\(\int \frac {A+B x}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx\) [41]

Optimal result

Integrand size = 33, antiderivative size = 438 \[ \int \frac {A+B x}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\frac {2 B \sqrt {\sqrt {b} c+\sqrt {-a} d} \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}} \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {-a} d}}\right ),\frac {\sqrt {b} c+\sqrt {-a} d}{\sqrt {b} c-\sqrt {-a} d}\right )}{\sqrt [4]{b} d f \sqrt {a+b x^2}}-\frac {2 \sqrt {\sqrt {b} c+\sqrt {-a} d} (B e-A f) \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c-\sqrt {-a} d}} \sqrt {1-\frac {\sqrt {b} (c+d x)}{\sqrt {b} c+\sqrt {-a} d}} \operatorname {EllipticPi}\left (-\frac {\left (c+\frac {\sqrt {-a} d}{\sqrt {b}}\right ) f}{d e-c f},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt {\sqrt {b} c+\sqrt {-a} d}}\right ),\frac {\sqrt {b} c+\sqrt {-a} d}{\sqrt {b} c-\sqrt {-a} d}\right )}{\sqrt [4]{b} f (d e-c f) \sqrt {a+b x^2}} \] Output:

2*B*(b^(1/2)*c+(-a)^(1/2)*d)^(1/2)*(1-b^(1/2)*(d*x+c)/(b^(1/2)*c-(-a)^(1/2 
)*d))^(1/2)*(1-b^(1/2)*(d*x+c)/(b^(1/2)*c+(-a)^(1/2)*d))^(1/2)*EllipticF(b 
^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+(-a)^(1/2)*d)^(1/2),((b^(1/2)*c+(-a)^(1/2) 
*d)/(b^(1/2)*c-(-a)^(1/2)*d))^(1/2))/b^(1/4)/d/f/(b*x^2+a)^(1/2)-2*(b^(1/2 
)*c+(-a)^(1/2)*d)^(1/2)*(-A*f+B*e)*(1-b^(1/2)*(d*x+c)/(b^(1/2)*c-(-a)^(1/2 
)*d))^(1/2)*(1-b^(1/2)*(d*x+c)/(b^(1/2)*c+(-a)^(1/2)*d))^(1/2)*EllipticPi( 
b^(1/4)*(d*x+c)^(1/2)/(b^(1/2)*c+(-a)^(1/2)*d)^(1/2),-(c+(-a)^(1/2)*d/b^(1 
/2))*f/(-c*f+d*e),((b^(1/2)*c+(-a)^(1/2)*d)/(b^(1/2)*c-(-a)^(1/2)*d))^(1/2 
))/b^(1/4)/f/(-c*f+d*e)/(b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.87 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x}{\sqrt {c+d x} (e+f x) \sqrt {a+b x^2}} \, dx=\frac {2 i \sqrt {\frac {d \left (\frac {i \sqrt {a}}{\sqrt {b}}+x\right )}{c+d x}} \sqrt {-\frac {\frac {i \sqrt {a} d}{\sqrt {b}}-d x}{c+d x}} (c+d x) \left ((-B c f+A d f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\frac {i \sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c-i \sqrt {a} d}{\sqrt {b} c+i \sqrt {a} d}\right )+d (B e-A f) \operatorname {EllipticPi}\left (\frac {\sqrt {b} (-d e+c f)}{\left (\sqrt {b} c+i \sqrt {a} d\right ) f},i \text {arcsinh}\left (\frac {\sqrt {-c-\frac {i \sqrt {a} d}{\sqrt {b}}}}{\sqrt {c+d x}}\right ),\frac {\sqrt {b} c-i \sqrt {a} d}{\sqrt {b} c+i \sqrt {a} d}\right )\right )}{d \sqrt {-c-\frac {i \sqrt {a} d}{\sqrt {b}}} f (d e-c f) \sqrt {a+b x^2}} \] Input:

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

Output:

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

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1103\) vs. \(2(438)=876\).

Time = 3.80 (sec) , antiderivative size = 1103, normalized size of antiderivative = 2.52, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2349, 27, 510, 729, 1416, 1540, 1416, 2220}

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

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 510

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

\(\Big \downarrow \) 729

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

\(\Big \downarrow \) 1416

\(\displaystyle 2 \left (A-\frac {B e}{f}\right ) \int \frac {1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}+\frac {B \sqrt [4]{a d^2+b c^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right ) \sqrt {\frac {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}{\left (a+\frac {b c^2}{d^2}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{\sqrt [4]{b} d f \sqrt {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}}\)

\(\Big \downarrow \) 1540

\(\displaystyle 2 \left (A-\frac {B e}{f}\right ) \left (\frac {f \sqrt {a d^2+b c^2} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt {b} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \int \frac {1}{\sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}\right )+\frac {B \sqrt [4]{a d^2+b c^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right ) \sqrt {\frac {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}{\left (a+\frac {b c^2}{d^2}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{\sqrt [4]{b} d f \sqrt {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}}\)

\(\Big \downarrow \) 1416

\(\displaystyle 2 \left (A-\frac {B e}{f}\right ) \left (\frac {f \sqrt {a d^2+b c^2} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \int \frac {\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1}{(d e-c f+f (c+d x)) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}d\sqrt {c+d x}}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{a d^2+b c^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right ) \sqrt {\frac {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}{\left (a+\frac {b c^2}{d^2}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right )^2}} \left (f \sqrt {a d^2+b c^2}+\sqrt {b} (d e-c f)\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \sqrt {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}} \left (a d f^2-b e (d e-2 c f)\right )}\right )+\frac {B \sqrt [4]{a d^2+b c^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right ) \sqrt {\frac {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}{\left (a+\frac {b c^2}{d^2}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {a d^2+b c^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{\sqrt [4]{b} d f \sqrt {a+\frac {b c^2}{d^2}-\frac {2 b c (c+d x)}{d^2}+\frac {b (c+d x)^2}{d^2}}}\)

\(\Big \downarrow \) 2220

\(\displaystyle \frac {B \sqrt [4]{b c^2+a d^2} \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{\sqrt [4]{b} d f \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}+2 \left (A-\frac {B e}{f}\right ) \left (\frac {\sqrt {b c^2+a d^2} f \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \left (\frac {\left (f-\frac {\sqrt {b} (d e-c f)}{\sqrt {b c^2+a d^2}}\right ) \arctan \left (\frac {\sqrt {b e^2+a f^2} \sqrt {c+d x}}{\sqrt {f} \sqrt {d e-c f} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{2 \sqrt {f} \sqrt {d e-c f} \sqrt {b e^2+a f^2}}+\frac {\left (\frac {\sqrt {b}}{f}+\frac {\sqrt {b c^2+a d^2}}{d e-c f}\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b c^2+a d^2} f-\sqrt {b} (d e-c f)\right )^2}{4 \sqrt {b} \sqrt {b c^2+a d^2} f (d e-c f)},2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{4 \sqrt [4]{b} \sqrt [4]{b c^2+a d^2} \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )}{d \left (a d f^2-b e (d e-2 c f)\right )}-\frac {\sqrt [4]{b} \sqrt [4]{b c^2+a d^2} \left (\sqrt {b c^2+a d^2} f+\sqrt {b} (d e-c f)\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right ) \sqrt {\frac {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}{\left (\frac {b c^2}{d^2}+a\right ) \left (\frac {\sqrt {b} (c+d x)}{\sqrt {b c^2+a d^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c+d x}}{\sqrt [4]{b c^2+a d^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {b c^2+a d^2}}+1\right )\right )}{2 d \left (a d f^2-b e (d e-2 c f)\right ) \sqrt {\frac {b c^2}{d^2}-\frac {2 b (c+d x) c}{d^2}+\frac {b (c+d x)^2}{d^2}+a}}\right )\)

Input:

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

Output:

(B*(b*c^2 + a*d^2)^(1/4)*(1 + (Sqrt[b]*(c + d*x))/Sqrt[b*c^2 + a*d^2])*Sqr 
t[(a + (b*c^2)/d^2 - (2*b*c*(c + d*x))/d^2 + (b*(c + d*x)^2)/d^2)/((a + (b 
*c^2)/d^2)*(1 + (Sqrt[b]*(c + d*x))/Sqrt[b*c^2 + a*d^2])^2)]*EllipticF[2*A 
rcTan[(b^(1/4)*Sqrt[c + d*x])/(b*c^2 + a*d^2)^(1/4)], (1 + (Sqrt[b]*c)/Sqr 
t[b*c^2 + a*d^2])/2])/(b^(1/4)*d*f*Sqrt[a + (b*c^2)/d^2 - (2*b*c*(c + d*x) 
)/d^2 + (b*(c + d*x)^2)/d^2]) + 2*(A - (B*e)/f)*(-1/2*(b^(1/4)*(b*c^2 + a* 
d^2)^(1/4)*(Sqrt[b*c^2 + a*d^2]*f + Sqrt[b]*(d*e - c*f))*(1 + (Sqrt[b]*(c 
+ d*x))/Sqrt[b*c^2 + a*d^2])*Sqrt[(a + (b*c^2)/d^2 - (2*b*c*(c + d*x))/d^2 
 + (b*(c + d*x)^2)/d^2)/((a + (b*c^2)/d^2)*(1 + (Sqrt[b]*(c + d*x))/Sqrt[b 
*c^2 + a*d^2])^2)]*EllipticF[2*ArcTan[(b^(1/4)*Sqrt[c + d*x])/(b*c^2 + a*d 
^2)^(1/4)], (1 + (Sqrt[b]*c)/Sqrt[b*c^2 + a*d^2])/2])/(d*(a*d*f^2 - b*e*(d 
*e - 2*c*f))*Sqrt[a + (b*c^2)/d^2 - (2*b*c*(c + d*x))/d^2 + (b*(c + d*x)^2 
)/d^2]) + (Sqrt[b*c^2 + a*d^2]*f*(Sqrt[b*c^2 + a*d^2]*f + Sqrt[b]*(d*e - c 
*f))*(((f - (Sqrt[b]*(d*e - c*f))/Sqrt[b*c^2 + a*d^2])*ArcTan[(Sqrt[b*e^2 
+ a*f^2]*Sqrt[c + d*x])/(Sqrt[f]*Sqrt[d*e - c*f]*Sqrt[a + (b*c^2)/d^2 - (2 
*b*c*(c + d*x))/d^2 + (b*(c + d*x)^2)/d^2])])/(2*Sqrt[f]*Sqrt[d*e - c*f]*S 
qrt[b*e^2 + a*f^2]) + ((Sqrt[b]/f + Sqrt[b*c^2 + a*d^2]/(d*e - c*f))*(1 + 
(Sqrt[b]*(c + d*x))/Sqrt[b*c^2 + a*d^2])*Sqrt[(a + (b*c^2)/d^2 - (2*b*c*(c 
 + d*x))/d^2 + (b*(c + d*x)^2)/d^2)/((a + (b*c^2)/d^2)*(1 + (Sqrt[b]*(c + 
d*x))/Sqrt[b*c^2 + a*d^2])^2)]*EllipticPi[-1/4*(Sqrt[b*c^2 + a*d^2]*f -...
 

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 510

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[2/d   Subst[Int[1/Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2 
)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a]
 

rule 729

Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) 
^2]), x_Symbol] :> Simp[2   Subst[Int[1/((d*e - c*f + f*x^2)*Sqrt[(b*c^2 + 
a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; 
FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1540

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S 
ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2)   Int[1 
/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2)   I 
nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, 
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 2220

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTan[Rt[-b + c*(d/e) + a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
-b + c*(d/e) + a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a 
+ b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*El 
lipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
 /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] & 
& EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[-b + c*(d/e) + a*(e/d)]
 

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 = 5.24 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.18

method	result	size
elliptic	\(\frac {\sqrt {\left (d x +c \right ) \left (b \,x^{2}+a \right )}\, \left (\frac {2 B \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) \sqrt {\frac {x +\frac {c}{d}}{-\frac {\sqrt {-a b}}{b}+\frac {c}{d}}}\, \sqrt {\frac {x -\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{-\frac {\sqrt {-a b}}{b}+\frac {c}{d}}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{b}}}\right )}{f \sqrt {b d \,x^{3}+b c \,x^{2}+a d x +a c}}+\frac {2 \left (A f -B e \right ) \left (-\frac {\sqrt {-a b}}{b}+\frac {c}{d}\right ) \sqrt {\frac {x +\frac {c}{d}}{-\frac {\sqrt {-a b}}{b}+\frac {c}{d}}}\, \sqrt {\frac {x -\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{b}}}\, \sqrt {\frac {x +\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {c}{d}}{-\frac {\sqrt {-a b}}{b}+\frac {c}{d}}}, \frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}+\frac {e}{f}}, \sqrt {\frac {-\frac {c}{d}+\frac {\sqrt {-a b}}{b}}{-\frac {c}{d}-\frac {\sqrt {-a b}}{b}}}\right )}{f^{2} \sqrt {b d \,x^{3}+b c \,x^{2}+a d x +a c}\, \left (-\frac {c}{d}+\frac {e}{f}\right )}\right )}{\sqrt {d x +c}\, \sqrt {b \,x^{2}+a}}\)	\(515\)
default	\(-\frac {2 \left (A \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {-a b}-b c}}, \frac {\left (b c -d \sqrt {-a b}\right ) f}{b \left (c f -d e \right )}, \sqrt {-\frac {d \sqrt {-a b}-b c}{d \sqrt {-a b}+b c}}\right ) b c d f -A \sqrt {-a b}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {-a b}-b c}}, \frac {\left (b c -d \sqrt {-a b}\right ) f}{b \left (c f -d e \right )}, \sqrt {-\frac {d \sqrt {-a b}-b c}{d \sqrt {-a b}+b c}}\right ) d^{2} f -B \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {-a b}-b c}}, \frac {\left (b c -d \sqrt {-a b}\right ) f}{b \left (c f -d e \right )}, \sqrt {-\frac {d \sqrt {-a b}-b c}{d \sqrt {-a b}+b c}}\right ) b c d e +B \sqrt {-a b}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {-a b}-b c}}, \frac {\left (b c -d \sqrt {-a b}\right ) f}{b \left (c f -d e \right )}, \sqrt {-\frac {d \sqrt {-a b}-b c}{d \sqrt {-a b}+b c}}\right ) d^{2} e -B \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {-a b}-b c}}, \sqrt {-\frac {d \sqrt {-a b}-b c}{d \sqrt {-a b}+b c}}\right ) b \,c^{2} f +B \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {-a b}-b c}}, \sqrt {-\frac {d \sqrt {-a b}-b c}{d \sqrt {-a b}+b c}}\right ) b c d e +B \sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {-a b}-b c}}, \sqrt {-\frac {d \sqrt {-a b}-b c}{d \sqrt {-a b}+b c}}\right ) c d f -B \sqrt {-a b}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {-a b}-b c}}, \sqrt {-\frac {d \sqrt {-a b}-b c}{d \sqrt {-a b}+b c}}\right ) d^{2} e \right ) \sqrt {\frac {d \left (b x +\sqrt {-a b}\right )}{d \sqrt {-a b}-b c}}\, \sqrt {\frac {\left (-b x +\sqrt {-a b}\right ) d}{d \sqrt {-a b}+b c}}\, \sqrt {-\frac {\left (d x +c \right ) b}{d \sqrt {-a b}-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 )}\)	\(788\)

Input:

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

Output:

((d*x+c)*(b*x^2+a))^(1/2)/(d*x+c)^(1/2)/(b*x^2+a)^(1/2)*(2*B/f*(-(-a*b)^(1 
/2)/b+c/d)*((x+c/d)/(-(-a*b)^(1/2)/b+c/d))^(1/2)*((x-(-a*b)^(1/2)/b)/(-c/d 
-(-a*b)^(1/2)/b))^(1/2)*((x+(-a*b)^(1/2)/b)/(-c/d+(-a*b)^(1/2)/b))^(1/2)/( 
b*d*x^3+b*c*x^2+a*d*x+a*c)^(1/2)*EllipticF(((x+c/d)/(-(-a*b)^(1/2)/b+c/d)) 
^(1/2),((-c/d+(-a*b)^(1/2)/b)/(-c/d-(-a*b)^(1/2)/b))^(1/2))+2*(A*f-B*e)/f^ 
2*(-(-a*b)^(1/2)/b+c/d)*((x+c/d)/(-(-a*b)^(1/2)/b+c/d))^(1/2)*((x-(-a*b)^( 
1/2)/b)/(-c/d-(-a*b)^(1/2)/b))^(1/2)*((x+(-a*b)^(1/2)/b)/(-c/d+(-a*b)^(1/2 
)/b))^(1/2)/(b*d*x^3+b*c*x^2+a*d*x+a*c)^(1/2)/(-c/d+e/f)*EllipticPi(((x+c/ 
d)/(-(-a*b)^(1/2)/b+c/d))^(1/2),(-c/d+(-a*b)^(1/2)/b)/(-c/d+e/f),((-c/d+(- 
a*b)^(1/2)/b)/(-c/d-(-a*b)^(1/2)/b))^(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="fric 
as")
 

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="maxi 
ma")
                                                                                    
                                                                                    
 

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="giac 
")
 

Output:

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

Mupad [F(-1)]

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