Optimal. Leaf size=133 \[ \frac {a \text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a^3-3 \text {$\#$1}^3 a^6+a^9+a b^5\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{3 b^2}+\frac {3 \left (a^3 x^3+b^2 x^2\right )^{2/3} \left (9 a^6 x^2-6 a^3 b^2 x+5 b^4\right )}{40 b^7 x^4} \]
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Rubi [B] time = 1.88, antiderivative size = 1430, normalized size of antiderivative = 10.75, number of steps used = 18, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2056, 6725, 129, 155, 12, 91} \begin {gather*} \frac {a^{2/3} \left (9 a^{16/3}-12 b^{5/3} a^{8/3}+20 b^{10/3}\right ) \left (x a^3+b^2\right )}{40 b^7 \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} b^{5/3} a^{8/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (x a^3+b^2\right )}{40 b^7 \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} b^{5/3} a^{8/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (x a^3+b^2\right )}{40 b^7 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (x a^3+b^2\right )}{20 b^5 x \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (x a^3+b^2\right )}{20 b^5 x \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (x a^3+b^2\right )}{20 b^5 x \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {3 \left (x a^3+b^2\right )}{8 b^3 x^2 \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x a^3+b^2}}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x a^3+b^2}}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) \sqrt [3]{x a^3+b^2}}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{x a^3+b^2}}{6 b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \log \left (\sqrt [3]{-1} \sqrt [3]{a} x+\sqrt [3]{b}\right ) \sqrt [3]{x a^3+b^2}}{6 b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {a^{8/9} x^{2/3} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{x a^3+b^2}}{6 b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+b^{5/3}}}-\sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}{2 b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}}}-\sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}{2 b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {a^{8/9} x^{2/3} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}}}-\sqrt [3]{x}\right ) \sqrt [3]{x a^3+b^2}}{2 b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 91
Rule 129
Rule 155
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (-b+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 {1}{x^{11/3} \sqrt [3]{b^2+a^3 x} \left (-b+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 {1}{3 b^{2/3} x^{11/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 b^{2/3} x^{11/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{3 b^{2/3} x^{11/3} \left (\sqrt [3]{b}-(-1)^{2/3} \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 (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{11/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{11/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{11/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{3} \sqrt [3]{a} \sqrt [3]{b} \left (3 a^{8/3}-4 b^{5/3}\right )-2 a^{10/3} x}{x^{8/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{8 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{3} \sqrt [3]{a} \sqrt [3]{b} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right )+2 \sqrt [3]{-1} a^{10/3} x}{x^{8/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{8 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{3} \sqrt [3]{a} \sqrt [3]{b} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right )-2 (-1)^{2/3} a^{10/3} x}{x^{8/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{8 b^3 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{9} a^{2/3} b^{2/3} \left (9 a^{16/3}-12 a^{8/3} b^{5/3}+20 b^{10/3}\right )-\frac {2}{3} a^{11/3} \sqrt [3]{b} \left (3 a^{8/3}-4 b^{5/3}\right ) x}{x^{5/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{40 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{9} a^{2/3} b^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right )+\frac {2}{3} \sqrt [3]{-1} a^{11/3} \sqrt [3]{b} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) x}{x^{5/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{40 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\frac {2}{9} a^{2/3} b^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right )-\frac {2}{3} (-1)^{2/3} a^{11/3} \sqrt [3]{b} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) x}{x^{5/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{40 b^{16/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {a^{2/3} \left (9 a^{16/3}-12 a^{8/3} b^{5/3}+20 b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int -\frac {80 a b^6}{27 x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{80 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int -\frac {80 a b^6}{27 x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{80 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int -\frac {80 a b^6}{27 x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{80 b^{23/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {a^{2/3} \left (9 a^{16/3}-12 a^{8/3} b^{5/3}+20 b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{5/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{5/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (a 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]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{3 b^{5/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {a^{2/3} \left (9 a^{16/3}-12 a^{8/3} b^{5/3}+20 b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}-12 (-1)^{2/3} a^{8/3} b^{5/3}-20 \sqrt [3]{-1} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{2/3} \left (9 a^{16/3}+12 \sqrt [3]{-1} a^{8/3} b^{5/3}+20 (-1)^{2/3} b^{10/3}\right ) \left (b^2+a^3 x\right )}{40 b^7 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 \left (b^2+a^3 x\right )}{8 b^3 x^2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}+4 \sqrt [3]{-1} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{a} \left (3 a^{8/3}-4 (-1)^{2/3} b^{5/3}\right ) \left (b^2+a^3 x\right )}{20 b^5 x \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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}+b^{5/3}} \sqrt [3]{x}}\right )}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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]{-1} b^{5/3}} \sqrt [3]{x}}\right )}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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} b^{5/3}} \sqrt [3]{x}}\right )}{\sqrt {3} b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{6 b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )}{6 b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {a^{8/9} x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{6 b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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}+b^{5/3}}}\right )}{2 b^2 \sqrt [3]{a^{8/3}+b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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]{-1} b^{5/3}}}\right )}{2 b^2 \sqrt [3]{a^{8/3}-\sqrt [3]{-1} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a^{8/9} 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} b^{5/3}}}\right )}{2 b^2 \sqrt [3]{a^{8/3}+(-1)^{2/3} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 232, normalized size = 1.74 \begin {gather*} \frac {27 a^9 x^3+9 a^6 b^2 x^2-3 a^3 b^4 x-40 a b^5 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x a^3+\left (-1-i \sqrt {3}\right ) b^{5/3} x \sqrt [3]{a}}{2 \left (x a^3+b^2\right )}\right )-40 a b^5 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x a^3+i \left (i+\sqrt {3}\right ) b^{5/3} x \sqrt [3]{a}}{2 \left (x a^3+b^2\right )}\right )-40 a b^5 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}+b^{5/3}\right ) x}{x a^3+b^2}\right )+15 b^6}{40 b^7 x^2 \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 = 133, normalized size = 1.00 \begin {gather*} \frac {3 \left (5 b^4-6 a^3 b^2 x+9 a^6 x^2\right ) \left (b^2 x^2+a^3 x^3\right )^{2/3}}{40 b^7 x^4}+\frac {a \text {RootSum}\left [a^9+a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b^2} \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^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{3} - b\right )} x^{3}}\,{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 \[\int \frac {1}{x^{3} \left (a \,x^{3}-b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]
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 {1}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x^{3} - b\right )} x^{3}}\,{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}{x^3\,{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-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 {1}{x^{3} \sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (a x^{3} - b\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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