\(\int \frac {(e+f x) (A+B x+C x^2)}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) [54]

Optimal result

Integrand size = 38, antiderivative size = 203 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\left (2 a^2 C f+2 b^2 (B e+A f)-a b (C e+B f)\right ) \sqrt {a+b x} \sqrt {a c-b c x}}{2 b^4 c}+\frac {(4 a C f-3 b (C e+B f)) (a+b x)^{3/2} \sqrt {a c-b c x}}{6 b^4 c}-\frac {C f (a+b x)^{5/2} \sqrt {a c-b c x}}{3 b^4 c}+\frac {\left (2 A b^2 e+a^2 (C e+B f)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{b^3 \sqrt {c}} \] Output:

-1/2*(2*a^2*C*f+2*b^2*(A*f+B*e)-a*b*(B*f+C*e))*(b*x+a)^(1/2)*(-b*c*x+a*c)^ 
(1/2)/b^4/c+1/6*(4*C*a*f-3*b*(B*f+C*e))*(b*x+a)^(3/2)*(-b*c*x+a*c)^(1/2)/b 
^4/c-1/3*C*f*(b*x+a)^(5/2)*(-b*c*x+a*c)^(1/2)/b^4/c+(2*A*b^2*e+a^2*(B*f+C* 
e))*arctan(c^(1/2)*(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2))/b^3/c^(1/2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.63 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {-\left ((a-b x) \sqrt {a+b x} \left (4 a^2 C f+b^2 \left (6 B e+6 A f+3 C e x+3 B f x+2 C f x^2\right )\right )\right )+6 b \left (2 A b^2 e+a^2 (C e+B f)\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{6 b^4 \sqrt {c (a-b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {2113, 2185, 25, 27, 676, 224, 216}

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.

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

\(\Big \downarrow \) 2113

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 676

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

(Sqrt[a^2*c - b^2*c*x^2]*(-1/3*(C*(e + f*x)^2*Sqrt[a^2*c - b^2*c*x^2])/(b^ 
2*c*f) + (-(((2*a^2*C*f^2 - b^2*(C*e^2 - 3*f*(B*e + A*f)))*Sqrt[a^2*c - b^ 
2*c*x^2])/(b^2*c)) + (f*(C*e - 3*B*f)*x*Sqrt[a^2*c - b^2*c*x^2])/(2*c) + ( 
3*f*(2*A*b^2*e + a^2*(C*e + B*f))*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c* 
x^2]])/(2*b*Sqrt[c]))/(3*b^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 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 676

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

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 = 0.68 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.86

method	result	size
risch	\(-\frac {\left (2 C f \,x^{2} b^{2}+3 B \,b^{2} f x +3 C \,b^{2} e x +6 A \,b^{2} f +6 B \,b^{2} e +4 a^{2} C f \right ) \left (-b x +a \right ) \sqrt {b x +a}}{6 b^{4} \sqrt {-c \left (b x -a \right )}}+\frac {\left (2 A \,b^{2} e +B \,a^{2} f +C \,a^{2} e \right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{2 b^{2} \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(175\)
default	\(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (6 A \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) b^{4} c e +3 B \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{2} c f +3 C \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{2} c e -2 C \,b^{2} f \,x^{2} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}-3 B \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{2} f x -3 C \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{2} e x -6 A \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{2} f -6 B \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{2} e -4 C \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{2} f \right )}{6 c \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{4} \sqrt {b^{2} c}}\)	\(343\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.49 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\left [-\frac {3 \, {\left (B a^{2} b f + {\left (C a^{2} b + 2 \, A b^{3}\right )} e\right )} \sqrt {-c} \log \left (2 \, b^{2} c x^{2} - 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) + 2 \, {\left (2 \, C b^{2} f x^{2} + 6 \, B b^{2} e + 2 \, {\left (2 \, C a^{2} + 3 \, A b^{2}\right )} f + 3 \, {\left (C b^{2} e + B b^{2} f\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{12 \, b^{4} c}, -\frac {3 \, {\left (B a^{2} b f + {\left (C a^{2} b + 2 \, A b^{3}\right )} e\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) + {\left (2 \, C b^{2} f x^{2} + 6 \, B b^{2} e + 2 \, {\left (2 \, C a^{2} + 3 \, A b^{2}\right )} f + 3 \, {\left (C b^{2} e + B b^{2} f\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{6 \, b^{4} c}\right ] \] Input:

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

Output:

[-1/12*(3*(B*a^2*b*f + (C*a^2*b + 2*A*b^3)*e)*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*(2*C*b^2*f*x^2 
 + 6*B*b^2*e + 2*(2*C*a^2 + 3*A*b^2)*f + 3*(C*b^2*e + B*b^2*f)*x)*sqrt(-b* 
c*x + a*c)*sqrt(b*x + a))/(b^4*c), -1/6*(3*(B*a^2*b*f + (C*a^2*b + 2*A*b^3 
)*e)*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)) + (2*C*b^2*f*x^2 + 6*B*b^2*e + 2*(2*C*a^2 + 3*A*b^2)*f + 3*(C* 
b^2*e + B*b^2*f)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(b^4*c)]
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.93 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {-b^{2} c x^{2} + a^{2} c} C f x^{2}}{3 \, b^{2} c} + \frac {A e \arcsin \left (\frac {b x}{a}\right )}{b \sqrt {c}} + \frac {{\left (C e + B f\right )} a^{2} \arcsin \left (\frac {b x}{a}\right )}{2 \, b^{3} \sqrt {c}} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} B e}{b^{2} c} - \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} C a^{2} f}{3 \, b^{4} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} A f}{b^{2} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e + B f\right )} x}{2 \, b^{2} c} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.94 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {{\left ({\left (\frac {2 \, {\left (b x + a\right )} C f}{c} + \frac {3 \, C b c^{2} e - 4 \, C a c^{2} f + 3 \, B b c^{2} f}{c^{3}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (C a b c^{2} e - 2 \, B b^{2} c^{2} e - 2 \, C a^{2} c^{2} f + B a b c^{2} f - 2 \, A b^{2} c^{2} f\right )}}{c^{3}}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} + \frac {6 \, {\left (C a^{2} b e + 2 \, A b^{3} e + B a^{2} b f\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}}}{6 \, b^{4}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 19.95 (sec) , antiderivative size = 1011, normalized size of antiderivative = 4.98 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx =\text {Too large to display} \] Input:

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

Output:

- ((2*B*a^2*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/((a + b*x)^(1/2) - a^ 
(1/2))^7 - (2*B*a^2*c^3*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/((a + b*x)^ 
(1/2) - a^(1/2)) - (14*B*a^2*c*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(( 
a + b*x)^(1/2) - a^(1/2))^5 + (14*B*a^2*c^2*f*((a*c - b*c*x)^(1/2) - (a*c) 
^(1/2))^3)/((a + b*x)^(1/2) - a^(1/2))^3)/(b^3*c^4 + (b^3*((a*c - b*c*x)^( 
1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^(1/2))^8 + (4*b^3*c^3*((a*c - 
b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (6*b^3*c^2* 
((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2))^4 + (4* 
b^3*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^(1/2) - a^(1/2))^6 
) - ((2*C*a^2*e*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/((a + b*x)^(1/2) - 
a^(1/2))^7 - (2*C*a^2*c^3*e*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/((a + b*x 
)^(1/2) - a^(1/2)) - (14*C*a^2*c*e*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/ 
((a + b*x)^(1/2) - a^(1/2))^5 + (14*C*a^2*c^2*e*((a*c - b*c*x)^(1/2) - (a* 
c)^(1/2))^3)/((a + b*x)^(1/2) - a^(1/2))^3)/(b^3*c^4 + (b^3*((a*c - b*c*x) 
^(1/2) - (a*c)^(1/2))^8)/((a + b*x)^(1/2) - a^(1/2))^8 + (4*b^3*c^3*((a*c 
- b*c*x)^(1/2) - (a*c)^(1/2))^2)/((a + b*x)^(1/2) - a^(1/2))^2 + (6*b^3*c^ 
2*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/((a + b*x)^(1/2) - a^(1/2))^4 + ( 
4*b^3*c*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/((a + b*x)^(1/2) - a^(1/2)) 
^6) - ((a*c - b*c*x)^(1/2)*((2*C*a^3*f)/(3*b^4*c) + (C*f*x^3)/(3*b*c) + (C 
*a*f*x^2)/(3*b^2*c) + (2*C*a^2*f*x)/(3*b^3*c)))/(a + b*x)^(1/2) - (4*A*...
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.03 \[ \int \frac {(e+f x) \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\sqrt {c}\, \left (-6 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{2} b^{2} f -6 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{2} b c e -12 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a \,b^{3} e -4 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} c f -6 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{2} f -6 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{3} e -3 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{3} f x -3 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{2} c e x -2 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{2} c f \,x^{2}\right )}{6 b^{4} c} \] Input:

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

Output:

(sqrt(c)*( - 6*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**2*b**2*f - 6*asin( 
sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**2*b*c*e - 12*asin(sqrt(a - b*x)/(sqrt( 
a)*sqrt(2)))*a*b**3*e - 4*sqrt(a + b*x)*sqrt(a - b*x)*a**2*c*f - 6*sqrt(a 
+ b*x)*sqrt(a - b*x)*a*b**2*f - 6*sqrt(a + b*x)*sqrt(a - b*x)*b**3*e - 3*s 
qrt(a + b*x)*sqrt(a - b*x)*b**3*f*x - 3*sqrt(a + b*x)*sqrt(a - b*x)*b**2*c 
*e*x - 2*sqrt(a + b*x)*sqrt(a - b*x)*b**2*c*f*x**2))/(6*b**4*c)