\(\int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx\) [2496]

Optimal result

Integrand size = 52, antiderivative size = 242 \[ \int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2 (c d-b e-c e x)}{\sqrt {3} \sqrt [3]{2 c d-b e} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}}\right )}{\sqrt {3} e (2 c d-b e)^{2/3}}-\frac {\log (d+e x)}{2 e (2 c d-b e)^{2/3}}+\frac {\log \left (3 c e^2 (c d-b e)-3 c^2 e^3 x-3 c e^2 \sqrt [3]{2 c d-b e} \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}\right )}{2 e (2 c d-b e)^{2/3}} \]

[Out]

-1/2*ln(e*x+d)/e/(-b*e+2*c*d)^(2/3)+1/2*ln(3*c*e^2*(-b*e+c*d)-3*c^2*e^3*x-3*c*e^2*(-b*e+2*c*d)^(1/3)*(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)^(1/3))/e/(-b*e+2*c*d)^(2/3)+1/3*arctan(-1/3*3^(1/2)-2/3*(-c*e*x-b*e
+c*d)/(-b*e+2*c*d)^(1/3)/(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)^(1/3)*3^(1/2))/e/(-b*e+2*c*d)^(2/
3)*3^(1/2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {764} \[ \int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx=-\frac {\arctan \left (\frac {2 (-b e+c d-c e x)}{\sqrt {3} \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} e (2 c d-b e)^{2/3}}+\frac {\log \left (-3 c e^2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}+3 c e^2 (c d-b e)-3 c^2 e^3 x\right )}{2 e (2 c d-b e)^{2/3}}-\frac {\log (d+e x)}{2 e (2 c d-b e)^{2/3}} \]

[In]

Int[1/((d + e*x)*(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)^(1/3)),x]

[Out]

-(ArcTan[1/Sqrt[3] + (2*(c*d - b*e - c*e*x))/(Sqrt[3]*(2*c*d - b*e)^(1/3)*(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)^(1/3))]/(Sqrt[3]*e*(2*c*d - b*e)^(2/3))) - Log[d + e*x]/(2*e*(2*c*d - b*e)^(2/3)) + Lo
g[3*c*e^2*(c*d - b*e) - 3*c^2*e^3*x - 3*c*e^2*(2*c*d - b*e)^(1/3)*(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)^(1/3)]/(2*e*(2*c*d - b*e)^(2/3))

Rule 764

Int[1/(((d_.) + (e_.)*(x_))*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[3*c*e^2*(2*c*
d - b*e), 3]}, Simp[(-Sqrt[3])*c*e*(ArcTan[1/Sqrt[3] + 2*((c*d - b*e - c*e*x)/(Sqrt[3]*q*(a + b*x + c*x^2)^(1/
3)))]/q^2), x] + (-Simp[3*c*e*(Log[d + e*x]/(2*q^2)), x] + Simp[3*c*e*(Log[c*d - b*e - c*e*x - q*(a + b*x + c*
x^2)^(1/3)]/(2*q^2)), x])] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && EqQ[c^2*d^2 - b*c*d*e + b^2*
e^2 - 3*a*c*e^2, 0] && PosQ[c*e^2*(2*c*d - b*e)]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Result contains complex when optimal does not.

Time = 1.32 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.89 \[ \int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx=\frac {\left (-6-2 i \sqrt {3}\right ) \text {arctanh}\left (\frac {\left (-i+\sqrt {3}\right ) c (d-e x)+i \left (b e+i \sqrt {3} b e+\sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}\right )}{\sqrt {3} \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}}\right )+i \left (i+\sqrt {3}\right ) \left (2 \log \left (\sqrt {e} \left (c d+i \sqrt {3} c d-i \left (-i+\sqrt {3}\right ) e (b+c x)+2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}\right )\right )-\log \left (e \left (-c d+b e+c e x+\sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}\right ) \left (-b e+i \sqrt {3} b e+\left (1-i \sqrt {3}\right ) c (d-e x)+2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2+b c e (-d+3 e x)+c^2 \left (d^2+3 e^2 x^2\right )}\right )\right )\right )}{12 e (2 c d-b e)^{2/3}} \]

[In]

Integrate[1/((d + e*x)*(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)^(1/3)),x]

[Out]

((-6 - (2*I)*Sqrt[3])*ArcTanh[((-I + Sqrt[3])*c*(d - e*x) + I*(b*e + I*Sqrt[3]*b*e + (2*c*d - b*e)^(1/3)*(b^2*
e^2 + b*c*e*(-d + 3*e*x) + c^2*(d^2 + 3*e^2*x^2))^(1/3)))/(Sqrt[3]*(2*c*d - b*e)^(1/3)*(b^2*e^2 + b*c*e*(-d +
3*e*x) + c^2*(d^2 + 3*e^2*x^2))^(1/3))] + I*(I + Sqrt[3])*(2*Log[Sqrt[e]*(c*d + I*Sqrt[3]*c*d - I*(-I + Sqrt[3
])*e*(b + c*x) + 2*(2*c*d - b*e)^(1/3)*(b^2*e^2 + b*c*e*(-d + 3*e*x) + c^2*(d^2 + 3*e^2*x^2))^(1/3))] - Log[e*
(-(c*d) + b*e + c*e*x + (2*c*d - b*e)^(1/3)*(b^2*e^2 + b*c*e*(-d + 3*e*x) + c^2*(d^2 + 3*e^2*x^2))^(1/3))*(-(b
*e) + I*Sqrt[3]*b*e + (1 - I*Sqrt[3])*c*(d - e*x) + 2*(2*c*d - b*e)^(1/3)*(b^2*e^2 + b*c*e*(-d + 3*e*x) + c^2*
(d^2 + 3*e^2*x^2))^(1/3))]))/(12*e*(2*c*d - b*e)^(2/3))

Maple [F]

\[\int \frac {1}{\left (e x +d \right ) \left (3 c^{2} e^{2} x^{2}+3 b c \,e^{2} x +b^{2} e^{2}-b c d e +c^{2} d^{2}\right )^{\frac {1}{3}}}d x\]

[In]

int(1/(e*x+d)/(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)^(1/3),x)

[Out]

int(1/(e*x+d)/(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)^(1/3),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*x+d)/(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)^(1/3),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx=\int \frac {1}{\left (d + e x\right ) \sqrt [3]{b^{2} e^{2} - b c d e + 3 b c e^{2} x + c^{2} d^{2} + 3 c^{2} e^{2} x^{2}}}\, dx \]

[In]

integrate(1/(e*x+d)/(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)**(1/3),x)

[Out]

Integral(1/((d + e*x)*(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)**(1/3)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx=\int { \frac {1}{{\left (3 \, c^{2} e^{2} x^{2} + 3 \, b c e^{2} x + c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \]

[In]

integrate(1/(e*x+d)/(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)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((3*c^2*e^2*x^2 + 3*b*c*e^2*x + c^2*d^2 - b*c*d*e + b^2*e^2)^(1/3)*(e*x + d)), x)

Giac [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx=\int { \frac {1}{{\left (3 \, c^{2} e^{2} x^{2} + 3 \, b c e^{2} x + c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \]

[In]

integrate(1/(e*x+d)/(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)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((3*c^2*e^2*x^2 + 3*b*c*e^2*x + c^2*d^2 - b*c*d*e + b^2*e^2)^(1/3)*(e*x + d)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (b^2\,e^2-b\,c\,d\,e+3\,b\,c\,e^2\,x+c^2\,d^2+3\,c^2\,e^2\,x^2\right )}^{1/3}} \,d x \]

[In]

int(1/((d + e*x)*(b^2*e^2 + c^2*d^2 + 3*c^2*e^2*x^2 + 3*b*c*e^2*x - b*c*d*e)^(1/3)),x)

[Out]

int(1/((d + e*x)*(b^2*e^2 + c^2*d^2 + 3*c^2*e^2*x^2 + 3*b*c*e^2*x - b*c*d*e)^(1/3)), x)