\(\int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx\) [42]

Optimal result

Integrand size = 37, antiderivative size = 122 \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}-\frac {(C e-B f) \arcsin (d x)}{d f^2}+\frac {\left (C e^2-B e f+A f^2\right ) \arctan \left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{f^2 \sqrt {d^2 e^2-f^2}} \] Output:

-C*(-d^2*x^2+1)^(1/2)/d^2/f-(-B*f+C*e)*arcsin(d*x)/d/f^2+(A*f^2-B*e*f+C*e^ 
2)*arctan((d^2*e*x+f)/(d^2*e^2-f^2)^(1/2)/(-d^2*x^2+1)^(1/2))/f^2/(d^2*e^2 
-f^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.28 \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\frac {-\frac {C f \sqrt {1-d^2 x^2}}{d^2}+\frac {2 (-C e+B f) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{d}-\frac {2 \sqrt {d^2 e^2-f^2} \left (C e^2+f (-B e+A f)\right ) \arctan \left (\frac {\sqrt {d^2 e^2-f^2} x}{e+f x-e \sqrt {1-d^2 x^2}}\right )}{(d e-f) (d e+f)}}{f^2} \] Input:

Integrate[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]
 

Output:

(-((C*f*Sqrt[1 - d^2*x^2])/d^2) + (2*(-(C*e) + B*f)*ArcTan[(d*x)/(-1 + Sqr 
t[1 - d^2*x^2])])/d - (2*Sqrt[d^2*e^2 - f^2]*(C*e^2 + f*(-(B*e) + A*f))*Ar 
cTan[(Sqrt[d^2*e^2 - f^2]*x)/(e + f*x - e*Sqrt[1 - d^2*x^2])])/((d*e - f)* 
(d*e + f)))/f^2
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {2112, 2185, 25, 27, 719, 223, 488, 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.

\(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {d x+1} (e+f x)} \, dx\)

\(\Big \downarrow \) 2112

\(\displaystyle \int \frac {A+B x+C x^2}{\sqrt {1-d^2 x^2} (e+f x)}dx\)

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {\int -\frac {d^2 f (A f-(C e-B f) x)}{(e+f x) \sqrt {1-d^2 x^2}}dx}{d^2 f^2}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {d^2 f (A f-(C e-B f) x)}{(e+f x) \sqrt {1-d^2 x^2}}dx}{d^2 f^2}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {A f-(C e-B f) x}{(e+f x) \sqrt {1-d^2 x^2}}dx}{f}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\)

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {\left (A f^2-B e f+C e^2\right ) \int \frac {1}{(e+f x) \sqrt {1-d^2 x^2}}dx}{f}-\frac {\arcsin (d x) (C e-B f)}{d f}}{f}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {\left (A f^2-B e f+C e^2\right ) \int \frac {1}{-d^2 e^2+f^2-\frac {\left (e x d^2+f\right )^2}{1-d^2 x^2}}d\frac {e x d^2+f}{\sqrt {1-d^2 x^2}}}{f}-\frac {\arcsin (d x) (C e-B f)}{d f}}{f}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {\left (A f^2-B e f+C e^2\right ) \arctan \left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right )}{f \sqrt {d^2 e^2-f^2}}-\frac {\arcsin (d x) (C e-B f)}{d f}}{f}-\frac {C \sqrt {1-d^2 x^2}}{d^2 f}\)

Input:

Int[(A + B*x + C*x^2)/(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(e + f*x)),x]
 

Output:

-((C*Sqrt[1 - d^2*x^2])/(d^2*f)) + (-(((C*e - B*f)*ArcSin[d*x])/(d*f)) + ( 
(C*e^2 - B*e*f + A*f^2)*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqrt[1 - 
 d^2*x^2])])/(f*Sqrt[d^2*e^2 - f^2]))/f
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 488

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

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2112

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; F 
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] & 
& EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(288\) vs. \(2(114)=228\).

Time = 1.13 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.37

method	result	size
risch	\(\frac {C \sqrt {x d +1}\, \left (x d -1\right ) \sqrt {\left (-x d +1\right ) \left (x d +1\right )}}{f \,d^{2} \sqrt {-\left (x d +1\right ) \left (x d -1\right )}\, \sqrt {-x d +1}}+\frac {\left (\frac {\left (B f -C e \right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{f \sqrt {d^{2}}}-\frac {\left (A \,f^{2}-B e f +C \,e^{2}\right ) \ln \left (\frac {-\frac {2 \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{x +\frac {e}{f}}\right )}{f^{2} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}\right ) \sqrt {\left (-x d +1\right ) \left (x d +1\right )}}{f \sqrt {-x d +1}\, \sqrt {x d +1}}\)	\(289\)
default	\(\frac {\left (-A \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} x e +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{2} f^{2}+B \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} x e +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{2} e f -C \,\operatorname {csgn}\left (d \right ) \ln \left (\frac {2 d^{2} x e +2 \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f +2 f}{f x +e}\right ) d^{2} e^{2}+B \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d \,f^{2} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}-C \,\operatorname {csgn}\left (d \right ) f^{2} \sqrt {-d^{2} x^{2}+1}\, \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}-C \arctan \left (\frac {\operatorname {csgn}\left (d \right ) d x}{\sqrt {-d^{2} x^{2}+1}}\right ) d e f \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\right ) \sqrt {-x d +1}\, \sqrt {x d +1}\, \operatorname {csgn}\left (d \right )}{\sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, f^{3} \sqrt {-d^{2} x^{2}+1}\, d^{2}}\)	\(373\)

Input:

int((C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)/(f*x+e),x,method=_RETURNVER 
BOSE)
 

Output:

C/f/d^2*(d*x+1)^(1/2)*(d*x-1)/(-(d*x+1)*(d*x-1))^(1/2)*((-d*x+1)*(d*x+1))^ 
(1/2)/(-d*x+1)^(1/2)+1/f*((B*f-C*e)/f/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d 
^2*x^2+1)^(1/2))-(A*f^2-B*e*f+C*e^2)/f^2/(-(d^2*e^2-f^2)/f^2)^(1/2)*ln((-2 
*(d^2*e^2-f^2)/f^2+2/f*d^2*e*(x+e/f)+2*(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*(x 
+e/f)^2+2/f*d^2*e*(x+e/f)-(d^2*e^2-f^2)/f^2)^(1/2))/(x+e/f)))*((-d*x+1)*(d 
*x+1))^(1/2)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (114) = 228\).

Time = 3.66 (sec) , antiderivative size = 493, normalized size of antiderivative = 4.04 \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\left [-\frac {{\left (C d^{2} e^{2} - B d^{2} e f + A d^{2} f^{2}\right )} \sqrt {-d^{2} e^{2} + f^{2}} \log \left (\frac {d^{2} e f x + f^{2} - \sqrt {-d^{2} e^{2} + f^{2}} {\left (d^{2} e x + f\right )} - {\left (\sqrt {-d^{2} e^{2} + f^{2}} \sqrt {-d x + 1} f + {\left (d^{2} e^{2} - f^{2}\right )} \sqrt {-d x + 1}\right )} \sqrt {d x + 1}}{f x + e}\right ) + {\left (C d^{2} e^{2} f - C f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} - 2 \, {\left (C d^{3} e^{3} - B d^{3} e^{2} f - C d e f^{2} + B d f^{3}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{4} e^{2} f^{2} - d^{2} f^{4}}, \frac {2 \, {\left (C d^{2} e^{2} - B d^{2} e f + A d^{2} f^{2}\right )} \sqrt {d^{2} e^{2} - f^{2}} \arctan \left (-\frac {\sqrt {d^{2} e^{2} - f^{2}} \sqrt {d x + 1} \sqrt {-d x + 1} e - \sqrt {d^{2} e^{2} - f^{2}} {\left (f x + e\right )}}{{\left (d^{2} e^{2} - f^{2}\right )} x}\right ) - {\left (C d^{2} e^{2} f - C f^{3}\right )} \sqrt {d x + 1} \sqrt {-d x + 1} + 2 \, {\left (C d^{3} e^{3} - B d^{3} e^{2} f - C d e f^{2} + B d f^{3}\right )} \arctan \left (\frac {\sqrt {d x + 1} \sqrt {-d x + 1} - 1}{d x}\right )}{d^{4} e^{2} f^{2} - d^{2} f^{4}}\right ] \] Input:

integrate((C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)/(f*x+e),x, algorithm= 
"fricas")
 

Output:

[-((C*d^2*e^2 - B*d^2*e*f + A*d^2*f^2)*sqrt(-d^2*e^2 + f^2)*log((d^2*e*f*x 
 + f^2 - sqrt(-d^2*e^2 + f^2)*(d^2*e*x + f) - (sqrt(-d^2*e^2 + f^2)*sqrt(- 
d*x + 1)*f + (d^2*e^2 - f^2)*sqrt(-d*x + 1))*sqrt(d*x + 1))/(f*x + e)) + ( 
C*d^2*e^2*f - C*f^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) - 2*(C*d^3*e^3 - B*d^3*e 
^2*f - C*d*e*f^2 + B*d*f^3)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x 
)))/(d^4*e^2*f^2 - d^2*f^4), (2*(C*d^2*e^2 - B*d^2*e*f + A*d^2*f^2)*sqrt(d 
^2*e^2 - f^2)*arctan(-(sqrt(d^2*e^2 - f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1)*e 
- sqrt(d^2*e^2 - f^2)*(f*x + e))/((d^2*e^2 - f^2)*x)) - (C*d^2*e^2*f - C*f 
^3)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*(C*d^3*e^3 - B*d^3*e^2*f - C*d*e*f^2 
+ B*d*f^3)*arctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^4*e^2*f^2 
- d^2*f^4)]
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\int \frac {A + B x + C x^{2}}{\left (e + f x\right ) \sqrt {- d x + 1} \sqrt {d x + 1}}\, dx \] Input:

integrate((C*x**2+B*x+A)/(-d*x+1)**(1/2)/(d*x+1)**(1/2)/(f*x+e),x)
 

Output:

Integral((A + B*x + C*x**2)/((e + f*x)*sqrt(-d*x + 1)*sqrt(d*x + 1)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)/(f*x+e),x, algorithm= 
"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>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)/(f*x+e),x, algorithm= 
"giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 17.47 (sec) , antiderivative size = 5803, normalized size of antiderivative = 47.57 \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\text {Too large to display} \] Input:

int((A + B*x + C*x^2)/((e + f*x)*(1 - d*x)^(1/2)*(d*x + 1)^(1/2)),x)
 

Output:

(4*C*e*atan((37748736*C^5*d^4*e^10*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2 
) - 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*d^2*e 
^8*f^2)) + (67108864*C^5*e^6*f^4*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/2) 
- 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*d^2*e^8 
*f^2)) - (100663296*C^5*d^2*e^8*f^2*((1 - d*x)^(1/2) - 1))/(((d*x + 1)^(1/ 
2) - 1)*(37748736*C^5*d^4*e^10 + 67108864*C^5*e^6*f^4 - 100663296*C^5*d^2* 
e^8*f^2))))/(d*f^2) - (4*B*atan((67108864*B^5*e*f^4*((1 - d*x)^(1/2) - 1)) 
/(((d*x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 10066 
3296*B^5*d^2*e^3*f^2)) + (37748736*B^5*d^4*e^5*((1 - d*x)^(1/2) - 1))/(((d 
*x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 100663296* 
B^5*d^2*e^3*f^2)) - (100663296*B^5*d^2*e^3*f^2*((1 - d*x)^(1/2) - 1))/(((d 
*x + 1)^(1/2) - 1)*(67108864*B^5*e*f^4 + 37748736*B^5*d^4*e^5 - 100663296* 
B^5*d^2*e^3*f^2))))/(d*f) - (8*C*((1 - d*x)^(1/2) - 1)^2)/(f*((d*x + 1)^(1 
/2) - 1)^2*(d^2 + (2*d^2*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 
+ (d^2*((1 - d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4)) - (A*atan((f^2*1 
i - d^2*e^2*1i - (f^2*((1 - d*x)^(1/2) - 1)^2*1i)/((d*x + 1)^(1/2) - 1)^2 
+ (d^2*e^2*((1 - d*x)^(1/2) - 1)^2*1i)/((d*x + 1)^(1/2) - 1)^2)/(f*(f + d* 
e)^(1/2)*(f - d*e)^(1/2) - (f*((1 - d*x)^(1/2) - 1)^2*(f + d*e)^(1/2)*(f - 
 d*e)^(1/2))/((d*x + 1)^(1/2) - 1)^2 + (2*d*e*((1 - d*x)^(1/2) - 1)*(f + d 
*e)^(1/2)*(f - d*e)^(1/2))/((d*x + 1)^(1/2) - 1)))*2i)/((f + d*e)^(1/2)...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 535, normalized size of antiderivative = 4.39 \[ \int \frac {A+B x+C x^2}{\sqrt {1-d x} \sqrt {1+d x} (e+f x)} \, dx=\frac {-2 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) b \,d^{3} e^{2} f +2 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) b d \,f^{3}+2 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) c \,d^{3} e^{3}-2 \mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right ) c d e \,f^{2}-2 \sqrt {d e +f}\, \sqrt {d e -f}\, \mathit {atan} \left (\frac {\sqrt {d e +f}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right )}{2}\right )-\sqrt {f}\, \sqrt {2}}{\sqrt {d e -f}}\right ) a \,d^{2} f^{2}+2 \sqrt {d e +f}\, \sqrt {d e -f}\, \mathit {atan} \left (\frac {\sqrt {d e +f}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right )}{2}\right )-\sqrt {f}\, \sqrt {2}}{\sqrt {d e -f}}\right ) b \,d^{2} e f -2 \sqrt {d e +f}\, \sqrt {d e -f}\, \mathit {atan} \left (\frac {\sqrt {d e +f}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right )}{2}\right )-\sqrt {f}\, \sqrt {2}}{\sqrt {d e -f}}\right ) c \,d^{2} e^{2}-2 \sqrt {d e +f}\, \sqrt {d e -f}\, \mathit {atan} \left (\frac {\sqrt {d e +f}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right )}{2}\right )+\sqrt {f}\, \sqrt {2}}{\sqrt {d e -f}}\right ) a \,d^{2} f^{2}+2 \sqrt {d e +f}\, \sqrt {d e -f}\, \mathit {atan} \left (\frac {\sqrt {d e +f}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right )}{2}\right )+\sqrt {f}\, \sqrt {2}}{\sqrt {d e -f}}\right ) b \,d^{2} e f -2 \sqrt {d e +f}\, \sqrt {d e -f}\, \mathit {atan} \left (\frac {\sqrt {d e +f}\, \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +1}}{\sqrt {2}}\right )}{2}\right )+\sqrt {f}\, \sqrt {2}}{\sqrt {d e -f}}\right ) c \,d^{2} e^{2}-\sqrt {d x +1}\, \sqrt {-d x +1}\, c \,d^{2} e^{2} f +\sqrt {d x +1}\, \sqrt {-d x +1}\, c \,f^{3}}{d^{2} f^{2} \left (d^{2} e^{2}-f^{2}\right )} \] Input:

int((C*x^2+B*x+A)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)/(f*x+e),x)
 

Output:

( - 2*asin(sqrt( - d*x + 1)/sqrt(2))*b*d**3*e**2*f + 2*asin(sqrt( - d*x + 
1)/sqrt(2))*b*d*f**3 + 2*asin(sqrt( - d*x + 1)/sqrt(2))*c*d**3*e**3 - 2*as 
in(sqrt( - d*x + 1)/sqrt(2))*c*d*e*f**2 - 2*sqrt(d*e + f)*sqrt(d*e - f)*at 
an((sqrt(d*e + f)*tan(asin(sqrt( - d*x + 1)/sqrt(2))/2) - sqrt(f)*sqrt(2)) 
/sqrt(d*e - f))*a*d**2*f**2 + 2*sqrt(d*e + f)*sqrt(d*e - f)*atan((sqrt(d*e 
 + f)*tan(asin(sqrt( - d*x + 1)/sqrt(2))/2) - sqrt(f)*sqrt(2))/sqrt(d*e - 
f))*b*d**2*e*f - 2*sqrt(d*e + f)*sqrt(d*e - f)*atan((sqrt(d*e + f)*tan(asi 
n(sqrt( - d*x + 1)/sqrt(2))/2) - sqrt(f)*sqrt(2))/sqrt(d*e - f))*c*d**2*e* 
*2 - 2*sqrt(d*e + f)*sqrt(d*e - f)*atan((sqrt(d*e + f)*tan(asin(sqrt( - d* 
x + 1)/sqrt(2))/2) + sqrt(f)*sqrt(2))/sqrt(d*e - f))*a*d**2*f**2 + 2*sqrt( 
d*e + f)*sqrt(d*e - f)*atan((sqrt(d*e + f)*tan(asin(sqrt( - d*x + 1)/sqrt( 
2))/2) + sqrt(f)*sqrt(2))/sqrt(d*e - f))*b*d**2*e*f - 2*sqrt(d*e + f)*sqrt 
(d*e - f)*atan((sqrt(d*e + f)*tan(asin(sqrt( - d*x + 1)/sqrt(2))/2) + sqrt 
(f)*sqrt(2))/sqrt(d*e - f))*c*d**2*e**2 - sqrt(d*x + 1)*sqrt( - d*x + 1)*c 
*d**2*e**2*f + sqrt(d*x + 1)*sqrt( - d*x + 1)*c*f**3)/(d**2*f**2*(d**2*e** 
2 - f**2))