\(\int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx\) [137]

Optimal result

Integrand size = 28, antiderivative size = 228 \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {\sqrt {c} f+\sqrt {-a} g} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f-\sqrt {-a} g}} \sqrt {1-\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \operatorname {EllipticPi}\left (\frac {e \left (f+\frac {\sqrt {-a} g}{\sqrt {c}}\right )}{e f-d g},\arcsin \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {-a} g}}\right ),\frac {\sqrt {c} f+\sqrt {-a} g}{\sqrt {c} f-\sqrt {-a} g}\right )}{\sqrt [4]{c} (e f-d g) \sqrt {a+c x^2}} \] 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)*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)/(-d*g+e*f)/(c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.09 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=-\frac {2 i \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x) \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-\operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i \sqrt {a} g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (e f-d g) \sqrt {a+c x^2}} \] Input:

Integrate[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]
 

Output:

((-2*I)*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g 
)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)*(EllipticF[I*ArcSinh[Sqrt[-f - (I*S 
qrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + 
I*Sqrt[a]*g)] - EllipticPi[(Sqrt[c]*(e*f - d*g))/(e*(Sqrt[c]*f + I*Sqrt[a] 
*g)), I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]* 
f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(Sqrt[-f - (I*Sqrt[a]*g)/Sqr 
t[c]]*(e*f - d*g)*Sqrt[a + c*x^2])
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(867\) vs. \(2(228)=456\).

Time = 2.73 (sec) , antiderivative size = 867, normalized size of antiderivative = 3.80, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {729, 25, 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.

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

\(\Big \downarrow \) 729

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1540

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 2222

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

Input:

Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a + c*x^2]),x]
 

Output:

2*(-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))/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 + (Sqr 
t[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 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + a*g^2])^2) 
]*EllipticPi[(Sqrt[c]*(e*f - d*g) + e*Sqrt[c*f^2 + a*g^2])^2/(4*Sqrt[c]*e* 
(e*f - d*g)*Sqrt[c*f^2 + a*g^2]), 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])/(4*c^(1/4)*(c*f 
^2 + a*g^2)^(1/4)*Sqrt[a + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g 
*x)^2)/g^2])))/(g*(a*e^2*g + c*d*(2*e*f - d*g))))
 

Defintions of rubi rules used

rule 25

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

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 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 = 4.62 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.03

method	result	size
default	\(\frac {2 \left (c f -\sqrt {-a c}\, g \right ) \operatorname {EllipticPi}\left (\sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}, \frac {\left (\sqrt {-a c}\, g -c f \right ) e}{c \left (d g -e f \right )}, \sqrt {-\frac {\sqrt {-a c}\, g -c f}{\sqrt {-a c}\, g +c f}}\right ) \sqrt {\frac {\left (c x +\sqrt {-a c}\right ) g}{\sqrt {-a c}\, g -c f}}\, \sqrt {\frac {\left (-c x +\sqrt {-a c}\right ) g}{\sqrt {-a c}\, g +c f}}\, \sqrt {-\frac {\left (g x +f \right ) c}{\sqrt {-a c}\, g -c f}}\, \sqrt {c \,x^{2}+a}\, \sqrt {g x +f}}{c \left (d g -e f \right ) \left (c g \,x^{3}+c f \,x^{2}+a g x +a f \right )}\)	\(235\)
elliptic	\(\frac {2 \sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}\, e \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +a f}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\)	\(291\)

Input:

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

Output:

2*(c*f-(-a*c)^(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*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)*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^ 
(1/2)*(c*x^2+a)^(1/2)*(g*x+f)^(1/2)/c/(d*g-e*f)/(c*g*x^3+c*f*x^2+a*g*x+a*f 
)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {a + c x^{2}} \left (d + e x\right ) \sqrt {f + g x}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} \sqrt {g x + f}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} \sqrt {g x + f}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+c x^2}} \, dx=\int \frac {1}{\left (e x +d \right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}d x \] Input:

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

Output:

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