Integrand size = 26, antiderivative size = 306 \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {c \left (-b+a x^3\right )^{2/3} \left (4 b x+3 a x^4\right )}{18 a^2}-\frac {2 b^2 c \arctan \left (\frac {\frac {x}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{a}}}{x}\right )}{9 \sqrt {3} a^{7/3}}+\frac {d \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{3 \sqrt [3]{b}}-\frac {2 b^2 c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{27 a^{7/3}}-\frac {d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{6 \sqrt [3]{b}}+\frac {b^2 c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{27 a^{7/3}} \]
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Time = 0.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.72, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1850, 1858, 272, 58, 631, 210, 31, 327, 245} \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {2 b^2 c \arctan \left (\frac {\frac {2 \sqrt [3]{a} x}{\sqrt [3]{a x^3-b}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} a^{7/3}}-\frac {b^2 c \log \left (\sqrt [3]{a x^3-b}-\sqrt [3]{a} x\right )}{9 a^{7/3}}+\frac {2 b c x \left (a x^3-b\right )^{2/3}}{9 a^2}+\frac {d \arctan \left (\frac {\sqrt [3]{b}-2 \sqrt [3]{a x^3-b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{b}}+\frac {c x^4 \left (a x^3-b\right )^{2/3}}{6 a}+\frac {d \log \left (\sqrt [3]{a x^3-b}+\sqrt [3]{b}\right )}{2 \sqrt [3]{b}}-\frac {d \log (x)}{2 \sqrt [3]{b}} \]
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Rule 31
Rule 58
Rule 210
Rule 245
Rule 272
Rule 327
Rule 631
Rule 1850
Rule 1858
Rubi steps \begin{align*} \text {integral}& = \frac {c x^4 \left (-b+a x^3\right )^{2/3}}{6 a}+\frac {\int \frac {-6 a d+4 b c x^4}{x \sqrt [3]{-b+a x^3}} \, dx}{6 a} \\ & = \frac {c x^4 \left (-b+a x^3\right )^{2/3}}{6 a}+\frac {\int \left (-\frac {6 a d}{x \sqrt [3]{-b+a x^3}}+\frac {4 b c x^3}{\sqrt [3]{-b+a x^3}}\right ) \, dx}{6 a} \\ & = \frac {c x^4 \left (-b+a x^3\right )^{2/3}}{6 a}+\frac {(2 b c) \int \frac {x^3}{\sqrt [3]{-b+a x^3}} \, dx}{3 a}-d \int \frac {1}{x \sqrt [3]{-b+a x^3}} \, dx \\ & = \frac {2 b c x \left (-b+a x^3\right )^{2/3}}{9 a^2}+\frac {c x^4 \left (-b+a x^3\right )^{2/3}}{6 a}+\frac {\left (2 b^2 c\right ) \int \frac {1}{\sqrt [3]{-b+a x^3}} \, dx}{9 a^2}-\frac {1}{3} d \text {Subst}\left (\int \frac {1}{x \sqrt [3]{-b+a x}} \, dx,x,x^3\right ) \\ & = \frac {2 b c x \left (-b+a x^3\right )^{2/3}}{9 a^2}+\frac {c x^4 \left (-b+a x^3\right )^{2/3}}{6 a}+\frac {2 b^2 c \arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{7/3}}-\frac {d \log (x)}{2 \sqrt [3]{b}}-\frac {b^2 c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{9 a^{7/3}}-\frac {1}{2} d \text {Subst}\left (\int \frac {1}{b^{2/3}-\sqrt [3]{b} x+x^2} \, dx,x,\sqrt [3]{-b+a x^3}\right )+\frac {d \text {Subst}\left (\int \frac {1}{\sqrt [3]{b}+x} \, dx,x,\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}} \\ & = \frac {2 b c x \left (-b+a x^3\right )^{2/3}}{9 a^2}+\frac {c x^4 \left (-b+a x^3\right )^{2/3}}{6 a}+\frac {2 b^2 c \arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{7/3}}-\frac {d \log (x)}{2 \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}-\frac {b^2 c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{9 a^{7/3}}-\frac {d \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}\right )}{\sqrt [3]{b}} \\ & = \frac {2 b c x \left (-b+a x^3\right )^{2/3}}{9 a^2}+\frac {c x^4 \left (-b+a x^3\right )^{2/3}}{6 a}+\frac {2 b^2 c \arctan \left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{-b+a x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{7/3}}+\frac {d \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {d \log (x)}{2 \sqrt [3]{b}}+\frac {d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{2 \sqrt [3]{b}}-\frac {b^2 c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{9 a^{7/3}} \\ \end{align*}
Time = 5.54 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.95 \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {1}{54} \left (\frac {3 c \left (-b+a x^3\right )^{2/3} \left (4 b x+3 a x^4\right )}{a^2}+\frac {18 \sqrt {3} d \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{b}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {4 \sqrt {3} b^2 c \arctan \left (\frac {1+\frac {2 \sqrt [3]{-b+a x^3}}{\sqrt [3]{a} x}}{\sqrt {3}}\right )}{a^{7/3}}+\frac {18 d \log \left (\sqrt [3]{b}+\sqrt [3]{-b+a x^3}\right )}{\sqrt [3]{b}}-\frac {4 b^2 c \log \left (-\sqrt [3]{a} x+\sqrt [3]{-b+a x^3}\right )}{a^{7/3}}-\frac {9 d \log \left (b^{2/3}-\sqrt [3]{b} \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{\sqrt [3]{b}}+\frac {2 b^2 c \log \left (a^{2/3} x^2+\sqrt [3]{a} x \sqrt [3]{-b+a x^3}+\left (-b+a x^3\right )^{2/3}\right )}{a^{7/3}}\right ) \]
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\[\int \frac {c \,x^{7}-d}{x \left (a \,x^{3}-b \right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\text {Timed out} \]
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Result contains complex when optimal does not.
Time = 2.70 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.27 \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\frac {c x^{7} e^{- \frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {a x^{3}}{b}} \right )}}{3 \sqrt [3]{b} \Gamma \left (\frac {10}{3}\right )} + \frac {d \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{3}}} \right )}}{3 \sqrt [3]{a} x \Gamma \left (\frac {4}{3}\right )} \]
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none
Time = 0.26 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.98 \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=-\frac {1}{54} \, {\left (\frac {4 \, \sqrt {3} b^{2} \arctan \left (\frac {\sqrt {3} {\left (a^{\frac {1}{3}} + \frac {2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {7}{3}}} - \frac {2 \, b^{2} \log \left (a^{\frac {2}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}} a^{\frac {1}{3}}}{x} + \frac {{\left (a x^{3} - b\right )}^{\frac {2}{3}}}{x^{2}}\right )}{a^{\frac {7}{3}}} + \frac {4 \, b^{2} \log \left (-a^{\frac {1}{3}} + \frac {{\left (a x^{3} - b\right )}^{\frac {1}{3}}}{x}\right )}{a^{\frac {7}{3}}} - \frac {3 \, {\left (\frac {7 \, {\left (a x^{3} - b\right )}^{\frac {2}{3}} a b^{2}}{x^{2}} - \frac {4 \, {\left (a x^{3} - b\right )}^{\frac {5}{3}} b^{2}}{x^{5}}\right )}}{a^{4} - \frac {2 \, {\left (a x^{3} - b\right )} a^{3}}{x^{3}} + \frac {{\left (a x^{3} - b\right )}^{2} a^{2}}{x^{6}}}\right )} c - \frac {1}{6} \, {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (a x^{3} - b\right )}^{\frac {1}{3}} - b^{\frac {1}{3}}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} + \frac {\log \left ({\left (a x^{3} - b\right )}^{\frac {2}{3}} - {\left (a x^{3} - b\right )}^{\frac {1}{3}} b^{\frac {1}{3}} + b^{\frac {2}{3}}\right )}{b^{\frac {1}{3}}} - \frac {2 \, \log \left ({\left (a x^{3} - b\right )}^{\frac {1}{3}} + b^{\frac {1}{3}}\right )}{b^{\frac {1}{3}}}\right )} d \]
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\[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=\int { \frac {c x^{7} - d}{{\left (a x^{3} - b\right )}^{\frac {1}{3}} x} \,d x } \]
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Timed out. \[ \int \frac {-d+c x^7}{x \sqrt [3]{-b+a x^3}} \, dx=-\int \frac {d-c\,x^7}{x\,{\left (a\,x^3-b\right )}^{1/3}} \,d x \]
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