\(\int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx\) [720]

Optimal result

Integrand size = 33, antiderivative size = 196 \[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=-\frac {\sqrt [3]{b^3 e-c^3 e x^3} \arctan \left (\frac {1-\frac {2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{e} \sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}}+\frac {\sqrt [3]{b^3 e-c^3 e x^3} \log \left (c \sqrt [3]{e} x+\sqrt [3]{b^3 e-c^3 e x^3}\right )}{2 c \sqrt [3]{e} \sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 11.15 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=-\frac {3 (e (b-c x))^{2/3} \sqrt [3]{\frac {b c-\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}} \sqrt [3]{\frac {b c+\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},\frac {1}{3},\frac {5}{3},\frac {2 c (b-c x)}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}},\frac {2 c (b-c x)}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right )}{2 c e \sqrt [3]{b^2+b c x+c^2 x^2}} \] Input:

Integrate[1/((b*e - c*e*x)^(1/3)*(b^2 + b*c*x + c^2*x^2)^(1/3)),x]
 

Output:

(-3*(e*(b - c*x))^(2/3)*((b*c - Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c 
 - Sqrt[3]*Sqrt[-(b^2*c^2)]))^(1/3)*((b*c + Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c 
^2*x)/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/ 
3, (2*c*(b - c*x))/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]), (2*c*(b - c*x))/(3* 
b*c - Sqrt[3]*Sqrt[-(b^2*c^2)])])/(2*c*e*(b^2 + b*c*x + c^2*x^2)^(1/3))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.74, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1151, 769}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 1151

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

\(\Big \downarrow \) 769

\(\displaystyle \frac {\sqrt [3]{b^3 e-c^3 e x^3} \left (\frac {\log \left (\sqrt [3]{b^3 e-c^3 e x^3}+c \sqrt [3]{e} x\right )}{2 c \sqrt [3]{e}}-\frac {\arctan \left (\frac {1-\frac {2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c \sqrt [3]{e}}\right )}{\sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}\)

Input:

Int[1/((b*e - c*e*x)^(1/3)*(b^2 + b*c*x + c^2*x^2)^(1/3)),x]
 

Output:

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

Defintions of rubi rules used

rule 769

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* 
(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 
3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
 

rule 1151

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy 
mbol] :> Simp[(d + e*x)^FracPart[p]*((a + b*x + c*x^2)^FracPart[p]/(a*d + c 
*e*x^3)^FracPart[p])   Int[(d + e*x)^(m - p)*(a*d + c*e*x^3)^p, x], x] /; F 
reeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*d + a*e, 0] && EqQ[c*d + b*e, 0] && 
 IGtQ[m - p + 1, 0] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 30.06 (sec) , antiderivative size = 523, normalized size of antiderivative = 2.67 \[ \int \frac {1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx=\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 ] \] Input:

integrate(1/(-c*e*x+b*e)^(1/3)/(c^2*x^2+b*c*x+b^2)^(1/3),x, algorithm="fri 
cas")
 

Output:

[1/6*(sqrt(3)*e*sqrt(-1/e^(2/3))*log(3*c^3*e*x^3 - b^3*e - 3*(c^2*x^2 + b* 
c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)*c*e^(1/3)*x + sqrt(3)*(2*(c^2*x^2 + 
b*c*x + b^2)^(1/3)*(-c*e*x + b*e)^(1/3)*c^2*e*x^2 + (c^2*x^2 + b*c*x + b^2 
)^(2/3)*(-c*e*x + b*e)^(2/3)*c*e^(2/3)*x + (c^3*e*x^3 - b^3*e)*e^(1/3))*sq 
rt(-1/e^(2/3))) - e^(2/3)*log(c^2*e*x^2 - (c^2*x^2 + b*c*x + b^2)^(1/3)*(- 
c*e*x + b*e)^(1/3)*c*e^(2/3)*x + (c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b 
*e)^(2/3)*e^(1/3)) + 2*e^(2/3)*log(c*e*x + (c^2*x^2 + b*c*x + b^2)^(1/3)*( 
-c*e*x + b*e)^(1/3)*e^(2/3)))/(c*e), -1/6*(2*sqrt(3)*e^(2/3)*arctan(1/3*sq 
rt(3)*(2*(c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)*c*e^(2/3)*x + 
(c^3*e*x^3 - b^3*e)*e^(1/3))/((c^3*e*x^3 - b^3*e)*e^(1/3))) + e^(2/3)*log( 
c^2*e*x^2 - (c^2*x^2 + b*c*x + b^2)^(1/3)*(-c*e*x + b*e)^(1/3)*c*e^(2/3)*x 
 + (c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)*e^(1/3)) - 2*e^(2/3) 
*log(c*e*x + (c^2*x^2 + b*c*x + b^2)^(1/3)*(-c*e*x + b*e)^(1/3)*e^(2/3)))/ 
(c*e)]
 

Sympy [F]

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

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

Output:

Integral(1/((-e*(-b + c*x))**(1/3)*(b**2 + b*c*x + c**2*x**2)**(1/3)), x)
 

Maxima [F]

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

integrate(1/(-c*e*x+b*e)^(1/3)/(c^2*x^2+b*c*x+b^2)^(1/3),x, algorithm="max 
ima")
 

Output:

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

Giac [F]

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

integrate(1/(-c*e*x+b*e)^(1/3)/(c^2*x^2+b*c*x+b^2)^(1/3),x, algorithm="gia 
c")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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