\(\int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x) (A+B x+C x^2) \, dx\) [47]

Optimal result

Integrand size = 38, antiderivative size = 255 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x) \left (A+B x+C x^2\right ) \, dx=\frac {1}{8} \left (4 A e+\frac {a^2 (C e+B f)}{b^2}\right ) x \sqrt {a+b x} \sqrt {a c-b c x}-\frac {\left (4 a^2 C f+4 b^2 (B e+A f)-3 a b (C e+B f)\right ) (a+b x)^{3/2} (a c-b c x)^{3/2}}{12 b^4 c}+\frac {(8 a C f-5 b (C e+B f)) (a+b x)^{5/2} (a c-b c x)^{3/2}}{20 b^4 c}-\frac {C f (a+b x)^{7/2} (a c-b c x)^{3/2}}{5 b^4 c}+\frac {a^2 \sqrt {c} \left (4 A b^2 e+a^2 (C e+B f)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{4 b^3} \] Output:

1/8*(4*A*e+a^2*(B*f+C*e)/b^2)*x*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)-1/12*(4*a 
^2*C*f+4*b^2*(A*f+B*e)-3*a*b*(B*f+C*e))*(b*x+a)^(3/2)*(-b*c*x+a*c)^(3/2)/b 
^4/c+1/20*(8*C*a*f-5*b*(B*f+C*e))*(b*x+a)^(5/2)*(-b*c*x+a*c)^(3/2)/b^4/c-1 
/5*C*f*(b*x+a)^(7/2)*(-b*c*x+a*c)^(3/2)/b^4/c+1/4*a^2*c^(1/2)*(4*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
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.72 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x) \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {c (a-b x)} \left (\sqrt {a-b x} \sqrt {a+b x} \left (-16 a^4 C f-a^2 b^2 (40 A f+5 B (8 e+3 f x)+C x (15 e+8 f x))+2 b^4 x (10 A (3 e+2 f x)+x (5 B (4 e+3 f x)+3 C x (5 e+4 f x)))\right )+30 a^2 b \left (4 A b^2 e+a^2 (C e+B f)\right ) \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )\right )}{120 b^4 \sqrt {a-b x}} \] Input:

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

Output:

(Sqrt[c*(a - b*x)]*(Sqrt[a - b*x]*Sqrt[a + b*x]*(-16*a^4*C*f - a^2*b^2*(40 
*A*f + 5*B*(8*e + 3*f*x) + C*x*(15*e + 8*f*x)) + 2*b^4*x*(10*A*(3*e + 2*f* 
x) + x*(5*B*(4*e + 3*f*x) + 3*C*x*(5*e + 4*f*x)))) + 30*a^2*b*(4*A*b^2*e + 
 a^2*(C*e + B*f))*ArcTan[Sqrt[a + b*x]/Sqrt[a - b*x]]))/(120*b^4*Sqrt[a - 
b*x])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2113, 2185, 25, 27, 676, 211, 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.

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

\(\Big \downarrow \) 2113

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (-\frac {\int -c f (e+f x) \left (\left (2 C a^2+5 A b^2\right ) f-b^2 (3 C e-5 B f) x\right ) \sqrt {a^2 c-b^2 c x^2}dx}{5 b^2 c f^2}-\frac {C (e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2}}{5 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\int c f (e+f x) \left (\left (2 C a^2+5 A b^2\right ) f-b^2 (3 C e-5 B f) x\right ) \sqrt {a^2 c-b^2 c x^2}dx}{5 b^2 c f^2}-\frac {C (e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2}}{5 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\int (e+f x) \left (\left (2 C a^2+5 A b^2\right ) f-b^2 (3 C e-5 B f) x\right ) \sqrt {a^2 c-b^2 c x^2}dx}{5 b^2 f}-\frac {C (e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2}}{5 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {5}{4} f \left (a^2 (B f+C e)+4 A b^2 e\right ) \int \sqrt {a^2 c-b^2 c x^2}dx-\frac {\left (a^2 c-b^2 c x^2\right )^{3/2} \left (2 a^2 C f^2-b^2 \left (3 C e^2-5 f (A f+B e)\right )\right )}{3 b^2 c}+\frac {f x \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2}}{5 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {5}{4} f \left (a^2 (B f+C e)+4 A b^2 e\right ) \left (\frac {1}{2} a^2 c \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}}dx+\frac {1}{2} x \sqrt {a^2 c-b^2 c x^2}\right )-\frac {\left (a^2 c-b^2 c x^2\right )^{3/2} \left (2 a^2 C f^2-b^2 \left (3 C e^2-5 f (A f+B e)\right )\right )}{3 b^2 c}+\frac {f x \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2}}{5 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {5}{4} f \left (a^2 (B f+C e)+4 A b^2 e\right ) \left (\frac {1}{2} a^2 c \int \frac {1}{\frac {b^2 c x^2}{a^2 c-b^2 c x^2}+1}d\frac {x}{\sqrt {a^2 c-b^2 c x^2}}+\frac {1}{2} x \sqrt {a^2 c-b^2 c x^2}\right )-\frac {\left (a^2 c-b^2 c x^2\right )^{3/2} \left (2 a^2 C f^2-b^2 \left (3 C e^2-5 f (A f+B e)\right )\right )}{3 b^2 c}+\frac {f x \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2}}{5 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {5}{4} f \left (\frac {a^2 \sqrt {c} \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{2 b}+\frac {1}{2} x \sqrt {a^2 c-b^2 c x^2}\right ) \left (a^2 (B f+C e)+4 A b^2 e\right )-\frac {\left (a^2 c-b^2 c x^2\right )^{3/2} \left (2 a^2 C f^2-b^2 \left (3 C e^2-5 f (A f+B e)\right )\right )}{3 b^2 c}+\frac {f x \left (a^2 c-b^2 c x^2\right )^{3/2} (3 C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^2 \left (a^2 c-b^2 c x^2\right )^{3/2}}{5 b^2 c f}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(-1/5*(C*(e + f*x)^2*(a^2*c - b^2*c*x^2)^ 
(3/2))/(b^2*c*f) + (-1/3*((2*a^2*C*f^2 - b^2*(3*C*e^2 - 5*f*(B*e + A*f)))* 
(a^2*c - b^2*c*x^2)^(3/2))/(b^2*c) + (f*(3*C*e - 5*B*f)*x*(a^2*c - b^2*c*x 
^2)^(3/2))/(4*c) + (5*f*(4*A*b^2*e + a^2*(C*e + B*f))*((x*Sqrt[a^2*c - b^2 
*c*x^2])/2 + (a^2*Sqrt[c]*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/( 
2*b)))/4)/(5*b^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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 = 0.61 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.99

method	result	size
risch	\(-\frac {\left (-24 C f \,x^{4} b^{4}-30 B \,b^{4} f \,x^{3}-30 C \,b^{4} e \,x^{3}-40 A \,b^{4} f \,x^{2}-40 B \,b^{4} e \,x^{2}+8 C \,a^{2} b^{2} f \,x^{2}-60 A \,b^{4} e x +15 B \,a^{2} b^{2} f x +15 C \,a^{2} b^{2} e x +40 A \,a^{2} f \,b^{2}+40 B \,a^{2} e \,b^{2}+16 C \,a^{4} f \right ) \left (-b x +a \right ) \sqrt {b x +a}\, c}{120 b^{4} \sqrt {-c \left (b x -a \right )}}+\frac {a^{2} \left (4 A \,b^{2} e +B \,a^{2} f +C \,a^{2} e \right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}\, c}{8 b^{2} \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(253\)
default	\(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (24 C \,b^{4} f \,x^{4} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+30 B \,b^{4} f \,x^{3} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+30 C \,b^{4} e \,x^{3} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+60 A \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{2} b^{4} c e +40 A \,b^{4} f \,x^{2} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+15 B \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{4} b^{2} c f +40 B \,b^{4} e \,x^{2} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+15 C \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) a^{4} b^{2} c e -8 C \,a^{2} b^{2} f \,x^{2} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+60 A \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{4} e x -15 B \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, a^{2} b^{2} f x -15 C \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, a^{2} b^{2} e x -40 A \,a^{2} b^{2} f \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}-40 B \,a^{2} b^{2} e \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}-16 C \,a^{4} f \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\right )}{120 \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{4}}\)	\(554\)

Input:

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

Output:

-1/120*(-24*C*b^4*f*x^4-30*B*b^4*f*x^3-30*C*b^4*e*x^3-40*A*b^4*f*x^2-40*B* 
b^4*e*x^2+8*C*a^2*b^2*f*x^2-60*A*b^4*e*x+15*B*a^2*b^2*f*x+15*C*a^2*b^2*e*x 
+40*A*a^2*b^2*f+40*B*a^2*b^2*e+16*C*a^4*f)/b^4*(-b*x+a)*(b*x+a)^(1/2)/(-c* 
(b*x-a))^(1/2)*c+1/8*a^2/b^2*(4*A*b^2*e+B*a^2*f+C*a^2*e)/(b^2*c)^(1/2)*arc 
tan((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 = 0.10 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.73 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x) \left (A+B x+C x^2\right ) \, dx=\left [\frac {15 \, {\left (B a^{4} b f + {\left (C a^{4} b + 4 \, A a^{2} b^{3}\right )} e\right )} \sqrt {-c} \log \left (2 \, b^{2} c x^{2} + 2 \, \sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {-c} x - a^{2} c\right ) + 2 \, {\left (24 \, C b^{4} f x^{4} - 40 \, B a^{2} b^{2} e + 30 \, {\left (C b^{4} e + B b^{4} f\right )} x^{3} + 8 \, {\left (5 \, B b^{4} e - {\left (C a^{2} b^{2} - 5 \, A b^{4}\right )} f\right )} x^{2} - 8 \, {\left (2 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f - 15 \, {\left (B a^{2} b^{2} f + {\left (C a^{2} b^{2} - 4 \, A b^{4}\right )} e\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{240 \, b^{4}}, -\frac {15 \, {\left (B a^{4} b f + {\left (C a^{4} b + 4 \, A a^{2} b^{3}\right )} e\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-b c x + a c} \sqrt {b x + a} b \sqrt {c} x}{b^{2} c x^{2} - a^{2} c}\right ) - {\left (24 \, C b^{4} f x^{4} - 40 \, B a^{2} b^{2} e + 30 \, {\left (C b^{4} e + B b^{4} f\right )} x^{3} + 8 \, {\left (5 \, B b^{4} e - {\left (C a^{2} b^{2} - 5 \, A b^{4}\right )} f\right )} x^{2} - 8 \, {\left (2 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f - 15 \, {\left (B a^{2} b^{2} f + {\left (C a^{2} b^{2} - 4 \, A b^{4}\right )} e\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{120 \, b^{4}}\right ] \] Input:

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

Output:

[1/240*(15*(B*a^4*b*f + (C*a^4*b + 4*A*a^2*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*(24*C*b^4 
*f*x^4 - 40*B*a^2*b^2*e + 30*(C*b^4*e + B*b^4*f)*x^3 + 8*(5*B*b^4*e - (C*a 
^2*b^2 - 5*A*b^4)*f)*x^2 - 8*(2*C*a^4 + 5*A*a^2*b^2)*f - 15*(B*a^2*b^2*f + 
 (C*a^2*b^2 - 4*A*b^4)*e)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^4, -1/120 
*(15*(B*a^4*b*f + (C*a^4*b + 4*A*a^2*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)) - (24*C*b^4*f*x^4 - 40 
*B*a^2*b^2*e + 30*(C*b^4*e + B*b^4*f)*x^3 + 8*(5*B*b^4*e - (C*a^2*b^2 - 5* 
A*b^4)*f)*x^2 - 8*(2*C*a^4 + 5*A*a^2*b^2)*f - 15*(B*a^2*b^2*f + (C*a^2*b^2 
 - 4*A*b^4)*e)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/b^4]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.97 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x) \left (A+B x+C x^2\right ) \, dx=\frac {A a^{2} \sqrt {c} e \arcsin \left (\frac {b x}{a}\right )}{2 \, b} + \frac {1}{2} \, \sqrt {-b^{2} c x^{2} + a^{2} c} A e x + \frac {{\left (C e + B f\right )} a^{4} \sqrt {c} \arcsin \left (\frac {b x}{a}\right )}{8 \, b^{3}} + \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e + B f\right )} a^{2} x}{8 \, b^{2}} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C f x^{2}}{5 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} B e}{3 \, b^{2} c} - \frac {2 \, {\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} C a^{2} f}{15 \, b^{4} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} A f}{3 \, b^{2} c} - \frac {{\left (-b^{2} c x^{2} + a^{2} c\right )}^{\frac {3}{2}} {\left (C e + B f\right )} x}{4 \, b^{2} c} \] Input:

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

Output:

1/2*A*a^2*sqrt(c)*e*arcsin(b*x/a)/b + 1/2*sqrt(-b^2*c*x^2 + a^2*c)*A*e*x + 
 1/8*(C*e + B*f)*a^4*sqrt(c)*arcsin(b*x/a)/b^3 + 1/8*sqrt(-b^2*c*x^2 + a^2 
*c)*(C*e + B*f)*a^2*x/b^2 - 1/5*(-b^2*c*x^2 + a^2*c)^(3/2)*C*f*x^2/(b^2*c) 
 - 1/3*(-b^2*c*x^2 + a^2*c)^(3/2)*B*e/(b^2*c) - 2/15*(-b^2*c*x^2 + a^2*c)^ 
(3/2)*C*a^2*f/(b^4*c) - 1/3*(-b^2*c*x^2 + a^2*c)^(3/2)*A*f/(b^2*c) - 1/4*( 
-b^2*c*x^2 + a^2*c)^(3/2)*(C*e + B*f)*x/(b^2*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1142 vs. \(2 (221) = 442\).

Time = 1.13 (sec) , antiderivative size = 1142, normalized size of antiderivative = 4.48 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x) \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/120*(120*(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^3*e - 6 
0*(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^2*e 
 - 60*(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^3 
*e - 60*(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*a 
*b^2*f + 20*(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*b*e + 20*(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^2*e + 20*(6*a^3*c*log(a 
bs(-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*a 
*b*f + 20*(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))*A*b^2*f - 5*(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...
 

Mupad [B] (verification not implemented)

Time = 26.59 (sec) , antiderivative size = 1765, normalized size of antiderivative = 6.92 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x) \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

((B*a^4*c^8*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)))/(2*((a + b*x)^(1/2) - a 
^(1/2))) - (B*a^4*c*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^15)/(2*((a + b*x 
)^(1/2) - a^(1/2))^15) - (35*B*a^4*c^7*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2 
))^3)/(2*((a + b*x)^(1/2) - a^(1/2))^3) + (273*B*a^4*c^6*f*((a*c - b*c*x)^ 
(1/2) - (a*c)^(1/2))^5)/(2*((a + b*x)^(1/2) - a^(1/2))^5) - (715*B*a^4*c^5 
*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^7)/(2*((a + b*x)^(1/2) - a^(1/2))^7 
) + (715*B*a^4*c^4*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^9)/(2*((a + b*x)^ 
(1/2) - a^(1/2))^9) - (273*B*a^4*c^3*f*((a*c - b*c*x)^(1/2) - (a*c)^(1/2)) 
^11)/(2*((a + b*x)^(1/2) - a^(1/2))^11) + (35*B*a^4*c^2*f*((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*c - 
b*c*x)^(1/2)*((2*C*a^4*f*(a + b*x)^(1/2))/(15*b^4) - (C*f*x^4*(a + b*x)...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.40 \[ \int \sqrt {a+b x} \sqrt {a c-b c x} (e+f x) \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {c}\, \left (-30 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{4} b^{2} f -30 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{4} b c e -120 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{3} b^{3} e -16 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{4} c f -40 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{3} b^{2} f -40 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{3} e -15 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{3} f x -15 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{2} c e x -8 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{2} c f \,x^{2}+60 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{4} e x +40 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{4} f \,x^{2}+40 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{5} e \,x^{2}+30 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{5} f \,x^{3}+30 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{4} c e \,x^{3}+24 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{4} c f \,x^{4}\right )}{120 b^{4}} \] Input:

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

Output:

(sqrt(c)*( - 30*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**4*b**2*f - 30*asi 
n(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**4*b*c*e - 120*asin(sqrt(a - b*x)/(sq 
rt(a)*sqrt(2)))*a**3*b**3*e - 16*sqrt(a + b*x)*sqrt(a - b*x)*a**4*c*f - 40 
*sqrt(a + b*x)*sqrt(a - b*x)*a**3*b**2*f - 40*sqrt(a + b*x)*sqrt(a - b*x)* 
a**2*b**3*e - 15*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b**3*f*x - 15*sqrt(a + b 
*x)*sqrt(a - b*x)*a**2*b**2*c*e*x - 8*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b** 
2*c*f*x**2 + 60*sqrt(a + b*x)*sqrt(a - b*x)*a*b**4*e*x + 40*sqrt(a + b*x)* 
sqrt(a - b*x)*a*b**4*f*x**2 + 40*sqrt(a + b*x)*sqrt(a - b*x)*b**5*e*x**2 + 
 30*sqrt(a + b*x)*sqrt(a - b*x)*b**5*f*x**3 + 30*sqrt(a + b*x)*sqrt(a - b* 
x)*b**4*c*e*x**3 + 24*sqrt(a + b*x)*sqrt(a - b*x)*b**4*c*f*x**4))/(120*b** 
4)