\(\int \frac {A+B x}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx\) [38]

Optimal result

Integrand size = 34, antiderivative size = 318 \[ \int \frac {A+B x}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx=-\frac {2 \sqrt {a} B \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {b} f \sqrt {c+d x} \sqrt {a-b x^2}}+\frac {2 \sqrt {a} (B e-A f) \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {a} d}} \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {a} f}{\sqrt {b} e+\sqrt {a} f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{f \left (\sqrt {b} e+\sqrt {a} f\right ) \sqrt {c+d x} \sqrt {a-b x^2}} \] Output:

-2*a^(1/2)*B*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*(1-b*x^2/a)^(1/ 
2)*EllipticF(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2^(1/2)*(a^(1/2)*d/(b 
^(1/2)*c+a^(1/2)*d))^(1/2))/b^(1/2)/f/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)+2*a^( 
1/2)*(-A*f+B*e)*(b^(1/2)*(d*x+c)/(b^(1/2)*c+a^(1/2)*d))^(1/2)*((-b*x^2+a)/ 
a)^(1/2)*EllipticPi(1/2*(1-b^(1/2)*x/a^(1/2))^(1/2)*2^(1/2),2*a^(1/2)*f/(b 
^(1/2)*e+a^(1/2)*f),2^(1/2)*(a^(1/2)*d/(b^(1/2)*c+a^(1/2)*d))^(1/2))/f/(b^ 
(1/2)*e+a^(1/2)*f)/(d*x+c)^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.86 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{\sqrt {c+d x} (e+f x) \sqrt {a-b x^2}} \, dx=\frac {2 i \sqrt {\frac {d \left (\frac {\sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {\sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x) \left ((-B c f+A d f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )+d (B e-A f) \operatorname {EllipticPi}\left (\frac {\sqrt {b} (-d e+c f)}{\left (\sqrt {b} c-\sqrt {a} d\right ) f},i \text {arcsinh}\left (\frac {\sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c+\sqrt {a} d}{\sqrt {b} c-\sqrt {a} d}\right )\right )}{d \sqrt {-c+\frac {\sqrt {a} d}{\sqrt {b}}} f (d e-c f) \sqrt {a-b x^2}} \] Input:

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

Output:

((2*I)*Sqrt[(d*(Sqrt[a]/Sqrt[b] + x))/(c + d*x)]*Sqrt[-(((Sqrt[a]*d)/Sqrt[ 
b] - d*x)/(c + d*x))]*(c + d*x)*((-(B*c*f) + A*d*f)*EllipticF[I*ArcSinh[Sq 
rt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt 
[b]*c - Sqrt[a]*d)] + d*(B*e - A*f)*EllipticPi[(Sqrt[b]*(-(d*e) + c*f))/(( 
Sqrt[b]*c - Sqrt[a]*d)*f), I*ArcSinh[Sqrt[-c + (Sqrt[a]*d)/Sqrt[b]]/Sqrt[c 
 + d*x]], (Sqrt[b]*c + Sqrt[a]*d)/(Sqrt[b]*c - Sqrt[a]*d)]))/(d*Sqrt[-c + 
(Sqrt[a]*d)/Sqrt[b]]*f*(d*e - c*f)*Sqrt[a - b*x^2])
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {2349, 27, 512, 511, 321, 731, 186, 413, 412}

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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 731

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

\(\Big \downarrow \) 186

\(\displaystyle -\frac {2 \sqrt {1-\frac {b x^2}{a}} \left (A-\frac {B e}{f}\right ) \int \frac {1}{\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {c+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f-f \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2}}-\frac {2 \sqrt {a} B \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} f \sqrt {a-b x^2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 413

\(\displaystyle -\frac {2 \sqrt {1-\frac {b x^2}{a}} \left (A-\frac {B e}{f}\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \int \frac {1}{\sqrt {\frac {\sqrt {b} x}{\sqrt {a}}+1} \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} c+\sqrt {a} d}} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f-f \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}d\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {a-b x^2} \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 \sqrt {a} B \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} f \sqrt {a-b x^2} \sqrt {c+d x}}\)

\(\Big \downarrow \) 412

\(\displaystyle -\frac {2 \sqrt {1-\frac {b x^2}{a}} \left (A-\frac {B e}{f}\right ) \sqrt {1-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticPi}\left (\frac {2 f}{\frac {\sqrt {b} e}{\sqrt {a}}+f},\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{\sqrt {b} c+\sqrt {a} d}\right )}{\sqrt {a-b x^2} \left (\frac {\sqrt {b} e}{\sqrt {a}}+f\right ) \sqrt {-\frac {\sqrt {a} d \left (1-\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {\sqrt {a} d}{\sqrt {b}}+c}}-\frac {2 \sqrt {a} B \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {\sqrt {b} (c+d x)}{\sqrt {a} d+\sqrt {b} c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {b} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 d}{\frac {\sqrt {b} c}{\sqrt {a}}+d}\right )}{\sqrt {b} f \sqrt {a-b x^2} \sqrt {c+d x}}\)

Input:

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

Output:

(-2*Sqrt[a]*B*Sqrt[(Sqrt[b]*(c + d*x))/(Sqrt[b]*c + Sqrt[a]*d)]*Sqrt[1 - ( 
b*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*d)/( 
(Sqrt[b]*c)/Sqrt[a] + d)])/(Sqrt[b]*f*Sqrt[c + d*x]*Sqrt[a - b*x^2]) - (2* 
(A - (B*e)/f)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqr 
t[a]))/(Sqrt[b]*c + Sqrt[a]*d)]*EllipticPi[(2*f)/((Sqrt[b]*e)/Sqrt[a] + f) 
, ArcSin[Sqrt[1 - (Sqrt[b]*x)/Sqrt[a]]/Sqrt[2]], (2*Sqrt[a]*d)/(Sqrt[b]*c 
+ Sqrt[a]*d)])/(((Sqrt[b]*e)/Sqrt[a] + f)*Sqrt[a - b*x^2]*Sqrt[c + (Sqrt[a 
]*d)/Sqrt[b] - (Sqrt[a]*d*(1 - (Sqrt[b]*x)/Sqrt[a]))/Sqrt[b]])
 

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 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

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

rule 731

Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) 
^2]), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[Sqrt[1 + b*(x^2/a)]/Sqrt[a 
+ b*x^2]   Int[1/((e + f*x)*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], 
 x]] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[b/a] &&  !GtQ[a, 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 = 4.94 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.58

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (d x +c \right )}\, \left (\frac {2 B \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{f \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}}+\frac {2 \left (A f -B e \right ) \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x -\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {a b}}{b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {\sqrt {a b}}{b}}}, \frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}+\frac {e}{f}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {a b}}{b}}}\right )}{f^{2} \sqrt {-b d \,x^{3}-b c \,x^{2}+a d x +a c}\, \left (-\frac {c}{d}+\frac {e}{f}\right )}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}\)	\(502\)
default	\(-\frac {2 \left (A \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{\sqrt {a b}\, d -b c}}, \frac {\left (-\sqrt {a b}\, d +b c \right ) f}{b \left (c f -d e \right )}, \sqrt {-\frac {\sqrt {a b}\, d -b c}{\sqrt {a b}\, d +b c}}\right ) b c d f -A \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{\sqrt {a b}\, d -b c}}, \frac {\left (-\sqrt {a b}\, d +b c \right ) f}{b \left (c f -d e \right )}, \sqrt {-\frac {\sqrt {a b}\, d -b c}{\sqrt {a b}\, d +b c}}\right ) \sqrt {a b}\, d^{2} f -B \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{\sqrt {a b}\, d -b c}}, \frac {\left (-\sqrt {a b}\, d +b c \right ) f}{b \left (c f -d e \right )}, \sqrt {-\frac {\sqrt {a b}\, d -b c}{\sqrt {a b}\, d +b c}}\right ) b c d e +B \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{\sqrt {a b}\, d -b c}}, \frac {\left (-\sqrt {a b}\, d +b c \right ) f}{b \left (c f -d e \right )}, \sqrt {-\frac {\sqrt {a b}\, d -b c}{\sqrt {a b}\, d +b c}}\right ) \sqrt {a b}\, d^{2} e -B \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{\sqrt {a b}\, d -b c}}, \sqrt {-\frac {\sqrt {a b}\, d -b c}{\sqrt {a b}\, d +b c}}\right ) b \,c^{2} f +B \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{\sqrt {a b}\, d -b c}}, \sqrt {-\frac {\sqrt {a b}\, d -b c}{\sqrt {a b}\, d +b c}}\right ) b c d e +B \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{\sqrt {a b}\, d -b c}}, \sqrt {-\frac {\sqrt {a b}\, d -b c}{\sqrt {a b}\, d +b c}}\right ) \sqrt {a b}\, c d f -B \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{\sqrt {a b}\, d -b c}}, \sqrt {-\frac {\sqrt {a b}\, d -b c}{\sqrt {a b}\, d +b c}}\right ) \sqrt {a b}\, d^{2} e \right ) \sqrt {\frac {\left (b x +\sqrt {a b}\right ) d}{\sqrt {a b}\, d -b c}}\, \sqrt {\frac {\left (-b x +\sqrt {a b}\right ) d}{\sqrt {a b}\, d +b c}}\, \sqrt {-\frac {\left (d x +c \right ) b}{\sqrt {a b}\, d -b c}}\, \sqrt {-b \,x^{2}+a}\, \sqrt {d x +c}}{d b f \left (c f -d e \right ) \left (-b d \,x^{3}-b c \,x^{2}+a d x +a c \right )}\)	\(754\)

Input:

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

Output:

((-b*x^2+a)*(d*x+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x+c)^(1/2)*(2*B/f*(c/d-1/b* 
(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x-1/b*(a*b)^(1/2))/(- 
c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d+1/b*(a*b)^(1/2)))^( 
1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(c/d-1/b*(a*b)^ 
(1/2)))^(1/2),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2))+2*(A* 
f-B*e)/f^2*(c/d-1/b*(a*b)^(1/2))*((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2)*((x 
-1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)*((x+1/b*(a*b)^(1/2))/(-c/d 
+1/b*(a*b)^(1/2)))^(1/2)/(-b*d*x^3-b*c*x^2+a*d*x+a*c)^(1/2)/(-c/d+e/f)*Ell 
ipticPi(((x+c/d)/(c/d-1/b*(a*b)^(1/2)))^(1/2),(-c/d+1/b*(a*b)^(1/2))/(-c/d 
+e/f),((-c/d+1/b*(a*b)^(1/2))/(-c/d-1/b*(a*b)^(1/2)))^(1/2)))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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