Optimal. Leaf size=310 \[ \frac {1}{6} \text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a^3-3 \text {$\#$1}^3 a^6+a^9+2 a b^5\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+a^3 \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )+2 b^2 \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )+a^3 (-\log (x))-2 b^2 \log (x)}{\text {$\#$1} a^3-\text {$\#$1}^4}\& \right ]-\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-a x\right )}{2 a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+b^2 x^2}+a x}\right )}{2 a}+\frac {\log \left (a x \sqrt [3]{a^3 x^3+b^2 x^2}+\left (a^3 x^3+b^2 x^2\right )^{2/3}+a^2 x^2\right )}{4 a} \]
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Rubi [B] time = 1.49, antiderivative size = 2011, normalized size of antiderivative = 6.49, number of steps used = 13, number of rules used = 6, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1593, 2056, 6725, 105, 59, 91}
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Warning: Unable to verify antiderivative.
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Rule 59
Rule 91
Rule 105
Rule 1593
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx &=\int \frac {x \left (-b+a x^2\right )}{\left (-b+2 a x^3\right ) \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 {\sqrt [3]{x} \left (-b+a x^2\right )}{\sqrt [3]{b^2+a^3 x} \left (-b+2 a x^3\right )} \, 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 \left (-\frac {\left (-\frac {\sqrt [3]{-1} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}\right ) \sqrt [3]{x}}{3 b \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {\left (\frac {\sqrt [3]{a} b}{2^{2/3}}-b^{4/3}\right ) \sqrt [3]{x}}{3 b \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {\left (\frac {(-1)^{2/3} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}\right ) \sqrt [3]{x}}{3 b \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x}}{\left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x}}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x}}{\left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) 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}{6 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{-\frac {1}{2}} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) 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}{6 \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) 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}{6 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \sqrt [3]{b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{-\frac {1}{2}} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \sqrt [3]{b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \sqrt [3]{b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) 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 )}{2 \sqrt {3} a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) 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 )}{2 \sqrt {3} a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) 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 )}{2 \sqrt {3} a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{12 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{12 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{12 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}}}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}}}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) 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 )}{4 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) 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 )}{4 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) 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 )}{4 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 268, normalized size = 0.86 \begin {gather*} \frac {3 \sqrt [3]{a} x \sqrt [3]{\frac {a^3 x}{b^2}+1} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )-x \left (\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x a^3+\sqrt [3]{2} b^{5/3} x \sqrt [3]{a}}{x a^3+b^2}\right )+\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}-\sqrt [3]{-2} b^{5/3}\right ) x}{x a^3+b^2}\right )+\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}\right ) x}{x a^3+b^2}\right )\right )}{2 \sqrt [3]{a} \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 313, normalized size = 1.01 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{2 a}-\frac {\log \left (a^2 x-a \sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\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 )}{4 a}+\frac {1}{6} \text {RootSum}\left [a^9+2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {a^3 \log (x)+2 b^2 \log (x)-a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-2 b^2 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^3+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a^3 \text {$\#$1}+\text {$\#$1}^4}\&\right ] \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(-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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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{3}-b x}{\left (2 a \,x^{3}-b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
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 x^{3} - b x}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a 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 = -0.00 \begin {gather*} \int \frac {b\,x-a\,x^3}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-2\,a\,x^3\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 {x \left (a x^{2} - b\right )}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{3} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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