Optimal. Leaf size=234 \[ -\frac {3 (d e-c f) \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d^2}+\frac {\sqrt {3} (d e-c f) \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d^2}-\frac {f \log \left (\sqrt [3]{d^3 x^3-c^3}-d x\right )}{2 d^2}+\frac {f \tan ^{-1}\left (\frac {\frac {2 d x}{\sqrt [3]{d^3 x^3-c^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2} \]
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Rubi [A] time = 0.22, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2152, 239, 2148} \[ -\frac {3 (d e-c f) \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d^2}+\frac {\sqrt {3} (d e-c f) \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d^2}-\frac {f \log \left (\sqrt [3]{d^3 x^3-c^3}-d x\right )}{2 d^2}+\frac {f \tan ^{-1}\left (\frac {\frac {2 d x}{\sqrt [3]{d^3 x^3-c^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2}+\frac {(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2} \]
Antiderivative was successfully verified.
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Rule 239
Rule 2148
Rule 2152
Rubi steps
\begin {align*} \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx &=\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*}
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Mathematica [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {f x + e}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {f x +e}{\left (d x +c \right ) \left (d^{3} x^{3}-c^{3}\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {f x + e}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {e+f\,x}{{\left (d^3\,x^3-c^3\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {e + f x}{\sqrt [3]{\left (- c + d x\right ) \left (c^{2} + c d x + d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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