\(\int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx\) [56]

Optimal result

Integrand size = 40, antiderivative size = 186 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=-\frac {C \sqrt {a+b x} \sqrt {a c-b c x}}{b^2 c f}-\frac {2 (C e-B f) \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{b \sqrt {c} f^2}+\frac {2 \left (C e^2-B e f+A f^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a c-b c x}}\right )}{\sqrt {c} f^2 \sqrt {b e-a f} \sqrt {b e+a f}} \] Output:

-C*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^2/c/f-2*(-B*f+C*e)*arctan(c^(1/2)*(b 
*x+a)^(1/2)/(-b*c*x+a*c)^(1/2))/b/c^(1/2)/f^2+2*(A*f^2-B*e*f+C*e^2)*arctan 
(c^(1/2)*(a*f+b*e)^(1/2)*(b*x+a)^(1/2)/(-a*f+b*e)^(1/2)/(-b*c*x+a*c)^(1/2) 
)/c^(1/2)/f^2/(-a*f+b*e)^(1/2)/(a*f+b*e)^(1/2)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.96 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\frac {\frac {C f (-a+b x) \sqrt {a+b x}}{b^2}-\frac {2 (C e-B f) \sqrt {a-b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{b}+\frac {2 \left (C e^2+f (-B e+A f)\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{\sqrt {b e-a f} \sqrt {b e+a f}}}{f^2 \sqrt {c (a-b x)}} \] Input:

Integrate[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)),x]
 

Output:

((C*f*(-a + b*x)*Sqrt[a + b*x])/b^2 - (2*(C*e - B*f)*Sqrt[a - b*x]*ArcTan[ 
Sqrt[a + b*x]/Sqrt[a - b*x]])/b + (2*(C*e^2 + f*(-(B*e) + A*f))*Sqrt[a - b 
*x]*ArcTan[(Sqrt[b*e + a*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]*Sqrt[a - b*x]) 
])/(Sqrt[b*e - a*f]*Sqrt[b*e + a*f]))/(f^2*Sqrt[c*(a - b*x)])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {2113, 2185, 25, 27, 719, 224, 216, 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.

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

\(\Big \downarrow \) 2113

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 217

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {\left (A f^2-B e f+C e^2\right ) \arctan \left (\frac {a^2 c f+b^2 c e x}{\sqrt {c} \sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{\sqrt {c} f \sqrt {b^2 e^2-a^2 f^2}}-\frac {(C e-B f) \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{b \sqrt {c} f}}{f}-\frac {C \sqrt {a^2 c-b^2 c x^2}}{b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

Input:

Int[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)),x]
 

Output:

(Sqrt[a^2*c - b^2*c*x^2]*(-((C*Sqrt[a^2*c - b^2*c*x^2])/(b^2*c*f)) + (-((( 
C*e - B*f)*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(b*Sqrt[c]*f)) + 
 ((C*e^2 - B*e*f + A*f^2)*ArcTan[(a^2*c*f + b^2*c*e*x)/(Sqrt[c]*Sqrt[b^2*e 
^2 - a^2*f^2]*Sqrt[a^2*c - b^2*c*x^2])])/(Sqrt[c]*f*Sqrt[b^2*e^2 - a^2*f^2 
]))/f))/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x])
 

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 216

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

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 224

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

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 2113

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

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 [A] (verified)

Time = 1.22 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.61

method	result	size
risch	\(-\frac {C \sqrt {b x +a}\, \left (-b x +a \right )}{f \,b^{2} \sqrt {-c \left (b x -a \right )}}+\frac {\left (\frac {\left (B f -C e \right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{f \sqrt {b^{2} c}}-\frac {\left (A \,f^{2}-B e f +C \,e^{2}\right ) \ln \left (\frac {\frac {2 c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}+\frac {2 b^{2} c e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {-\left (x +\frac {e}{f}\right )^{2} b^{2} c +\frac {2 b^{2} c e \left (x +\frac {e}{f}\right )}{f}+\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}}{x +\frac {e}{f}}\right )}{f^{2} \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{f \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(299\)
default	\(\frac {\left (-A \ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, f}{f x +e}\right ) b^{2} c \,f^{2} \sqrt {b^{2} c}+B \ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, f}{f x +e}\right ) b^{2} c e f \sqrt {b^{2} c}+B \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) b^{2} c \,f^{2} \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}-C \ln \left (\frac {2 b^{2} c e x +2 a^{2} c f +2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, f}{f x +e}\right ) b^{2} c \,e^{2} \sqrt {b^{2} c}-C \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) b^{2} c e f \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}-C \,f^{2} \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\right ) \sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}}{b^{2} \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {b^{2} c}\, f^{3} c \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\)	\(487\)

Input:

int((C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e),x,method=_RETUR 
NVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\text {Timed out} \] Input:

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e),x, algori 
thm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((A + B*x + C*x**2)/(sqrt(-c*(-a + b*x))*sqrt(a + b*x)*(e + f*x)), 
 x)
 

Maxima [F(-2)]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e),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((4*b^2*c>0)', see `assume?` for 
more detai
 

Giac [F(-2)]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e),x, algori 
thm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 28.70 (sec) , antiderivative size = 9298, normalized size of antiderivative = 49.99 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx=\text {Too large to display} \] Input:

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

Output:

(B*a*e*atan(((B*a*e*((4096*(32*B^3*a^(17/2)*c^3*e*f^2*(a*c)^(5/2) + 24*B^3 
*a^(15/2)*b^2*c^4*e^3*(a*c)^(3/2)))/(a^6*b^8*e^6) - (4096*(32*B^3*a^(17/2) 
*c^2*e*f^2*(a*c)^(5/2) - 96*B^3*a^(15/2)*b^2*c^3*e^3*(a*c)^(3/2))*((a*c - 
b*c*x)^(1/2) - (a*c)^(1/2))^2)/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2) 
 - (B*a*e*((4096*(16*B^2*a^12*c^6*f^4 + 9*B^2*a^8*b^4*c^6*e^4))/(a^6*b^8*e 
^6) + (B*a*e*((4096*(24*B*a^(17/2)*b^2*c^4*e*f^4*(a*c)^(5/2) - 30*B*a^(15/ 
2)*b^4*c^5*e^3*f^2*(a*c)^(3/2)))/(a^6*b^8*e^6) + (16384*(20*B*a^12*c^6*f^5 
 - 22*B*a^10*b^2*c^6*e^2*f^3)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6*b^ 
7*e^6*((a + b*x)^(1/2) - a^(1/2))) + (B*a*e*((4096*(9*a^8*b^6*c^7*e^4*f^2 
- 7*a^10*b^4*c^7*e^2*f^4))/(a^6*b^8*e^6) + (4096*(9*a^8*b^6*c^6*e^4*f^2 - 
11*a^10*b^4*c^6*e^2*f^4)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(a^6*b^8*e 
^6*((a + b*x)^(1/2) - a^(1/2))^2) - (16384*(5*a^(17/2)*b^2*c^4*e*f^5*(a*c) 
^(5/2) - 6*a^(15/2)*b^4*c^5*e^3*f^3*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a 
*c)^(1/2)))/(a^6*b^7*e^6*((a + b*x)^(1/2) - a^(1/2)))))/(f*(a^4*c*f^2 - a^ 
2*b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(96*B*a^ 
(17/2)*b^2*c^3*e*f^4*(a*c)^(5/2) - 90*B*a^(15/2)*b^4*c^4*e^3*f^2*(a*c)^(3/ 
2)))/(a^6*b^8*e^6*((a + b*x)^(1/2) - a^(1/2))^2)))/(f*(a^4*c*f^2 - a^2*b^2 
*c*e^2)^(1/2)) + (16384*(8*B^2*a^(17/2)*c^3*e*f^3*(a*c)^(5/2) + 3*B^2*a^(1 
5/2)*b^2*c^4*e^3*f*(a*c)^(3/2))*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(a^6* 
b^7*e^6*((a + b*x)^(1/2) - a^(1/2))) + (4096*((a*c - b*c*x)^(1/2) - (a*...
 

Reduce [B] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 3194, normalized size of antiderivative = 17.17 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(c)*( - 2*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**2*f**4 + 2*as 
in(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b*c*e*f**3 - 2*asin(sqrt(a - b*x) 
/(sqrt(a)*sqrt(2)))*a**2*b**3*e*f**3 + 2*asin(sqrt(a - b*x)/(sqrt(a)*sqrt( 
2)))*a**2*b**2*c*e**2*f**2 + 2*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a*b** 
4*e**2*f**2 - 2*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a*b**3*c*e**3*f + 2* 
asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*b**5*e**3*f - 2*asin(sqrt(a - b*x)/( 
sqrt(a)*sqrt(2)))*b**4*c*e**4 - 2*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*s 
qrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)*sqrt(a*f - b*e)*sqrt 
(2)*atan((tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt 
(a - b*x)/(sqrt(a)*sqrt(2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b* 
e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)))*a*b**2*f**2 + 2*sqrt(f)*sqrt(a 
)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + 
 b*e)*sqrt(a*f - b*e)*sqrt(2)*atan((tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2 
)))/2)*a*f + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*b*e)/(sqrt(2*sqr 
t(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)))*b**3 
*e*f - 2*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 
3*a*f + b*e)*sqrt(a*f + b*e)*sqrt(a*f - b*e)*sqrt(2)*atan((tan(asin(sqrt(a 
 - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2 
)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e) 
*sqrt(a*f + b*e)))*b**2*c*e**2 - 2*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*...