+\sqrt {3}+i} F\left (\sin ^{-1}\left (\sqrt {\frac {\left (i+\sqrt {3}\right ) \sqrt [3]{b} x-2 i \sqrt [3]{a}}{\left (-3 i+\sqrt {3}\right ) \sqrt [3]{a}}}\right )|\frac {1}{2} \left (1+i \sqrt {3}\right )\right )}{2 \sqrt {2}}\right )}{\left (3-(2-i) \sqrt {3}\right ) b^{2/3} \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x^{3} + a} {\left (2 \, {\left (2 \, b x^{4} + 2 \, a x - \sqrt {3} {\left (b x^{4} - 2 \, a x\right )}\right )} a^{\frac {2}{3}} - {\left (b x^{5} - 8 \, a x^{2} - \sqrt {3} {\left (b x^{5} + 4 \, a x^{2}\right )}\right )} a^{\frac {1}{3}} b^{\frac {1}{3}} + {\left (b x^{6} + 6 \, \sqrt {3} a x^{3} + 10 \, a x^{3}\right )} b^{\frac {2}{3}}\right )}}{b^{3} x^{9} + 21 \, a b^{2} x^{6} + 12 \, a^{2} b x^{3} - 8 \, a^{3}}, x\right ) \]
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 = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (b^{\frac {1}{3}} x +\left (1-\sqrt {3}\right ) a^{\frac {1}{3}}\right ) \sqrt {b \,x^{3}+a}}\, dx \]
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 \[ \int \frac {x}{\sqrt {b x^{3} + a} {\left (b^{\frac {1}{3}} x - a^{\frac {1}{3}} {\left (\sqrt {3} - 1\right )}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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 \[ \int \frac {x}{\sqrt {a + b x^{3}} \left (- \sqrt {3} \sqrt [3]{a} + \sqrt [3]{a} + \sqrt [3]{b} x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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