Optimal. Leaf size=425 \[ -\frac {\log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-a x\right )}{a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3-b^2 x^2}+a x}\right )}{a}-\frac {i \left (\sqrt {3}-i\right ) \log \left (\sqrt [3]{-1} \sqrt [3]{a^3 x^3-b^2 x^2}+\sqrt [3]{a} x \sqrt [3]{a^2+b}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b}}+\frac {\log \left (a x \sqrt [3]{a^3 x^3-b^2 x^2}+\left (a^3 x^3-b^2 x^2\right )^{2/3}+a^2 x^2\right )}{2 a}+\frac {\sqrt {6 \left (-1+i \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a} x \sqrt [3]{a^2+b}}{\sqrt [3]{a} x \sqrt [3]{a^2+b}-2 \sqrt [3]{-1} \sqrt [3]{a^3 x^3-b^2 x^2}}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b}}+\frac {\left (1+i \sqrt {3}\right ) \log \left ((-1)^{2/3} \left (a^3 x^3-b^2 x^2\right )^{2/3}+a^{2/3} x^2 \left (a^2+b\right )^{2/3}-\sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{a^2+b} \sqrt [3]{a^3 x^3-b^2 x^2}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+b}} \]
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Rubi [A] time = 0.23, antiderivative size = 477, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2056, 157, 59, 91} \begin {gather*} -\frac {x^{2/3} \log (x) \sqrt [3]{a^3 x-b^2}}{2 a \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}}{a \sqrt [3]{x}}-1\right )}{2 a \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} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{a^3 x^3-b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b^2} \log (a x+b)}{\sqrt [3]{a} \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 )}{\sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{a^3 x^3-b^2 x^2}}+\frac {2 \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} \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 59
Rule 91
Rule 157
Rule 2056
Rubi steps
\begin {align*} \int \frac {-b+a x}{(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 {-b+a x}{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 {\left (x^{2/3} \sqrt [3]{-b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b^2+a^3 x}} \, dx}{\sqrt [3]{-b^2 x^2+a^3 x^3}}-\frac {\left (2 b 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} a \sqrt [3]{x}}\right )}{a \sqrt [3]{-b^2 x^2+a^3 x^3}}+\frac {2 \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} \sqrt [3]{-b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b^2+a^3 x} \log (x)}{2 a \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)}{\sqrt [3]{a} \sqrt [3]{a^2+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 )}{\sqrt [3]{a} \sqrt [3]{a^2+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 (-1+\frac {\sqrt [3]{-b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 89, normalized size = 0.21 \begin {gather*} \frac {3 x \sqrt [3]{1-\frac {a^3 x}{b^2}} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {a^3 x}{b^2}\right )-6 x \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2+b\right ) x}{a^3 x-b^2}\right )}{\sqrt [3]{x^2 \left (a^3 x-b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.15, size = 475, normalized size = 1.12 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3-b^2 x^2}+a x}\right )}{a}-\frac {i \left (\sqrt {3}-i\right ) \log \left (2 \sqrt [3]{a} x \sqrt [3]{a^2+b}+\left (1+i \sqrt {3}\right ) \sqrt [3]{a^3 x^3-b^2 x^2}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b}}-\frac {\log \left (a^2 x-a \sqrt [3]{a^3 x^3-b^2 x^2}\right )}{a}+\frac {\log \left (a x \sqrt [3]{a^3 x^3-b^2 x^2}+\left (a^3 x^3-b^2 x^2\right )^{2/3}+a^2 x^2\right )}{2 a}+\frac {\sqrt {6 \left (-1+i \sqrt {3}\right )} \tan ^{-1}\left (\frac {3 \sqrt [3]{a} x \sqrt [3]{a^2+b}}{-\sqrt {3} \sqrt [3]{a^3 x^3-b^2 x^2}-3 i \sqrt [3]{a^3 x^3-b^2 x^2}+\sqrt {3} \sqrt [3]{a} x \sqrt [3]{a^2+b}}\right )}{\sqrt [3]{a} \sqrt [3]{a^2+b}}+\frac {\left (1+i \sqrt {3}\right ) \log \left (\left (\sqrt {3}+i\right ) \left (a^3 x^3-b^2 x^2\right )^{2/3}-2 i a^{2/3} x^2 \left (a^2+b\right )^{2/3}+\sqrt [3]{a} \left (-\sqrt {3} x+i x\right ) \sqrt [3]{a^2+b} \sqrt [3]{a^3 x^3-b^2 x^2}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 806, normalized size = 1.90 \begin {gather*} \left [\frac {2 \, \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 \, \sqrt {3} {\left (a^{2} + b\right )} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) - 2 \, {\left (a^{2} + b\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + {\left (a^{2} + b\right )} \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 4 \, {\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 ) - 2 \, {\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} + a b\right )}}, -\frac {2 \, \sqrt {3} {\left (a^{2} + b\right )} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) - 4 \, \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^{2} + b\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a^{2} + b\right )} \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 4 \, {\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 ) + 2 \, {\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} + a b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 143.52, size = 255, normalized size = 0.60 \begin {gather*} \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} + a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} + a b\right )}^{\frac {1}{3}}}\right )}{{\left (a^{3} + a b\right )}^{\frac {1}{3}}} - \frac {\log \left ({\left (a^{3} + a b\right )}^{\frac {2}{3}} + {\left (a^{3} + a b\right )}^{\frac {1}{3}} {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{{\left (a^{3} + a b\right )}^{\frac {1}{3}}} + \frac {2 \, \log \left ({\left | -{\left (a^{3} + a b\right )}^{\frac {1}{3}} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{{\left (a^{3} + a b\right )}^{\frac {1}{3}}} - \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a} + \frac {\log \left (a^{2} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, a} - \frac {\log \left ({\left | -a + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x -b}{\left (a x +b \right ) \left (a^{3} x^{3}-b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx \end {gather*}
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 {a x - b}{{\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 {b-a\,x}{{\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 {a x - b}{\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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