\(\int \frac {\sqrt {a+b x} \sqrt {a c-b c x} (A+B x+C x^2)}{(e+f x)^3} \, dx\) [51]

Optimal result

Integrand size = 40, antiderivative size = 390 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{(e+f x)^3} \, dx=\frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (2 (b e-a f) (b e+a f) (3 C e-B f)-f^2 \left (2 a^2 C f+b^2 \left (B e-\frac {3 C e^2}{f}-A f\right )\right ) x\right )}{2 f^3 \left (b^2 e^2-a^2 f^2\right ) (e+f x)}+\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x} \left (a^2-b^2 x^2\right )}{2 f \left (b^2 e^2-a^2 f^2\right ) (e+f x)^2}+\frac {2 b \sqrt {c} (3 C e-B f) \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{f^4}-\frac {\sqrt {c} \left (2 a^4 C f^4+2 b^4 e^3 (3 C e-B f)-a^2 b^2 f^2 \left (9 C e^2-f (3 B e-A f)\right )\right ) \arctan \left (\frac {\sqrt {c} \sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a c-b c x}}\right )}{f^4 (b e-a f)^{3/2} (b e+a f)^{3/2}} \] Output:

1/2*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(2*(-a*f+b*e)*(a*f+b*e)*(-B*f+3*C*e)- 
f^2*(2*a^2*C*f+b^2*(B*e-3*C*e^2/f-A*f))*x)/f^3/(-a^2*f^2+b^2*e^2)/(f*x+e)+ 
1/2*(A*f^2-B*e*f+C*e^2)*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(-b^2*x^2+a^2)/f/ 
(-a^2*f^2+b^2*e^2)/(f*x+e)^2+2*b*c^(1/2)*(-B*f+3*C*e)*arctan(c^(1/2)*(b*x+ 
a)^(1/2)/(-b*c*x+a*c)^(1/2))/f^4-c^(1/2)*(2*a^4*C*f^4+2*b^4*e^3*(-B*f+3*C* 
e)-a^2*b^2*f^2*(9*C*e^2-f*(-A*f+3*B*e)))*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/(-a*f+b*e)^(3/2)/(a*f 
+b*e)^(3/2)
 

Mathematica [A] (verified)

Time = 1.43 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{(e+f x)^3} \, dx=\frac {\sqrt {c (a-b x)} \left (-\frac {f \sqrt {a+b x} \left (a^2 f^2 \left (f (B e+A f+2 B f x)-C \left (5 e^2+8 e f x+2 f^2 x^2\right )\right )+b^2 e \left (A f^3 x-B e f (2 e+3 f x)+C e \left (6 e^2+9 e f x+2 f^2 x^2\right )\right )\right )}{2 (-b e+a f) (b e+a f) (e+f x)^2}+\frac {2 b (3 C e-B f) \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{\sqrt {a-b x}}-\frac {\left (2 a^4 C f^4+2 b^4 e^3 (3 C e-B f)-a^2 b^2 f^2 \left (9 C e^2+f (-3 B e+A f)\right )\right ) \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{(b e-a f)^{3/2} (b e+a f)^{3/2} \sqrt {a-b x}}\right )}{f^4} \] Input:

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

Output:

(Sqrt[c*(a - b*x)]*(-1/2*(f*Sqrt[a + b*x]*(a^2*f^2*(f*(B*e + A*f + 2*B*f*x 
) - C*(5*e^2 + 8*e*f*x + 2*f^2*x^2)) + b^2*e*(A*f^3*x - B*e*f*(2*e + 3*f*x 
) + C*e*(6*e^2 + 9*e*f*x + 2*f^2*x^2))))/((-(b*e) + a*f)*(b*e + a*f)*(e + 
f*x)^2) + (2*b*(3*C*e - B*f)*ArcTan[Sqrt[a + b*x]/Sqrt[a - b*x]])/Sqrt[a - 
 b*x] - ((2*a^4*C*f^4 + 2*b^4*e^3*(3*C*e - B*f) - a^2*b^2*f^2*(9*C*e^2 + f 
*(-3*B*e + A*f)))*ArcTan[(Sqrt[b*e + a*f]*Sqrt[a + b*x])/(Sqrt[b*e - a*f]* 
Sqrt[a - b*x])])/((b*e - a*f)^(3/2)*(b*e + a*f)^(3/2)*Sqrt[a - b*x])))/f^4
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2113, 2182, 27, 681, 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.

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

\(\Big \downarrow \) 2113

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 681

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 (b e-a f) (a f+b e) (3 C e-B f)-f^2 x \left (2 a^2 C f+b^2 \left (-A f+B e-\frac {3 C e^2}{f}\right )\right )\right )}{f^3 (e+f x)}-\frac {c \left (\frac {\left (2 a^4 C f^4-a^2 b^2 f^2 \left (9 C e^2-f (3 B e-A f)\right )+2 b^4 e^3 (3 C e-B f)\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{f}-\frac {2 b^2 (b e-a f) (a f+b e) (3 C e-B f) \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}}dx}{f}\right )}{f^3}}{2 \left (b^2 e^2-a^2 f^2\right )}+\frac {f \left (a^2 c-b^2 c x^2\right )^{3/2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 (b e-a f) (a f+b e) (3 C e-B f)-f^2 x \left (2 a^2 C f+b^2 \left (-A f+B e-\frac {3 C e^2}{f}\right )\right )\right )}{f^3 (e+f x)}-\frac {c \left (\frac {\left (2 a^4 C f^4-a^2 b^2 f^2 \left (9 C e^2-f (3 B e-A f)\right )+2 b^4 e^3 (3 C e-B f)\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{f}-\frac {2 b^2 (b e-a f) (a f+b e) (3 C e-B f) \int \frac {1}{\frac {b^2 c x^2}{a^2 c-b^2 c x^2}+1}d\frac {x}{\sqrt {a^2 c-b^2 c x^2}}}{f}\right )}{f^3}}{2 \left (b^2 e^2-a^2 f^2\right )}+\frac {f \left (a^2 c-b^2 c x^2\right )^{3/2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 (b e-a f) (a f+b e) (3 C e-B f)-f^2 x \left (2 a^2 C f+b^2 \left (-A f+B e-\frac {3 C e^2}{f}\right )\right )\right )}{f^3 (e+f x)}-\frac {c \left (\frac {\left (2 a^4 C f^4-a^2 b^2 f^2 \left (9 C e^2-f (3 B e-A f)\right )+2 b^4 e^3 (3 C e-B f)\right ) \int \frac {1}{(e+f x) \sqrt {a^2 c-b^2 c x^2}}dx}{f}-\frac {2 b (b e-a f) (a f+b e) (3 C e-B f) \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{\sqrt {c} f}\right )}{f^3}}{2 \left (b^2 e^2-a^2 f^2\right )}+\frac {f \left (a^2 c-b^2 c x^2\right )^{3/2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 (b e-a f) (a f+b e) (3 C e-B f)-f^2 x \left (2 a^2 C f+b^2 \left (-A f+B e-\frac {3 C e^2}{f}\right )\right )\right )}{f^3 (e+f x)}-\frac {c \left (-\frac {\left (2 a^4 C f^4-a^2 b^2 f^2 \left (9 C e^2-f (3 B e-A f)\right )+2 b^4 e^3 (3 C e-B f)\right ) \int \frac {1}{-b^2 c e^2+a^2 c f^2-\frac {\left (c f a^2+b^2 c e x\right )^2}{a^2 c-b^2 c x^2}}d\frac {c f a^2+b^2 c e x}{\sqrt {a^2 c-b^2 c x^2}}}{f}-\frac {2 b (b e-a f) (a f+b e) (3 C e-B f) \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{\sqrt {c} f}\right )}{f^3}}{2 \left (b^2 e^2-a^2 f^2\right )}+\frac {f \left (a^2 c-b^2 c x^2\right )^{3/2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (\frac {f \left (a^2 c-b^2 c x^2\right )^{3/2} \left (A+\frac {e (C e-B f)}{f^2}\right )}{2 c (e+f x)^2 \left (b^2 e^2-a^2 f^2\right )}+\frac {\frac {\sqrt {a^2 c-b^2 c x^2} \left (2 (b e-a f) (a f+b e) (3 C e-B f)-f^2 x \left (2 a^2 C f+b^2 \left (-A f+B e-\frac {3 C e^2}{f}\right )\right )\right )}{f^3 (e+f x)}-\frac {c \left (\frac {\left (2 a^4 C f^4-a^2 b^2 f^2 \left (9 C e^2-f (3 B e-A f)\right )+2 b^4 e^3 (3 C e-B f)\right ) \arctan \left (\frac {a^2 c f+b^2 c e x}{\sqrt {c} \sqrt {a^2 c-b^2 c x^2} \sqrt {b^2 e^2-a^2 f^2}}\right )}{\sqrt {c} f \sqrt {b^2 e^2-a^2 f^2}}-\frac {2 b (b e-a f) (a f+b e) (3 C e-B f) \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right )}{\sqrt {c} f}\right )}{f^3}}{2 \left (b^2 e^2-a^2 f^2\right )}\right )}{\sqrt {a^2 c-b^2 c x^2}}\)

Input:

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

Output:

(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*((f*(A + (e*(C*e - B*f))/f^2)*(a^2*c - b^ 
2*c*x^2)^(3/2))/(2*c*(b^2*e^2 - a^2*f^2)*(e + f*x)^2) + (((2*(b*e - a*f)*( 
b*e + a*f)*(3*C*e - B*f) - f^2*(2*a^2*C*f + b^2*(B*e - (3*C*e^2)/f - A*f)) 
*x)*Sqrt[a^2*c - b^2*c*x^2])/(f^3*(e + f*x)) - (c*((-2*b*(b*e - a*f)*(b*e 
+ a*f)*(3*C*e - B*f)*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(Sqrt[ 
c]*f) + ((2*a^4*C*f^4 + 2*b^4*e^3*(3*C*e - B*f) - a^2*b^2*f^2*(9*C*e^2 - f 
*(3*B*e - A*f)))*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^3) 
/(2*(b^2*e^2 - a^2*f^2))))/Sqrt[a^2*c - b^2*c*x^2]
 

Defintions of rubi rules used

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 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[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 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1277\) vs. \(2(354)=708\).

Time = 1.05 (sec) , antiderivative size = 1278, normalized size of antiderivative = 3.28

method	result	size
risch	\(\text {Expression too large to display}\)	\(1278\)
default	\(\text {Expression too large to display}\)	\(3116\)

Input:

int((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^3,x,method=_RET 
URNVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{(e+f x)^3} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^3,x, algo 
rithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{(e+f x)^3} \, dx=\int \frac {\sqrt {- c \left (- a + b x\right )} \sqrt {a + b x} \left (A + B x + C x^{2}\right )}{\left (e + f x\right )^{3}}\, dx \] Input:

integrate((b*x+a)**(1/2)*(-b*c*x+a*c)**(1/2)*(C*x**2+B*x+A)/(f*x+e)**3,x)
 

Output:

Integral(sqrt(-c*(-a + b*x))*sqrt(a + b*x)*(A + B*x + C*x**2)/(e + f*x)**3 
, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{(e+f x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^3,x, algo 
rithm="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((a*f-b*e)>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1592 vs. \(2 (354) = 708\).

Time = 0.74 (sec) , antiderivative size = 1592, normalized size of antiderivative = 4.08 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{(e+f x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^3,x, algo 
rithm="giac")
 

Output:

(sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a)*C*b/f^3 + (6*C*b^5*sqrt(-c)*c*e^ 
4 - 2*B*b^5*sqrt(-c)*c*e^3*f - 9*C*a^2*b^3*sqrt(-c)*c*e^2*f^2 + 3*B*a^2*b^ 
3*sqrt(-c)*c*e*f^3 + 2*C*a^4*b*sqrt(-c)*c*f^4 - A*a^2*b^3*sqrt(-c)*c*f^4)* 
arctan(-1/2*(2*b*c*e - (sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c 
))^2*f)/(sqrt(-b^2*e^2 + a^2*f^2)*c))/((b^2*e^2*f^4 - a^2*f^6)*sqrt(-b^2*e 
^2 + a^2*f^2)*c) + 2*(20*C*b^6*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c 
 + 2*a*c))^4*sqrt(-c)*c^2*e^5 - 6*C*b^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b 
*x + a)*c + 2*a*c))^6*sqrt(-c)*c*e^4*f - 12*B*b^6*(sqrt(b*x + a)*sqrt(-c) 
- sqrt(-(b*x + a)*c + 2*a*c))^4*sqrt(-c)*c^2*e^4*f - 56*C*a^2*b^5*(sqrt(b* 
x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*c^3*e^4*f + 4*B*b 
^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^6*sqrt(-c)*c*e^3* 
f^2 - 6*C*a^2*b^4*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4* 
sqrt(-c)*c^2*e^3*f^2 + 4*A*b^6*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c 
 + 2*a*c))^4*sqrt(-c)*c^2*e^3*f^2 + 32*B*a^2*b^5*(sqrt(b*x + a)*sqrt(-c) - 
 sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*c^3*e^3*f^2 + 40*C*a^4*b^4*sqrt(-c 
)*c^4*e^3*f^2 + 5*C*a^2*b^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 
2*a*c))^6*sqrt(-c)*c*e^2*f^3 - 2*A*b^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b* 
x + a)*c + 2*a*c))^6*sqrt(-c)*c*e^2*f^3 + 2*B*a^2*b^4*(sqrt(b*x + a)*sqrt( 
-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*sqrt(-c)*c^2*e^2*f^3 + 44*C*a^4*b^3*(s 
qrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*c^3*e^2*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{(e+f x)^3} \, dx=\text {Hanged} \] Input:

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

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 9.97 (sec) , antiderivative size = 20805, normalized size of antiderivative = 53.35 \[ \int \frac {\sqrt {a+b x} \sqrt {a c-b c x} \left (A+B x+C x^2\right )}{(e+f x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(c)*(4*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**5*b**2*e**2*f**6 + 8* 
asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**5*b**2*e*f**7*x + 4*asin(sqrt(a - 
 b*x)/(sqrt(a)*sqrt(2)))*a**5*b**2*f**8*x**2 - 12*asin(sqrt(a - b*x)/(sqrt 
(a)*sqrt(2)))*a**5*b*c*e**3*f**5 - 24*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)) 
)*a**5*b*c*e**2*f**6*x - 12*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**5*b*c 
*e*f**7*x**2 + 4*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**4*b**3*e**3*f**5 
 + 8*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**4*b**3*e**2*f**6*x + 4*asin( 
sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**4*b**3*e*f**7*x**2 - 12*asin(sqrt(a - 
b*x)/(sqrt(a)*sqrt(2)))*a**4*b**2*c*e**4*f**4 - 24*asin(sqrt(a - b*x)/(sqr 
t(a)*sqrt(2)))*a**4*b**2*c*e**3*f**5*x - 12*asin(sqrt(a - b*x)/(sqrt(a)*sq 
rt(2)))*a**4*b**2*c*e**2*f**6*x**2 - 8*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2) 
))*a**3*b**4*e**4*f**4 - 16*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b** 
4*e**3*f**5*x - 8*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**4*e**2*f** 
6*x**2 + 24*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**3*c*e**5*f**3 + 
48*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**3*c*e**4*f**4*x + 24*asin 
(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**3*c*e**3*f**5*x**2 - 8*asin(sqrt 
(a - b*x)/(sqrt(a)*sqrt(2)))*a**2*b**5*e**5*f**3 - 16*asin(sqrt(a - b*x)/( 
sqrt(a)*sqrt(2)))*a**2*b**5*e**4*f**4*x - 8*asin(sqrt(a - b*x)/(sqrt(a)*sq 
rt(2)))*a**2*b**5*e**3*f**5*x**2 + 24*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)) 
)*a**2*b**4*c*e**6*f**2 + 48*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**2...