Integrand size = 30, antiderivative size = 234 \[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\frac {f \arctan \left (\frac {1+\frac {2 d x}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {\sqrt {3} (d e-c f) \arctan \left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d^2}+\frac {(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2}-\frac {f \log \left (-d x+\sqrt [3]{-c^3+d^3 x^3}\right )}{2 d^2}-\frac {3 (d e-c f) \log \left (d (c-d x)+2^{2/3} d \sqrt [3]{-c^3+d^3 x^3}\right )}{4 \sqrt [3]{2} c d^2} \]
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Time = 0.15 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2177, 245, 2174} \[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt {3}}\right ) (d e-c f)}{2 \sqrt [3]{2} c d^2}+\frac {f \arctan \left (\frac {\frac {2 d x}{\sqrt [3]{d^3 x^3-c^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2}-\frac {3 (d e-c f) \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d^2}-\frac {f \log \left (\sqrt [3]{d^3 x^3-c^3}-d x\right )}{2 d^2}+\frac {(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2} \]
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Rule 245
Rule 2174
Rule 2177
Rubi steps \begin{align*} \text {integral}& = \frac {f \int \frac {1}{\sqrt [3]{-c^3+d^3 x^3}} \, dx}{d}+\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx}{d} \\ & = \frac {f \tan ^{-1}\left (\frac {1+\frac {2 d x}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {\sqrt {3} (d e-c f) \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d^2}+\frac {(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2}-\frac {f \log \left (-d x+\sqrt [3]{-c^3+d^3 x^3}\right )}{2 d^2}-\frac {3 (d e-c f) \log \left (d (c-d x)+2^{2/3} d \sqrt [3]{-c^3+d^3 x^3}\right )}{4 \sqrt [3]{2} c d^2} \\ \end{align*}
\[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx \]
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\[\int \frac {f x +e}{\left (d x +c \right ) \left (d^{3} x^{3}-c^{3}\right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\text {Timed out} \]
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\[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int \frac {e + f x}{\sqrt [3]{\left (- c + d x\right ) \left (c^{2} + c d x + d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \]
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\[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int { \frac {f x + e}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int { \frac {f x + e}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int \frac {e+f\,x}{{\left (d^3\,x^3-c^3\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \]
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