Optimal. Leaf size=300 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6-a b^3\& ,\frac {\text {$\#$1}^3 a^2 \log (x)-\text {$\#$1}^3 a^2 \log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-\text {$\#$1} x\right )-b^3 \log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-\text {$\#$1} x\right )+a^5 \log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-\text {$\#$1} x\right )+a^5 (-\log (x))+b^3 \log (x)}{\text {$\#$1}^2 a^3-\text {$\#$1}^5}\& \right ]-\log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-a x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3-b^2 x^2}+a x}\right )+\frac {1}{2} \log \left (a x \sqrt [3]{a^3 x^3-b^2 x^2}+\left (a^3 x^3-b^2 x^2\right )^{2/3}+a^2 x^2\right ) \]
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Rubi [B] time = 1.11, antiderivative size = 839, normalized size of antiderivative = 2.80, number of steps used = 7, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2056, 906, 59, 6725, 91} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{a^3 x^3-b^2 x^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{x} a}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {\sqrt {3} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \tan ^{-1}\left (\frac {2 \sqrt [6]{a} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{2 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {\sqrt {3} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \tan ^{-1}\left (\frac {2 \sqrt [6]{a} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{2 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt {b}-\sqrt {a} x\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt {a} x+\sqrt {b}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^3 x^3-b^2 x^2} \log \left (a^3 x-b^2\right )}{2 x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {3 \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}-1\right )}{2 x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {3 \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt [6]{a} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {3 \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt [6]{a} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{a^3 x-b^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 91
Rule 906
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{-b+a x^2} \, dx &=\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \int \frac {x^{2/3} \sqrt [3]{-b^2+a^3 x}}{-b+a x^2} \, dx}{x^{2/3} \sqrt [3]{-b^2+a^3 x}}\\ &=\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \int \frac {a^3 b-a b^2 x}{\sqrt [3]{x} \left (-b^2+a^3 x\right )^{2/3} \left (-b+a x^2\right )} \, dx}{a x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {\left (a^2 \sqrt [3]{-b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} \left (-b^2+a^3 x\right )^{2/3}} \, dx}{x^{2/3} \sqrt [3]{-b^2+a^3 x}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-b^2 x^2+a^3 x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-b^2+a^3 x\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {3 \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-1+\frac {a \sqrt [3]{x}}{\sqrt [3]{-b^2+a^3 x}}\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \int \left (-\frac {a^3 b^{3/2}-\sqrt {a} b^3}{2 b \sqrt [3]{x} \left (\sqrt {b}-\sqrt {a} x\right ) \left (-b^2+a^3 x\right )^{2/3}}-\frac {a^3 b^{3/2}+\sqrt {a} b^3}{2 b \sqrt [3]{x} \left (\sqrt {b}+\sqrt {a} x\right ) \left (-b^2+a^3 x\right )^{2/3}}\right ) \, dx}{a x^{2/3} \sqrt [3]{-b^2+a^3 x}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-b^2 x^2+a^3 x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-b^2+a^3 x\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {3 \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-1+\frac {a \sqrt [3]{x}}{\sqrt [3]{-b^2+a^3 x}}\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\left (\sqrt {b} \left (a^{5/2}-b^{3/2}\right ) \sqrt [3]{-b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} \left (\sqrt {b}-\sqrt {a} x\right ) \left (-b^2+a^3 x\right )^{2/3}} \, dx}{2 \sqrt {a} x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\left (\sqrt {b} \left (a^{5/2}+b^{3/2}\right ) \sqrt [3]{-b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} \left (\sqrt {b}+\sqrt {a} x\right ) \left (-b^2+a^3 x\right )^{2/3}} \, dx}{2 \sqrt {a} x^{2/3} \sqrt [3]{-b^2+a^3 x}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-b^2 x^2+a^3 x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {\sqrt {3} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{a} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{2 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {\sqrt {3} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [6]{a} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-b^2+a^3 x}}\right )}{2 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (\sqrt {b}-\sqrt {a} x\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (\sqrt {b}+\sqrt {a} x\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {\sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-b^2+a^3 x\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}-\frac {3 \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (-1+\frac {a \sqrt [3]{x}}{\sqrt [3]{-b^2+a^3 x}}\right )}{2 x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {3 \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (\sqrt [6]{a} \sqrt [3]{a^{5/2}-b^{3/2}} \sqrt [3]{x}-\sqrt [3]{-b^2+a^3 x}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}+\frac {3 \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{-b^2 x^2+a^3 x^3} \log \left (\sqrt [6]{a} \sqrt [3]{a^{5/2}+b^{3/2}} \sqrt [3]{x}-\sqrt [3]{-b^2+a^3 x}\right )}{4 a^{5/6} x^{2/3} \sqrt [3]{-b^2+a^3 x}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 179, normalized size = 0.60 \begin {gather*} \frac {3 x^2 \left (2 a^{5/2} \left (1-\frac {a^3 x}{b^2}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {a^3 x}{b^2}\right )+\left (b^{3/2}-a^{5/2}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {a} \left (a^{5/2}-b^{3/2}\right ) x}{a^3 x-b^2}\right )-\left (a^{5/2}+b^{3/2}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {a} \left (a^{5/2}+b^{3/2}\right ) x}{a^3 x-b^2}\right )\right )}{4 \sqrt {a} \left (x^2 \left (a^3 x-b^2\right )\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 299, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b^2 x^2+a^3 x^3}}\right )-\log \left (-a x+\sqrt [3]{-b^2 x^2+a^3 x^3}\right )+\frac {1}{2} \log \left (a^2 x^2+a x \sqrt [3]{-b^2 x^2+a^3 x^3}+\left (-b^2 x^2+a^3 x^3\right )^{2/3}\right )+\frac {1}{2} \text {RootSum}\left [a^6-a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {a^5 \log (x)-b^3 \log (x)-a^5 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+b^3 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-a^2 \log (x) \text {$\#$1}^3+a^2 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a^3 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 909, normalized size = 3.03
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 {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} - b}\,{d x} \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 \[\int \frac {\left (a^{3} x^{3}-b^{2} x^{2}\right )^{\frac {1}{3}}}{a \,x^{2}-b}\, dx\]
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 {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} - b}\,{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 {{\left (a^3\,x^3-b^2\,x^2\right )}^{1/3}}{b-a\,x^2} \,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 {\sqrt [3]{x^{2} \left (a^{3} x - b^{2}\right )}}{a x^{2} - b}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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