Optimal. Leaf size=242 \[ -\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 (c d-b e-c e x)}{\sqrt {3} \sqrt [3]{2 c d-b e} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}}\right )}{\sqrt {3} e (2 c d-b e)^{2/3}}-\frac {\log (d+e x)}{2 e (2 c d-b e)^{2/3}}+\frac {\log \left (3 c e^2 (c d-b e)-3 c^2 e^3 x-3 c e^2 \sqrt [3]{2 c d-b e} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}\right )}{2 e (2 c d-b e)^{2/3}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {764}
\begin {gather*} -\frac {\text {ArcTan}\left (\frac {2 (-b e+c d-c e x)}{\sqrt {3} \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} e (2 c d-b e)^{2/3}}+\frac {\log \left (-3 c e^2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}+3 c e^2 (c d-b e)-3 c^2 e^3 x\right )}{2 e (2 c d-b e)^{2/3}}-\frac {\log (d+e x)}{2 e (2 c d-b e)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 764
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx &=-\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 (c d-b e-c e x)}{\sqrt {3} \sqrt [3]{2 c d-b e} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}}\right )}{\sqrt {3} e (2 c d-b e)^{2/3}}-\frac {\log (d+e x)}{2 e (2 c d-b e)^{2/3}}+\frac {\log \left (3 c e^2 (c d-b e)-3 c^2 e^3 x-3 c e^2 \sqrt [3]{2 c d-b e} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}\right )}{2 e (2 c d-b e)^{2/3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 1.43, size = 457, normalized size = 1.89 \begin {gather*} \frac {\left (-6-2 i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {\left (-i+\sqrt {3}\right ) c (d-e x)+i \left (b e+i \sqrt {3} b e+\sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}\right )}{\sqrt {3} \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}}\right )+i \left (i+\sqrt {3}\right ) \left (2 \log \left (\sqrt {e} \left (c d+i \sqrt {3} c d-i \left (-i+\sqrt {3}\right ) e (b+c x)+2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}\right )\right )-\log \left (e \left (-c d+b e+c e x+\sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}\right ) \left (-b e+i \sqrt {3} b e+\left (1-i \sqrt {3}\right ) c (d-e x)+2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}\right )\right )\right )}{12 e (2 c d-b e)^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.59, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e x +d \right ) \left (3 c^{2} e^{2} x^{2}+3 b c \,e^{2} x +b^{2} e^{2}-b c d e +c^{2} d^{2}\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right ) \sqrt [3]{b^{2} e^{2} - b c d e + 3 b c e^{2} x + c^{2} d^{2} + 3 c^{2} e^{2} x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{\left (d+e\,x\right )\,{\left (b^2\,e^2-b\,c\,d\,e+3\,b\,c\,e^2\,x+c^2\,d^2+3\,c^2\,e^2\,x^2\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________