Optimal. Leaf size=81 \[ \frac {\text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a^3-3 \text {$\#$1}^3 a^6+a^9-a^3 b^2\& ,\frac {\log \left (\sqrt [3]{a^3 x^3-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{3 b} \]
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Rubi [B] time = 0.99, antiderivative size = 946, normalized size of antiderivative = 11.68, number of steps used = 6, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2056, 6725, 91} \begin {gather*} \frac {x^{2/3} \sqrt [3]{a^3 x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt [3]{b}-a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt [3]{-1} a x+\sqrt [3]{b}\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt [3]{b}-(-1)^{2/3} a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 91
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{\left (-b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (-b+a^3 x^3\right )} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \left (-\frac {1}{3 b^{2/3} x^{2/3} \left (\sqrt [3]{b}-a x\right ) \sqrt [3]{-b+a^3 x}}-\frac {1}{3 b^{2/3} x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{-b+a^3 x}}-\frac {1}{3 b^{2/3} x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{-b+a^3 x}}\right ) \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-a x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{-b x^2+a^3 x^3}}\\ &=\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt [3]{b}-a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt [3]{b}+\sqrt [3]{-1} a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt [3]{b}-(-1)^{2/3} a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}}}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}}}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}}}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 132, normalized size = 1.63 \begin {gather*} -\frac {x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2-b^{2/3}\right ) x}{a^3 x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2+\sqrt [3]{-1} b^{2/3}\right ) x}{a^3 x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2-(-1)^{2/3} b^{2/3}\right ) x}{a^3 x-b}\right )\right )}{b \sqrt [3]{x^2 \left (a^3 x-b\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 81, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [a^9-a^3 b^2-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b} \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 {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{3} - b\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{3} x^{3}-b \right ) \left (a^{3} x^{3}-b \,x^{2}\right )^{\frac {1}{3}}}\, dx\]
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 x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{3} - 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 = -0.01 \begin {gather*} -\int \frac {1}{\left (b-a^3\,x^3\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,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\right )} \left (a^{3} x^{3} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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