Optimal. Leaf size=564 \[ -\frac {\log (d+e x) \sqrt [3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x}}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}}+\frac {3 \sqrt [3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x} \log \left (-\frac {\sqrt [3]{\frac {3}{2}} \left (3 c^2 e^2 x-c e (c d-2 b e)\right )^{2/3}}{\sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e}}-\sqrt [3]{6} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x}\right )}{4 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}}-\frac {\sqrt {3} \sqrt [3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} \left (3 c^2 e^2 x-c e (c d-2 b e)\right )^{2/3}}{\sqrt {3} \sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x}}\right )}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}} \]
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Rubi [A] time = 0.41, antiderivative size = 564, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {755, 123} \[ -\frac {\log (d+e x) \sqrt [3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x}}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}}+\frac {3 \sqrt [3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x} \log \left (-\frac {\sqrt [3]{\frac {3}{2}} \left (3 c^2 e^2 x-c e (c d-2 b e)\right )^{2/3}}{\sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e}}-\sqrt [3]{6} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x}\right )}{4 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}}-\frac {\sqrt {3} \sqrt [3]{3 c^2 e^2 x-c e (c d-2 b e)} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} \left (3 c^2 e^2 x-c e (c d-2 b e)\right )^{2/3}}{\sqrt {3} \sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e} \sqrt [3]{c e (b e+c d)+3 c^2 e^2 x}}\right )}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (b e+c d)+9 b c e^2 x+9 c^2 e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 123
Rule 755
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}} \, dx &=\frac {\left (\sqrt [3]{9 b c e^2-3 c e (2 c d-b e)+18 c^2 e^2 x} \sqrt [3]{9 b c e^2+3 c e (2 c d-b e)+18 c^2 e^2 x}\right ) \int \frac {1}{(d+e x) \sqrt [3]{9 b c e^2-3 c e (2 c d-b e)+18 c^2 e^2 x} \sqrt [3]{9 b c e^2+3 c e (2 c d-b e)+18 c^2 e^2 x}} \, dx}{\sqrt [3]{-c^2 d^2+b c d e+2 b^2 e^2+9 b c e^2 x+9 c^2 e^2 x^2}}\\ &=-\frac {\sqrt {3} \sqrt [3]{-c e (c d-2 b e)+3 c^2 e^2 x} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{2} \left (-c e (c d-2 b e)+3 c^2 e^2 x\right )^{2/3}}{\sqrt {3} \sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x}}\right )}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (c d+b e)+9 b c e^2 x+9 c^2 e^2 x^2}}-\frac {\sqrt [3]{-c e (c d-2 b e)+3 c^2 e^2 x} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x} \log (d+e x)}{2 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (c d+b e)+9 b c e^2 x+9 c^2 e^2 x^2}}+\frac {3 \sqrt [3]{-c e (c d-2 b e)+3 c^2 e^2 x} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x} \log \left (-\frac {\sqrt [3]{\frac {3}{2}} \left (-c e (c d-2 b e)+3 c^2 e^2 x\right )^{2/3}}{\sqrt [3]{c} \sqrt [3]{e} \sqrt [3]{2 c d-b e}}-\sqrt [3]{6} \sqrt [3]{c e (c d+b e)+3 c^2 e^2 x}\right )}{4 \sqrt [3]{2} c^{2/3} e^{5/3} (2 c d-b e)^{2/3} \sqrt [3]{-(c d-2 b e) (c d+b e)+9 b c e^2 x+9 c^2 e^2 x^2}}\\ \end {align*}
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Mathematica [C] time = 0.41, size = 290, normalized size = 0.51 \[ -\frac {\sqrt [3]{3} \sqrt [3]{\frac {-\sqrt {c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} \sqrt [3]{\frac {\sqrt {c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};-\frac {-6 d e c^2+3 b e^2 c+\sqrt {c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)},\frac {6 d e c^2-3 b e^2 c+\sqrt {c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)}\right )}{2\ 2^{2/3} e \sqrt [3]{2 b^2 e^2+b c e (d+9 e x)-\left (c^2 \left (d^2-9 e^2 x^2\right )\right )}} \]
Warning: Unable to verify antiderivative.
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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 \[ \int \frac {1}{{\left (9 \, c^{2} e^{2} x^{2} + 9 \, b c e^{2} x - c^{2} d^{2} + b c d e + 2 \, b^{2} e^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.71, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x +d \right ) \left (9 c^{2} e^{2} x^{2}+9 b c \,e^{2} x +2 b^{2} e^{2}+b c d e -c^{2} d^{2}\right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {1}{{\left (9 \, c^{2} e^{2} x^{2} + 9 \, b c e^{2} x - c^{2} d^{2} + b c d e + 2 \, b^{2} e^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (d+e\,x\right )\,{\left (2\,b^2\,e^2+b\,c\,d\,e+9\,b\,c\,e^2\,x-c^2\,d^2+9\,c^2\,e^2\,x^2\right )}^{1/3}} \,d x \]
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 \[ \int \frac {1}{\sqrt [3]{\left (b e + c d + 3 c e x\right ) \left (2 b e - c d + 3 c e x\right )} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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