Optimal. Leaf size=67 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6+a^3 b\& ,\frac {\log \left (\sqrt [3]{a^3 x^3-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{2 b} \]
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Rubi [B] time = 0.29, antiderivative size = 651, normalized size of antiderivative = 9.72, number of steps used = 5, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2056, 912, 91} \begin {gather*} \frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {b}-\sqrt {-a^3} x\right )}{4 b \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {-a^3} x+\sqrt {b}\right )}{4 b \sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3} \sqrt [3]{a^3 x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3} \sqrt [3]{a^3 x^3-b x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3} \sqrt [3]{a^3 x^3-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 91
Rule 912
Rule 2056
Rubi steps
\begin {align*} \int \frac {1}{\left (b+a^3 x^2\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^2\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}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x} \left (\sqrt {b}-\sqrt {-a^3} x\right )}+\frac {1}{2 \sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x} \left (\sqrt {b}+\sqrt {-a^3} x\right )}\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} \sqrt [3]{-b+a^3 x} \left (\sqrt {b}-\sqrt {-a^3} x\right )} \, dx}{2 \sqrt {b} \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} \sqrt [3]{-b+a^3 x} \left (\sqrt {b}+\sqrt {-a^3} x\right )} \, dx}{2 \sqrt {b} \sqrt [3]{-b x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} 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^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{x}}\right )}{2 \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {\sqrt {3} 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^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{x}}\right )}{2 \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}-\sqrt {-a^3} x\right )}{4 \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}+\sqrt {-a^3} x\right )}{4 \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}}}\right )}{4 \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}}}\right )}{4 \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} b \sqrt [3]{-b x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 107, normalized size = 1.60 \begin {gather*} \frac {3 x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3-\sqrt {-a^3} \sqrt {b}\right ) x}{a^3 x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3+\sqrt {-a^3} \sqrt {b}\right ) x}{a^3 x-b}\right )\right )}{2 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 = 67, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [a^6+a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 2070, normalized size = 30.90
result too large to display
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^{2} + 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 \[\int \frac {1}{\left (a^{3} x^{2}+b \right ) \left (a^{3} x^{3}-b \,x^{2}\right )^{\frac {1}{3}}}\, dx\]
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^{2} + 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 (a^3\,x^2+b\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^{2} + b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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