\(\int \frac {\sqrt {f+g x}}{(d+e x) \sqrt {a+c x^2}} \, dx\) [128]

Optimal result

Integrand size = 28, antiderivative size = 415 \[ \int \frac {\sqrt {f+g x}}{(d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{\sqrt [4]{c} e \sqrt {a+c x^2}}-\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticPi}\left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g},\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{\sqrt [4]{c} e \sqrt {a+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.05 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {f+g x}}{(d+e x) \sqrt {a+c x^2}} \, dx=-\frac {2 i \sqrt {\frac {g \left (\sqrt {a}+i \sqrt {c} x\right )}{-i \sqrt {c} f+\sqrt {a} g}} \sqrt {f+g x} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-\operatorname {EllipticPi}\left (\frac {e \left (f-\frac {i \sqrt {a} g}{\sqrt {c}}\right )}{e f-d g},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-i \sqrt {a} g}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{e \sqrt {\frac {\sqrt {c} (f+g x)}{g \left (i \sqrt {a}+\sqrt {c} x\right )}} \sqrt {a+c x^2}} \] Input:

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

Output:

((-2*I)*Sqrt[(g*(Sqrt[a] + I*Sqrt[c]*x))/((-I)*Sqrt[c]*f + Sqrt[a]*g)]*Sqr 
t[f + g*x]*(EllipticF[I*ArcSinh[Sqrt[-((Sqrt[c]*(f + g*x))/(Sqrt[c]*f - I* 
Sqrt[a]*g))]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] - Elli 
pticPi[(e*(f - (I*Sqrt[a]*g)/Sqrt[c]))/(e*f - d*g), I*ArcSinh[Sqrt[-((Sqrt 
[c]*(f + g*x))/(Sqrt[c]*f - I*Sqrt[a]*g))]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sq 
rt[c]*f + I*Sqrt[a]*g)]))/(e*Sqrt[(Sqrt[c]*(f + g*x))/(g*(I*Sqrt[a] + Sqrt 
[c]*x))]*Sqrt[a + c*x^2])
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1112\) vs. \(2(415)=830\).

Time = 3.31 (sec) , antiderivative size = 1112, normalized size of antiderivative = 2.68, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {736, 510, 729, 25, 1416, 1540, 1416, 2222}

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.

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

\(\Big \downarrow \) 736

\(\displaystyle \frac {(e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e}+\frac {g \int \frac {1}{\sqrt {f+g x} \sqrt {c x^2+a}}dx}{e}\)

\(\Big \downarrow \) 510

\(\displaystyle \frac {(e f-d g) \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {c x^2+a}}dx}{e}+\frac {2 \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}\)

\(\Big \downarrow \) 729

\(\displaystyle \frac {2 (e f-d g) \int -\frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}+\frac {2 \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}-\frac {2 (e f-d g) \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+a}}d\sqrt {f+g x}}{e}\)

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1540

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 2222

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

Input:

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

Output:

((c*f^2 + a*g^2)^(1/4)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[ 
(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f 
^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*EllipticF[2*Arc 
Tan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[ 
c*f^2 + a*g^2])/2])/(c^(1/4)*e*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^ 
2 + (c*(f + g*x)^2)/g^2]) + (2*(e*f - d*g)*(-1/2*(c^(1/4)*(c*e*f^2 + a*e*g 
^2 - Sqrt[c]*(e*f - d*g)*Sqrt[c*f^2 + a*g^2])*(1 + (Sqrt[c]*(f + g*x))/Sqr 
t[c*f^2 + a*g^2])*Sqrt[(a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + 
g*x)^2)/g^2)/((a + (c*f^2)/g^2)*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^ 
2])^2)]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + a*g^2)^(1/4)], 
 (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + a*g^2])/2])/(g*(c*f^2 + a*g^2)^(1/4)*(a*e^2 
*g + c*d*(2*e*f - d*g))*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c* 
(f + g*x)^2)/g^2]) + (e*Sqrt[c*f^2 + a*g^2]*(Sqrt[c]*(e*f - d*g) - e*Sqrt[ 
c*f^2 + a*g^2])*(((e + (Sqrt[c]*(e*f - d*g))/Sqrt[c*f^2 + a*g^2])*ArcTanh[ 
(Sqrt[c*d^2 + a*e^2]*Sqrt[f + g*x])/(Sqrt[e]*Sqrt[e*f - d*g]*Sqrt[a + (c*f 
^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])])/(2*Sqrt[e]*Sqrt[ 
c*d^2 + a*e^2]*Sqrt[e*f - d*g]) - ((Sqrt[c]/e - Sqrt[c*f^2 + a*g^2]/(e*f - 
 d*g))*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])*Sqrt[(a + (c*f^2)/g^2 
 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((a + (c*f^2)/g^2)*(1 + (S 
qrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2)]*EllipticPi[(Sqrt[c]*(e*f - d...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 736

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

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 2222

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 
rcTanh[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]))*Ell 
ipticPi[-(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] && NegQ[-b + c*(d/e) + a*(e/d)]
 

Maple [A] (verified)

Time = 2.80 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.06

method	result	size
default	\(\frac {2 \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, \sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{\sqrt {-a c}\, g +c f}}\, \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{\sqrt {-a c}\, g -c f}}\, \left (f \operatorname {EllipticF}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) c -\sqrt {-a c}\, \operatorname {EllipticF}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) g -\operatorname {EllipticPi}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \frac {\left (\sqrt {-a c}\, g -c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) c f +\sqrt {-a c}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \frac {\left (\sqrt {-a c}\, g -c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) g \right )}{e c \left (c g \,x^{3}+c f \,x^{2}+a g x +a f \right )}\)	\(439\)
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 g \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{e \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}}-\frac {2 \left (d g -e f \right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{e^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\)	\(515\)

Input:

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

Output:

2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*(( 
-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((- 
a*c)^(1/2)*g-c*f))^(1/2)*(f*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1 
/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*c-(-a*c)^(1/2)*Ell 
ipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a 
*c)^(1/2)*g+c*f))^(1/2))*g-EllipticPi((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1 
/2),((-a*c)^(1/2)*g-c*f)*e/c/(d*g-e*f),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2 
)*g+c*f))^(1/2))*c*f+(-a*c)^(1/2)*EllipticPi((-(g*x+f)*c/((-a*c)^(1/2)*g-c 
*f))^(1/2),((-a*c)^(1/2)*g-c*f)*e/c/(d*g-e*f),(-((-a*c)^(1/2)*g-c*f)/((-a* 
c)^(1/2)*g+c*f))^(1/2))*g)/e/c/(c*g*x^3+c*f*x^2+a*g*x+a*f)
 

Fricas [F(-1)]

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

integrate((g*x+f)^(1/2)/(e*x+d)/(c*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((g*x+f)**(1/2)/(e*x+d)/(c*x**2+a)**(1/2),x)
 

Output:

Integral(sqrt(f + g*x)/(sqrt(a + c*x**2)*(d + e*x)), x)
 

Maxima [F]

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

integrate((g*x+f)^(1/2)/(e*x+d)/(c*x^2+a)^(1/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

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

Giac [F]

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

integrate((g*x+f)^(1/2)/(e*x+d)/(c*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((f + g*x)^(1/2)/((a + c*x^2)^(1/2)*(d + e*x)),x)
 

Output:

int((f + g*x)^(1/2)/((a + c*x^2)^(1/2)*(d + e*x)), x)
 

Reduce [F]

\[ \int \frac {\sqrt {f+g x}}{(d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}{c e \,x^{3}+c d \,x^{2}+a e x +a d}d x \] Input:

int((g*x+f)^(1/2)/(e*x+d)/(c*x^2+a)^(1/2),x)
 

Output:

int((sqrt(f + g*x)*sqrt(a + c*x**2))/(a*d + a*e*x + c*d*x**2 + c*e*x**3),x 
)