Optimal. Leaf size=196 \[ \frac{\sqrt [3]{b^3 e-c^3 e x^3} \log \left (\sqrt [3]{b^3 e-c^3 e x^3}+c \sqrt [3]{e} x\right )}{2 c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}-\frac{\sqrt [3]{b^3 e-c^3 e x^3} \tan ^{-1}\left (\frac{1-\frac{2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt{3}}\right )}{\sqrt{3} c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}} \]
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Rubi [A] time = 0.0585334, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {713, 239} \[ \frac{\sqrt [3]{b^3 e-c^3 e x^3} \log \left (\sqrt [3]{b^3 e-c^3 e x^3}+c \sqrt [3]{e} x\right )}{2 c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}-\frac{\sqrt [3]{b^3 e-c^3 e x^3} \tan ^{-1}\left (\frac{1-\frac{2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt{3}}\right )}{\sqrt{3} c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}} \]
Antiderivative was successfully verified.
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Rule 713
Rule 239
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx &=\frac{\sqrt [3]{b^3 e-c^3 e x^3} \int \frac{1}{\sqrt [3]{b^3 e-c^3 e x^3}} \, dx}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}}\\ &=-\frac{\sqrt [3]{b^3 e-c^3 e x^3} \tan ^{-1}\left (\frac{1-\frac{2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt{3}}\right )}{\sqrt{3} c \sqrt [3]{e} \sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}}+\frac{\sqrt [3]{b^3 e-c^3 e x^3} \log \left (c \sqrt [3]{e} x+\sqrt [3]{b^3 e-c^3 e x^3}\right )}{2 c \sqrt [3]{e} \sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}}\\ \end{align*}
Mathematica [C] time = 0.190699, size = 241, normalized size = 1.23 \[ -\frac{3 \sqrt [3]{\frac{-\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}} \sqrt [3]{\frac{\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{\sqrt{3} \sqrt{-b^2 c^2}+3 b c}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 c (b-c x)}{3 b c+\sqrt{3} \sqrt{-b^2 c^2}},\frac{2 c (b-c x)}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}\right ) (e (b-c x))^{2/3}}{2 c e \sqrt [3]{b^2+b c x+c^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 3.069, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [3]{-cex+be}}}{\frac{1}{\sqrt [3]{{c}^{2}{x}^{2}+bcx+{b}^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{1}{3}}{\left (-c e x + b e\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt [3]{- e \left (- b + c x\right )} \sqrt [3]{b^{2} + b c x + c^{2} x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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