\(\int \sqrt {a+b x} \sqrt {a c-b c x} (A+B x+C x^2) \, dx\) [48]

Optimal result

Integrand size = 33, antiderivative size = 170 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right ) \, dx=\frac {1}{8} \left (4 A+\frac {a^2 C}{b^2}\right ) x \sqrt {a+b x} \sqrt {a c-b c x}-\frac {(4 b B-3 a C) (a+b x)^{3/2} (a c-b c x)^{3/2}}{12 b^3 c}-\frac {C (a+b x)^{5/2} (a c-b c x)^{3/2}}{4 b^3 c}+\frac {a^2 \sqrt {c} \left (4 A b^2+a^2 C\right ) \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{4 b^3} \] Output:

1/8*(4*A+a^2*C/b^2)*x*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)-1/12*(4*B*b-3*C*a)* 
(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/b^3/c-1/4*C*(b*x+a)^(5/2)*(-b*c*x+a*c)^(3 
/2)/b^3/c+1/4*a^2*c^(1/2)*(4*A*b^2+C*a^2)*arctan(c^(1/2)*(b*x+a)^(1/2)/(-b 
*c*x+a*c)^(1/2))/b^3
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.72 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {c (a-b x)} \left (b \sqrt {a-b x} \sqrt {a+b x} \left (-a^2 (8 B+3 C x)+2 b^2 x (6 A+x (4 B+3 C x))\right )+6 a^2 \left (4 A b^2+a^2 C\right ) \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{24 b^3 \sqrt {a-b x}} \] Input:

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

Output:

(Sqrt[c*(a - b*x)]*(b*Sqrt[a - b*x]*Sqrt[a + b*x]*(-(a^2*(8*B + 3*C*x)) + 
2*b^2*x*(6*A + x*(4*B + 3*C*x))) + 6*a^2*(4*A*b^2 + a^2*C)*ArcTan[Sqrt[a + 
 b*x]/Sqrt[a - b*x]]))/(24*b^3*Sqrt[a - b*x])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.90, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1189, 83, 646, 40, 45, 218}

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.

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

\(\Big \downarrow \) 1189

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

\(\Big \downarrow \) 83

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

\(\Big \downarrow \) 646

\(\displaystyle \frac {1}{4} \left (\frac {a^2 C}{b^2}+4 A\right ) \int \sqrt {a+b x} \sqrt {a c-b c x}dx-\frac {B (a+b x)^{3/2} (a c-b c x)^{3/2}}{3 b^2 c}-\frac {C x (a+b x)^{3/2} (a c-b c x)^{3/2}}{4 b^2 c}\)

\(\Big \downarrow \) 40

\(\displaystyle \frac {1}{4} \left (\frac {a^2 C}{b^2}+4 A\right ) \left (\frac {1}{2} a^2 c \int \frac {1}{\sqrt {a+b x} \sqrt {a c-b c x}}dx+\frac {1}{2} x \sqrt {a+b x} \sqrt {a c-b c x}\right )-\frac {B (a+b x)^{3/2} (a c-b c x)^{3/2}}{3 b^2 c}-\frac {C x (a+b x)^{3/2} (a c-b c x)^{3/2}}{4 b^2 c}\)

\(\Big \downarrow \) 45

\(\displaystyle \frac {1}{4} \left (\frac {a^2 C}{b^2}+4 A\right ) \left (a^2 c \int \frac {1}{\frac {c (a+b x) b}{a c-b c x}+b}d\frac {\sqrt {a+b x}}{\sqrt {a c-b c x}}+\frac {1}{2} x \sqrt {a+b x} \sqrt {a c-b c x}\right )-\frac {B (a+b x)^{3/2} (a c-b c x)^{3/2}}{3 b^2 c}-\frac {C x (a+b x)^{3/2} (a c-b c x)^{3/2}}{4 b^2 c}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{4} \left (\frac {a^2 C}{b^2}+4 A\right ) \left (\frac {a^2 \sqrt {c} \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{b}+\frac {1}{2} x \sqrt {a+b x} \sqrt {a c-b c x}\right )-\frac {B (a+b x)^{3/2} (a c-b c x)^{3/2}}{3 b^2 c}-\frac {C x (a+b x)^{3/2} (a c-b c x)^{3/2}}{4 b^2 c}\)

Input:

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

Output:

-1/3*(B*(a + b*x)^(3/2)*(a*c - b*c*x)^(3/2))/(b^2*c) - (C*x*(a + b*x)^(3/2 
)*(a*c - b*c*x)^(3/2))/(4*b^2*c) + ((4*A + (a^2*C)/b^2)*((x*Sqrt[a + b*x]* 
Sqrt[a*c - b*c*x])/2 + (a^2*Sqrt[c]*ArcTan[(Sqrt[c]*Sqrt[a + b*x])/Sqrt[a* 
c - b*c*x]])/b))/4
 

Defintions of rubi rules used

rule 40

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x* 
(a + b*x)^m*((c + d*x)^m/(2*m + 1)), x] + Simp[2*a*c*(m/(2*m + 1))   Int[(a 
 + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[ 
b*c + a*d, 0] && IGtQ[m + 1/2, 0]
 

rule 45

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] &&  !GtQ[c, 0]
 

rule 83

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f 
*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
 

rule 218

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

rule 646

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2), x_Symbol] :> Simp[b*x*(c + d*x)^(m + 1)*((e + f*x)^(n + 1)/(d*f*(2*m + 
 3))), x] - Simp[(b*c*e - a*d*f*(2*m + 3))/(d*f*(2*m + 3))   Int[(c + d*x)^ 
m*(e + f*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m, n] && 
 EqQ[d*e + c*f, 0] &&  !LtQ[m, -1]
 

rule 1189

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

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {\left (6 C \,b^{2} x^{3}+8 b^{2} B \,x^{2}+12 A \,b^{2} x -3 C \,a^{2} x -8 a^{2} B \right ) \left (-b x +a \right ) \sqrt {b x +a}\, c}{24 b^{2} \sqrt {-c \left (b x -a \right )}}+\frac {a^{2} \left (4 b^{2} A +a^{2} C \right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c}{8 b^{2} \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(163\)
default	\(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (6 C \,b^{2} x^{3} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}+12 A \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{2} c +8 B \,b^{2} x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}+3 C \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{4} c +12 A \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{2} x -3 C \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{2} x -8 B \,a^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\right )}{24 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{2} \sqrt {b^{2} c}}\)	\(269\)

Input:

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

Output:

1/24*(6*C*b^2*x^3+8*B*b^2*x^2+12*A*b^2*x-3*C*a^2*x-8*B*a^2)/b^2*(-b*x+a)*( 
b*x+a)^(1/2)/(-c*(b*x-a))^(1/2)*c+1/8*a^2*(4*A*b^2+C*a^2)/b^2/(b^2*c)^(1/2 
)*arctan((b^2*c)^(1/2)*x/(-b^2*c*x^2+a^2*c)^(1/2))*(-(b*x+a)*c*(b*x-a))^(1 
/2)/(b*x+a)^(1/2)/(-c*(b*x-a))^(1/2)*c
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.56 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right ) \, dx=\left [\frac {3 \, {\left (C a^{4} + 4 \, A a^{2} b^{2}\right )} \sqrt {-c} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) + 2 \, {\left (6 \, C b^{3} x^{3} + 8 \, B b^{3} x^{2} - 8 \, B a^{2} b - 3 \, {\left (C a^{2} b - 4 \, A b^{3}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{48 \, b^{3}}, -\frac {3 \, {\left (C a^{4} + 4 \, A a^{2} b^{2}\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) - {\left (6 \, C b^{3} x^{3} + 8 \, B b^{3} x^{2} - 8 \, B a^{2} b - 3 \, {\left (C a^{2} b - 4 \, A b^{3}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{24 \, b^{3}}\right ] \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(C*x^2+B*x+A),x, algorithm="fri 
cas")
 

Output:

[1/48*(3*(C*a^4 + 4*A*a^2*b^2)*sqrt(-c)*log(2*b^2*c*x^2 + 2*sqrt(-b*c*x + 
a*c)*sqrt(b*x + a)*b*sqrt(-c)*x - a^2*c) + 2*(6*C*b^3*x^3 + 8*B*b^3*x^2 - 
8*B*a^2*b - 3*(C*a^2*b - 4*A*b^3)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^3 
, -1/24*(3*(C*a^4 + 4*A*a^2*b^2)*sqrt(c)*arctan(sqrt(-b*c*x + a*c)*sqrt(b* 
x + a)*b*sqrt(c)*x/(b^2*c*x^2 - a^2*c)) - (6*C*b^3*x^3 + 8*B*b^3*x^2 - 8*B 
*a^2*b - 3*(C*a^2*b - 4*A*b^3)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^3]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.82 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right ) \, dx=\frac {C a^{4} \sqrt {c} \arcsin \left (\frac {b x}{a}\right )}{8 \, b^{3}} + \frac {A a^{2} \sqrt {c} \arcsin \left (\frac {b x}{a}\right )}{2 \, b} + \frac {1}{2} \, \sqrt {-b^{2} c x^{2} + a^{2} c} A x + \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} C a^{2} x}{8 \, b^{2}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C x}{4 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} B}{3 \, b^{2} c} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(C*x^2+B*x+A),x, algorithm="max 
ima")
 

Output:

1/8*C*a^4*sqrt(c)*arcsin(b*x/a)/b^3 + 1/2*A*a^2*sqrt(c)*arcsin(b*x/a)/b + 
1/2*sqrt(-b^2*c*x^2 + a^2*c)*A*x + 1/8*sqrt(-b^2*c*x^2 + a^2*c)*C*a^2*x/b^ 
2 - 1/4*(-b^2*c*x^2 + a^2*c)^(3/2)*C*x/(b^2*c) - 1/3*(-b^2*c*x^2 + a^2*c)^ 
(3/2)*B/(b^2*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (142) = 284\).

Time = 0.61 (sec) , antiderivative size = 527, normalized size of antiderivative = 3.10 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right ) \, dx=-\frac {24 \, {\left (\frac {2 \, a c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} A a b^{2} - 12 \, {\left (\frac {2 \, a^{2} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} {\left (b x - 2 \, a\right )}\right )} B a b - 12 \, {\left (\frac {2 \, a^{2} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} {\left (b x - 2 \, a\right )}\right )} A b^{2} + 4 \, {\left (\frac {6 \, a^{3} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - {\left ({\left (2 \, b x - 5 \, a\right )} {\left (b x + a\right )} + 9 \, a^{2}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} C a + 4 \, {\left (\frac {6 \, a^{3} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - {\left ({\left (2 \, b x - 5 \, a\right )} {\left (b x + a\right )} + 9 \, a^{2}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} B b - {\left (\frac {18 \, a^{4} c \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}} - {\left (39 \, a^{3} - {\left (2 \, {\left (3 \, b x - 10 \, a\right )} {\left (b x + a\right )} + 43 \, a^{2}\right )} {\left (b x + a\right )}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a}\right )} C}{24 \, b^{3}} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(C*x^2+B*x+A),x, algorithm="gia 
c")
 

Output:

-1/24*(24*(2*a*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a 
*c)))/sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*A*a*b^2 - 12*(2 
*a^2*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt 
(-c) + sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*(b*x - 2*a))*B*a*b - 12*(2 
*a^2*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt 
(-c) + sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*(b*x - 2*a))*A*b^2 + 4*(6* 
a^3*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt( 
-c) - ((2*b*x - 5*a)*(b*x + a) + 9*a^2)*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b* 
x + a))*C*a + 4*(6*a^3*c*log(abs(-sqrt(b*x + a)*sqrt(-c) + sqrt(-(b*x + a) 
*c + 2*a*c)))/sqrt(-c) - ((2*b*x - 5*a)*(b*x + a) + 9*a^2)*sqrt(-(b*x + a) 
*c + 2*a*c)*sqrt(b*x + a))*B*b - (18*a^4*c*log(abs(-sqrt(b*x + a)*sqrt(-c) 
 + sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c) - (39*a^3 - (2*(3*b*x - 10*a)*(b* 
x + a) + 43*a^2)*(b*x + a))*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a))*C)/b 
^3
 

Mupad [B] (verification not implemented)

Time = 15.29 (sec) , antiderivative size = 876, normalized size of antiderivative = 5.15 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

((C*a^4*c^8*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(2*((a + b*x)^(1/2) - a^( 
1/2))) - (C*a^4*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^15)/(2*((a + b*x)^(1 
/2) - a^(1/2))^15) - (35*C*a^4*c^7*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3)/ 
(2*((a + b*x)^(1/2) - a^(1/2))^3) + (273*C*a^4*c^6*((a*c - b*c*x)^(1/2) - 
(a*c)^(1/2))^5)/(2*((a + b*x)^(1/2) - a^(1/2))^5) - (715*C*a^4*c^5*((a*c - 
 b*c*x)^(1/2) - (a*c)^(1/2))^7)/(2*((a + b*x)^(1/2) - a^(1/2))^7) + (715*C 
*a^4*c^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^9)/(2*((a + b*x)^(1/2) - a^(1 
/2))^9) - (273*C*a^4*c^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^11)/(2*((a + 
b*x)^(1/2) - a^(1/2))^11) + (35*C*a^4*c^2*((a*c - b*c*x)^(1/2) - (a*c)^(1/ 
2))^13)/(2*((a + b*x)^(1/2) - a^(1/2))^13))/(b^3*c^8 + (b^3*((a*c - b*c*x) 
^(1/2) - (a*c)^(1/2))^16)/((a + b*x)^(1/2) - a^(1/2))^16 + (8*b^3*c^7*((a* 
c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (28*b^3 
*c^6*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2))^4 
+ (56*b^3*c^5*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^(1/2) - a^ 
(1/2))^6 + (70*b^3*c^4*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/((a + b*x)^( 
1/2) - a^(1/2))^8 + (56*b^3*c^3*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^10)/(( 
a + b*x)^(1/2) - a^(1/2))^10 + (28*b^3*c^2*((a*c - b*c*x)^(1/2) - (a*c)^(1 
/2))^12)/((a + b*x)^(1/2) - a^(1/2))^12 + (8*b^3*c*((a*c - b*c*x)^(1/2) - 
(a*c)^(1/2))^14)/((a + b*x)^(1/2) - a^(1/2))^14) + (A*x*(a*c - b*c*x)^(1/2 
)*(a + b*x)^(1/2))/2 - (B*(a^2 - b^2*x^2)*(a*c - b*c*x)^(1/2)*(a + b*x)...
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.95 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {c}\, \left (-6 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{4} c -24 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{3} b^{2}-8 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{2}-3 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b c x +12 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{3} x +8 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{4} x^{2}+6 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{3} c \,x^{3}\right )}{24 b^{3}} \] Input:

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

Output:

(sqrt(c)*( - 6*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**4*c - 24*asin(sqrt 
(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**2 - 8*sqrt(a + b*x)*sqrt(a - b*x)*a** 
2*b**2 - 3*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b*c*x + 12*sqrt(a + b*x)*sqrt( 
a - b*x)*a*b**3*x + 8*sqrt(a + b*x)*sqrt(a - b*x)*b**4*x**2 + 6*sqrt(a + b 
*x)*sqrt(a - b*x)*b**3*c*x**3))/(24*b**3)