Optimal. Leaf size=66 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {3} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{3 \sqrt {b}} \]
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Rubi [A] time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {446, 83, 63, 205, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {3} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{3 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 83
Rule 203
Rule 205
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {-b+a x^3}}{x \left (2 b+a x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {-b+a x}}{x (2 b+a x)} \, dx,x,x^3\right )\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {-b+a x}} \, dx,x,x^3\right )\right )+\frac {1}{2} a \operatorname {Subst}\left (\int \frac {1}{\sqrt {-b+a x} (2 b+a x)} \, dx,x,x^3\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {-b+a x^3}\right )}{3 a}+\operatorname {Subst}\left (\int \frac {1}{3 b+x^2} \, dx,x,\sqrt {-b+a x^3}\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-b+a x^3}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {3} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.94 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {3} \sqrt {b}}\right )}{3 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 66, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-b+a x^3}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {3} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 151, normalized size = 2.29 \begin {gather*} \left [-\frac {\sqrt {3} \sqrt {-b} \log \left (\frac {a x^{3} - 2 \, \sqrt {3} \sqrt {a x^{3} - b} \sqrt {-b} - 4 \, b}{a x^{3} + 2 \, b}\right ) + \sqrt {-b} \log \left (\frac {a x^{3} + 2 \, \sqrt {a x^{3} - b} \sqrt {-b} - 2 \, b}{x^{3}}\right )}{6 \, b}, \frac {\sqrt {3} \sqrt {b} \arctan \left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \, \sqrt {b}}\right ) - \sqrt {b} \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, b}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 50, normalized size = 0.76 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \, \sqrt {b}}\right )}{3 \, \sqrt {b}} - \frac {\arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.46, size = 458, normalized size = 6.94
method | result | size |
default | \(-\frac {a \left (\frac {2 \sqrt {a \,x^{3}-b}}{3 a}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\frac {\left (a^{2} b \right )^{\frac {1}{3}} \sqrt {-\frac {i a \left (2 x +\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+\left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{2 \left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{-3 \left (a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {-i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+\left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (-i \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (a^{2} b \right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{3}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \sqrt {a \,x^{3}-b}}\right )}{3 a^{3}}\right )}{2 b}+\frac {\frac {2 \sqrt {a \,x^{3}-b}}{3}+\frac {2 b \arctanh \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}}{2 b}\) | \(458\) |
elliptic | \(\frac {\sqrt {2}\, \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {3}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha \right )}{12 a b \sqrt {a \,x^{3}-b}}-\frac {\sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {3}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right )\right )}{12 \sqrt {a \,x^{3}-b}}+\frac {i \sqrt {2}\, \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}\right )}{6 b \sqrt {a \,x^{3}-b}}-\frac {i \sqrt {2}\, \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha \right )}{12 a b \sqrt {a \,x^{3}-b}}-\frac {i \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right )\right )}{12 \sqrt {a \,x^{3}-b}}+\frac {\arctanh \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\) | \(1430\) |
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 {\sqrt {a x^{3} - b}}{{\left (a x^{3} + 2 \, b\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 96, normalized size = 1.45 \begin {gather*} \frac {\ln \left (\frac {2\,b-a\,x^3+\sqrt {b}\,\sqrt {a\,x^3-b}\,2{}\mathrm {i}}{x^3}\right )\,1{}\mathrm {i}}{6\,\sqrt {b}}+\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}\,b\,4{}\mathrm {i}+6\,\sqrt {b}\,\sqrt {a\,x^3-b}-\sqrt {3}\,a\,x^3\,1{}\mathrm {i}}{2\,a\,x^3+4\,b}\right )\,1{}\mathrm {i}}{6\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.49, size = 63, normalized size = 0.95 \begin {gather*} \frac {2 \left (- \frac {a \operatorname {atan}{\left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}} \right )}}{6 \sqrt {b}} + \frac {\sqrt {3} a \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \sqrt {b}} \right )}}{6 \sqrt {b}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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