Optimal. Leaf size=319 \[ -\frac {i \left (\sqrt {3}-i\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{a^3 x^3-b^2 x^2}+\sqrt [3]{a} x \sqrt [3]{a^2-b}\right )}{2 \sqrt [3]{a} b \sqrt [3]{a^2-b}}+\frac {\sqrt {-3+3 i \sqrt {3}} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x \sqrt [3]{a^2-b}}{\sqrt [3]{a} x \sqrt [3]{a^2-b}-2 \sqrt [3]{-1} \sqrt [3]{a^3 x^3-b^2 x^2}}\right )}{\sqrt {2} \sqrt [3]{a} b \sqrt [3]{a^2-b}}+\frac {\left (1+i \sqrt {3}\right ) \log \left ((-1)^{2/3} \left (a^3 x^3-b^2 x^2\right )^{2/3}+a^{2/3} x^2 \left (a^2-b\right )^{2/3}-\sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3-b^2 x^2}\right )}{4 \sqrt [3]{a} b \sqrt [3]{a^2-b}} \]
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Rubi [A] time = 0.14, antiderivative size = 291, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2056, 91} \begin {gather*} -\frac {x^{2/3} \sqrt [3]{a^3 x-b^2} \log (a x-b)}{2 \sqrt [3]{a} b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3-b^2 x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b^2} \log \left (\frac {\sqrt [3]{a^3 x-b^2}}{\sqrt [3]{a} \sqrt [3]{a^2-b}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3-b^2 x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x-b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2-b}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a} b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3-b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 91
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} (-b+a x) \sqrt [3]{-b^2+a^3 x}} \, dx}{\sqrt [3]{-b^2 x^2+a^3 x^3}}\\ &=\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}}\right )}{\sqrt [3]{a} \sqrt [3]{a^2-b} b \sqrt [3]{-b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b^2+a^3 x} \log (-b+a x)}{2 \sqrt [3]{a} \sqrt [3]{a^2-b} b \sqrt [3]{-b^2 x^2+a^3 x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-b}}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-b} b \sqrt [3]{-b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 56, normalized size = 0.18 \begin {gather*} -\frac {3 x \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3-a b\right ) x}{a^3 x-b^2}\right )}{b \sqrt [3]{x^2 \left (a^3 x-b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.97, size = 365, normalized size = 1.14 \begin {gather*} \frac {\sqrt {-3+3 i \sqrt {3}} \tan ^{-1}\left (\frac {3 \sqrt [3]{a} \sqrt [3]{a^2-b} x}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b} x-3 i \sqrt [3]{-b^2 x^2+a^3 x^3}-\sqrt {3} \sqrt [3]{-b^2 x^2+a^3 x^3}}\right )}{\sqrt {2} \sqrt [3]{a} \sqrt [3]{a^2-b} b}-\frac {i \left (-i+\sqrt {3}\right ) \log \left (2 \sqrt [3]{a} \sqrt [3]{a^2-b} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{-b^2 x^2+a^3 x^3}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-b} b}+\frac {\left (1+i \sqrt {3}\right ) \log \left (-2 i a^{2/3} \left (a^2-b\right )^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{a^2-b} \left (i x-\sqrt {3} x\right ) \sqrt [3]{-b^2 x^2+a^3 x^3}+\left (i+\sqrt {3}\right ) \left (-b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 \sqrt [3]{a} \sqrt [3]{a^2-b} b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 549, normalized size = 1.72 \begin {gather*} \left [\frac {\sqrt {3} {\left (a^{3} - a b\right )} \sqrt {-\frac {1}{{\left (a^{3} - a b\right )}^{\frac {2}{3}}}} \log \left (-\frac {2 \, b^{2} x - {\left (3 \, a^{3} - a b\right )} x^{2} + 3 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )}^{\frac {2}{3}} x + \sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {4}{3}} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )} x - 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}} {\left (a^{3} - a b\right )}^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{{\left (a^{3} - a b\right )}^{\frac {2}{3}}}}}{a x^{2} - b x}\right ) + 2 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (\frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )}^{\frac {1}{3}} x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, {\left (a^{3} b - a b^{2}\right )}}, \frac {2 \, \sqrt {3} {\left (a^{3} - a b\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {1}{3}} x + 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} - a b\right )}^{\frac {1}{3}} x}\right ) + 2 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (\frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )}^{\frac {1}{3}} x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, {\left (a^{3} b - a b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a x -b \right ) \left (a^{3} x^{3}-b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
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 {1}{{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x - b\right )}}\,{d x} \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 {1}{{\left (a^3\,x^3-b^2\,x^2\right )}^{1/3}\,\left (b-a\,x\right )} \,d x \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 {1}{\sqrt [3]{x^{2} \left (a^{3} x - b^{2}\right )} \left (a x - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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