\(\int \frac {2 e-f x}{(e+f x) \sqrt {d e^3+3 d e^2 f x+c f^3 x^2+d f^3 x^3}} \, dx\) [88]

Optimal result

Integrand size = 50, antiderivative size = 147 \[ \int \frac {2 e-f x}{(e+f x) \sqrt {d e^3+3 d e^2 f x+c f^3 x^2+d f^3 x^3}} \, dx=\frac {2 (e+f x)^{3/2} \sqrt {d-\frac {f^2 (3 d e-c f) x^2}{(e+f x)^3}} \arctan \left (\frac {f \sqrt {3 d e-c f} x}{(e+f x)^{3/2} \sqrt {d-\frac {f^2 (3 d e-c f) x^2}{(e+f x)^3}}}\right )}{f \sqrt {3 d e-c f} \sqrt {d e^3+3 d e^2 f x+c f^3 x^2+d f^3 x^3}} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.46 \[ \int \frac {2 e-f x}{(e+f x) \sqrt {d e^3+3 d e^2 f x+c f^3 x^2+d f^3 x^3}} \, dx=\frac {2 \arctan \left (\frac {f \sqrt {3 d e-c f} x}{\sqrt {c f^3 x^2+d \left (e^3+3 e^2 f x+f^3 x^3\right )}}\right )}{f \sqrt {3 d e-c f}} \] Input:

Integrate[(2*e - f*x)/((e + f*x)*Sqrt[d*e^3 + 3*d*e^2*f*x + c*f^3*x^2 + d* 
f^3*x^3]),x]
 

Output:

(2*ArcTan[(f*Sqrt[3*d*e - c*f]*x)/Sqrt[c*f^3*x^2 + d*(e^3 + 3*e^2*f*x + f^ 
3*x^3)]])/(f*Sqrt[3*d*e - c*f])
 

Rubi [F]

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.

\(\displaystyle \int \frac {2 e-f x}{(e+f x) \sqrt {c f^3 x^2+d e^3+3 d e^2 f x+d f^3 x^3}} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 e}{(e+f x) \sqrt {c f^3 x^2+d e^3+3 d e^2 f x+d f^3 x^3}}-\frac {1}{\sqrt {c f^3 x^2+d e^3+3 d e^2 f x+d f^3 x^3}}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (\frac {3 e}{(e+f x) \sqrt {c f^3 x^2+d e^3+3 d e^2 f x+d f^3 x^3}}-\frac {1}{\sqrt {c f^3 x^2+d e^3+3 d e^2 f x+d f^3 x^3}}\right )dx\)

Input:

Int[(2*e - f*x)/((e + f*x)*Sqrt[d*e^3 + 3*d*e^2*f*x + c*f^3*x^2 + d*f^3*x^ 
3]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [A] (verified)

Time = 3.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.47

method	result	size
default	\(-\frac {2 \arctan \left (\frac {\sqrt {d \left (x^{3} f^{3}+3 x \,e^{2} f +e^{3}\right )+c \,f^{3} x^{2}}}{x \sqrt {-f^{2} \left (c f -3 d e \right )}}\right )}{\sqrt {-f^{2} \left (c f -3 d e \right )}}\)	\(69\)
pseudoelliptic	\(-\frac {2 \arctan \left (\frac {\sqrt {d \left (x^{3} f^{3}+3 x \,e^{2} f +e^{3}\right )+c \,f^{3} x^{2}}}{x \sqrt {-f^{2} \left (c f -3 d e \right )}}\right )}{\sqrt {-f^{2} \left (c f -3 d e \right )}}\)	\(69\)
elliptic	\(\text {Expression too large to display}\)	\(14975\)

Input:

int((-f*x+2*e)/(f*x+e)/(d*f^3*x^3+c*f^3*x^2+3*d*e^2*f*x+d*e^3)^(1/2),x,met 
hod=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 525, normalized size of antiderivative = 3.57 \[ \int \frac {2 e-f x}{(e+f x) \sqrt {d e^3+3 d e^2 f x+c f^3 x^2+d f^3 x^3}} \, dx=\left [-\frac {\sqrt {-3 \, d e + c f} \log \left (\frac {d^{2} f^{6} x^{6} + 6 \, d^{2} e^{5} f x + d^{2} e^{6} - 2 \, {\left (9 \, d^{2} e f^{5} - 4 \, c d f^{6}\right )} x^{5} + {\left (15 \, d^{2} e^{2} f^{4} - 24 \, c d e f^{5} + 8 \, c^{2} f^{6}\right )} x^{4} - 4 \, {\left (13 \, d^{2} e^{3} f^{3} - 6 \, c d e^{2} f^{4}\right )} x^{3} - {\left (9 \, d^{2} e^{4} f^{2} - 8 \, c d e^{3} f^{3}\right )} x^{2} + 4 \, {\left (d f^{4} x^{4} + 3 \, d e^{2} f^{2} x^{2} + d e^{3} f x - {\left (3 \, d e f^{3} - 2 \, c f^{4}\right )} x^{3}\right )} \sqrt {d f^{3} x^{3} + c f^{3} x^{2} + 3 \, d e^{2} f x + d e^{3}} \sqrt {-3 \, d e + c f}}{f^{6} x^{6} + 6 \, e f^{5} x^{5} + 15 \, e^{2} f^{4} x^{4} + 20 \, e^{3} f^{3} x^{3} + 15 \, e^{4} f^{2} x^{2} + 6 \, e^{5} f x + e^{6}}\right )}{2 \, {\left (3 \, d e f - c f^{2}\right )}}, \frac {\sqrt {3 \, d e - c f} \arctan \left (-\frac {\sqrt {d f^{3} x^{3} + c f^{3} x^{2} + 3 \, d e^{2} f x + d e^{3}} {\left (d f^{3} x^{3} + 3 \, d e^{2} f x + d e^{3} - {\left (3 \, d e f^{2} - 2 \, c f^{3}\right )} x^{2}\right )} \sqrt {3 \, d e - c f}}{2 \, {\left ({\left (3 \, d^{2} e f^{4} - c d f^{5}\right )} x^{4} + {\left (3 \, c d e f^{4} - c^{2} f^{5}\right )} x^{3} + 3 \, {\left (3 \, d^{2} e^{3} f^{2} - c d e^{2} f^{3}\right )} x^{2} + {\left (3 \, d^{2} e^{4} f - c d e^{3} f^{2}\right )} x\right )}}\right )}{3 \, d e f - c f^{2}}\right ] \] Input:

integrate((-f*x+2*e)/(f*x+e)/(d*f^3*x^3+c*f^3*x^2+3*d*e^2*f*x+d*e^3)^(1/2) 
,x, algorithm="fricas")
 

Output:

[-1/2*sqrt(-3*d*e + c*f)*log((d^2*f^6*x^6 + 6*d^2*e^5*f*x + d^2*e^6 - 2*(9 
*d^2*e*f^5 - 4*c*d*f^6)*x^5 + (15*d^2*e^2*f^4 - 24*c*d*e*f^5 + 8*c^2*f^6)* 
x^4 - 4*(13*d^2*e^3*f^3 - 6*c*d*e^2*f^4)*x^3 - (9*d^2*e^4*f^2 - 8*c*d*e^3* 
f^3)*x^2 + 4*(d*f^4*x^4 + 3*d*e^2*f^2*x^2 + d*e^3*f*x - (3*d*e*f^3 - 2*c*f 
^4)*x^3)*sqrt(d*f^3*x^3 + c*f^3*x^2 + 3*d*e^2*f*x + d*e^3)*sqrt(-3*d*e + c 
*f))/(f^6*x^6 + 6*e*f^5*x^5 + 15*e^2*f^4*x^4 + 20*e^3*f^3*x^3 + 15*e^4*f^2 
*x^2 + 6*e^5*f*x + e^6))/(3*d*e*f - c*f^2), sqrt(3*d*e - c*f)*arctan(-1/2* 
sqrt(d*f^3*x^3 + c*f^3*x^2 + 3*d*e^2*f*x + d*e^3)*(d*f^3*x^3 + 3*d*e^2*f*x 
 + d*e^3 - (3*d*e*f^2 - 2*c*f^3)*x^2)*sqrt(3*d*e - c*f)/((3*d^2*e*f^4 - c* 
d*f^5)*x^4 + (3*c*d*e*f^4 - c^2*f^5)*x^3 + 3*(3*d^2*e^3*f^2 - c*d*e^2*f^3) 
*x^2 + (3*d^2*e^4*f - c*d*e^3*f^2)*x))/(3*d*e*f - c*f^2)]
 

Sympy [F]

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

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

Output:

-Integral(-2*e/(e*sqrt(c*f**3*x**2 + d*e**3 + 3*d*e**2*f*x + d*f**3*x**3) 
+ f*x*sqrt(c*f**3*x**2 + d*e**3 + 3*d*e**2*f*x + d*f**3*x**3)), x) - Integ 
ral(f*x/(e*sqrt(c*f**3*x**2 + d*e**3 + 3*d*e**2*f*x + d*f**3*x**3) + f*x*s 
qrt(c*f**3*x**2 + d*e**3 + 3*d*e**2*f*x + d*f**3*x**3)), x)
 

Maxima [F]

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

integrate((-f*x+2*e)/(f*x+e)/(d*f^3*x^3+c*f^3*x^2+3*d*e^2*f*x+d*e^3)^(1/2) 
,x, algorithm="maxima")
 

Output:

-integrate((f*x - 2*e)/(sqrt(d*f^3*x^3 + c*f^3*x^2 + 3*d*e^2*f*x + d*e^3)* 
(f*x + e)), x)
 

Giac [F]

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

integrate((-f*x+2*e)/(f*x+e)/(d*f^3*x^3+c*f^3*x^2+3*d*e^2*f*x+d*e^3)^(1/2) 
,x, algorithm="giac")
 

Output:

integrate(-(f*x - 2*e)/(sqrt(d*f^3*x^3 + c*f^3*x^2 + 3*d*e^2*f*x + d*e^3)* 
(f*x + e)), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {2 e-f x}{(e+f x) \sqrt {d e^3+3 d e^2 f x+c f^3 x^2+d f^3 x^3}} \, dx=2 \left (\int \frac {\sqrt {d \,f^{3} x^{3}+c \,f^{3} x^{2}+3 d \,e^{2} f x +d \,e^{3}}}{d \,f^{4} x^{4}+c \,f^{4} x^{3}+d e \,f^{3} x^{3}+c e \,f^{3} x^{2}+3 d \,e^{2} f^{2} x^{2}+4 d \,e^{3} f x +d \,e^{4}}d x \right ) e -\left (\int \frac {\sqrt {d \,f^{3} x^{3}+c \,f^{3} x^{2}+3 d \,e^{2} f x +d \,e^{3}}\, x}{d \,f^{4} x^{4}+c \,f^{4} x^{3}+d e \,f^{3} x^{3}+c e \,f^{3} x^{2}+3 d \,e^{2} f^{2} x^{2}+4 d \,e^{3} f x +d \,e^{4}}d x \right ) f \] Input:

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

Output:

2*int(sqrt(c*f**3*x**2 + d*e**3 + 3*d*e**2*f*x + d*f**3*x**3)/(c*e*f**3*x* 
*2 + c*f**4*x**3 + d*e**4 + 4*d*e**3*f*x + 3*d*e**2*f**2*x**2 + d*e*f**3*x 
**3 + d*f**4*x**4),x)*e - int((sqrt(c*f**3*x**2 + d*e**3 + 3*d*e**2*f*x + 
d*f**3*x**3)*x)/(c*e*f**3*x**2 + c*f**4*x**3 + d*e**4 + 4*d*e**3*f*x + 3*d 
*e**2*f**2*x**2 + d*e*f**3*x**3 + d*f**4*x**4),x)*f