\(\int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx\) [27]

Optimal result

Integrand size = 46, antiderivative size = 232 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=-\frac {2 C \sqrt {b c^2-b d^2 x^2}}{b d^2 f \sqrt {c+d x}}-\frac {\sqrt {2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c^2-b d^2 x^2}}{\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}\right )}{\sqrt {b} \sqrt {c} d^2 (d e-c f)}+\frac {2 \left (C e^2-B e f+A f^2\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c^2-b d^2 x^2}}{\sqrt {b} \sqrt {d e+c f} \sqrt {c+d x}}\right )}{\sqrt {b} f^{3/2} (d e-c f) \sqrt {d e+c f}} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.06 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {c^2-d^2 x^2} \left (-\frac {2 C \sqrt {c^2-d^2 x^2}}{d^2 f \sqrt {c+d x}}+\frac {2 \left (C e^2+f (-B e+A f)\right ) \arctan \left (\frac {\sqrt {-d e-c f} \sqrt {c^2-d^2 x^2}}{\sqrt {f} (c-d x) \sqrt {c+d x}}\right )}{f^{3/2} \sqrt {-d e-c f} (d e-c f)}+\frac {\sqrt {2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {c+d x}}{\sqrt {c^2-d^2 x^2}}\right )}{\sqrt {c} d^2 (-d e+c f)}\right )}{\sqrt {b \left (c^2-d^2 x^2\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.32, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.196, Rules used = {2349, 600, 458, 471, 221, 718, 97, 73, 221}

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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 458

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

\(\Big \downarrow \) 471

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 718

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

\(\Big \downarrow \) 97

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {c+d x} \sqrt {b c-b d x} \left (A+\frac {e (C e-B f)}{f^2}\right ) \left (\frac {2 \sqrt {f} \text {arctanh}\left (\frac {\sqrt {f} \sqrt {b c-b d x}}{\sqrt {b} \sqrt {c f+d e}}\right )}{\sqrt {b} (d e-c f) \sqrt {c f+d e}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c-b d x}}{\sqrt {2} \sqrt {b} \sqrt {c}}\right )}{\sqrt {b} \sqrt {c} (d e-c f)}\right )}{\sqrt {b c^2-b d^2 x^2}}+\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {b c^2-b d^2 x^2}}{\sqrt {2} \sqrt {b} \sqrt {c} \sqrt {c+d x}}\right ) (-B d f+c C f+C d e)}{\sqrt {b} \sqrt {c} d^2 f^2}-\frac {2 C \sqrt {b c^2-b d^2 x^2}}{b d^2 f \sqrt {c+d x}}\)

Input:

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

Output:

(-2*C*Sqrt[b*c^2 - b*d^2*x^2])/(b*d^2*f*Sqrt[c + d*x]) + ((A + (e*(C*e - B 
*f))/f^2)*Sqrt[c + d*x]*Sqrt[b*c - b*d*x]*(-((Sqrt[2]*ArcTanh[Sqrt[b*c - b 
*d*x]/(Sqrt[2]*Sqrt[b]*Sqrt[c])])/(Sqrt[b]*Sqrt[c]*(d*e - c*f))) + (2*Sqrt 
[f]*ArcTanh[(Sqrt[f]*Sqrt[b*c - b*d*x])/(Sqrt[b]*Sqrt[d*e + c*f])])/(Sqrt[ 
b]*(d*e - c*f)*Sqrt[d*e + c*f])))/Sqrt[b*c^2 - b*d^2*x^2] + (Sqrt[2]*(C*d* 
e + c*C*f - B*d*f)*ArcTanh[Sqrt[b*c^2 - b*d^2*x^2]/(Sqrt[2]*Sqrt[b]*Sqrt[c 
]*Sqrt[c + d*x])])/(Sqrt[b]*Sqrt[c]*d^2*f^2)
 

Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 97

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[b/(b*c - a*d)   Int[(e + f*x)^p/(a + b*x), x], x] - Simp[d/(b*c 
 - a*d)   Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 
 x] &&  !IntegerQ[p]
 

rule 221

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

rule 458

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c 
, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
 

rule 471

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2*d   Subst[Int[1/(2*b*c + d^2*x^2), x], x, Sqrt[a + b*x^2]/Sqrt[c + d*x] 
], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 718

Int[((d_) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(a + c*x^2)^FracPart[p]/((d + e*x)^FracPart[p]* 
(a/d + (c*x)/e)^FracPart[p])   Int[(d + e*x)^(m + p)*(f + g*x)^n*(a/d + (c/ 
e)*x)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && EqQ[c*d^2 + a*e^2, 
 0]
 

rule 2349

Int[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. 
)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, c + d*x, x]*(c + d 
*x)^(m + 1)*(e + f*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, c 
+ d*x, x]   Int[(c + d*x)^m*(e + f*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, n, p}, x] && PolynomialQ[Px, x] && LtQ[m, 0] &&  !IntegerQ[n 
] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.15

method	result	size
risch	\(-\frac {2 C \left (-d x +c \right ) \sqrt {-\frac {b \left (d^{2} x^{2}-c^{2}\right )}{d x +c}}\, \sqrt {d x +c}}{d^{2} f \sqrt {-b \left (d x -c \right )}\, \sqrt {-b \left (d^{2} x^{2}-c^{2}\right )}}+\frac {2 \left (-\frac {d^{2} \left (A \,f^{2}-B e f +C \,e^{2}\right ) \operatorname {arctanh}\left (\frac {f \sqrt {-b d x +b c}}{\sqrt {b \left (c f +d e \right ) f}}\right )}{\left (c f -d e \right ) \sqrt {b \left (c f +d e \right ) f}}+\frac {f \left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-b d x +b c}\, \sqrt {2}}{2 \sqrt {b c}}\right )}{2 \left (c f -d e \right ) \sqrt {b c}}\right ) \sqrt {-\frac {b \left (d^{2} x^{2}-c^{2}\right )}{d x +c}}\, \sqrt {d x +c}}{d^{2} f \sqrt {-b \left (d^{2} x^{2}-c^{2}\right )}}\)	\(266\)
default	\(\frac {\sqrt {b \left (-d^{2} x^{2}+c^{2}\right )}\, \left (A \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) \sqrt {b \left (c f +d e \right ) f}\, b \,d^{2} f -2 A \sqrt {b c}\, \operatorname {arctanh}\left (\frac {f \sqrt {\left (-d x +c \right ) b}}{\sqrt {b \left (c f +d e \right ) f}}\right ) b \,d^{2} f^{2}-B \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) \sqrt {b \left (c f +d e \right ) f}\, b c d f +2 B \sqrt {b c}\, \operatorname {arctanh}\left (\frac {f \sqrt {\left (-d x +c \right ) b}}{\sqrt {b \left (c f +d e \right ) f}}\right ) b \,d^{2} e f +C \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {\left (-d x +c \right ) b}\, \sqrt {2}}{2 \sqrt {b c}}\right ) \sqrt {b \left (c f +d e \right ) f}\, b \,c^{2} f -2 C \sqrt {b c}\, \operatorname {arctanh}\left (\frac {f \sqrt {\left (-d x +c \right ) b}}{\sqrt {b \left (c f +d e \right ) f}}\right ) b \,d^{2} e^{2}-2 C \sqrt {\left (-d x +c \right ) b}\, \sqrt {b c}\, \sqrt {b \left (c f +d e \right ) f}\, c f +2 C \sqrt {\left (-d x +c \right ) b}\, \sqrt {b c}\, \sqrt {b \left (c f +d e \right ) f}\, d e \right )}{\sqrt {d x +c}\, \sqrt {\left (-d x +c \right ) b}\, b \,d^{2} f \left (c f -d e \right ) \sqrt {b c}\, \sqrt {b \left (c f +d e \right ) f}}\)	\(384\)

Input:

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

Output:

-2*C*(-d*x+c)/d^2/f/(-b*(d*x-c))^(1/2)*(-1/(d*x+c)*b*(d^2*x^2-c^2))^(1/2)* 
(d*x+c)^(1/2)/(-b*(d^2*x^2-c^2))^(1/2)+2/d^2/f*(-d^2*(A*f^2-B*e*f+C*e^2)/( 
c*f-d*e)/(b*(c*f+d*e)*f)^(1/2)*arctanh(f*(-b*d*x+b*c)^(1/2)/(b*(c*f+d*e)*f 
)^(1/2))+1/2*f*(A*d^2-B*c*d+C*c^2)/(c*f-d*e)*2^(1/2)/(b*c)^(1/2)*arctanh(1 
/2*(-b*d*x+b*c)^(1/2)*2^(1/2)/(b*c)^(1/2)))*(-1/(d*x+c)*b*(d^2*x^2-c^2))^( 
1/2)*(d*x+c)^(1/2)/(-b*(d^2*x^2-c^2))^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (197) = 394\).

Time = 1.89 (sec) , antiderivative size = 1836, normalized size of antiderivative = 7.91 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*(sqrt(2)*((C*b*c^3*d - B*b*c^2*d^2 + A*b*c*d^3)*e*f^2 + (C*b*c^4 - B 
*b*c^3*d + A*b*c^2*d^2)*f^3 + ((C*b*c^2*d^2 - B*b*c*d^3 + A*b*d^4)*e*f^2 + 
 (C*b*c^3*d - B*b*c^2*d^2 + A*b*c*d^3)*f^3)*x)*sqrt(1/(b*c))*log(-(d^2*x^2 
 - 2*c*d*x - 2*sqrt(2)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*c*sqrt(1/(b* 
c)) - 3*c^2)/(d^2*x^2 + 2*c*d*x + c^2)) + 2*(C*c*d^2*e^2 - B*c*d^2*e*f + A 
*c*d^2*f^2 + (C*d^3*e^2 - B*d^3*e*f + A*d^3*f^2)*x)*sqrt(b*d*e*f + b*c*f^2 
)*log(-(b*d^2*f*x^2 - b*c*d*e - 2*b*c^2*f - (b*d^2*e + b*c*d*f)*x + 2*sqrt 
(-b*d^2*x^2 + b*c^2)*sqrt(b*d*e*f + b*c*f^2)*sqrt(d*x + c))/(d*f*x^2 + c*e 
 + (d*e + c*f)*x)) + 4*(C*d^2*e^2*f - C*c^2*f^3)*sqrt(-b*d^2*x^2 + b*c^2)* 
sqrt(d*x + c))/(b*c*d^4*e^2*f^2 - b*c^3*d^2*f^4 + (b*d^5*e^2*f^2 - b*c^2*d 
^3*f^4)*x), -(sqrt(2)*((C*b*c^3*d - B*b*c^2*d^2 + A*b*c*d^3)*e*f^2 + (C*b* 
c^4 - B*b*c^3*d + A*b*c^2*d^2)*f^3 + ((C*b*c^2*d^2 - B*b*c*d^3 + A*b*d^4)* 
e*f^2 + (C*b*c^3*d - B*b*c^2*d^2 + A*b*c*d^3)*f^3)*x)*sqrt(-1/(b*c))*arcta 
n(sqrt(2)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*c*sqrt(-1/(b*c))/(d^2*x^2 
 - c^2)) + (C*c*d^2*e^2 - B*c*d^2*e*f + A*c*d^2*f^2 + (C*d^3*e^2 - B*d^3*e 
*f + A*d^3*f^2)*x)*sqrt(b*d*e*f + b*c*f^2)*log(-(b*d^2*f*x^2 - b*c*d*e - 2 
*b*c^2*f - (b*d^2*e + b*c*d*f)*x + 2*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(b*d*e*f 
 + b*c*f^2)*sqrt(d*x + c))/(d*f*x^2 + c*e + (d*e + c*f)*x)) + 2*(C*d^2*e^2 
*f - C*c^2*f^3)*sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c))/(b*c*d^4*e^2*f^2 - 
 b*c^3*d^2*f^4 + (b*d^5*e^2*f^2 - b*c^2*d^3*f^4)*x), -1/2*(sqrt(2)*((C*...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)/(-b*d^2*x^2+b*c^2)^(1/2),x, 
algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/(sqrt(-b*d^2*x^2 + b*c^2)*sqrt(d*x + c)*(f*x + 
 e)), x)
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\frac {\sqrt {2} {\left (C c^{2} - B c d + A d^{2}\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-{\left (d x + c\right )} b + 2 \, b c}}{2 \, \sqrt {-b c}}\right )}{{\left (d^{2} e - c d f\right )} \sqrt {-b c}} - \frac {2 \, {\left (C d e^{2} - B d e f + A d f^{2}\right )} \arctan \left (\frac {\sqrt {-{\left (d x + c\right )} b + 2 \, b c} f}{\sqrt {-b d e f - b c f^{2}}}\right )}{\sqrt {-b d e f - b c f^{2}} {\left (d e f - c f^{2}\right )}} - \frac {2 \, \sqrt {-{\left (d x + c\right )} b + 2 \, b c} C}{b d f}}{d} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 541, normalized size of antiderivative = 2.33 \[ \int \frac {A+B x+C x^2}{\sqrt {c+d x} (e+f x) \sqrt {b c^2-b d^2 x^2}} \, dx=\frac {\sqrt {b}\, \left (4 \sqrt {f}\, \sqrt {c f +d e}\, \mathit {atan} \left (\frac {\sqrt {-d x +c}\, f i}{\sqrt {f}\, \sqrt {c f +d e}}\right ) a c \,d^{2} f^{2} i -4 \sqrt {f}\, \sqrt {c f +d e}\, \mathit {atan} \left (\frac {\sqrt {-d x +c}\, f i}{\sqrt {f}\, \sqrt {c f +d e}}\right ) b c \,d^{2} e f i +4 \sqrt {f}\, \sqrt {c f +d e}\, \mathit {atan} \left (\frac {\sqrt {-d x +c}\, f i}{\sqrt {f}\, \sqrt {c f +d e}}\right ) c^{2} d^{2} e^{2} i -4 \sqrt {-d x +c}\, c^{4} f^{3}+4 \sqrt {-d x +c}\, c^{2} d^{2} e^{2} f -\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a c \,d^{2} f^{3}-\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) a \,d^{3} e \,f^{2}+\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b \,c^{2} d \,f^{3}+\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) b c \,d^{2} e \,f^{2}-\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c^{4} f^{3}-\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}-\sqrt {c}\, \sqrt {2}\right ) c^{3} d e \,f^{2}+\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a c \,d^{2} f^{3}+\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) a \,d^{3} e \,f^{2}-\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b \,c^{2} d \,f^{3}-\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) b c \,d^{2} e \,f^{2}+\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c^{4} f^{3}+\sqrt {c}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {-d x +c}+\sqrt {c}\, \sqrt {2}\right ) c^{3} d e \,f^{2}\right )}{2 b c \,d^{2} f^{2} \left (c^{2} f^{2}-d^{2} e^{2}\right )} \] Input:

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

Output:

(sqrt(b)*(4*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt 
(c*f + d*e)))*a*c*d**2*f**2*i - 4*sqrt(f)*sqrt(c*f + d*e)*atan((sqrt(c - d 
*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*b*c*d**2*e*f*i + 4*sqrt(f)*sqrt(c*f + 
d*e)*atan((sqrt(c - d*x)*f*i)/(sqrt(f)*sqrt(c*f + d*e)))*c**2*d**2*e**2*i 
- 4*sqrt(c - d*x)*c**4*f**3 + 4*sqrt(c - d*x)*c**2*d**2*e**2*f - sqrt(c)*s 
qrt(2)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*c*d**2*f**3 - sqrt(c)*sqrt(2 
)*log(sqrt(c - d*x) - sqrt(c)*sqrt(2))*a*d**3*e*f**2 + sqrt(c)*sqrt(2)*log 
(sqrt(c - d*x) - sqrt(c)*sqrt(2))*b*c**2*d*f**3 + sqrt(c)*sqrt(2)*log(sqrt 
(c - d*x) - sqrt(c)*sqrt(2))*b*c*d**2*e*f**2 - sqrt(c)*sqrt(2)*log(sqrt(c 
- d*x) - sqrt(c)*sqrt(2))*c**4*f**3 - sqrt(c)*sqrt(2)*log(sqrt(c - d*x) - 
sqrt(c)*sqrt(2))*c**3*d*e*f**2 + sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt( 
c)*sqrt(2))*a*c*d**2*f**3 + sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sq 
rt(2))*a*d**3*e*f**2 - sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2) 
)*b*c**2*d*f**3 - sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*b*c 
*d**2*e*f**2 + sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**4*f 
**3 + sqrt(c)*sqrt(2)*log(sqrt(c - d*x) + sqrt(c)*sqrt(2))*c**3*d*e*f**2)) 
/(2*b*c*d**2*f**2*(c**2*f**2 - d**2*e**2))