Optimal. Leaf size=103 \[ \frac {(-a+b x)^{2/3}}{a x}-\frac {b \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{-a+b x}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{2 a^{4/3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {44, 58, 631,
210, 31} \begin {gather*} -\frac {b \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b x-a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3}}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \log \left (\sqrt [3]{b x-a}+\sqrt [3]{a}\right )}{2 a^{4/3}}+\frac {(b x-a)^{2/3}}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 44
Rule 58
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt [3]{-a+b x}} \, dx &=\frac {(-a+b x)^{2/3}}{a x}+\frac {b \int \frac {1}{x \sqrt [3]{-a+b x}} \, dx}{3 a}\\ &=\frac {(-a+b x)^{2/3}}{a x}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+x} \, dx,x,\sqrt [3]{-a+b x}\right )}{2 a^{4/3}}+\frac {b \text {Subst}\left (\int \frac {1}{a^{2/3}-\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{-a+b x}\right )}{2 a}\\ &=\frac {(-a+b x)^{2/3}}{a x}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{2 a^{4/3}}+\frac {b \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-a+b x}}{\sqrt [3]{a}}\right )}{a^{4/3}}\\ &=\frac {(-a+b x)^{2/3}}{a x}-\frac {b \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3}}+\frac {b \log (x)}{6 a^{4/3}}-\frac {b \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )}{2 a^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 128, normalized size = 1.24 \begin {gather*} \frac {6 \sqrt [3]{a} (-a+b x)^{2/3}-2 \sqrt {3} b x \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{-a+b x}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 b x \log \left (\sqrt [3]{a}+\sqrt [3]{-a+b x}\right )+b x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{-a+b x}+(-a+b x)^{2/3}\right )}{6 a^{4/3} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 113, normalized size = 1.10
method | result | size |
risch | \(-\frac {-b x +a}{a x \left (b x -a \right )^{\frac {1}{3}}}-\frac {b \ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )}{3 a^{\frac {4}{3}}}+\frac {b \ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {4}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x -a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{\frac {4}{3}}}\) | \(110\) |
derivativedivides | \(3 b \left (\frac {\left (b x -a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x -a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )\) | \(113\) |
default | \(3 b \left (\frac {\left (b x -a \right )^{\frac {2}{3}}}{3 a b x}+\frac {-\frac {\ln \left (a^{\frac {1}{3}}+\left (b x -a \right )^{\frac {1}{3}}\right )}{3 a^{\frac {1}{3}}}+\frac {\ln \left (\left (b x -a \right )^{\frac {2}{3}}-a^{\frac {1}{3}} \left (b x -a \right )^{\frac {1}{3}}+a^{\frac {2}{3}}\right )}{6 a^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b x -a \right )^{\frac {1}{3}}}{a^{\frac {1}{3}}}-1\right )}{3}\right )}{3 a^{\frac {1}{3}}}}{3 a}\right )\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 116, normalized size = 1.13 \begin {gather*} \frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{3 \, a^{\frac {4}{3}}} + \frac {{\left (b x - a\right )}^{\frac {2}{3}} b}{{\left (b x - a\right )} a + a^{2}} + \frac {b \log \left ({\left (b x - a\right )}^{\frac {2}{3}} - {\left (b x - a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{6 \, a^{\frac {4}{3}}} - \frac {b \log \left ({\left (b x - a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}{3 \, a^{\frac {4}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.66, size = 328, normalized size = 3.18 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a b x \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x - a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} - 3 \, a}{x}\right ) + \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x - a\right )}^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x - a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right ) + 6 \, {\left (b x - a\right )}^{\frac {2}{3}} a}{6 \, a^{2} x}, \frac {6 \, \sqrt {\frac {1}{3}} a b x \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) + \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x - a\right )}^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} b x \log \left ({\left (b x - a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right ) + 6 \, {\left (b x - a\right )}^{\frac {2}{3}} a}{6 \, a^{2} x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.24, size = 838, normalized size = 8.14 \begin {gather*} - \frac {2 a^{\frac {5}{3}} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{- \frac {a}{b} + x} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{9 a^{3} b^{\frac {4}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right ) + 9 a^{2} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right )} - \frac {2 a^{\frac {5}{3}} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{- \frac {a}{b} + x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{9 a^{3} b^{\frac {4}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right ) + 9 a^{2} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right )} - \frac {2 a^{\frac {5}{3}} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{- \frac {a}{b} + x} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{9 a^{3} b^{\frac {4}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right ) + 9 a^{2} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right )} - \frac {2 a^{\frac {2}{3}} b^{\frac {10}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{- \frac {a}{b} + x} e^{\frac {i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{9 a^{3} b^{\frac {4}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right ) + 9 a^{2} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right )} - \frac {2 a^{\frac {2}{3}} b^{\frac {10}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} e^{\frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{- \frac {a}{b} + x} e^{i \pi }}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{9 a^{3} b^{\frac {4}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right ) + 9 a^{2} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right )} - \frac {2 a^{\frac {2}{3}} b^{\frac {10}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} e^{- \frac {2 i \pi }{3}} \log {\left (1 - \frac {\sqrt [3]{b} \sqrt [3]{- \frac {a}{b} + x} e^{\frac {5 i \pi }{3}}}{\sqrt [3]{a}} \right )} \Gamma \left (\frac {2}{3}\right )}{9 a^{3} b^{\frac {4}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right ) + 9 a^{2} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right )} + \frac {6 a b^{3} \left (- \frac {a}{b} + x\right )^{2} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right )}{9 a^{3} b^{\frac {4}{3}} \left (- \frac {a}{b} + x\right )^{\frac {4}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right ) + 9 a^{2} b^{\frac {7}{3}} \left (- \frac {a}{b} + x\right )^{\frac {7}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {5}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.45, size = 144, normalized size = 1.40 \begin {gather*} \frac {\frac {2 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x - a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right )}}{3 \, \left (-a\right )^{\frac {1}{3}}}\right )}{\left (-a\right )^{\frac {1}{3}} a} - \frac {b^{2} \log \left ({\left (b x - a\right )}^{\frac {2}{3}} + {\left (b x - a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right )}{\left (-a\right )^{\frac {1}{3}} a} - \frac {2 \, \left (-a\right )^{\frac {2}{3}} b^{2} \log \left ({\left | {\left (b x - a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}} \right |}\right )}{a^{2}} + \frac {6 \, {\left (b x - a\right )}^{\frac {2}{3}} b}{a x}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 133, normalized size = 1.29 \begin {gather*} \frac {{\left (b\,x-a\right )}^{2/3}}{a\,x}-\frac {b\,\ln \left ({\left (b\,x-a\right )}^{1/3}+a^{1/3}\right )}{3\,a^{4/3}}+\frac {\ln \left (\frac {{\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{4\,a^{5/3}}+\frac {b^2\,{\left (b\,x-a\right )}^{1/3}}{a^2}\right )\,\left (b-\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{4/3}}+\frac {\ln \left (\frac {{\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}^2}{4\,a^{5/3}}+\frac {b^2\,{\left (b\,x-a\right )}^{1/3}}{a^2}\right )\,\left (b+\sqrt {3}\,b\,1{}\mathrm {i}\right )}{6\,a^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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