Optimal. Leaf size=129 \[ \frac {c \text {RootSum}\left [\text {$\#$1}^9 (-d)+3 \text {$\#$1}^6 a d-3 \text {$\#$1}^3 a^2 d+a^3 d+b^3 c\& ,\frac {\log \left (\sqrt [3]{a x^3-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{3 d^2}+\frac {3 \left (a x^3-b x^2\right )^{2/3} \left (9 a^2 x^2+6 a b x+5 b^2\right )}{40 b^3 d x^4} \]
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Rubi [B] time = 2.51, antiderivative size = 1402, normalized size of antiderivative = 10.87, number of steps used = 18, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2056, 6725, 129, 155, 12, 91} \begin {gather*} \frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{a x^3-b x^2}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{a x^3-b x^2}}-\frac {\left (3 \sqrt [3]{d} a+4 \sqrt [3]{-1} b \sqrt [3]{c}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{a x^3-b x^2}}+\frac {\left (-9 d^{2/3} a^2+12 (-1)^{2/3} b \sqrt [3]{c} \sqrt [3]{d} a+20 \sqrt [3]{-1} b^2 c^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 d^{2/3} a^2-12 b \sqrt [3]{c} \sqrt [3]{d} a+20 b^2 c^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{a x^3-b x^2}}-\frac {\left (9 d^{2/3} a^2+12 \sqrt [3]{-1} b \sqrt [3]{c} \sqrt [3]{d} a+20 (-1)^{2/3} b^2 c^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{a x^3-b x^2}}-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} b \sqrt [3]{c}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \tan ^{-1}\left (\frac {2 \sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{\sqrt [3]{d} a+(-1)^{2/3} b \sqrt [3]{c}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{\sqrt [3]{d} a+(-1)^{2/3} b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}-\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (-\sqrt [3]{c} x-\sqrt [3]{d}\right )}{6 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}-\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [3]{-1} \sqrt [3]{c} x-\sqrt [3]{d}\right )}{6 \sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}-\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (-(-1)^{2/3} \sqrt [3]{c} x-\sqrt [3]{d}\right )}{6 \sqrt [3]{\sqrt [3]{d} a+(-1)^{2/3} b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} b \sqrt [3]{c}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a \sqrt [3]{d}-\sqrt [3]{-1} b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b x^2}}+\frac {c x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [9]{d} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+(-1)^{2/3} b \sqrt [3]{c}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{\sqrt [3]{d} a+(-1)^{2/3} b \sqrt [3]{c}} d^{17/9} \sqrt [3]{a x^3-b 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 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{-b+a x} \left (d+c x^3\right )} \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (-\frac {1}{3 d^{2/3} x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}-\frac {1}{3 d^{2/3} x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}-\frac {1}{3 d^{2/3} x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}\right ) \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{3 d^{2/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{3 d^{2/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{11/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{3 d^{2/3} \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\frac {2}{3} \left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right )-2 a \sqrt [3]{c} x}{x^{8/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{8 b d \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {-\frac {2}{3} \left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right )+2 \sqrt [3]{-1} a \sqrt [3]{c} x}{x^{8/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{8 b d \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\frac {2}{3} \left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right )-2 (-1)^{2/3} a \sqrt [3]{c} x}{x^{8/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{8 b d \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\frac {2}{9} \left (20 b^2 c^{2/3}-12 a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right )-\frac {2}{3} a \sqrt [3]{c} \left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) x}{x^{5/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{40 b^2 d^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {-\frac {2}{9} \left (20 \sqrt [3]{-1} b^2 c^{2/3}+12 (-1)^{2/3} a b \sqrt [3]{c} \sqrt [3]{d}-9 a^2 d^{2/3}\right )-\frac {2}{3} (-1)^{2/3} a \sqrt [3]{c} \left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) x}{x^{5/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{40 b^2 d^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (3 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\frac {2}{9} \left (20 (-1)^{2/3} b^2 c^{2/3}+12 \sqrt [3]{-1} a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right )-\frac {2}{3} \sqrt [3]{-1} a \sqrt [3]{c} \left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) x}{x^{5/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{40 b^2 d^{4/3} \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (20 \sqrt [3]{-1} b^2 c^{2/3}+12 (-1)^{2/3} a b \sqrt [3]{c} \sqrt [3]{d}-9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 b^2 c^{2/3}-12 a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 (-1)^{2/3} b^2 c^{2/3}+12 \sqrt [3]{-1} a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {80 b^3 c}{27 x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{80 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {80 b^3 c}{27 x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{80 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (9 x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {80 b^3 c}{27 x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{80 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (20 \sqrt [3]{-1} b^2 c^{2/3}+12 (-1)^{2/3} a b \sqrt [3]{c} \sqrt [3]{d}-9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 b^2 c^{2/3}-12 a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 (-1)^{2/3} b^2 c^{2/3}+12 \sqrt [3]{-1} a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (c x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )} \, dx}{3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (c x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )} \, dx}{3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (c x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x} \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )} \, dx}{3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (20 \sqrt [3]{-1} b^2 c^{2/3}+12 (-1)^{2/3} a b \sqrt [3]{c} \sqrt [3]{d}-9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 b^2 c^{2/3}-12 a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (20 (-1)^{2/3} b^2 c^{2/3}+12 \sqrt [3]{-1} a b \sqrt [3]{c} \sqrt [3]{d}+9 a^2 d^{2/3}\right ) (b-a x)}{40 b^3 d^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {3 (b-a x)}{8 b d x^2 \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {\left (4 (-1)^{2/3} b \sqrt [3]{c}-3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}-\frac {\left (4 \sqrt [3]{-1} b \sqrt [3]{c}+3 a \sqrt [3]{d}\right ) (b-a x)}{20 b^2 d^{4/3} x \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}-\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{d}-\sqrt [3]{c} x\right )}{6 \sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}-\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{6 \sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}-\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c} x\right )}{6 \sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}}}\right )}{2 \sqrt [3]{b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}}}\right )}{2 \sqrt [3]{-\sqrt [3]{-1} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}+\frac {c x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [9]{d} \sqrt [3]{-b+a x}}{\sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}}}\right )}{2 \sqrt [3]{(-1)^{2/3} b \sqrt [3]{c}+a \sqrt [3]{d}} d^{17/9} \sqrt [3]{-b x^2+a x^3}}\\ \end {align*}
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Mathematica [C] time = 0.66, size = 228, normalized size = 1.77 \begin {gather*} \frac {27 a^3 d x^3-9 a^2 b d x^2-40 b^3 c x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {\left (a+\frac {b \sqrt [3]{c}}{\sqrt [3]{d}}\right ) x}{b-a x}\right )-40 b^3 c x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (\left (1-i \sqrt {3}\right ) b \sqrt [3]{c}-2 a \sqrt [3]{d}\right ) x}{2 \sqrt [3]{d} (b-a x)}\right )-40 b^3 c x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (\left (1+i \sqrt {3}\right ) b \sqrt [3]{c}-2 a \sqrt [3]{d}\right ) x}{2 \sqrt [3]{d} (b-a x)}\right )-3 a b^2 d x-15 b^3 d}{40 b^3 d^2 x^2 \sqrt [3]{x^2 (a x-b)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.70, size = 129, normalized size = 1.00 \begin {gather*} \frac {3 \left (5 b^2+6 a b x+9 a^2 x^2\right ) \left (-b x^2+a x^3\right )^{2/3}}{40 b^3 d x^4}+\frac {c \text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 d^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(-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 = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{3} \left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{3}} \left (x^{3} c +d \right )}\, dx\]
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 x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (c x^{3} + d\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 (c\,x^3+d\right )\,{\left (a\,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}{x^{3} \sqrt [3]{x^{2} \left (a x - b\right )} \left (c x^{3} + d\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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