Optimal. Leaf size=225 \[ -\frac {\tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a^3 x^3-b^3}}{\sqrt {3} a}+\frac {x}{\sqrt {3}}}{x}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a^3 x^3-b^3}}{\sqrt {3} b}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (\frac {-a x-b}{2 \sqrt [3]{a^3 x^3-b^3}-a x+b}\right )-\frac {1}{6} \log \left (-b \sqrt [3]{a^3 x^3-b^3}+\left (a^3 x^3-b^3\right )^{2/3}+b^2\right )+\frac {1}{6} \log \left (a x \sqrt [3]{a^3 x^3-b^3}+\left (a^3 x^3-b^3\right )^{2/3}+a^2 x^2\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 131, normalized size of antiderivative = 0.58, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1844, 239, 266, 56, 617, 204, 31} \begin {gather*} \frac {1}{2} \log \left (\sqrt [3]{a^3 x^3-b^3}+b\right )-\frac {1}{2} \log \left (\sqrt [3]{a^3 x^3-b^3}-a x\right )+\frac {\tan ^{-1}\left (\frac {\frac {2 a x}{\sqrt [3]{a^3 x^3-b^3}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {b-2 \sqrt [3]{a^3 x^3-b^3}}{\sqrt {3} b}\right )}{\sqrt {3}}-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 56
Rule 204
Rule 239
Rule 266
Rule 617
Rule 1844
Rubi steps
\begin {align*} \int \frac {-b+a x}{x \sqrt [3]{-b^3+a^3 x^3}} \, dx &=\int \left (\frac {a}{\sqrt [3]{-b^3+a^3 x^3}}-\frac {b}{x \sqrt [3]{-b^3+a^3 x^3}}\right ) \, dx\\ &=a \int \frac {1}{\sqrt [3]{-b^3+a^3 x^3}} \, dx-b \int \frac {1}{x \sqrt [3]{-b^3+a^3 x^3}} \, dx\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 a x}{\sqrt [3]{-b^3+a^3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (-a x+\sqrt [3]{-b^3+a^3 x^3}\right )-\frac {1}{3} b \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{-b^3+a^3 x}} \, dx,x,x^3\right )\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 a x}{\sqrt [3]{-b^3+a^3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}-\frac {1}{2} \log \left (-a x+\sqrt [3]{-b^3+a^3 x^3}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{b+x} \, dx,x,\sqrt [3]{-b^3+a^3 x^3}\right )-\frac {1}{2} b \operatorname {Subst}\left (\int \frac {1}{b^2-b x+x^2} \, dx,x,\sqrt [3]{-b^3+a^3 x^3}\right )\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 a x}{\sqrt [3]{-b^3+a^3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (b+\sqrt [3]{-b^3+a^3 x^3}\right )-\frac {1}{2} \log \left (-a x+\sqrt [3]{-b^3+a^3 x^3}\right )-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-b^3+a^3 x^3}}{b}\right )\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 a x}{\sqrt [3]{-b^3+a^3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-b^3+a^3 x^3}}{b}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\log (x)}{2}+\frac {1}{2} \log \left (b+\sqrt [3]{-b^3+a^3 x^3}\right )-\frac {1}{2} \log \left (-a x+\sqrt [3]{-b^3+a^3 x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.18, size = 158, normalized size = 0.70 \begin {gather*} \frac {1}{6} \left (-2 \log \left (1-\frac {a x}{\sqrt [3]{a^3 x^3-b^3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 a x}{\sqrt [3]{a^3 x^3-b^3}}+1}{\sqrt {3}}\right )-\frac {3 \left (a^3 x^3-b^3\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-\frac {a^3 x^3}{b^3}\right )}{b^2}+\log \left (\frac {a x}{\sqrt [3]{a^3 x^3-b^3}}+\frac {a^2 x^2}{\left (a^3 x^3-b^3\right )^{2/3}}+1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.56, size = 225, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\frac {x}{\sqrt {3}}+\frac {2 \sqrt [3]{-b^3+a^3 x^3}}{\sqrt {3} a}}{x}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b^3+a^3 x^3}}{\sqrt {3} b}\right )}{\sqrt {3}}-\frac {2}{3} \tanh ^{-1}\left (\frac {-b-a x}{b-a x+2 \sqrt [3]{-b^3+a^3 x^3}}\right )-\frac {1}{6} \log \left (b^2-b \sqrt [3]{-b^3+a^3 x^3}+\left (-b^3+a^3 x^3\right )^{2/3}\right )+\frac {1}{6} \log \left (a^2 x^2+a x \sqrt [3]{-b^3+a^3 x^3}+\left (-b^3+a^3 x^3\right )^{2/3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x - b}{{\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {a x -b}{x \left (a^{3} x^{3}-b^{3}\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 227, normalized size = 1.01 \begin {gather*} -\frac {1}{6} \, a {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + \frac {2 \, {\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a}\right )}{a} - \frac {\log \left (a^{2} + \frac {{\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}} a}{x} + \frac {{\left (a^{3} x^{3} - b^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{a} + \frac {2 \, \log \left (-a + \frac {{\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}}}{x}\right )}{a}\right )} - \frac {1}{6} \, b {\left (\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (b - 2 \, {\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, b}\right )}{b} + \frac {\log \left (b^{2} - {\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}} b + {\left (a^{3} x^{3} - b^{3}\right )}^{\frac {2}{3}}\right )}{b} - \frac {2 \, \log \left (b + {\left (a^{3} x^{3} - b^{3}\right )}^{\frac {1}{3}}\right )}{b}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.02, size = 166, normalized size = 0.74 \begin {gather*} \frac {\ln \left (b^3+b^2\,{\left (a^3\,x^3-b^3\right )}^{1/3}\right )}{3}+\ln \left (9\,b^3\,{\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+b^2\,{\left (a^3\,x^3-b^3\right )}^{1/3}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )-\ln \left (9\,b^3\,{\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}^2+b^2\,{\left (a^3\,x^3-b^3\right )}^{1/3}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\frac {a\,x\,{\left (1-\frac {a^3\,x^3}{b^3}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ \frac {a^3\,x^3}{b^3}\right )}{{\left (a^3\,x^3-b^3\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.55, size = 80, normalized size = 0.36 \begin {gather*} \frac {a x e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {a^{3} x^{3}}{b^{3}}} \right )}}{3 b \Gamma \left (\frac {4}{3}\right )} + \frac {b \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b^{3} e^{2 i \pi }}{a^{3} x^{3}}} \right )}}{3 a x \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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