\(\int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx\) [58]

Optimal result

Integrand size = 40, antiderivative size = 301 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x}}{2 c f (b e-a f) (b e+a f) (e+f x)^2}+\frac {\left (2 a^2 f^2 (2 C e-B f)-b^2 \left (C e^3+e f (B e-3 A f)\right )\right ) \sqrt {a+b x} \sqrt {a c-b c x}}{2 c f (b e-a f)^2 (b e+a f)^2 (e+f x)}+\frac {\left (2 a^4 C f^2+a^2 b^2 e (C e-3 B f)+A \left (2 b^4 e^2+a^2 b^2 f^2\right )\right ) \arctan \left (\frac {\sqrt {c} \sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a c-b c x}}\right )}{\sqrt {c} (b e-a f)^{5/2} (b e+a f)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.99 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\frac {\frac {(-a+b x) \sqrt {a+b x} \left (b^2 e \left (C e^2 x+B e (2 e+f x)-A f (4 e+3 f x)\right )+a^2 f (-C e (3 e+4 f x)+f (A f+B (e+2 f x)))\right )}{2 (b e-a f)^2 (b e+a f)^2 (e+f x)^2}+\frac {\left (2 a^4 C f^2+a^2 b^2 e (C e-3 B f)+A \left (2 b^4 e^2+a^2 b^2 f^2\right )\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{(b e-a f)^{5/2} (b e+a f)^{5/2}}}{\sqrt {c (a-b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2113, 2182, 27, 679, 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.

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

\(\Big \downarrow \) 2113

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 679

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 217

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

Input:

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

Output:

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

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 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 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 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 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. \(1793\) vs. \(2(273)=546\).

Time = 1.32 (sec) , antiderivative size = 1794, normalized size of antiderivative = 5.96

method	result	size
default	\(\text {Expression too large to display}\)	\(1794\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 661 vs. \(2 (273) = 546\).

Time = 38.13 (sec) , antiderivative size = 1355, normalized size of antiderivative = 4.50 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(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. 1425 vs. \(2 (273) = 546\).

Time = 0.64 (sec) , antiderivative size = 1425, normalized size of antiderivative = 4.73 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

-((C*a^2*b^3*sqrt(-c)*e^2 + 2*A*b^5*sqrt(-c)*e^2 - 3*B*a^2*b^3*sqrt(-c)*e* 
f + 2*C*a^4*b*sqrt(-c)*f^2 + A*a^2*b^3*sqrt(-c)*f^2)*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^4*e^4 - 2*a^2*b^2*e^2*f^2 + a^4*f^4)*sqrt(-b^2*e^2 + 
a^2*f^2)*c) + 2*(4*C*b^6*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a 
*c))^4*sqrt(-c)*c*e^5 - 2*C*b^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)* 
c + 2*a*c))^6*sqrt(-c)*e^4*f + 4*B*b^6*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b* 
x + a)*c + 2*a*c))^4*sqrt(-c)*c*e^4*f - 8*C*a^2*b^5*(sqrt(b*x + a)*sqrt(-c 
) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*c^2*e^4*f - 14*C*a^2*b^4*(sqrt( 
b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*sqrt(-c)*c*e^3*f^2 - 12* 
A*b^6*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^4*sqrt(-c)*c*e 
^3*f^2 - 16*B*a^2*b^5*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c) 
)^2*sqrt(-c)*c^2*e^3*f^2 + 8*C*a^4*b^4*sqrt(-c)*c^3*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)*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)* 
e^2*f^3 + 10*B*a^2*b^4*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c 
))^4*sqrt(-c)*c*e^2*f^3 + 44*C*a^4*b^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b* 
x + a)*c + 2*a*c))^2*sqrt(-c)*c^2*e^2*f^3 + 40*A*a^2*b^5*(sqrt(b*x + a)*sq 
rt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*c^2*e^2*f^3 + 8*B*a^4*b^4* 
sqrt(-c)*c^3*e^2*f^3 - 3*B*a^2*b^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x...
 

Mupad [B] (verification not implemented)

Time = 67.45 (sec) , antiderivative size = 9344, normalized size of antiderivative = 31.04 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

((((a*c - b*c*x)^(1/2) - (a*c)^(1/2))*(4*C*a^4*c^3*f^2 + 2*C*a^2*b^2*c^3*e 
^2))/(((a + b*x)^(1/2) - a^(1/2))*(b^5*e^5 - 2*a^2*b^3*e^3*f^2 + a^4*b*e*f 
^4)) + (((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^3*(68*C*a^4*c^2*f^2 - 14*C*a^2 
*b^2*c^2*e^2))/(((a + b*x)^(1/2) - a^(1/2))^3*(b^5*e^5 - 2*a^2*b^3*e^3*f^2 
 + a^4*b*e*f^4)) - ((68*C*a^4*c*f^2 - 14*C*a^2*b^2*c*e^2)*((a*c - b*c*x)^( 
1/2) - (a*c)^(1/2))^5)/(((a + b*x)^(1/2) - a^(1/2))^5*(b^5*e^5 - 2*a^2*b^3 
*e^3*f^2 + a^4*b*e*f^4)) - ((4*C*a^4*f^2 + 2*C*a^2*b^2*e^2)*((a*c - b*c*x) 
^(1/2) - (a*c)^(1/2))^7)/(((a + b*x)^(1/2) - a^(1/2))^7*(b^5*e^5 - 2*a^2*b 
^3*e^3*f^2 + a^4*b*e*f^4)) - (a^(1/2)*(a*c)^(1/2)*(48*C*a^4*c*f^3 - 24*C*a 
^2*b^2*c*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^4)/(((a + b*x)^(1/2) - 
 a^(1/2))^4*(b^6*e^6 - 2*a^2*b^4*e^4*f^2 + a^4*b^2*e^2*f^4)) + (a^(1/2)*(a 
*c)^(1/2)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^6*(24*C*a^4*f^3 + 12*C*a^2*b 
^2*e^2*f))/(((a + b*x)^(1/2) - a^(1/2))^6*(b^6*e^6 - 2*a^2*b^4*e^4*f^2 + a 
^4*b^2*e^2*f^4)) + (a^(1/2)*(a*c)^(1/2)*(24*C*a^4*c^2*f^3 + 12*C*a^2*b^2*c 
^2*e^2*f)*((a*c - b*c*x)^(1/2) - (a*c)^(1/2))^2)/(((a + b*x)^(1/2) - a^(1/ 
2))^2*(b^6*e^6 - 2*a^2*b^4*e^4*f^2 + a^4*b^2*e^2*f^4)))/(((a*c - b*c*x)^(1 
/2) - (a*c)^(1/2))^8/((a + b*x)^(1/2) - a^(1/2))^8 + c^4 + (((a*c - b*c*x) 
^(1/2) - (a*c)^(1/2))^6*(16*a^2*c*f^2 + 4*b^2*c*e^2))/(b^2*e^2*((a + b*x)^ 
(1/2) - a^(1/2))^6) + ((16*a^2*c^3*f^2 + 4*b^2*c^3*e^2)*((a*c - b*c*x)^(1/ 
2) - (a*c)^(1/2))^2)/(b^2*e^2*((a + b*x)^(1/2) - a^(1/2))^2) - ((32*a^2...
 

Reduce [B] (verification not implemented)

Time = 3.09 (sec) , antiderivative size = 15212, normalized size of antiderivative = 50.54 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(c)*( - 4*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt 
(2) - 3*a*f + b*e)*sqrt(a*f + b*e)*sqrt(a*f - b*e)*sqrt(2)*atan((tan(asin( 
sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt(a - b*x)/(sqrt(a)* 
sqrt(2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f 
+ b*e)*sqrt(a*f + b*e)))*a**4*c*e**2*f**2 - 8*sqrt(f)*sqrt(a)*sqrt(2*sqrt( 
f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)*sqrt(a*f 
 - b*e)*sqrt(2)*atan((tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + t 
an(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)*s 
qrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)))*a**4*c*e*f**3*x - 
4*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + 
 b*e)*sqrt(a*f + b*e)*sqrt(a*f - b*e)*sqrt(2)*atan((tan(asin(sqrt(a - b*x) 
/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)* 
b*e)/(sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a 
*f + b*e)))*a**4*c*f**4*x**2 - 2*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sq 
rt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)*sqrt(a*f - b*e)*sqrt( 
2)*atan((tan(asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))/2)*a*f + tan(asin(sqrt( 
a - b*x)/(sqrt(a)*sqrt(2)))/2)*b*e)/(sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e 
)*sqrt(2) - 3*a*f + b*e)*sqrt(a*f + b*e)))*a**3*b**2*e**2*f**2 - 4*sqrt(f) 
*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqr 
t(a*f + b*e)*sqrt(a*f - b*e)*sqrt(2)*atan((tan(asin(sqrt(a - b*x)/(sqrt...