Optimal. Leaf size=328 \[ \frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \left (1-\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a+b x^3}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt [6]{a} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+2 \sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a+b x^3}}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{3} \left (1+\sqrt {3}\right ) \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt {2} \sqrt {-a+b x^3}}\right )}{2 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt {-a+b x^3}}{\sqrt {2} 3^{3/4} \sqrt {a}}\right )}{3 \sqrt {2} 3^{3/4} a^{5/6} b^{2/3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {501}
\begin {gather*} \frac {\left (2-\sqrt {3}\right ) \text {ArcTan}\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 {b x^3-a}}\right )}{6 \sqrt {2} \sqrt [4]{3} a^{5/6} b^{2/3}}+\frac {\left (2-\sqrt {3}\right ) \text {ArcTan}\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 {b x^3-a}}\right )}{3 \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 {b x^3-a}}\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 {b x^3-a}}{\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 501
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} \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 ) \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 ) \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] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 10.06, size = 85, normalized size = 0.26 \begin {gather*} -\frac {x^2 \sqrt {1-\frac {b x^3}{a}} 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*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.36, size = 510, normalized size = 1.55
method | result | size |
default | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (b \,\textit {\_Z}^{3}-6 \sqrt {3}\, a -10 a \right )}{\sum }\frac {\left (a \,b^{2}\right )^{\frac {1}{3}} \sqrt {-\frac {i b \left (2 x +\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}+\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {-i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}+\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (3 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b +4 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}-3 i \left (a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-6 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -2 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b -6 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+6 i \left (a \,b^{2}\right )^{\frac {2}{3}}-2 \left (a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}+3 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b +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 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, b +i \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+4 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b -2 i \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +2 \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, a b -3 \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -2 i a b -2 \sqrt {3}\, a b +3 a b}{6 b a}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \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 )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {b \,x^{3}-a}}\right )}{27 b^{3} a}\) | \(510\) |
elliptic | \(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (b \,\textit {\_Z}^{3}-6 \sqrt {3}\, a -10 a \right )}{\sum }\frac {\left (a \,b^{2}\right )^{\frac {1}{3}} \sqrt {-\frac {i b \left (2 x +\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}+\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{2 \left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \left (x -\frac {\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{-3 \left (a \,b^{2}\right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {2}\, \sqrt {\frac {i b \left (2 x +\frac {-i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}+\left (a \,b^{2}\right )^{\frac {1}{3}}}{b}\right )}{\left (a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (3 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b +4 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}-3 i \left (a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}-6 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b -2 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b -6 b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+6 i \left (a \,b^{2}\right )^{\frac {2}{3}}-2 \left (a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}+3 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha b +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 \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, b +i \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+4 i \left (a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} b -2 i \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +2 \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, a b -3 \left (a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -2 i a b -2 \sqrt {3}\, a b +3 a b}{6 b a}, \sqrt {-\frac {i \sqrt {3}\, \left (a \,b^{2}\right )^{\frac {1}{3}}}{b \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 )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha \sqrt {b \,x^{3}-a}}\right )}{27 b^{3} a}\) | \(510\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
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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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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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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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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