\(\int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx\) [57]

Optimal result

Integrand size = 40, antiderivative size = 235 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\frac {\left (C e^2-B e f+A f^2\right ) \sqrt {a+b x} \sqrt {a c-b c x}}{c f (b e-a f) (b e+a f) (e+f x)}+\frac {2 C \arctan \left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a c-b c x}}\right )}{b \sqrt {c} f^2}+\frac {2 \left (a^2 f^2 (2 C e-B f)-b^2 \left (C e^3-A e f^2\right )\right ) \arctan \left (\frac {\sqrt {c} \sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a c-b c x}}\right )}{\sqrt {c} f^2 (b e-a f)^{3/2} (b e+a f)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\frac {2 \left (\frac {f \left (C e^2+f (-B e+A f)\right ) (-a+b x) \sqrt {a+b x}}{2 (-b e+a f) (b e+a f) (e+f x)}+\frac {C \sqrt {a-b x} \arctan \left (\frac {\sqrt {a+b x}}{\sqrt {a-b x}}\right )}{b}-\frac {\left (a^2 f^2 (-2 C e+B f)+b^2 \left (C e^3-A e f^2\right )\right ) \sqrt {a-b x} \arctan \left (\frac {\sqrt {b e+a f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {a-b x}}\right )}{(b e-a f)^{3/2} (b e+a f)^{3/2}}\right )}{f^2 \sqrt {c (a-b x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.31, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2113, 2182, 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.

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

\(\Big \downarrow \) 2113

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 217

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

Input:

Int[(A + B*x + C*x^2)/(Sqrt[a + b*x]*Sqrt[a*c - b*c*x]*(e + f*x)^2),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])/(c*(b^2*e^2 - a^2*f^2)*(e + f*x)) + ((C*(b^2*e^2 - a^2*f^2)*ArcTan[( 
b*Sqrt[c]*x)/Sqrt[a^2*c - b^2*c*x^2]])/(b*Sqrt[c]*f^2) + ((a^2*f^2*(2*C*e 
- B*f) - b^2*(C*e^3 - A*e*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]*f^2*Sqrt[b^2*e^2 - 
 a^2*f^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 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 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. \(1165\) vs. \(2(207)=414\).

Time = 1.23 (sec) , antiderivative size = 1166, normalized size of antiderivative = 4.96

method	result	size
default	\(\text {Expression too large to display}\)	\(1166\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((C*x^2+B*x+A)/(b*x+a)^(1/2)/(-b*c*x+a*c)^(1/2)/(f*x+e)^2,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((4*b^2*c>0)', see `assume?` for 
more detai
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (207) = 414\).

Time = 0.30 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.24 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\frac {\frac {2 \, {\left (C b^{3} \sqrt {-c} e^{3} - 2 \, C a^{2} b \sqrt {-c} e f^{2} - A b^{3} \sqrt {-c} e f^{2} + B a^{2} b \sqrt {-c} f^{3}\right )} \arctan \left (-\frac {2 \, b c e - {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} f}{2 \, \sqrt {-b^{2} e^{2} + a^{2} f^{2}} c}\right )}{{\left (b^{2} e^{2} f^{2} - a^{2} f^{4}\right )} \sqrt {-b^{2} e^{2} + a^{2} f^{2}} c} - \frac {C \log \left ({\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2}\right )}{\sqrt {-c} f^{2}} + \frac {4 \, {\left (C b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} e^{3} - B b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} e^{2} f - 2 \, C a^{2} b^{2} \sqrt {-c} c e^{2} f + A b^{3} {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} \sqrt {-c} e f^{2} + 2 \, B a^{2} b^{2} \sqrt {-c} c e f^{2} - 2 \, A a^{2} b^{2} \sqrt {-c} c f^{3}\right )}}{{\left (b^{2} e^{2} f^{2} - a^{2} f^{4}\right )} {\left (4 \, b {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{2} c e - {\left (\sqrt {b x + a} \sqrt {-c} - \sqrt {-{\left (b x + a\right )} c + 2 \, a c}\right )}^{4} f - 4 \, a^{2} c^{2} f\right )}}}{b} \] Input:

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

Output:

(2*(C*b^3*sqrt(-c)*e^3 - 2*C*a^2*b*sqrt(-c)*e*f^2 - A*b^3*sqrt(-c)*e*f^2 + 
 B*a^2*b*sqrt(-c)*f^3)*arctan(-1/2*(2*b*c*e - (sqrt(b*x + a)*sqrt(-c) - sq 
rt(-(b*x + a)*c + 2*a*c))^2*f)/(sqrt(-b^2*e^2 + a^2*f^2)*c))/((b^2*e^2*f^2 
 - a^2*f^4)*sqrt(-b^2*e^2 + a^2*f^2)*c) - C*log((sqrt(b*x + a)*sqrt(-c) - 
sqrt(-(b*x + a)*c + 2*a*c))^2)/(sqrt(-c)*f^2) + 4*(C*b^3*(sqrt(b*x + a)*sq 
rt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*e^3 - B*b^3*(sqrt(b*x + a) 
*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c))^2*sqrt(-c)*e^2*f - 2*C*a^2*b^2*sqr 
t(-c)*c*e^2*f + A*b^3*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a*c) 
)^2*sqrt(-c)*e*f^2 + 2*B*a^2*b^2*sqrt(-c)*c*e*f^2 - 2*A*a^2*b^2*sqrt(-c)*c 
*f^3)/((b^2*e^2*f^2 - a^2*f^4)*(4*b*(sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + 
 a)*c + 2*a*c))^2*c*e - (sqrt(b*x + a)*sqrt(-c) - sqrt(-(b*x + a)*c + 2*a* 
c))^4*f - 4*a^2*c^2*f)))/b
 

Mupad [B] (verification not implemented)

Time = 121.92 (sec) , antiderivative size = 106511, normalized size of antiderivative = 453.24 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.93 (sec) , antiderivative size = 8629, normalized size of antiderivative = 36.72 \[ \int \frac {A+B x+C x^2}{\sqrt {a+b x} \sqrt {a c-b c x} (e+f x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(c)*( - 2*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**5*c*e*f**5 - 2*asi 
n(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**5*c*f**6*x - 2*asin(sqrt(a - b*x)/(s 
qrt(a)*sqrt(2)))*a**4*b*c*e**2*f**4 - 2*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2 
)))*a**4*b*c*e*f**5*x + 4*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**2* 
c*e**3*f**3 + 4*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**3*b**2*c*e**2*f** 
4*x + 4*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**2*b**3*c*e**4*f**2 + 4*as 
in(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*a**2*b**3*c*e**3*f**3*x - 2*asin(sqrt( 
a - b*x)/(sqrt(a)*sqrt(2)))*a*b**4*c*e**5*f - 2*asin(sqrt(a - b*x)/(sqrt(a 
)*sqrt(2)))*a*b**4*c*e**4*f**2*x - 2*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2))) 
*b**5*c*e**6 - 2*asin(sqrt(a - b*x)/(sqrt(a)*sqrt(2)))*b**5*c*e**5*f*x - 2 
*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**2*b**2*e*f**3 - 2*sqrt(f)*sqrt(a)*sqrt(2*sqrt(f)*sqrt(a)*sqr 
t(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**2*b**2*f**4*x + 4*sqrt(f)*sqr 
t(a)*sqrt(2*sqrt(f)*sqrt(a)*sqrt(a*f - b*e)*sqrt(2) - 3*a*f + b*e)*sqrt...