Optimal. Leaf size=263 \[ \frac {c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{a x^3-b}+\left (a x^3-b\right )^{2/3}\right )}{6 \sqrt [3]{a}}-\frac {d \log \left (-\sqrt [3]{b} \sqrt [3]{a x^3-b}+\left (a x^3-b\right )^{2/3}+b^{2/3}\right )}{6 \sqrt [3]{b}}-\frac {c \log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a}}-\frac {c \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{a}}+\frac {x}{\sqrt {3}}}{x}\right )}{\sqrt {3} \sqrt [3]{a}}+\frac {d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{3 \sqrt [3]{b}}+\frac {d \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.15, antiderivative size = 163, normalized size of antiderivative = 0.62, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1844, 239, 266, 56, 617, 204, 31} \begin {gather*} -\frac {c \log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{2 \sqrt [3]{a}}+\frac {c \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}+\frac {d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{2 \sqrt [3]{b}}+\frac {d \tan ^{-1}\left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {d \log (x)}{2 \sqrt [3]{b}} \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 {-d+c x}{x \sqrt [3]{-b+a x^3}} \, dx &=\int \left (\frac {c}{\sqrt [3]{-b+a x^3}}-\frac {d}{x \sqrt [3]{-b+a x^3}}\right ) \, dx\\ &=c \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx-d \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx\\ &=\frac {c \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{a}}-\frac {1}{3} d \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{-b+a x}} \, dx,x,x^3\right )\\ &=\frac {c \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {d \log (x)}{2 \sqrt [3]{b}}-\frac {c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{a}}-\frac {1}{2} d \operatorname {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{-b+a x^3}\right )+\frac {d \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{b}+x} \, dx,x,\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}\\ &=\frac {c \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}-\frac {d \log (x)}{2 \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}-\frac {c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{a}}-\frac {d \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}\right )}{\sqrt [3]{b}}\\ &=\frac {c \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a}}+\frac {d \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {d \log (x)}{2 \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}-\frac {c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{a}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 159, normalized size = 0.60 \begin {gather*} \frac {1}{6} \left (\frac {c \left (\log \left (\frac {a^{2/3} x^2}{\left (a x^3-b\right )^{2/3}}+\frac {\sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1\right )-2 \log \left (1-\frac {\sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )\right )}{\sqrt [3]{a}}-\frac {3 d \left (a x^3-b\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-\frac {a x^3}{b}\right )}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.62, size = 263, normalized size = 1.00 \begin {gather*} -\frac {c \tan ^{-1}\left (\frac {\frac {x}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{a}}}{x}\right )}{\sqrt {3} \sqrt [3]{a}}+\frac {d \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{3 \sqrt [3]{b}}-\frac {c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{3 \sqrt [3]{a}}-\frac {d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}}+\frac {c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{a}} \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 {c x - d}{{\left (a x^{3} - b\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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {c x -d}{x \left (a \,x^{3}-b \right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 209, normalized size = 0.79 \begin {gather*} -\frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - \frac {\log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{a^{\frac {1}{3}}} + \frac {2 \, \log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{a^{\frac {1}{3}}}\right )} c - \frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} + \frac {\log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}{b^{\frac {1}{3}}}\right )} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.13, size = 165, normalized size = 0.63 \begin {gather*} \frac {d\,\ln \left (d^2\,{\left (a\,x^3-b\right )}^{1/3}+b^{1/3}\,d^2\right )}{3\,b^{1/3}}-\frac {\ln \left (d^2\,{\left (a\,x^3-b\right )}^{1/3}+\frac {b^{1/3}\,{\left (d-\sqrt {3}\,d\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (d-\sqrt {3}\,d\,1{}\mathrm {i}\right )}{6\,b^{1/3}}-\frac {\ln \left (d^2\,{\left (a\,x^3-b\right )}^{1/3}+\frac {b^{1/3}\,{\left (d+\sqrt {3}\,d\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (d+\sqrt {3}\,d\,1{}\mathrm {i}\right )}{6\,b^{1/3}}+\frac {c\,x\,{\left (1-\frac {a\,x^3}{b}\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {1}{3};\ \frac {4}{3};\ \frac {a\,x^3}{b}\right )}{{\left (a\,x^3-b\right )}^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.11, size = 80, normalized size = 0.30 \begin {gather*} \frac {c 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 x^{3}}{b}} \right )}}{3 \sqrt [3]{b} \Gamma \left (\frac {4}{3}\right )} + \frac {d \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 e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [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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