\(\int \frac {(e+f x)^3 (A+B x+C x^2)}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx\) [52]

Optimal result

Integrand size = 40, antiderivative size = 465 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\left (136 a^4 C f^3+120 b^4 e^2 (B e+3 A f)-45 a^3 b f^2 (3 C e+B f)-60 a b^3 e \left (C e^2+3 f (B e+A f)\right )+80 a^2 b^2 f \left (3 C e^2+f (3 B e+A f)\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x}}{120 b^6 c}+\frac {f \left (14 a^2 C f^2-5 b^2 \left (3 C e^2+f (3 B e+A f)\right )\right ) x^2 \sqrt {a+b x} \sqrt {a c-b c x}}{15 b^4 c}-\frac {f^2 (15 b C e+5 b B f-16 a C f) x^3 \sqrt {a+b x} \sqrt {a c-b c x}}{20 b^3 c}+\frac {\left (32 a^3 C f^3-15 a^2 b f^2 (3 C e+B f)-20 b^3 \left (C e^3+3 e f (B e+A f)\right )\right ) (a+b x)^{3/2} \sqrt {a c-b c x}}{40 b^6 c}-\frac {C f^3 (a+b x)^{9/2} \sqrt {a c-b c x}}{5 b^6 c}+\frac {\left (3 a^4 f^2 (3 C e+B f)+4 a^2 b^2 e^2 (C e+3 B f)+4 A \left (2 b^4 e^3+3 a^2 b^2 e f^2\right )\right ) \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{4 b^5 \sqrt {c}} \] Output:

-1/120*(136*a^4*C*f^3+120*b^4*e^2*(3*A*f+B*e)-45*a^3*b*f^2*(B*f+3*C*e)-60* 
a*b^3*e*(C*e^2+3*f*(A*f+B*e))+80*a^2*b^2*f*(3*C*e^2+f*(A*f+3*B*e)))*(b*x+a 
)^(1/2)*(-b*c*x+a*c)^(1/2)/b^6/c+1/15*f*(14*a^2*C*f^2-5*b^2*(3*C*e^2+f*(A* 
f+3*B*e)))*x^2*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^4/c-1/20*f^2*(5*B*b*f-16 
*C*a*f+15*C*b*e)*x^3*(b*x+a)^(1/2)*(-b*c*x+a*c)^(1/2)/b^3/c+1/40*(32*a^3*C 
*f^3-15*a^2*b*f^2*(B*f+3*C*e)-20*b^3*(C*e^3+3*e*f*(A*f+B*e)))*(b*x+a)^(3/2 
)*(-b*c*x+a*c)^(1/2)/b^6/c-1/5*C*f^3*(b*x+a)^(9/2)*(-b*c*x+a*c)^(1/2)/b^6/ 
c+1/4*(3*a^4*f^2*(B*f+3*C*e)+4*a^2*b^2*e^2*(3*B*f+C*e)+4*A*(3*a^2*b^2*e*f^ 
2+2*b^4*e^3))*arctan(c^(1/2)*(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2))/b^5/c^(1/2)
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.61 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {-\left ((a-b x) \sqrt {a+b x} \left (64 a^4 C f^3+a^2 b^2 f \left (5 f (48 B e+16 A f+9 B f x)+C \left (240 e^2+135 e f x+32 f^2 x^2\right )\right )+2 b^4 \left (10 A f \left (18 e^2+9 e f x+2 f^2 x^2\right )+15 B \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )+3 C x \left (10 e^3+20 e^2 f x+15 e f^2 x^2+4 f^3 x^3\right )\right )\right )\right )+30 b \left (3 a^4 f^2 (3 C e+B f)+4 a^2 b^2 e^2 (C e+3 B f)+4 A \left (2 b^4 e^3+3 a^2 b^2 e f^2\right )\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{120 b^6 \sqrt {c (a-b x)}} \] Input:

Integrate[((e + f*x)^3*(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]*(64*a^4*C*f^3 + a^2*b^2*f*(5*f*(48*B*e + 16*A*f 
 + 9*B*f*x) + C*(240*e^2 + 135*e*f*x + 32*f^2*x^2)) + 2*b^4*(10*A*f*(18*e^ 
2 + 9*e*f*x + 2*f^2*x^2) + 15*B*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3 
) + 3*C*x*(10*e^3 + 20*e^2*f*x + 15*e*f^2*x^2 + 4*f^3*x^3)))) + 30*b*(3*a^ 
4*f^2*(3*C*e + B*f) + 4*a^2*b^2*e^2*(C*e + 3*B*f) + 4*A*(2*b^4*e^3 + 3*a^2 
*b^2*e*f^2))*Sqrt[a - b*x]*ArcTan[Sqrt[a + b*x]/Sqrt[a - b*x]])/(120*b^6*S 
qrt[c*(a - b*x)])
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2113, 2185, 25, 27, 687, 25, 27, 687, 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.

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

\(\Big \downarrow \) 2113

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 687

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {(e+f x)^3 \sqrt {a^2 c-b^2 c x^2} (C e-5 B f)}{4 c}-\frac {\int -\frac {b^2 c (e+f x)^2 \left (f \left ((13 C e+15 B f) a^2+20 A b^2 e\right )+\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) x\right )}{\sqrt {a^2 c-b^2 c x^2}}dx}{4 b^2 c}}{5 b^2 f}-\frac {C (e+f x)^4 \sqrt {a^2 c-b^2 c x^2}}{5 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {\int \frac {b^2 c (e+f x)^2 \left (f \left ((13 C e+15 B f) a^2+20 A b^2 e\right )+\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) x\right )}{\sqrt {a^2 c-b^2 c x^2}}dx}{4 b^2 c}+\frac {(e+f x)^3 \sqrt {a^2 c-b^2 c x^2} (C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^4 \sqrt {a^2 c-b^2 c x^2}}{5 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {1}{4} \int \frac {(e+f x)^2 \left (f \left ((13 C e+15 B f) a^2+20 A b^2 e\right )+\left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (3 B e+4 A f)\right )\right ) x\right )}{\sqrt {a^2 c-b^2 c x^2}}dx+\frac {(e+f x)^3 \sqrt {a^2 c-b^2 c x^2} (C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^4 \sqrt {a^2 c-b^2 c x^2}}{5 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {1}{4} \left (-\frac {\int -\frac {c (e+f x) \left (\left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x b^2+f \left (32 C f^2 a^4+3 b^2 e (11 C e+25 B f) a^2+20 A \left (3 e^2 b^4+2 a^2 f^2 b^2\right )\right )\right )}{\sqrt {a^2 c-b^2 c x^2}}dx}{3 b^2 c}-\frac {(e+f x)^2 \sqrt {a^2 c-b^2 c x^2} \left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (4 A f+3 B e)\right )\right )}{3 b^2 c}\right )+\frac {(e+f x)^3 \sqrt {a^2 c-b^2 c x^2} (C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^4 \sqrt {a^2 c-b^2 c x^2}}{5 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {1}{4} \left (\frac {\int \frac {c (e+f x) \left (\left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x b^2+f \left (32 C f^2 a^4+3 b^2 e (11 C e+25 B f) a^2+20 A \left (3 e^2 b^4+2 a^2 f^2 b^2\right )\right )\right )}{\sqrt {a^2 c-b^2 c x^2}}dx}{3 b^2 c}-\frac {(e+f x)^2 \sqrt {a^2 c-b^2 c x^2} \left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (4 A f+3 B e)\right )\right )}{3 b^2 c}\right )+\frac {(e+f x)^3 \sqrt {a^2 c-b^2 c x^2} (C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^4 \sqrt {a^2 c-b^2 c x^2}}{5 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {1}{4} \left (\frac {\int \frac {(e+f x) \left (\left (a^2 f^2 (71 C e+45 B f)-b^2 \left (6 C e^3-10 e f (3 B e+10 A f)\right )\right ) x b^2+f \left (32 C f^2 a^4+3 b^2 e (11 C e+25 B f) a^2+20 A \left (3 e^2 b^4+2 a^2 f^2 b^2\right )\right )\right )}{\sqrt {a^2 c-b^2 c x^2}}dx}{3 b^2}-\frac {(e+f x)^2 \sqrt {a^2 c-b^2 c x^2} \left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (4 A f+3 B e)\right )\right )}{3 b^2 c}\right )+\frac {(e+f x)^3 \sqrt {a^2 c-b^2 c x^2} (C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^4 \sqrt {a^2 c-b^2 c x^2}}{5 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {1}{4} \left (\frac {\frac {15}{2} f \left (3 a^4 f^2 (B f+3 C e)+4 A \left (3 a^2 b^2 e f^2+2 b^4 e^3\right )+4 a^2 b^2 e^2 (3 B f+C e)\right ) \int \frac {1}{\sqrt {a^2 c-b^2 c x^2}}dx-\frac {f x \sqrt {a^2 c-b^2 c x^2} \left (a^2 f^2 (45 B f+71 C e)-b^2 \left (6 C e^3-10 e f (10 A f+3 B e)\right )\right )}{2 c}-\frac {2 \sqrt {a^2 c-b^2 c x^2} \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (5 f (A f+3 B e)+13 C e^2\right )-\left (b^4 \left (3 C e^4-5 e^2 f (16 A f+3 B e)\right )\right )\right )}{b^2 c}}{3 b^2}-\frac {(e+f x)^2 \sqrt {a^2 c-b^2 c x^2} \left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (4 A f+3 B e)\right )\right )}{3 b^2 c}\right )+\frac {(e+f x)^3 \sqrt {a^2 c-b^2 c x^2} (C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^4 \sqrt {a^2 c-b^2 c x^2}}{5 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {1}{4} \left (\frac {\frac {15}{2} f \left (3 a^4 f^2 (B f+3 C e)+4 A \left (3 a^2 b^2 e f^2+2 b^4 e^3\right )+4 a^2 b^2 e^2 (3 B f+C e)\right ) \int \frac {1}{\frac {b^2 c x^2}{a^2 c-b^2 c x^2}+1}d\frac {x}{\sqrt {a^2 c-b^2 c x^2}}-\frac {f x \sqrt {a^2 c-b^2 c x^2} \left (a^2 f^2 (45 B f+71 C e)-b^2 \left (6 C e^3-10 e f (10 A f+3 B e)\right )\right )}{2 c}-\frac {2 \sqrt {a^2 c-b^2 c x^2} \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (5 f (A f+3 B e)+13 C e^2\right )-\left (b^4 \left (3 C e^4-5 e^2 f (16 A f+3 B e)\right )\right )\right )}{b^2 c}}{3 b^2}-\frac {(e+f x)^2 \sqrt {a^2 c-b^2 c x^2} \left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (4 A f+3 B e)\right )\right )}{3 b^2 c}\right )+\frac {(e+f x)^3 \sqrt {a^2 c-b^2 c x^2} (C e-5 B f)}{4 c}}{5 b^2 f}-\frac {C (e+f x)^4 \sqrt {a^2 c-b^2 c x^2}}{5 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {a^2 c-b^2 c x^2} \left (\frac {\frac {(e+f x)^3 \sqrt {a^2 c-b^2 c x^2} (C e-5 B f)}{4 c}+\frac {1}{4} \left (\frac {-\frac {f x \sqrt {a^2 c-b^2 c x^2} \left (a^2 f^2 (45 B f+71 C e)-b^2 \left (6 C e^3-10 e f (10 A f+3 B e)\right )\right )}{2 c}+\frac {15 f \arctan \left (\frac {b \sqrt {c} x}{\sqrt {a^2 c-b^2 c x^2}}\right ) \left (3 a^4 f^2 (B f+3 C e)+4 A \left (3 a^2 b^2 e f^2+2 b^4 e^3\right )+4 a^2 b^2 e^2 (3 B f+C e)\right )}{2 b \sqrt {c}}-\frac {2 \sqrt {a^2 c-b^2 c x^2} \left (16 a^4 C f^4+4 a^2 b^2 f^2 \left (5 f (A f+3 B e)+13 C e^2\right )-\left (b^4 \left (3 C e^4-5 e^2 f (16 A f+3 B e)\right )\right )\right )}{b^2 c}}{3 b^2}-\frac {(e+f x)^2 \sqrt {a^2 c-b^2 c x^2} \left (16 a^2 C f^2-b^2 \left (3 C e^2-5 f (4 A f+3 B e)\right )\right )}{3 b^2 c}\right )}{5 b^2 f}-\frac {C (e+f x)^4 \sqrt {a^2 c-b^2 c x^2}}{5 b^2 c f}\right )}{\sqrt {a+b x} \sqrt {a c-b c x}}\)

Input:

Int[((e + f*x)^3*(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]*(-1/5*(C*(e + f*x)^4*Sqrt[a^2*c - b^2*c*x^2])/(b^ 
2*c*f) + (((C*e - 5*B*f)*(e + f*x)^3*Sqrt[a^2*c - b^2*c*x^2])/(4*c) + (-1/ 
3*((16*a^2*C*f^2 - b^2*(3*C*e^2 - 5*f*(3*B*e + 4*A*f)))*(e + f*x)^2*Sqrt[a 
^2*c - b^2*c*x^2])/(b^2*c) + ((-2*(16*a^4*C*f^4 + 4*a^2*b^2*f^2*(13*C*e^2 
+ 5*f*(3*B*e + A*f)) - b^4*(3*C*e^4 - 5*e^2*f*(3*B*e + 16*A*f)))*Sqrt[a^2* 
c - b^2*c*x^2])/(b^2*c) - (f*(a^2*f^2*(71*C*e + 45*B*f) - b^2*(6*C*e^3 - 1 
0*e*f*(3*B*e + 10*A*f)))*x*Sqrt[a^2*c - b^2*c*x^2])/(2*c) + (15*f*(3*a^4*f 
^2*(3*C*e + B*f) + 4*a^2*b^2*e^2*(C*e + 3*B*f) + 4*A*(2*b^4*e^3 + 3*a^2*b^ 
2*e*f^2))*ArcTan[(b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(2*b*Sqrt[c]))/(3 
*b^2))/4)/(5*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 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 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 = 0.77 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.84

method	result	size
risch	\(-\frac {\left (24 C \,f^{3} x^{4} b^{4}+30 B \,b^{4} f^{3} x^{3}+90 C \,b^{4} e \,f^{2} x^{3}+40 A \,b^{4} f^{3} x^{2}+120 B \,b^{4} e \,f^{2} x^{2}+32 C \,a^{2} b^{2} f^{3} x^{2}+120 C \,b^{4} e^{2} f \,x^{2}+180 A \,b^{4} e \,f^{2} x +45 B \,a^{2} b^{2} f^{3} x +180 B \,b^{4} e^{2} f x +135 C \,a^{2} b^{2} e \,f^{2} x +60 C \,b^{4} e^{3} x +80 A \,a^{2} b^{2} f^{3}+360 A \,b^{4} e^{2} f +240 B \,a^{2} b^{2} e \,f^{2}+120 B \,b^{4} e^{3}+64 a^{4} C \,f^{3}+240 C \,a^{2} b^{2} e^{2} f \right ) \left (-b x +a \right ) \sqrt {b x +a}}{120 b^{6} \sqrt {-c \left (b x -a \right )}}+\frac {\left (12 A \,a^{2} b^{2} e \,f^{2}+8 A \,b^{4} e^{3}+3 B \,a^{4} f^{3}+12 B \,a^{2} e^{2} f \,b^{2}+9 C \,a^{4} e \,f^{2}+4 C \,a^{2} e^{3} b^{2}\right ) \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {-b^{2} c \,x^{2}+a^{2} c}}\right ) \sqrt {-\left (b x +a \right ) c \left (b x -a \right )}}{8 b^{4} \sqrt {b^{2} c}\, \sqrt {b x +a}\, \sqrt {-c \left (b x -a \right )}}\)	\(390\)
default	\(\frac {\sqrt {b x +a}\, \sqrt {c \left (-b x +a \right )}\, \left (-24 C \,b^{4} f^{3} x^{4} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}-30 B \,b^{4} f^{3} x^{3} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}-90 C \,b^{4} e \,f^{2} x^{3} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+180 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 \,f^{2}+120 A \arctan \left (\frac {\sqrt {b^{2} c}\, x}{\sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}}\right ) b^{6} c \,e^{3}-40 A \,b^{4} f^{3} x^{2} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+45 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^{3}+180 B \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^{2} f -120 B \,b^{4} e \,f^{2} x^{2} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}+135 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 \,f^{2}+60 C \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^{3}-32 C \,a^{2} b^{2} f^{3} x^{2} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}-120 C \,b^{4} e^{2} f \,x^{2} \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}-180 A \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{4} e \,f^{2} x -45 B \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, a^{2} b^{2} f^{3} x -180 B \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{4} e^{2} f x -135 C \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, a^{2} b^{2} e \,f^{2} x -60 C \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}\, b^{4} e^{3} x -80 A \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{2} b^{2} f^{3}-360 A \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{4} e^{2} f -240 B \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{2} b^{2} e \,f^{2}-120 B \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, b^{4} e^{3}-64 C \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{4} f^{3}-240 C \sqrt {b^{2} c}\, \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, a^{2} b^{2} e^{2} f \right )}{120 c \,b^{6} \sqrt {c \left (-b^{2} x^{2}+a^{2}\right )}\, \sqrt {b^{2} c}}\)	\(913\)

Input:

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

Output:

-1/120*(24*C*b^4*f^3*x^4+30*B*b^4*f^3*x^3+90*C*b^4*e*f^2*x^3+40*A*b^4*f^3* 
x^2+120*B*b^4*e*f^2*x^2+32*C*a^2*b^2*f^3*x^2+120*C*b^4*e^2*f*x^2+180*A*b^4 
*e*f^2*x+45*B*a^2*b^2*f^3*x+180*B*b^4*e^2*f*x+135*C*a^2*b^2*e*f^2*x+60*C*b 
^4*e^3*x+80*A*a^2*b^2*f^3+360*A*b^4*e^2*f+240*B*a^2*b^2*e*f^2+120*B*b^4*e^ 
3+64*C*a^4*f^3+240*C*a^2*b^2*e^2*f)/b^6*(-b*x+a)*(b*x+a)^(1/2)/(-c*(b*x-a) 
)^(1/2)+1/8*(12*A*a^2*b^2*e*f^2+8*A*b^4*e^3+3*B*a^4*f^3+12*B*a^2*b^2*e^2*f 
+9*C*a^4*e*f^2+4*C*a^2*b^2*e^3)/b^4/(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 = 0.12 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.51 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\left [-\frac {15 \, {\left (12 \, B a^{2} b^{3} e^{2} f + 3 \, B a^{4} b f^{3} + 4 \, {\left (C a^{2} b^{3} + 2 \, A b^{5}\right )} e^{3} + 3 \, {\left (3 \, C a^{4} b + 4 \, A a^{2} b^{3}\right )} e f^{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 (24 \, C b^{4} f^{3} x^{4} + 120 \, B b^{4} e^{3} + 240 \, B a^{2} b^{2} e f^{2} + 120 \, {\left (2 \, C a^{2} b^{2} + 3 \, A b^{4}\right )} e^{2} f + 16 \, {\left (4 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f^{3} + 30 \, {\left (3 \, C b^{4} e f^{2} + B b^{4} f^{3}\right )} x^{3} + 8 \, {\left (15 \, C b^{4} e^{2} f + 15 \, B b^{4} e f^{2} + {\left (4 \, C a^{2} b^{2} + 5 \, A b^{4}\right )} f^{3}\right )} x^{2} + 15 \, {\left (4 \, C b^{4} e^{3} + 12 \, B b^{4} e^{2} f + 3 \, B a^{2} b^{2} f^{3} + 3 \, {\left (3 \, C a^{2} b^{2} + 4 \, A b^{4}\right )} e f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{240 \, b^{6} c}, -\frac {15 \, {\left (12 \, B a^{2} b^{3} e^{2} f + 3 \, B a^{4} b f^{3} + 4 \, {\left (C a^{2} b^{3} + 2 \, A b^{5}\right )} e^{3} + 3 \, {\left (3 \, C a^{4} b + 4 \, A a^{2} b^{3}\right )} e f^{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 (24 \, C b^{4} f^{3} x^{4} + 120 \, B b^{4} e^{3} + 240 \, B a^{2} b^{2} e f^{2} + 120 \, {\left (2 \, C a^{2} b^{2} + 3 \, A b^{4}\right )} e^{2} f + 16 \, {\left (4 \, C a^{4} + 5 \, A a^{2} b^{2}\right )} f^{3} + 30 \, {\left (3 \, C b^{4} e f^{2} + B b^{4} f^{3}\right )} x^{3} + 8 \, {\left (15 \, C b^{4} e^{2} f + 15 \, B b^{4} e f^{2} + {\left (4 \, C a^{2} b^{2} + 5 \, A b^{4}\right )} f^{3}\right )} x^{2} + 15 \, {\left (4 \, C b^{4} e^{3} + 12 \, B b^{4} e^{2} f + 3 \, B a^{2} b^{2} f^{3} + 3 \, {\left (3 \, C a^{2} b^{2} + 4 \, A b^{4}\right )} e f^{2}\right )} x\right )} \sqrt {-b c x + a c} \sqrt {b x + a}}{120 \, b^{6} c}\right ] \] Input:

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

Output:

[-1/240*(15*(12*B*a^2*b^3*e^2*f + 3*B*a^4*b*f^3 + 4*(C*a^2*b^3 + 2*A*b^5)* 
e^3 + 3*(3*C*a^4*b + 4*A*a^2*b^3)*e*f^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*(24*C*b^4*f^3*x^4 + 
 120*B*b^4*e^3 + 240*B*a^2*b^2*e*f^2 + 120*(2*C*a^2*b^2 + 3*A*b^4)*e^2*f + 
 16*(4*C*a^4 + 5*A*a^2*b^2)*f^3 + 30*(3*C*b^4*e*f^2 + B*b^4*f^3)*x^3 + 8*( 
15*C*b^4*e^2*f + 15*B*b^4*e*f^2 + (4*C*a^2*b^2 + 5*A*b^4)*f^3)*x^2 + 15*(4 
*C*b^4*e^3 + 12*B*b^4*e^2*f + 3*B*a^2*b^2*f^3 + 3*(3*C*a^2*b^2 + 4*A*b^4)* 
e*f^2)*x)*sqrt(-b*c*x + a*c)*sqrt(b*x + a))/(b^6*c), -1/120*(15*(12*B*a^2* 
b^3*e^2*f + 3*B*a^4*b*f^3 + 4*(C*a^2*b^3 + 2*A*b^5)*e^3 + 3*(3*C*a^4*b + 4 
*A*a^2*b^3)*e*f^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)) + (24*C*b^4*f^3*x^4 + 120*B*b^4*e^3 + 240*B*a^2* 
b^2*e*f^2 + 120*(2*C*a^2*b^2 + 3*A*b^4)*e^2*f + 16*(4*C*a^4 + 5*A*a^2*b^2) 
*f^3 + 30*(3*C*b^4*e*f^2 + B*b^4*f^3)*x^3 + 8*(15*C*b^4*e^2*f + 15*B*b^4*e 
*f^2 + (4*C*a^2*b^2 + 5*A*b^4)*f^3)*x^2 + 15*(4*C*b^4*e^3 + 12*B*b^4*e^2*f 
 + 3*B*a^2*b^2*f^3 + 3*(3*C*a^2*b^2 + 4*A*b^4)*e*f^2)*x)*sqrt(-b*c*x + a*c 
)*sqrt(b*x + a))/(b^6*c)]
 

Sympy [F(-1)]

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

integrate((f*x+e)**3*(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 = 0.12 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.01 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {\sqrt {-b^{2} c x^{2} + a^{2} c} C f^{3} x^{4}}{5 \, b^{2} c} - \frac {4 \, \sqrt {-b^{2} c x^{2} + a^{2} c} C a^{2} f^{3} x^{2}}{15 \, b^{4} c} + \frac {A e^{3} \arcsin \left (\frac {b x}{a}\right )}{b \sqrt {c}} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} B e^{3}}{b^{2} c} - \frac {3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} A e^{2} f}{b^{2} c} - \frac {8 \, \sqrt {-b^{2} c x^{2} + a^{2} c} C a^{4} f^{3}}{15 \, b^{6} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e f^{2} + B f^{3}\right )} x^{3}}{4 \, b^{2} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} x^{2}}{3 \, b^{2} c} + \frac {3 \, {\left (3 \, C e f^{2} + B f^{3}\right )} a^{4} \arcsin \left (\frac {b x}{a}\right )}{8 \, b^{5} \sqrt {c}} + \frac {{\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} a^{2} \arcsin \left (\frac {b x}{a}\right )}{2 \, b^{3} \sqrt {c}} - \frac {3 \, \sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e f^{2} + B f^{3}\right )} a^{2} x}{8 \, b^{4} c} - \frac {\sqrt {-b^{2} c x^{2} + a^{2} c} {\left (C e^{3} + 3 \, B e^{2} f + 3 \, A e f^{2}\right )} x}{2 \, b^{2} c} - \frac {2 \, \sqrt {-b^{2} c x^{2} + a^{2} c} {\left (3 \, C e^{2} f + 3 \, B e f^{2} + A f^{3}\right )} a^{2}}{3 \, b^{4} c} \] Input:

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

Output:

-1/5*sqrt(-b^2*c*x^2 + a^2*c)*C*f^3*x^4/(b^2*c) - 4/15*sqrt(-b^2*c*x^2 + a 
^2*c)*C*a^2*f^3*x^2/(b^4*c) + A*e^3*arcsin(b*x/a)/(b*sqrt(c)) - sqrt(-b^2* 
c*x^2 + a^2*c)*B*e^3/(b^2*c) - 3*sqrt(-b^2*c*x^2 + a^2*c)*A*e^2*f/(b^2*c) 
- 8/15*sqrt(-b^2*c*x^2 + a^2*c)*C*a^4*f^3/(b^6*c) - 1/4*sqrt(-b^2*c*x^2 + 
a^2*c)*(3*C*e*f^2 + B*f^3)*x^3/(b^2*c) - 1/3*sqrt(-b^2*c*x^2 + a^2*c)*(3*C 
*e^2*f + 3*B*e*f^2 + A*f^3)*x^2/(b^2*c) + 3/8*(3*C*e*f^2 + B*f^3)*a^4*arcs 
in(b*x/a)/(b^5*sqrt(c)) + 1/2*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)*a^2*arcsin(b 
*x/a)/(b^3*sqrt(c)) - 3/8*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e*f^2 + B*f^3)*a^2 
*x/(b^4*c) - 1/2*sqrt(-b^2*c*x^2 + a^2*c)*(C*e^3 + 3*B*e^2*f + 3*A*e*f^2)* 
x/(b^2*c) - 2/3*sqrt(-b^2*c*x^2 + a^2*c)*(3*C*e^2*f + 3*B*e*f^2 + A*f^3)*a 
^2/(b^4*c)
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 571, normalized size of antiderivative = 1.23 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=-\frac {{\left ({\left (2 \, {\left (3 \, {\left (\frac {4 \, {\left (b x + a\right )} C f^{3}}{c} + \frac {15 \, C b c^{4} e f^{2} - 16 \, C a c^{4} f^{3} + 5 \, B b c^{4} f^{3}}{c^{5}}\right )} {\left (b x + a\right )} + \frac {60 \, C b^{2} c^{4} e^{2} f - 135 \, C a b c^{4} e f^{2} + 60 \, B b^{2} c^{4} e f^{2} + 88 \, C a^{2} c^{4} f^{3} - 45 \, B a b c^{4} f^{3} + 20 \, A b^{2} c^{4} f^{3}}{c^{5}}\right )} {\left (b x + a\right )} + \frac {5 \, {\left (12 \, C b^{3} c^{4} e^{3} - 48 \, C a b^{2} c^{4} e^{2} f + 36 \, B b^{3} c^{4} e^{2} f + 81 \, C a^{2} b c^{4} e f^{2} - 48 \, B a b^{2} c^{4} e f^{2} + 36 \, A b^{3} c^{4} e f^{2} - 32 \, C a^{3} c^{4} f^{3} + 27 \, B a^{2} b c^{4} f^{3} - 16 \, A a b^{2} c^{4} f^{3}\right )}}{c^{5}}\right )} {\left (b x + a\right )} - \frac {15 \, {\left (4 \, C a b^{3} c^{4} e^{3} - 8 \, B b^{4} c^{4} e^{3} - 24 \, C a^{2} b^{2} c^{4} e^{2} f + 12 \, B a b^{3} c^{4} e^{2} f - 24 \, A b^{4} c^{4} e^{2} f + 15 \, C a^{3} b c^{4} e f^{2} - 24 \, B a^{2} b^{2} c^{4} e f^{2} + 12 \, A a b^{3} c^{4} e f^{2} - 8 \, C a^{4} c^{4} f^{3} + 5 \, B a^{3} b c^{4} f^{3} - 8 \, A a^{2} b^{2} c^{4} f^{3}\right )}}{c^{5}}\right )} \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \sqrt {b x + a} + \frac {30 \, {\left (4 \, C a^{2} b^{3} e^{3} + 8 \, A b^{5} e^{3} + 12 \, B a^{2} b^{3} e^{2} f + 9 \, C a^{4} b e f^{2} + 12 \, A a^{2} b^{3} e f^{2} + 3 \, B a^{4} b f^{3}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {-c} + \sqrt {-{\left (b x + a\right )} c + 2 \, a c} \right |}\right )}{\sqrt {-c}}}{120 \, b^{6}} \] Input:

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

Output:

-1/120*(((2*(3*(4*(b*x + a)*C*f^3/c + (15*C*b*c^4*e*f^2 - 16*C*a*c^4*f^3 + 
 5*B*b*c^4*f^3)/c^5)*(b*x + a) + (60*C*b^2*c^4*e^2*f - 135*C*a*b*c^4*e*f^2 
 + 60*B*b^2*c^4*e*f^2 + 88*C*a^2*c^4*f^3 - 45*B*a*b*c^4*f^3 + 20*A*b^2*c^4 
*f^3)/c^5)*(b*x + a) + 5*(12*C*b^3*c^4*e^3 - 48*C*a*b^2*c^4*e^2*f + 36*B*b 
^3*c^4*e^2*f + 81*C*a^2*b*c^4*e*f^2 - 48*B*a*b^2*c^4*e*f^2 + 36*A*b^3*c^4* 
e*f^2 - 32*C*a^3*c^4*f^3 + 27*B*a^2*b*c^4*f^3 - 16*A*a*b^2*c^4*f^3)/c^5)*( 
b*x + a) - 15*(4*C*a*b^3*c^4*e^3 - 8*B*b^4*c^4*e^3 - 24*C*a^2*b^2*c^4*e^2* 
f + 12*B*a*b^3*c^4*e^2*f - 24*A*b^4*c^4*e^2*f + 15*C*a^3*b*c^4*e*f^2 - 24* 
B*a^2*b^2*c^4*e*f^2 + 12*A*a*b^3*c^4*e*f^2 - 8*C*a^4*c^4*f^3 + 5*B*a^3*b*c 
^4*f^3 - 8*A*a^2*b^2*c^4*f^3)/c^5)*sqrt(-(b*x + a)*c + 2*a*c)*sqrt(b*x + a 
) + 30*(4*C*a^2*b^3*e^3 + 8*A*b^5*e^3 + 12*B*a^2*b^3*e^2*f + 9*C*a^4*b*e*f 
^2 + 12*A*a^2*b^3*e*f^2 + 3*B*a^4*b*f^3)*log(abs(-sqrt(b*x + a)*sqrt(-c) + 
 sqrt(-(b*x + a)*c + 2*a*c)))/sqrt(-c))/b^6
 

Mupad [B] (verification not implemented)

Time = 95.58 (sec) , antiderivative size = 4167, normalized size of antiderivative = 8.96 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\text {Too large to display} \] Input:

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

Output:

- ((((23*B*a^4*c*f^3)/2 - 18*B*a^2*b^2*c*e^2*f)*((a*c - b*c*x)^(1/2) - (a* 
c)^(1/2))^13)/(b^5*((a + b*x)^(1/2) - a^(1/2))^13) + (((a*c - b*c*x)^(1/2) 
 - (a*c)^(1/2))^15*((3*B*a^4*f^3)/2 + 6*B*a^2*b^2*e^2*f))/(b^5*((a + b*x)^ 
(1/2) - a^(1/2))^15) - (((3*B*a^4*c^7*f^3)/2 + 6*B*a^2*b^2*c^7*e^2*f)*((a* 
c - b*c*x)^(1/2) - (a*c)^(1/2)))/(b^5*((a + b*x)^(1/2) - a^(1/2))) - (((23 
*B*a^4*c^6*f^3)/2 - 18*B*a^2*b^2*c^6*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^( 
1/2))^3)/(b^5*((a + b*x)^(1/2) - a^(1/2))^3) + (((333*B*a^4*c^5*f^3)/2 + 9 
0*B*a^2*b^2*c^5*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^5)/(b^5*((a + b 
*x)^(1/2) - a^(1/2))^5) - (((333*B*a^4*c^2*f^3)/2 + 90*B*a^2*b^2*c^2*e^2*f 
)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^11)/(b^5*((a + b*x)^(1/2) - a^(1/2)) 
^11) - (((671*B*a^4*c^4*f^3)/2 - 66*B*a^2*b^2*c^4*e^2*f)*((a*c - b*c*x)^(1 
/2) - (a*c)^(1/2))^7)/(b^5*((a + b*x)^(1/2) - a^(1/2))^7) + (((671*B*a^4*c 
^3*f^3)/2 - 66*B*a^2*b^2*c^3*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^9) 
/(b^5*((a + b*x)^(1/2) - a^(1/2))^9) + (a^(1/2)*(a*c)^(1/2)*(48*B*b^2*c^5* 
e^3 + 192*B*a^2*c^5*e*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/(b^4*((a 
 + b*x)^(1/2) - a^(1/2))^4) + (a^(1/2)*(a*c)^(1/2)*(160*B*b^2*c^3*e^3 + 12 
8*B*a^2*c^3*e*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^8)/(b^4*((a + b*x)^ 
(1/2) - a^(1/2))^8) + (a^(1/2)*(a*c)^(1/2)*(120*B*b^2*c^4*e^3 + 256*B*a^2* 
c^4*e*f^2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6)/(b^4*((a + b*x)^(1/2) - 
a^(1/2))^6) + (a^(1/2)*(a*c)^(1/2)*(120*B*b^2*c^2*e^3 + 256*B*a^2*c^2*e...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 622, normalized size of antiderivative = 1.34 \[ \int \frac {(e+f x)^3 \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {a c-b c x}} \, dx=\frac {\sqrt {c}\, \left (-90 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{4} b^{2} f^{3}-270 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{4} b c e \,f^{2}-360 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{3} b^{3} e \,f^{2}-360 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{2} b^{4} e^{2} f -120 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a^{2} b^{3} c \,e^{3}-240 \mathit {asin} \left (\frac {\sqrt {-b x +a}}{\sqrt {a}\, \sqrt {2}}\right ) a \,b^{5} e^{3}-64 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{4} c \,f^{3}-80 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{3} b^{2} f^{3}-240 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{3} e \,f^{2}-45 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{3} f^{3} x -240 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{2} c \,e^{2} f -135 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{2} c e \,f^{2} x -32 \sqrt {b x +a}\, \sqrt {-b x +a}\, a^{2} b^{2} c \,f^{3} x^{2}-360 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{4} e^{2} f -180 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{4} e \,f^{2} x -40 \sqrt {b x +a}\, \sqrt {-b x +a}\, a \,b^{4} f^{3} x^{2}-120 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{5} e^{3}-180 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{5} e^{2} f x -120 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{5} e \,f^{2} x^{2}-30 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{5} f^{3} x^{3}-60 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{4} c \,e^{3} x -120 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{4} c \,e^{2} f \,x^{2}-90 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{4} c e \,f^{2} x^{3}-24 \sqrt {b x +a}\, \sqrt {-b x +a}\, b^{4} c \,f^{3} x^{4}\right )}{120 b^{6} c} \] Input:

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

Output:

(sqrt(c)*( - 90*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**4*b**2*f**3 - 270 
*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**4*b*c*e*f**2 - 360*asin(sqrt(a - 
 b*x)/(sqrt(a)*sqrt(2)))*a**3*b**3*e*f**2 - 360*asin(sqrt(a - b*x)/(sqrt(a 
)*sqrt(2)))*a**2*b**4*e**2*f - 120*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a 
**2*b**3*c*e**3 - 240*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a*b**5*e**3 - 
64*sqrt(a + b*x)*sqrt(a - b*x)*a**4*c*f**3 - 80*sqrt(a + b*x)*sqrt(a - b*x 
)*a**3*b**2*f**3 - 240*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b**3*e*f**2 - 45*s 
qrt(a + b*x)*sqrt(a - b*x)*a**2*b**3*f**3*x - 240*sqrt(a + b*x)*sqrt(a - b 
*x)*a**2*b**2*c*e**2*f - 135*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b**2*c*e*f** 
2*x - 32*sqrt(a + b*x)*sqrt(a - b*x)*a**2*b**2*c*f**3*x**2 - 360*sqrt(a + 
b*x)*sqrt(a - b*x)*a*b**4*e**2*f - 180*sqrt(a + b*x)*sqrt(a - b*x)*a*b**4* 
e*f**2*x - 40*sqrt(a + b*x)*sqrt(a - b*x)*a*b**4*f**3*x**2 - 120*sqrt(a + 
b*x)*sqrt(a - b*x)*b**5*e**3 - 180*sqrt(a + b*x)*sqrt(a - b*x)*b**5*e**2*f 
*x - 120*sqrt(a + b*x)*sqrt(a - b*x)*b**5*e*f**2*x**2 - 30*sqrt(a + b*x)*s 
qrt(a - b*x)*b**5*f**3*x**3 - 60*sqrt(a + b*x)*sqrt(a - b*x)*b**4*c*e**3*x 
 - 120*sqrt(a + b*x)*sqrt(a - b*x)*b**4*c*e**2*f*x**2 - 90*sqrt(a + b*x)*s 
qrt(a - b*x)*b**4*c*e*f**2*x**3 - 24*sqrt(a + b*x)*sqrt(a - b*x)*b**4*c*f* 
*3*x**4))/(120*b**6*c)