Optimal. Leaf size=190 \[ \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6-a^3 b\& ,\frac {\log \left (\sqrt [3]{a^3 x^3-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {\log \left (\sqrt [3]{a^3 x^3-b x^2}-a x\right )}{a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3-b x^2}+a x}\right )}{a}+\frac {\log \left (a x \sqrt [3]{a^3 x^3-b x^2}+\left (a^3 x^3-b x^2\right )^{2/3}+a^2 x^2\right )}{2 a} \]
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Rubi [B] time = 1.02, antiderivative size = 788, normalized size of antiderivative = 4.15, number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2056, 6725, 59, 912, 91} \begin {gather*} -\frac {x^{2/3} \log (x) \sqrt [3]{a^3 x-b}}{2 a \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}}{a \sqrt [3]{x}}-1\right )}{2 a \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} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {b}-a^{3/2} x\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (a^{3/2} x+\sqrt {b}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\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 {a} \sqrt [3]{a^{3/2}-\sqrt {b}}}-\sqrt [3]{x}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\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 {a} \sqrt [3]{a^{3/2}+\sqrt {b}}}-\sqrt [3]{x}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\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 {a} \sqrt [3]{x} \sqrt [3]{a^{3/2}-\sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {a} \sqrt [3]{a^{3/2}-\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 {a} \sqrt [3]{x} \sqrt [3]{a^{3/2}+\sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 91
Rule 912
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {b+a^3 x^2}{\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 {b+a^3 x^2}{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}{x^{2/3} \sqrt [3]{-b+a^3 x}}+\frac {2 b}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (-b+a^3 x^2\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}} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}+\frac {\left (2 b 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 {\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} a \sqrt [3]{x}}\right )}{a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log (x)}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-1+\frac {\sqrt [3]{-b+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b x^2+a^3 x^3}}+\frac {\left (2 b x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \left (-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}-a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}}-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}}\right ) \, dx}{\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} a \sqrt [3]{x}}\right )}{a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log (x)}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-1+\frac {\sqrt [3]{-b+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{\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} a \sqrt [3]{x}}\right )}{a \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 {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{x}}\right )}{\sqrt {a} \sqrt [3]{a^{3/2}-\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 {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{x}}\right )}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log (x)}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}-a^{3/2} x\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}+a^{3/2} x\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {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 {a} \sqrt [3]{a^{3/2}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {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 {a} \sqrt [3]{a^{3/2}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-1+\frac {\sqrt [3]{-b+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 135, normalized size = 0.71 \begin {gather*} \frac {3 x \left (\sqrt [3]{1-\frac {a^3 x}{b}} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {a^3 x}{b}\right )-\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a^{3/2} \left (a^{3/2}-\sqrt {b}\right ) x}{a^3 x-b}\right )-\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a^{3/2} \left (a^{3/2}+\sqrt {b}\right ) x}{a^3 x-b}\right )\right )}{\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 = 194, normalized size = 1.02 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b x^2+a^3 x^3}}\right )}{a}-\frac {\log \left (a^2 x-a \sqrt [3]{-b x^2+a^3 x^3}\right )}{a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{-b x^2+a^3 x^3}+\left (-b x^2+a^3 x^3\right )^{2/3}\right )}{2 a}+\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 ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 1787, normalized size = 9.41
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 {a^{3} x^{2} + b}{{\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 \[\int \frac {a^{3} x^{2}+b}{\left (a^{3} x^{2}-b \right ) \left (a^{3} x^{3}-b \,x^{2}\right )^{\frac {1}{3}}}\, dx\]
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 {a^{3} x^{2} + b}{{\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 {a^3\,x^2+b}{\left (b-a^3\,x^2\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 {a^{3} x^{2} + b}{\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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