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 = 0.06, antiderivative size = 196, normalized size of antiderivative = 1.00, 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 239
Rule 713
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*}
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Mathematica [C] time = 0.19, 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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fricas [A] time = 114.69, size = 523, normalized size = 2.67 \[ \left [\frac {\sqrt {3} e \sqrt {-\frac {1}{e^{\frac {2}{3}}}} \log \left (3 \, c^{3} e x^{3} - b^{3} e - 3 \, {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} c e^{\frac {1}{3}} x + \sqrt {3} {\left (2 \, {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} c^{2} e x^{2} + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} c e^{\frac {2}{3}} x + {\left (c^{3} e x^{3} - b^{3} e\right )} e^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{e^{\frac {2}{3}}}}\right ) - e^{\frac {2}{3}} \log \left (c^{2} e x^{2} - {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} c e^{\frac {2}{3}} x + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} e^{\frac {1}{3}}\right ) + 2 \, e^{\frac {2}{3}} \log \left (c e x + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} e^{\frac {2}{3}}\right )}{6 \, c e}, -\frac {2 \, \sqrt {3} e^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} c e^{\frac {2}{3}} x + {\left (c^{3} e x^{3} - b^{3} e\right )} e^{\frac {1}{3}}\right )}}{3 \, {\left (c^{3} e x^{3} - b^{3} e\right )} e^{\frac {1}{3}}}\right ) + e^{\frac {2}{3}} \log \left (c^{2} e x^{2} - {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} c e^{\frac {2}{3}} x + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {2}{3}} {\left (-c e x + b e\right )}^{\frac {2}{3}} e^{\frac {1}{3}}\right ) - 2 \, e^{\frac {2}{3}} \log \left (c e x + {\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}} e^{\frac {2}{3}}\right )}{6 \, c e}\right ] \]
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 \[ \int \frac {1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.20, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (-c e x +b e \right )^{\frac {1}{3}} \left (c^{2} x^{2}+b c x +b^{2}\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac {1}{3}} {\left (-c e x + b e\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,e-c\,e\,x\right )}^{1/3}\,{\left (b^2+b\,c\,x+c^2\,x^2\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{\sqrt [3]{- e \left (- b + c x\right )} \sqrt [3]{b^{2} + b c x + c^{2} x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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