\(\int \frac {(d+e x) (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx\) [175]

Optimal result

Integrand size = 42, antiderivative size = 144 \[ \int \frac {(d+e x) (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {(3 b e g-4 c (e f+d g)-2 c e g x) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 c^2 e^2}+\frac {(2 c d-b e) (4 c e f+2 c d g-3 b e g) \arctan \left (\frac {\sqrt {c} (d+e x)}{\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right )}{4 c^{5/2} e^2} \] Output:

1/4*(3*b*e*g-4*c*(d*g+e*f)-2*c*e*g*x)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/ 
2)/c^2/e^2+1/4*(-b*e+2*c*d)*(-3*b*e*g+2*c*d*g+4*c*e*f)*arctan(c^(1/2)*(e*x 
+d)/(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2))/c^(5/2)/e^2
 

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.26 \[ \int \frac {(d+e x) (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {\sqrt {d+e x} \left (-\frac {\sqrt {d+e x} (-b e+c (d-e x)) (-3 b e g+2 c (2 e f+2 d g+e g x))}{c^2}+\left (-\frac {1}{c}\right )^{7/2} c (2 c d-b e) (4 c e f+2 c d g-3 b e g) \sqrt {c d-b e-c e x} \log \left (\sqrt {d+e x}+\left (-\frac {1}{c}\right )^{3/2} c \sqrt {c d-b e-c e x}\right )\right )}{4 e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

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

Output:

(Sqrt[d + e*x]*(-((Sqrt[d + e*x]*(-(b*e) + c*(d - e*x))*(-3*b*e*g + 2*c*(2 
*e*f + 2*d*g + e*g*x)))/c^2) + (-c^(-1))^(7/2)*c*(2*c*d - b*e)*(4*c*e*f + 
2*c*d*g - 3*b*e*g)*Sqrt[c*d - b*e - c*e*x]*Log[Sqrt[d + e*x] + (-c^(-1))^( 
3/2)*c*Sqrt[c*d - b*e - c*e*x]]))/(4*e^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e 
*x))])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1225, 1092, 217}

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.

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {(2 c d-b e) (-3 b e g+2 c d g+4 c e f) \int \frac {1}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx}{8 c^2 e}+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-4 c (d g+e f)-2 c e g x)}{4 c^2 e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {(2 c d-b e) (-3 b e g+2 c d g+4 c e f) \int \frac {1}{-\frac {(b+2 c x)^2 e^4}{-c x^2 e^2-b x e^2+d (c d-b e)}-4 c e^2}d\left (-\frac {e^2 (b+2 c x)}{\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}\right )}{4 c^2 e}+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-4 c (d g+e f)-2 c e g x)}{4 c^2 e^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(2 c d-b e) \arctan \left (\frac {e (b+2 c x)}{2 \sqrt {c} \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}\right ) (-3 b e g+2 c d g+4 c e f)}{8 c^{5/2} e^2}+\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2} (3 b e g-4 c (d g+e f)-2 c e g x)}{4 c^2 e^2}\)

Input:

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

Output:

((3*b*e*g - 4*c*(e*f + d*g) - 2*c*e*g*x)*Sqrt[d*(c*d - b*e) - b*e^2*x - c* 
e^2*x^2])/(4*c^2*e^2) + ((2*c*d - b*e)*(4*c*e*f + 2*c*d*g - 3*b*e*g)*ArcTa 
n[(e*(b + 2*c*x))/(2*Sqrt[c]*Sqrt[d*(c*d - b*e) - b*e^2*x - c*e^2*x^2])])/ 
(8*c^(5/2)*e^2)
 

Defintions of rubi rules used

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1225

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( 
x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 
 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), 
 x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p 
+ 3))/(2*c^2*(2*p + 3))   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, p}, x] &&  !LeQ[p, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(379\) vs. \(2(132)=264\).

Time = 2.48 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.64

method	result	size
default	\(\frac {d f \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{\sqrt {c \,e^{2}}}+\left (d g +e f \right ) \left (-\frac {\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{c \,e^{2}}-\frac {b \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \sqrt {c \,e^{2}}}\right )+e g \left (-\frac {x \sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{2 c \,e^{2}}-\frac {3 b \left (-\frac {\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}{c \,e^{2}}-\frac {b \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \sqrt {c \,e^{2}}}\right )}{4 c}+\frac {\left (-b d e +c \,d^{2}\right ) \arctan \left (\frac {\sqrt {c \,e^{2}}\, \left (x +\frac {b}{2 c}\right )}{\sqrt {-x^{2} c \,e^{2}-x b \,e^{2}-b d e +c \,d^{2}}}\right )}{2 c \,e^{2} \sqrt {c \,e^{2}}}\right )\)	\(380\)

Input:

int((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x,method=_RETUR 
NVERBOSE)
 

Output:

d*f/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d 
*e+c*d^2)^(1/2))+(d*g+e*f)*(-1/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2 
)-1/2*b/c/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2 
*x-b*d*e+c*d^2)^(1/2)))+e*g*(-1/2*x/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2) 
^(1/2)-3/4*b/c*(-1/c/e^2*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)-1/2*b/c/(c 
*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d 
^2)^(1/2)))+1/2*(-b*d*e+c*d^2)/c/e^2/(c*e^2)^(1/2)*arctan((c*e^2)^(1/2)*(x 
+1/2*b/c)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.80 \[ \int \frac {(d+e x) (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\left [-\frac {{\left (4 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} f + {\left (4 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} g\right )} \sqrt {-c} \log \left (8 \, c^{2} e^{2} x^{2} + 8 \, b c e^{2} x - 4 \, c^{2} d^{2} + 4 \, b c d e + b^{2} e^{2} - 4 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {-c}\right ) + 4 \, {\left (2 \, c^{2} e g x + 4 \, c^{2} e f + {\left (4 \, c^{2} d - 3 \, b c e\right )} g\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{16 \, c^{3} e^{2}}, -\frac {{\left (4 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} f + {\left (4 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} g\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (2 \, c e x + b e\right )} \sqrt {c}}{2 \, {\left (c^{2} e^{2} x^{2} + b c e^{2} x - c^{2} d^{2} + b c d e\right )}}\right ) + 2 \, {\left (2 \, c^{2} e g x + 4 \, c^{2} e f + {\left (4 \, c^{2} d - 3 \, b c e\right )} g\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}}{8 \, c^{3} e^{2}}\right ] \] Input:

integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algori 
thm="fricas")
 

Output:

[-1/16*((4*(2*c^2*d*e - b*c*e^2)*f + (4*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*g 
)*sqrt(-c)*log(8*c^2*e^2*x^2 + 8*b*c*e^2*x - 4*c^2*d^2 + 4*b*c*d*e + b^2*e 
^2 - 4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(-c) 
) + 4*(2*c^2*e*g*x + 4*c^2*e*f + (4*c^2*d - 3*b*c*e)*g)*sqrt(-c*e^2*x^2 - 
b*e^2*x + c*d^2 - b*d*e))/(c^3*e^2), -1/8*((4*(2*c^2*d*e - b*c*e^2)*f + (4 
*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*g)*sqrt(c)*arctan(1/2*sqrt(-c*e^2*x^2 - 
b*e^2*x + c*d^2 - b*d*e)*(2*c*e*x + b*e)*sqrt(c)/(c^2*e^2*x^2 + b*c*e^2*x 
- c^2*d^2 + b*c*d*e)) + 2*(2*c^2*e*g*x + 4*c^2*e*f + (4*c^2*d - 3*b*c*e)*g 
)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))/(c^3*e^2)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (136) = 272\).

Time = 2.07 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.89 \[ \int \frac {(d+e x) (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\begin {cases} \left (- \frac {g x}{2 c e} - \frac {- \frac {3 b e g}{4 c} + d g + e f}{c e^{2}}\right ) \sqrt {- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}} + \left (- \frac {b \left (- \frac {3 b e g}{4 c} + d g + e f\right )}{2 c} + d f + \frac {g \left (- b d e + c d^{2}\right )}{2 c e}\right ) \left (\begin {cases} \frac {\log {\left (- b e^{2} - 2 c e^{2} x + 2 \sqrt {- c e^{2}} \sqrt {- b d e - b e^{2} x + c d^{2} - c e^{2} x^{2}} \right )}}{\sqrt {- c e^{2}}} & \text {for}\: \frac {b^{2} e^{2}}{4 c} - b d e + c d^{2} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {- c e^{2} \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) & \text {for}\: c e^{2} \neq 0 \\- \frac {2 \left (\frac {g \left (- b d e - b e^{2} x + c d^{2}\right )^{\frac {5}{2}}}{5 b^{2} e^{3}} + \frac {\left (- b d e - b e^{2} x + c d^{2}\right )^{\frac {3}{2}} \left (b d e g - b e^{2} f - 2 c d^{2} g\right )}{3 b^{2} e^{3}} + \frac {\sqrt {- b d e - b e^{2} x + c d^{2}} \left (- b c d^{3} e g + b c d^{2} e^{2} f + c^{2} d^{4} g\right )}{b^{2} e^{3}}\right )}{b e^{2}} & \text {for}\: b e^{2} \neq 0 \\\frac {d f x + \frac {e g x^{3}}{3} + \frac {x^{2} \left (d g + e f\right )}{2}}{\sqrt {- b d e + c d^{2}}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise(((-g*x/(2*c*e) - (-3*b*e*g/(4*c) + d*g + e*f)/(c*e**2))*sqrt(-b* 
d*e - b*e**2*x + c*d**2 - c*e**2*x**2) + (-b*(-3*b*e*g/(4*c) + d*g + e*f)/ 
(2*c) + d*f + g*(-b*d*e + c*d**2)/(2*c*e))*Piecewise((log(-b*e**2 - 2*c*e* 
*2*x + 2*sqrt(-c*e**2)*sqrt(-b*d*e - b*e**2*x + c*d**2 - c*e**2*x**2))/sqr 
t(-c*e**2), Ne(b**2*e**2/(4*c) - b*d*e + c*d**2, 0)), ((b/(2*c) + x)*log(b 
/(2*c) + x)/sqrt(-c*e**2*(b/(2*c) + x)**2), True)), Ne(c*e**2, 0)), (-2*(g 
*(-b*d*e - b*e**2*x + c*d**2)**(5/2)/(5*b**2*e**3) + (-b*d*e - b*e**2*x + 
c*d**2)**(3/2)*(b*d*e*g - b*e**2*f - 2*c*d**2*g)/(3*b**2*e**3) + sqrt(-b*d 
*e - b*e**2*x + c*d**2)*(-b*c*d**3*e*g + b*c*d**2*e**2*f + c**2*d**4*g)/(b 
**2*e**3))/(b*e**2), Ne(b*e**2, 0)), ((d*f*x + e*g*x**3/3 + x**2*(d*g + e* 
f)/2)/sqrt(-b*d*e + c*d**2), True))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+e x) (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algori 
thm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e*(b*e-2*c*d)>0)', see `assume?` 
 for more
 

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x) (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=-\frac {1}{4} \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (\frac {2 \, g x}{c e} + \frac {4 \, c e^{2} f + 4 \, c d e g - 3 \, b e^{2} g}{c^{2} e^{3}}\right )} - \frac {{\left (8 \, c^{2} d e f - 4 \, b c e^{2} f + 4 \, c^{2} d^{2} g - 8 \, b c d e g + 3 \, b^{2} e^{2} g\right )} \log \left ({\left | -b e^{2} + 2 \, {\left (\sqrt {-c e^{2}} x - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e}\right )} \sqrt {-c} {\left | e \right |} \right |}\right )}{8 \, \sqrt {-c} c^{2} e {\left | e \right |}} \] Input:

integrate((e*x+d)*(g*x+f)/(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2),x, algori 
thm="giac")
 

Output:

-1/4*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(2*g*x/(c*e) + (4*c*e^2*f 
+ 4*c*d*e*g - 3*b*e^2*g)/(c^2*e^3)) - 1/8*(8*c^2*d*e*f - 4*b*c*e^2*f + 4*c 
^2*d^2*g - 8*b*c*d*e*g + 3*b^2*e^2*g)*log(abs(-b*e^2 + 2*(sqrt(-c*e^2)*x - 
 sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e))*sqrt(-c)*abs(e)))/(sqrt(-c)*c 
^2*e*abs(e))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 492, normalized size of antiderivative = 3.42 \[ \int \frac {(d+e x) (f+g x)}{\sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}} \, dx=\frac {i \left (3 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b^{3} e^{3} g -14 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b^{2} c d \,e^{2} g -4 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b^{2} c \,e^{3} f +20 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b \,c^{2} d^{2} e g +16 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) b \,c^{2} d \,e^{2} f -8 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) c^{3} d^{3} g -16 \sqrt {c}\, \mathit {asinh} \left (\frac {\sqrt {-c e x -b e +c d}\, i}{\sqrt {-b e +2 c d}}\right ) c^{3} d^{2} e f +3 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, b c e g -4 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, c^{2} d g -4 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, c^{2} e f -2 \sqrt {e x +d}\, \sqrt {b e -2 c d}\, \sqrt {-b e +2 c d}\, \sqrt {-c e x -b e +c d}\, c^{2} e g x \right )}{4 c^{3} e^{2} \left (b e -2 c d \right )} \] Input:

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

Output:

(i*(3*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b 
**3*e**3*g - 14*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 
 2*c*d))*b**2*c*d*e**2*g - 4*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/ 
sqrt( - b*e + 2*c*d))*b**2*c*e**3*f + 20*sqrt(c)*asinh((sqrt( - b*e + c*d 
- c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*d**2*e*g + 16*sqrt(c)*asinh((sqrt 
( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*b*c**2*d*e**2*f - 8*sqrt(c 
)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c**3*d**3*g - 
 16*sqrt(c)*asinh((sqrt( - b*e + c*d - c*e*x)*i)/sqrt( - b*e + 2*c*d))*c** 
3*d**2*e*f + 3*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( 
- b*e + c*d - c*e*x)*b*c*e*g - 4*sqrt(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b 
*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c**2*d*g - 4*sqrt(d + e*x)*sqrt(b*e 
 - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e*x)*c**2*e*f - 2*sqr 
t(d + e*x)*sqrt(b*e - 2*c*d)*sqrt( - b*e + 2*c*d)*sqrt( - b*e + c*d - c*e* 
x)*c**2*e*g*x))/(4*c**3*e**2*(b*e - 2*c*d))