\(\int \frac {\sqrt {1-d x} \sqrt {1+d x} (A+B x+C x^2)}{(e+f x)^2} \, dx\) [36]

Optimal result

Integrand size = 37, antiderivative size = 322 \[ \int \frac {\sqrt {1-d x} \sqrt {1+d x} \left (A+B x+C x^2\right )}{(e+f x)^2} \, dx=\frac {\left (2 \left (2 B d^2 e^2-\frac {3 C d^2 e^3}{f}+2 C e f-A d^2 e f-B f^2\right )-\left (2 d^2 f (B e-A f)-C \left (3 d^2 e^2-f^2\right )\right ) x\right ) \sqrt {1-d^2 x^2}}{2 f^2 \left (d^2 e^2-f^2\right )}+\frac {\left (C e^2-B e f+A f^2\right ) \left (1-d^2 x^2\right )^{3/2}}{f \left (d^2 e^2-f^2\right ) (e+f x)}+\frac {\left (2 d^2 f (2 B e-A f)-C \left (6 d^2 e^2-f^2\right )\right ) \arcsin (d x)}{2 d f^4}+\frac {\left (3 C d^2 e^3-2 B d^2 e^2 f-2 C e f^2+A d^2 e f^2+B f^3\right ) \arctan \left (\frac {f+d^2 e x}{\sqrt {d^2 e^2-f^2} \sqrt {1-d^2 x^2}}\right )}{f^4 \sqrt {d^2 e^2-f^2}} \] Output:

1/2*(4*B*d^2*e^2-6*C*d^2*e^3/f+4*C*e*f-2*A*d^2*e*f-2*B*f^2-(2*d^2*f*(-A*f+ 
B*e)-C*(3*d^2*e^2-f^2))*x)*(-d^2*x^2+1)^(1/2)/f^2/(d^2*e^2-f^2)+(A*f^2-B*e 
*f+C*e^2)*(-d^2*x^2+1)^(3/2)/f/(d^2*e^2-f^2)/(f*x+e)+1/2*(2*d^2*f*(-A*f+2* 
B*e)-C*(6*d^2*e^2-f^2))*arcsin(d*x)/d/f^4+(A*d^2*e*f^2-2*B*d^2*e^2*f+3*C*d 
^2*e^3+B*f^3-2*C*e*f^2)*arctan((d^2*e*x+f)/(d^2*e^2-f^2)^(1/2)/(-d^2*x^2+1 
)^(1/2))/f^4/(d^2*e^2-f^2)^(1/2)
 

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {1-d x} \sqrt {1+d x} \left (A+B x+C x^2\right )}{(e+f x)^2} \, dx=\frac {\frac {f \sqrt {1-d^2 x^2} \left (2 f (2 B e-A f+B f x)+C \left (-6 e^2-3 e f x+f^2 x^2\right )\right )}{e+f x}+\frac {2 \left (-2 d^2 f (-2 B e+A f)+C \left (-6 d^2 e^2+f^2\right )\right ) \arctan \left (\frac {d x}{-1+\sqrt {1-d^2 x^2}}\right )}{d}-\frac {4 \sqrt {d^2 e^2-f^2} \left (3 C d^2 e^3-2 B d^2 e^2 f-2 C e f^2+A d^2 e f^2+B f^3\right ) \arctan \left (\frac {\sqrt {d^2 e^2-f^2} x}{e+f x-e \sqrt {1-d^2 x^2}}\right )}{(d e-f) (d e+f)}}{2 f^4} \] Input:

Integrate[(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(A + B*x + C*x^2))/(e + f*x)^2,x]
 

Output:

((f*Sqrt[1 - d^2*x^2]*(2*f*(2*B*e - A*f + B*f*x) + C*(-6*e^2 - 3*e*f*x + f 
^2*x^2)))/(e + f*x) + (2*(-2*d^2*f*(-2*B*e + A*f) + C*(-6*d^2*e^2 + f^2))* 
ArcTan[(d*x)/(-1 + Sqrt[1 - d^2*x^2])])/d - (4*Sqrt[d^2*e^2 - f^2]*(3*C*d^ 
2*e^3 - 2*B*d^2*e^2*f - 2*C*e*f^2 + A*d^2*e*f^2 + B*f^3)*ArcTan[(Sqrt[d^2* 
e^2 - f^2]*x)/(e + f*x - e*Sqrt[1 - d^2*x^2])])/((d*e - f)*(d*e + f)))/(2* 
f^4)
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.216, Rules used = {2112, 2182, 682, 27, 719, 223, 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.

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

\(\Big \downarrow \) 2112

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

\(\Big \downarrow \) 2182

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

\(\Big \downarrow \) 682

\(\displaystyle \frac {-\frac {\int \frac {d^2 \left ((d e-f) f (d e+f) (3 C e-2 B f)-\left (d^2 e^2-f^2\right ) \left (2 d^2 f (2 B e-A f)-C \left (6 d^2 e^2-f^2\right )\right ) x\right )}{f (e+f x) \sqrt {1-d^2 x^2}}dx}{2 d^2 f^2}-\frac {\sqrt {1-d^2 x^2} \left (x \left (2 d^2 f (B e-A f)-C \left (3 d^2 e^2-f^2\right )\right )+2 \left (A d^2 e f-B \left (2 d^2 e^2-f^2\right )+\frac {3 C d^2 e^3}{f}-2 C e f\right )\right )}{2 f^2}}{d^2 e^2-f^2}+\frac {\left (1-d^2 x^2\right )^{3/2} \left (A f^2-B e f+C e^2\right )}{f \left (d^2 e^2-f^2\right ) (e+f x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {(d e-f) f (d e+f) (3 C e-2 B f)-\left (d^2 e^2-f^2\right ) \left (2 d^2 f (2 B e-A f)-C \left (6 d^2 e^2-f^2\right )\right ) x}{(e+f x) \sqrt {1-d^2 x^2}}dx}{2 f^3}-\frac {\sqrt {1-d^2 x^2} \left (x \left (2 d^2 f (B e-A f)-C \left (3 d^2 e^2-f^2\right )\right )+2 \left (A d^2 e f-B \left (2 d^2 e^2-f^2\right )+\frac {3 C d^2 e^3}{f}-2 C e f\right )\right )}{2 f^2}}{d^2 e^2-f^2}+\frac {\left (1-d^2 x^2\right )^{3/2} \left (A f^2-B e f+C e^2\right )}{f \left (d^2 e^2-f^2\right ) (e+f x)}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {-\frac {-\frac {\left (d^2 e^2-f^2\right ) \left (2 d^2 f (2 B e-A f)-C \left (6 d^2 e^2-f^2\right )\right ) \int \frac {1}{\sqrt {1-d^2 x^2}}dx}{f}-\frac {2 (d e-f) (d e+f) \left (A d^2 e f^2-2 B d^2 e^2 f+B f^3+3 C d^2 e^3-2 C e f^2\right ) \int \frac {1}{(e+f x) \sqrt {1-d^2 x^2}}dx}{f}}{2 f^3}-\frac {\sqrt {1-d^2 x^2} \left (x \left (2 d^2 f (B e-A f)-C \left (3 d^2 e^2-f^2\right )\right )+2 \left (A d^2 e f-B \left (2 d^2 e^2-f^2\right )+\frac {3 C d^2 e^3}{f}-2 C e f\right )\right )}{2 f^2}}{d^2 e^2-f^2}+\frac {\left (1-d^2 x^2\right )^{3/2} \left (A f^2-B e f+C e^2\right )}{f \left (d^2 e^2-f^2\right ) (e+f x)}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {-\frac {-\frac {2 (d e-f) (d e+f) \left (A d^2 e f^2-2 B d^2 e^2 f+B f^3+3 C d^2 e^3-2 C e f^2\right ) \int \frac {1}{(e+f x) \sqrt {1-d^2 x^2}}dx}{f}-\frac {\arcsin (d x) \left (d^2 e^2-f^2\right ) \left (2 d^2 f (2 B e-A f)-C \left (6 d^2 e^2-f^2\right )\right )}{d f}}{2 f^3}-\frac {\sqrt {1-d^2 x^2} \left (x \left (2 d^2 f (B e-A f)-C \left (3 d^2 e^2-f^2\right )\right )+2 \left (A d^2 e f-B \left (2 d^2 e^2-f^2\right )+\frac {3 C d^2 e^3}{f}-2 C e f\right )\right )}{2 f^2}}{d^2 e^2-f^2}+\frac {\left (1-d^2 x^2\right )^{3/2} \left (A f^2-B e f+C e^2\right )}{f \left (d^2 e^2-f^2\right ) (e+f x)}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {\frac {2 (d e-f) (d e+f) \left (A d^2 e f^2-2 B d^2 e^2 f+B f^3+3 C d^2 e^3-2 C e f^2\right ) \int \frac {1}{-d^2 e^2+f^2-\frac {\left (e x d^2+f\right )^2}{1-d^2 x^2}}d\frac {e x d^2+f}{\sqrt {1-d^2 x^2}}}{f}-\frac {\arcsin (d x) \left (d^2 e^2-f^2\right ) \left (2 d^2 f (2 B e-A f)-C \left (6 d^2 e^2-f^2\right )\right )}{d f}}{2 f^3}-\frac {\sqrt {1-d^2 x^2} \left (x \left (2 d^2 f (B e-A f)-C \left (3 d^2 e^2-f^2\right )\right )+2 \left (A d^2 e f-B \left (2 d^2 e^2-f^2\right )+\frac {3 C d^2 e^3}{f}-2 C e f\right )\right )}{2 f^2}}{d^2 e^2-f^2}+\frac {\left (1-d^2 x^2\right )^{3/2} \left (A f^2-B e f+C e^2\right )}{f \left (d^2 e^2-f^2\right ) (e+f x)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {-\frac {-\frac {\arcsin (d x) \left (d^2 e^2-f^2\right ) \left (2 d^2 f (2 B e-A f)-C \left (6 d^2 e^2-f^2\right )\right )}{d f}-\frac {2 (d e-f) (d e+f) \arctan \left (\frac {d^2 e x+f}{\sqrt {1-d^2 x^2} \sqrt {d^2 e^2-f^2}}\right ) \left (A d^2 e f^2-2 B d^2 e^2 f+B f^3+3 C d^2 e^3-2 C e f^2\right )}{f \sqrt {d^2 e^2-f^2}}}{2 f^3}-\frac {\sqrt {1-d^2 x^2} \left (x \left (2 d^2 f (B e-A f)-C \left (3 d^2 e^2-f^2\right )\right )+2 \left (A d^2 e f-B \left (2 d^2 e^2-f^2\right )+\frac {3 C d^2 e^3}{f}-2 C e f\right )\right )}{2 f^2}}{d^2 e^2-f^2}+\frac {\left (1-d^2 x^2\right )^{3/2} \left (A f^2-B e f+C e^2\right )}{f \left (d^2 e^2-f^2\right ) (e+f x)}\)

Input:

Int[(Sqrt[1 - d*x]*Sqrt[1 + d*x]*(A + B*x + C*x^2))/(e + f*x)^2,x]
 

Output:

((C*e^2 - B*e*f + A*f^2)*(1 - d^2*x^2)^(3/2))/(f*(d^2*e^2 - f^2)*(e + f*x) 
) + (-1/2*((2*((3*C*d^2*e^3)/f - 2*C*e*f + A*d^2*e*f - B*(2*d^2*e^2 - f^2) 
) + (2*d^2*f*(B*e - A*f) - C*(3*d^2*e^2 - f^2))*x)*Sqrt[1 - d^2*x^2])/f^2 
- (-(((d^2*e^2 - f^2)*(2*d^2*f*(2*B*e - A*f) - C*(6*d^2*e^2 - f^2))*ArcSin 
[d*x])/(d*f)) - (2*(d*e - f)*(d*e + f)*(3*C*d^2*e^3 - 2*B*d^2*e^2*f - 2*C* 
e*f^2 + A*d^2*e*f^2 + B*f^3)*ArcTan[(f + d^2*e*x)/(Sqrt[d^2*e^2 - f^2]*Sqr 
t[1 - d^2*x^2])])/(f*Sqrt[d^2*e^2 - f^2]))/(2*f^3))/(d^2*e^2 - f^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 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 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

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 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[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 2112

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.), x_Symbol] :> Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; F 
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] & 
& EqQ[m, n] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
 

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. \(647\) vs. \(2(305)=610\).

Time = 0.78 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.01

method	result	size
risch	\(-\frac {\left (C f x +2 B f -4 C e \right ) \sqrt {x d +1}\, \left (x d -1\right ) \sqrt {\left (-x d +1\right ) \left (x d +1\right )}}{2 f^{3} \sqrt {-\left (x d +1\right ) \left (x d -1\right )}\, \sqrt {-x d +1}}-\frac {\left (\frac {\left (2 A \,d^{2} f^{2}-4 B \,d^{2} e f +6 C \,d^{2} e^{2}-C \,f^{2}\right ) \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+1}}\right )}{f \sqrt {d^{2}}}+\frac {2 \left (2 A \,d^{2} e \,f^{2}-3 B \,d^{2} e^{2} f +4 C \,d^{2} e^{3}+B \,f^{3}-2 C e \,f^{2}\right ) \ln \left (\frac {-\frac {2 \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{x +\frac {e}{f}}\right )}{f^{2} \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}+\frac {2 \left (A \,d^{2} e^{2} f^{2}-e^{3} B \,d^{2} f +e^{4} d^{2} C -A \,f^{4}+B e \,f^{3}-C \,e^{2} f^{2}\right ) \left (\frac {f^{2} \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{\left (d^{2} e^{2}-f^{2}\right ) \left (x +\frac {e}{f}\right )}-\frac {f \,d^{2} e \ln \left (\frac {-\frac {2 \left (d^{2} e^{2}-f^{2}\right )}{f^{2}}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}+2 \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}\, \sqrt {-d^{2} \left (x +\frac {e}{f}\right )^{2}+\frac {2 d^{2} e \left (x +\frac {e}{f}\right )}{f}-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}{x +\frac {e}{f}}\right )}{\left (d^{2} e^{2}-f^{2}\right ) \sqrt {-\frac {d^{2} e^{2}-f^{2}}{f^{2}}}}\right )}{f^{3}}\right ) \sqrt {\left (-x d +1\right ) \left (x d +1\right )}}{2 f^{3} \sqrt {-x d +1}\, \sqrt {x d +1}}\)	\(648\)
default	\(\text {Expression too large to display}\)	\(1347\)

Input:

int((-d*x+1)^(1/2)*(d*x+1)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^2,x,method=_RETURNV 
ERBOSE)
 

Output:

-1/2*(C*f*x+2*B*f-4*C*e)*(d*x+1)^(1/2)*(d*x-1)/f^3/(-(d*x+1)*(d*x-1))^(1/2 
)*((-d*x+1)*(d*x+1))^(1/2)/(-d*x+1)^(1/2)-1/2/f^3*((2*A*d^2*f^2-4*B*d^2*e* 
f+6*C*d^2*e^2-C*f^2)/f/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+1)^(1/2) 
)+2/f^2*(2*A*d^2*e*f^2-3*B*d^2*e^2*f+4*C*d^2*e^3+B*f^3-2*C*e*f^2)/(-(d^2*e 
^2-f^2)/f^2)^(1/2)*ln((-2*(d^2*e^2-f^2)/f^2+2/f*d^2*e*(x+e/f)+2*(-(d^2*e^2 
-f^2)/f^2)^(1/2)*(-d^2*(x+e/f)^2+2/f*d^2*e*(x+e/f)-(d^2*e^2-f^2)/f^2)^(1/2 
))/(x+e/f))+2*(A*d^2*e^2*f^2-B*d^2*e^3*f+C*d^2*e^4-A*f^4+B*e*f^3-C*e^2*f^2 
)/f^3*(1/(d^2*e^2-f^2)*f^2/(x+e/f)*(-d^2*(x+e/f)^2+2/f*d^2*e*(x+e/f)-(d^2* 
e^2-f^2)/f^2)^(1/2)-f*d^2*e/(d^2*e^2-f^2)/(-(d^2*e^2-f^2)/f^2)^(1/2)*ln((- 
2*(d^2*e^2-f^2)/f^2+2/f*d^2*e*(x+e/f)+2*(-(d^2*e^2-f^2)/f^2)^(1/2)*(-d^2*( 
x+e/f)^2+2/f*d^2*e*(x+e/f)-(d^2*e^2-f^2)/f^2)^(1/2))/(x+e/f))))*((-d*x+1)* 
(d*x+1))^(1/2)/(-d*x+1)^(1/2)/(d*x+1)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 667 vs. \(2 (305) = 610\).

Time = 26.66 (sec) , antiderivative size = 1358, normalized size of antiderivative = 4.22 \[ \int \frac {\sqrt {1-d x} \sqrt {1+d x} \left (A+B x+C x^2\right )}{(e+f x)^2} \, dx=\text {Too large to display} \] Input:

integrate((-d*x+1)^(1/2)*(d*x+1)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^2,x, algorith 
m="fricas")
 

Output:

[-1/2*(6*C*d^3*e^5*f - 4*B*d^3*e^4*f^2 + 4*B*d*e^2*f^4 - 2*A*d*e*f^5 + 2*( 
A*d^3 - 3*C*d)*e^3*f^3 + 2*(3*C*d^3*e^5 - 2*B*d^3*e^4*f + B*d*e^2*f^3 + (A 
*d^3 - 2*C*d)*e^3*f^2 + (3*C*d^3*e^4*f - 2*B*d^3*e^3*f^2 + B*d*e*f^4 + (A* 
d^3 - 2*C*d)*e^2*f^3)*x)*sqrt(-d^2*e^2 + f^2)*log((d^2*e*f*x + f^2 - sqrt( 
-d^2*e^2 + f^2)*(d^2*e*x + f) - (sqrt(-d^2*e^2 + f^2)*sqrt(-d*x + 1)*f + ( 
d^2*e^2 - f^2)*sqrt(-d*x + 1))*sqrt(d*x + 1))/(f*x + e)) + (6*C*d^3*e^5*f 
- 4*B*d^3*e^4*f^2 + 4*B*d*e^2*f^4 - 2*A*d*e*f^5 + 2*(A*d^3 - 3*C*d)*e^3*f^ 
3 - (C*d^3*e^3*f^3 - C*d*e*f^5)*x^2 + (3*C*d^3*e^4*f^2 - 2*B*d^3*e^3*f^3 - 
 3*C*d*e^2*f^4 + 2*B*d*e*f^5)*x)*sqrt(d*x + 1)*sqrt(-d*x + 1) + 2*(3*C*d^3 
*e^4*f^2 - 2*B*d^3*e^3*f^3 + 2*B*d*e*f^5 - A*d*f^6 + (A*d^3 - 3*C*d)*e^2*f 
^4)*x - 2*(6*C*d^4*e^6 - 4*B*d^4*e^5*f + 4*B*d^2*e^3*f^3 + (2*A*d^4 - 7*C* 
d^2)*e^4*f^2 - (2*A*d^2 - C)*e^2*f^4 + (6*C*d^4*e^5*f - 4*B*d^4*e^4*f^2 + 
4*B*d^2*e^2*f^4 + (2*A*d^4 - 7*C*d^2)*e^3*f^3 - (2*A*d^2 - C)*e*f^5)*x)*ar 
ctan((sqrt(d*x + 1)*sqrt(-d*x + 1) - 1)/(d*x)))/(d^3*e^4*f^4 - d*e^2*f^6 + 
 (d^3*e^3*f^5 - d*e*f^7)*x), -1/2*(6*C*d^3*e^5*f - 4*B*d^3*e^4*f^2 + 4*B*d 
*e^2*f^4 - 2*A*d*e*f^5 + 2*(A*d^3 - 3*C*d)*e^3*f^3 - 4*(3*C*d^3*e^5 - 2*B* 
d^3*e^4*f + B*d*e^2*f^3 + (A*d^3 - 2*C*d)*e^3*f^2 + (3*C*d^3*e^4*f - 2*B*d 
^3*e^3*f^2 + B*d*e*f^4 + (A*d^3 - 2*C*d)*e^2*f^3)*x)*sqrt(d^2*e^2 - f^2)*a 
rctan(-(sqrt(d^2*e^2 - f^2)*sqrt(d*x + 1)*sqrt(-d*x + 1)*e - sqrt(d^2*e^2 
- f^2)*(f*x + e))/((d^2*e^2 - f^2)*x)) + (6*C*d^3*e^5*f - 4*B*d^3*e^4*f...
 

Sympy [F]

\[ \int \frac {\sqrt {1-d x} \sqrt {1+d x} \left (A+B x+C x^2\right )}{(e+f x)^2} \, dx=\int \frac {\sqrt {- d x + 1} \sqrt {d x + 1} \left (A + B x + C x^{2}\right )}{\left (e + f x\right )^{2}}\, dx \] Input:

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

Output:

Integral(sqrt(-d*x + 1)*sqrt(d*x + 1)*(A + B*x + C*x**2)/(e + f*x)**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {1-d x} \sqrt {1+d x} \left (A+B x+C x^2\right )}{(e+f x)^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-d*x+1)^(1/2)*(d*x+1)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^2,x, algorith 
m="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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {1-d x} \sqrt {1+d x} \left (A+B x+C x^2\right )}{(e+f x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-d*x+1)^(1/2)*(d*x+1)^(1/2)*(C*x^2+B*x+A)/(f*x+e)^2,x, algorith 
m="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable 
to make series expansion Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 38.55 (sec) , antiderivative size = 11177, normalized size of antiderivative = 34.71 \[ \int \frac {\sqrt {1-d x} \sqrt {1+d x} \left (A+B x+C x^2\right )}{(e+f x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((4*B*((1 - d*x)^(1/2) - 1))/(f*((d*x + 1)^(1/2) - 1)) + (36*B*((1 - d*x)^ 
(1/2) - 1)^3)/(f*((d*x + 1)^(1/2) - 1)^3) - (36*B*((1 - d*x)^(1/2) - 1)^5) 
/(f*((d*x + 1)^(1/2) - 1)^5) - (4*B*((1 - d*x)^(1/2) - 1)^7)/(f*((d*x + 1) 
^(1/2) - 1)^7) + (16*B*d*e*((1 - d*x)^(1/2) - 1)^2)/(f^2*((d*x + 1)^(1/2) 
- 1)^2) + (32*B*d*e*((1 - d*x)^(1/2) - 1)^4)/(f^2*((d*x + 1)^(1/2) - 1)^4) 
 + (16*B*d*e*((1 - d*x)^(1/2) - 1)^6)/(f^2*((d*x + 1)^(1/2) - 1)^6))/(d*e 
+ (4*f*((1 - d*x)^(1/2) - 1)^3)/((d*x + 1)^(1/2) - 1)^3 - (4*f*((1 - d*x)^ 
(1/2) - 1)^5)/((d*x + 1)^(1/2) - 1)^5 - (4*f*((1 - d*x)^(1/2) - 1)^7)/((d* 
x + 1)^(1/2) - 1)^7 + (4*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 1) + 
(4*d*e*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (6*d*e*((1 - d*x 
)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4 + (4*d*e*((1 - d*x)^(1/2) - 1)^6)/ 
((d*x + 1)^(1/2) - 1)^6 + (d*e*((1 - d*x)^(1/2) - 1)^8)/((d*x + 1)^(1/2) - 
 1)^8) - ((4*A*((1 - d*x)^(1/2) - 1))/(e*((d*x + 1)^(1/2) - 1)) - (4*A*((1 
 - d*x)^(1/2) - 1)^3)/(e*((d*x + 1)^(1/2) - 1)^3) + (8*A*d*((1 - d*x)^(1/2 
) - 1)^2)/(f*((d*x + 1)^(1/2) - 1)^2))/(d*e - (4*f*((1 - d*x)^(1/2) - 1)^3 
)/((d*x + 1)^(1/2) - 1)^3 + (4*f*((1 - d*x)^(1/2) - 1))/((d*x + 1)^(1/2) - 
 1) + (2*d*e*((1 - d*x)^(1/2) - 1)^2)/((d*x + 1)^(1/2) - 1)^2 + (d*e*((1 - 
 d*x)^(1/2) - 1)^4)/((d*x + 1)^(1/2) - 1)^4) - ((8*((1 - d*x)^(1/2) - 1)^2 
*(C*f^2 + 3*C*d^2*e^2))/(f^3*((d*x + 1)^(1/2) - 1)^2) - (32*((1 - d*x)^(1/ 
2) - 1)^4*(2*C*f^2 - 3*C*d^2*e^2))/(f^3*((d*x + 1)^(1/2) - 1)^4) - (32*...
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 1966, normalized size of antiderivative = 6.11 \[ \int \frac {\sqrt {1-d x} \sqrt {1+d x} \left (A+B x+C x^2\right )}{(e+f x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*asin(sqrt( - d*x + 1)/sqrt(2))*a*d**4*e**3*f**2 + 4*asin(sqrt( - d*x + 
1)/sqrt(2))*a*d**4*e**2*f**3*x - 4*asin(sqrt( - d*x + 1)/sqrt(2))*a*d**2*e 
*f**4 - 4*asin(sqrt( - d*x + 1)/sqrt(2))*a*d**2*f**5*x - 8*asin(sqrt( - d* 
x + 1)/sqrt(2))*b*d**4*e**4*f - 8*asin(sqrt( - d*x + 1)/sqrt(2))*b*d**4*e* 
*3*f**2*x + 8*asin(sqrt( - d*x + 1)/sqrt(2))*b*d**2*e**2*f**3 + 8*asin(sqr 
t( - d*x + 1)/sqrt(2))*b*d**2*e*f**4*x + 12*asin(sqrt( - d*x + 1)/sqrt(2)) 
*c*d**4*e**5 + 12*asin(sqrt( - d*x + 1)/sqrt(2))*c*d**4*e**4*f*x - 14*asin 
(sqrt( - d*x + 1)/sqrt(2))*c*d**2*e**3*f**2 - 14*asin(sqrt( - d*x + 1)/sqr 
t(2))*c*d**2*e**2*f**3*x + 2*asin(sqrt( - d*x + 1)/sqrt(2))*c*e*f**4 + 2*a 
sin(sqrt( - d*x + 1)/sqrt(2))*c*f**5*x - 4*sqrt(d*e + f)*sqrt(d*e - f)*ata 
n((sqrt(d*e + f)*tan(asin(sqrt( - d*x + 1)/sqrt(2))/2) - sqrt(f)*sqrt(2))/ 
sqrt(d*e - f))*a*d**3*e**2*f**2 - 4*sqrt(d*e + f)*sqrt(d*e - f)*atan((sqrt 
(d*e + f)*tan(asin(sqrt( - d*x + 1)/sqrt(2))/2) - sqrt(f)*sqrt(2))/sqrt(d* 
e - f))*a*d**3*e*f**3*x + 8*sqrt(d*e + f)*sqrt(d*e - f)*atan((sqrt(d*e + f 
)*tan(asin(sqrt( - d*x + 1)/sqrt(2))/2) - sqrt(f)*sqrt(2))/sqrt(d*e - f))* 
b*d**3*e**3*f + 8*sqrt(d*e + f)*sqrt(d*e - f)*atan((sqrt(d*e + f)*tan(asin 
(sqrt( - d*x + 1)/sqrt(2))/2) - sqrt(f)*sqrt(2))/sqrt(d*e - f))*b*d**3*e** 
2*f**2*x - 4*sqrt(d*e + f)*sqrt(d*e - f)*atan((sqrt(d*e + f)*tan(asin(sqrt 
( - d*x + 1)/sqrt(2))/2) - sqrt(f)*sqrt(2))/sqrt(d*e - f))*b*d*e*f**3 - 4* 
sqrt(d*e + f)*sqrt(d*e - f)*atan((sqrt(d*e + f)*tan(asin(sqrt( - d*x + ...