Optimal. Leaf size=310 \[ -\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {a+b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}} \]
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Rubi [A] time = 0.05, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {487} \begin {gather*} -\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {a+b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 487
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a+b x^3\right )} \, dx &=-\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}-\frac {\left (2+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {a+b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}+\frac {\left (2+\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt {a+b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 83, normalized size = 0.27 \begin {gather*} \frac {x^2 \sqrt {\frac {b x^3}{a}+1} F_1\left (\frac {2}{3};\frac {1}{2},1;\frac {5}{3};-\frac {b x^3}{a},-\frac {b x^3}{10 a-6 \sqrt {3} a}\right )}{\left (20 a-12 \sqrt {3} a\right ) \sqrt {a+b x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 31.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {a+b x^3} \left (2 \left (5-3 \sqrt {3}\right ) a+b x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.42, size = 538, normalized size = 1.74 \begin {gather*} \frac {i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {\left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i \left (2 x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{b}\right ) b}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (4 \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right )^{2} b^{2}+6 \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right )^{2} b^{2}+3 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right ) b -2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right ) b +6 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right ) b -3 \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right ) b -3 i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}-6 i \left (-a \,b^{2}\right )^{\frac {2}{3}}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right )^{2} b +4 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right )^{2} b +i \sqrt {3}\, a b -2 \sqrt {3}\, a b +2 i a b -3 a b -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right )-2 \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right )-2 i \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right )-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right )}{6 a b}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{\left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 b}\right ) b}}\right )}{27 a \,b^{3} \sqrt {b \,x^{3}+a}\, \RootOf \left (\textit {\_Z}^{3} b -6 \sqrt {3}\, a +10 a \right )} \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 {x}{{\left (b x^{3} - 2 \, a {\left (3 \, \sqrt {3} - 5\right )}\right )} \sqrt {b x^{3} + a}}\,{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 {x}{\sqrt {b\,x^3+a}\,\left (b\,x^3-2\,a\,\left (3\,\sqrt {3}-5\right )\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 {x}{\sqrt {a + b x^{3}} \left (- 6 \sqrt {3} a + 10 a + b x^{3}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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