\(\int \frac {\sqrt {a+b x} \sqrt {a c-b c x} (A+B x+C x^2)}{e+f x} \, dx\) [49]

Optimal result

Integrand size = 40, antiderivative size = 316 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{e+f x} \, dx=\frac {\left (2 A b f^2+(2 b e+a f) (C e-B f)\right ) \sqrt {a+b x} \sqrt {a c-b c x}}{2 b f^3}-\frac {(C e-B f) (a+b x)^{3/2} \sqrt {a c-b c x}}{2 b f^2}-\frac {C (a+b x)^{3/2} (a c-b c x)^{3/2}}{3 b^2 c f}-\frac {\sqrt {c} \left (a^2 f^2 (C e-B f)-2 b^2 \left (C e^3-e f (B e-A f)\right )\right ) \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{b f^4}-\frac {2 \sqrt {c} \sqrt {b e-a f} \sqrt {b e+a f} \left (C e^2-B e f+A f^2\right ) \arctan \left (\frac {\sqrt {c} \sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a c-b c x}}\right )}{f^4} \] Output:

1/2*(2*A*b*f^2+(a*f+2*b*e)*(-B*f+C*e))*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b/ 
f^3-1/2*(-B*f+C*e)*(b*x+a)^(3/2)*(-b*c*x+a*c)^(1/2)/b/f^2-1/3*C*(b*x+a)^(3 
/2)*(-b*c*x+a*c)^(3/2)/b^2/c/f-c^(1/2)*(a^2*f^2*(-B*f+C*e)-2*b^2*(C*e^3-e* 
f*(-A*f+B*e)))*arctan(c^(1/2)*(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2))/b/f^4-2*c^ 
(1/2)*(-a*f+b*e)^(1/2)*(a*f+b*e)^(1/2)*(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))/f^4
 

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{e+f x} \, dx=\frac {\sqrt {c (a-b x)} \left (\frac {f \sqrt {a+b x} \left (-2 a^2 C f^2+b^2 \left (3 f (-2 B e+2 A f+B f x)+C \left (6 e^2-3 e f x+2 f^2 x^2\right )\right )\right )}{b^2}+\frac {6 \left (a^2 f^2 (-C e+B f)+2 b^2 \left (C e^3+e f (-B e+A f)\right )\right ) \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{b \sqrt {a-b x}}-\frac {12 \left (b^2 e^2-a^2 f^2\right ) \left (C e^2+f (-B e+A f)\right ) \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{\sqrt {b e-a f} \sqrt {b e+a f} \sqrt {a-b x}}\right )}{6 f^4} \] Input:

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

Output:

(Sqrt[c*(a - b*x)]*((f*Sqrt[a + b*x]*(-2*a^2*C*f^2 + b^2*(3*f*(-2*B*e + 2* 
A*f + B*f*x) + C*(6*e^2 - 3*e*f*x + 2*f^2*x^2))))/b^2 + (6*(a^2*f^2*(-(C*e 
) + B*f) + 2*b^2*(C*e^3 + e*f*(-(B*e) + A*f)))*ArcTan[Sqrt[a + b*x]/Sqrt[a 
 - b*x]])/(b*Sqrt[a - b*x]) - (12*(b^2*e^2 - a^2*f^2)*(C*e^2 + f*(-(B*e) + 
 A*f))*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]*Sqrt[a - b*x])))/(6*f^4)
 

Rubi [A] (verified)

Time = 0.89 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2113, 2185, 27, 682, 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.

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

\(\Big \downarrow \) 2113

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (-\frac {\int -\frac {3 b^2 c f (A f-(C e-B f) x) \sqrt {a^2 c-b^2 c x^2}}{e+f x}dx}{3 b^2 c f^2}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\int \frac {(A f-(C e-B f) x) \sqrt {a^2 c-b^2 c x^2}}{e+f x}dx}{f}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 682

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 \left (A f^2-B e f+C e^2\right )-f x (C e-B f)\right )}{2 f^2}-\frac {\int -\frac {b^2 c^2 \left (f \left (C e^2-f (B e-2 A f)\right ) a^2+\left (2 A b^2 e f^2+(C e-B f) \left (2 b^2 e^2-a^2 f^2\right )\right ) x\right )}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{2 b^2 c f^2}}{f}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {\int \frac {b^2 c^2 \left (f \left (C e^2-f (B e-2 A f)\right ) a^2+\left (2 A b^2 e f^2+(C e-B f) \left (2 b^2 e^2-a^2 f^2\right )\right ) x\right )}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{2 b^2 c f^2}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 \left (A f^2-B e f+C e^2\right )-f x (C e-B f)\right )}{2 f^2}}{f}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {c \int \frac {f \left (C e^2-f (B e-2 A f)\right ) a^2+\left (2 A b^2 e f^2+(C e-B f) \left (2 b^2 e^2-a^2 f^2\right )\right ) x}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{2 f^2}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 \left (A f^2-B e f+C e^2\right )-f x (C e-B f)\right )}{2 f^2}}{f}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {c \left (\frac {\left (\left (2 b^2 e^2-a^2 f^2\right ) (C e-B f)+2 A b^2 e f^2\right ) \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}}dx}{f}-\frac {2 (b e-a f) (a f+b e) \left (A f^2-B e f+C e^2\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{f}\right )}{2 f^2}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 \left (A f^2-B e f+C e^2\right )-f x (C e-B f)\right )}{2 f^2}}{f}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {c \left (\frac {\left (\left (2 b^2 e^2-a^2 f^2\right ) (C e-B f)+2 A b^2 e f^2\right ) \int \frac {1}{\frac {b^2 c x^2}{a^2 c-b^2 c x^2}+1}d\frac {x}{\sqrt {a^2 c-b^2 c x^2}}}{f}-\frac {2 (b e-a f) (a f+b e) \left (A f^2-B e f+C e^2\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{f}\right )}{2 f^2}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 \left (A f^2-B e f+C e^2\right )-f x (C e-B f)\right )}{2 f^2}}{f}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {c \left (\frac {\arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (\left (2 b^2 e^2-a^2 f^2\right ) (C e-B f)+2 A b^2 e f^2\right )}{b \sqrt {c} f}-\frac {2 (b e-a f) (a f+b e) \left (A f^2-B e f+C e^2\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{f}\right )}{2 f^2}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 \left (A f^2-B e f+C e^2\right )-f x (C e-B f)\right )}{2 f^2}}{f}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {c \left (\frac {2 (b e-a f) (a f+b e) \left (A f^2-B e f+C e^2\right ) \int \frac {1}{-b^2 c e^2+a^2 c f^2-\frac {\left (c f a^2+b^2 c e x\right )^2}{a^2 c-b^2 c x^2}}d\frac {c f a^2+b^2 c e x}{\sqrt {a^2 c-b^2 c x^2}}}{f}+\frac {\arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (\left (2 b^2 e^2-a^2 f^2\right ) (C e-B f)+2 A b^2 e f^2\right )}{b \sqrt {c} f}\right )}{2 f^2}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 \left (A f^2-B e f+C e^2\right )-f x (C e-B f)\right )}{2 f^2}}{f}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {c \left (\frac {\arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (\left (2 b^2 e^2-a^2 f^2\right ) (C e-B f)+2 A b^2 e f^2\right )}{b \sqrt {c} f}-\frac {2 (b e-a f) (a f+b e) \left (A f^2-B e f+C e^2\right ) \arctan \left (\frac {a^2 c f+b^2 c e x}{\sqrt {c} \sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{\sqrt {c} f \sqrt {b^2 e^2-a^2 f^2}}\right )}{2 f^2}+\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 \left (A f^2-B e f+C e^2\right )-f x (C e-B f)\right )}{2 f^2}}{f}-\frac {C \left (a^2 c-b^2 c x^2\right )^{3/2}}{3 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(-1/3*(C*(a^2*c - b^2*c*x^2)^(3/2))/(b^2* 
c*f) + (((2*(C*e^2 - B*e*f + A*f^2) - f*(C*e - B*f)*x)*Sqrt[a^2*c - b^2*c* 
x^2])/(2*f^2) + (c*(((2*A*b^2*e*f^2 + (C*e - B*f)*(2*b^2*e^2 - a^2*f^2))*A 
rcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(b*Sqrt[c]*f) - (2*(b*e - a* 
f)*(b*e + a*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])))/(2*f^2))/f))/Sqrt[a^2*c - b^2*c*x^2]
 

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 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

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 = 1.05 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.43

method	result	size
risch	\(\frac {\left (2 C \,x^{2} f^{2} b^{2}+3 B \,b^{2} f^{2} x -3 C \,b^{2} e f x +6 A \,b^{2} f^{2}-6 e B \,b^{2} f -2 a^{2} C \,f^{2}+6 e^{2} b^{2} C \right ) \left (-b x +a \right ) \sqrt {b x +a}\, c}{6 b^{2} f^{3} \sqrt {-c \left (b x -a \right )}}+\frac {\left (\frac {\left (2 A \,b^{2} e \,f^{2}+a^{2} B \,f^{3}-2 e^{2} B \,b^{2} f -C \,a^{2} e \,f^{2}+2 e^{3} b^{2} C \right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right )}{f \sqrt {b^{2} c}}-\frac {2 \left (a^{2} A \,f^{4}-A \,b^{2} e^{2} f^{2}-B \,a^{2} e \,f^{3}+e^{3} B \,b^{2} f +C \,a^{2} e^{2} f^{2}-e^{4} b^{2} C \right ) \ln \left (\frac {\frac {2 c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}+\frac {2 b^{2} c e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}\, \sqrt {-\left (x +\frac {e}{f}\right )^{2} b^{2} c +\frac {2 b^{2} c e \left (x +\frac {e}{f}\right )}{f}+\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}}{x +\frac {e}{f}}\right )}{f^{2} \sqrt {\frac {c \left (a^{2} f^{2}-b^{2} e^{2}\right )}{f^{2}}}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c}{2 f^{3} \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(452\)
default	\(\text {Expression too large to display}\)	\(1284\)

Input:

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

Output:

1/6*(2*C*b^2*f^2*x^2+3*B*b^2*f^2*x-3*C*b^2*e*f*x+6*A*b^2*f^2-6*B*b^2*e*f-2 
*C*a^2*f^2+6*C*b^2*e^2)/b^2*(-b*x+a)*(b*x+a)^(1/2)/f^3/(-c*(b*x-a))^(1/2)* 
c+1/2/f^3*((2*A*b^2*e*f^2+B*a^2*f^3-2*B*b^2*e^2*f-C*a^2*e*f^2+2*C*b^2*e^3) 
/f/(b^2*c)^(1/2)*arctan((b^2*c)^(1/2)*x/(-b^2*c*x^2+a^2*c)^(1/2))-2*(A*a^2 
*f^4-A*b^2*e^2*f^2-B*a^2*e*f^3+B*b^2*e^3*f+C*a^2*e^2*f^2-C*b^2*e^4)/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)*c
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Giac [F(-2)]

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

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(C*x^2+B*x+A)/(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 = 71.00 (sec) , antiderivative size = 24910, normalized size of antiderivative = 78.83 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{e+f x} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((((C*b^2*c*e^4 - C*a^2*c*e^2*f^2)*(((C*b^2*c*e^4 - C*a^2*c*e^2*f^2)* 
((4096*(4*C^2*a^10*c^8*f^14 + 9*C^2*a^2*b^8*c^8*e^8*f^6 - 18*C^2*a^4*b^6*c 
^8*e^6*f^8 + 18*C^2*a^6*b^4*c^8*e^4*f^10 - 11*C^2*a^8*b^2*c^8*e^2*f^12))/( 
b^8*e^4*f^12) + ((C*b^2*c*e^4 - C*a^2*c*e^2*f^2)*((4096*(12*C*a^(9/2)*b^2* 
c^5*f^15*(a*c)^(5/2) + 30*C*a^(3/2)*b^6*c^6*e^4*f^11*(a*c)^(3/2) - 24*C*a^ 
(5/2)*b^4*c^5*e^2*f^13*(a*c)^(5/2) - 18*C*a^(7/2)*b^4*c^6*e^2*f^13*(a*c)^( 
3/2)))/(b^8*e^4*f^12) - ((C*b^2*c*e^4 - C*a^2*c*e^2*f^2)*((4096*(7*a^4*b^4 
*c^7*f^16 - 9*a^2*b^6*c^7*e^2*f^14))/(b^8*e^4*f^12) + (16384*((a*c - b*c*x 
)^(1/2) - (a*c)^(1/2))*(5*a^(5/2)*b^2*c^4*f^16*(a*c)^(5/2) - 6*a^(3/2)*b^4 
*c^5*e^2*f^14*(a*c)^(3/2)))/(b^7*e^5*f^11*((a + b*x)^(1/2) - a^(1/2))) + ( 
4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(11*a^4*b^4*c^6*f^16 - 9*a^2*b^ 
6*c^6*e^2*f^14))/(b^8*e^4*f^12*((a + b*x)^(1/2) - a^(1/2))^2)))/(f^4*(a^2* 
c*f^2 - b^2*c*e^2)^(1/2)) + (4096*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2*(6 
8*C*a^(9/2)*b^2*c^4*f^15*(a*c)^(5/2) + 90*C*a^(3/2)*b^6*c^5*e^4*f^11*(a*c) 
^(3/2) - 96*C*a^(5/2)*b^4*c^4*e^2*f^13*(a*c)^(5/2) - 62*C*a^(7/2)*b^4*c^5* 
e^2*f^13*(a*c)^(3/2)))/(b^8*e^4*f^12*((a + b*x)^(1/2) - a^(1/2))^2) + (163 
84*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(10*C*a^8*c^7*f^15 + 22*C*a^4*b^4*c 
^7*e^4*f^11 - 32*C*a^6*b^2*c^7*e^2*f^13))/(b^7*e^5*f^11*((a + b*x)^(1/2) - 
 a^(1/2)))))/(f^4*(a^2*c*f^2 - b^2*c*e^2)^(1/2)) + (16384*((a*c - b*c*x)^( 
1/2) - (a*c)^(1/2))*(5*C^2*a^(13/2)*c^5*e^2*f^12*(a*c)^(5/2) + 3*C^2*a^...
 

Reduce [B] (verification not implemented)

Time = 5.83 (sec) , antiderivative size = 3341, normalized size of antiderivative = 10.57 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{e+f x} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(c)*( - 6*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**2*f**4 + 6*as 
in(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b*c*e*f**3 - 18*asin(sqrt(a - b*x 
)/(sqrt(a)*sqrt(2)))*a**2*b**3*e*f**3 + 6*asin(sqrt(a - b*x)/(sqrt(a)*sqrt 
(2)))*a**2*b**2*c*e**2*f**2 - 12*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a*b 
**3*c*e**3*f + 12*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*b**5*e**3*f - 12*a 
sin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*b**4*c*e**4 - 12*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)*s 
qrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)))*a*b**2*f** 
2 + 12*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)*s 
qrt(a*f + b*e)))*b**3*e*f - 12*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 - 12*sqrt(2*sqrt(f)*s 
qrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)*atan((tan...