\(\int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx\) [149]

Optimal result

Integrand size = 30, antiderivative size = 985 \[ \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=\frac {2 \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \sqrt {c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}} \sqrt {1-\frac {\left (c d^2+a e^2\right ) (f+g x)}{\left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) (d+e x)}} \sqrt {1-\frac {\left (c d^2+a e^2\right ) (f+g x)}{\left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) (d+e x)}} E\left (\arcsin \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)} \sqrt {d+e x}}\right )|\frac {c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)}{c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)}\right )}{\left (c d^2+a e^2\right )^{3/2} (e f-d g) \sqrt {a+c x^2} \sqrt {1-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}}-\frac {2 \left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) \sqrt {c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)} (d+e x) \sqrt {\frac {(e f-d g)^2 \left (a+c x^2\right )}{\left (c f^2+a g^2\right ) (d+e x)^2}} \sqrt {1-\frac {\left (c d^2+a e^2\right ) (f+g x)}{\left (c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)\right ) (d+e x)}} \sqrt {1-\frac {\left (c d^2+a e^2\right ) (f+g x)}{\left (c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c d^2+a e^2} \sqrt {f+g x}}{\sqrt {c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)} \sqrt {d+e x}}\right ),\frac {c d f+a e g+\sqrt {-a} \sqrt {c} (e f-d g)}{c d f+a e g-\sqrt {-a} \sqrt {c} (e f-d g)}\right )}{\left (c d^2+a e^2\right )^{3/2} (e f-d g) \sqrt {a+c x^2} \sqrt {1-\frac {2 (c d f+a e g) (f+g x)}{\left (c f^2+a g^2\right ) (d+e x)}+\frac {\left (c d^2+a e^2\right ) (f+g x)^2}{\left (c f^2+a g^2\right ) (d+e x)^2}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 30.06 (sec) , antiderivative size = 777, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a+c x^2}} \, dx=\frac {\left (i \sqrt {a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {\frac {\left (i \sqrt {c} d+\sqrt {a} e\right ) (f+g x)}{\left (i \sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \left (2 e \left (\sqrt {c} f+i \sqrt {a} g\right ) \sqrt {\frac {(e f-d g) \left (-i \sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f+i \sqrt {a} g\right ) (d+e x)}} E\left (\arcsin \left (\sqrt {\frac {\left (\sqrt {c} d-i \sqrt {a} e\right ) (f+g x)}{\left (\sqrt {c} f-i \sqrt {a} g\right ) (d+e x)}}\right )|\frac {\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )}\right )+\sqrt {2} \left (\sqrt {c} d+i \sqrt {a} e\right ) g \sqrt {\frac {d-\frac {i \sqrt {a} e}{\sqrt {c}}+\frac {i \sqrt {c} d x}{\sqrt {a}}+e x}{d+e x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(e f-d g) \left (i \sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-i \sqrt {a} g\right ) (d+e x)}}\right ),-\frac {\frac {i \sqrt {c} d f}{\sqrt {a}}-e f+d g+\frac {i \sqrt {a} e g}{\sqrt {c}}}{2 e f-2 d g}\right )+2 \sqrt {c} (-e f+d g) \sqrt {\frac {(e f-d g) \left (-i \sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f+i \sqrt {a} g\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\sqrt {c} d-i \sqrt {a} e\right ) (f+g x)}{\left (\sqrt {c} f-i \sqrt {a} g\right ) (d+e x)}}\right ),\frac {\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )}{\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )}\right )\right )}{\left (c d^2 e+a e^3\right ) \sqrt {\frac {(e f-d g) \left (i \sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-i \sqrt {a} g\right ) (d+e x)}} \sqrt {f+g x} \sqrt {a+c x^2}} \] Input:

Integrate[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + c*x^2]),x]
 

Output:

((I*Sqrt[a] + Sqrt[c]*x)*Sqrt[d + e*x]*Sqrt[((I*Sqrt[c]*d + Sqrt[a]*e)*(f 
+ g*x))/((I*Sqrt[c]*f + Sqrt[a]*g)*(d + e*x))]*(2*e*(Sqrt[c]*f + I*Sqrt[a] 
*g)*Sqrt[((e*f - d*g)*((-I)*Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f + I*Sqrt[a]* 
g)*(d + e*x))]*EllipticE[ArcSin[Sqrt[((Sqrt[c]*d - I*Sqrt[a]*e)*(f + g*x)) 
/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]], ((Sqrt[c]*d + I*Sqrt[a]*e)*(Sqrt 
[c]*f - I*Sqrt[a]*g))/((Sqrt[c]*d - I*Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g) 
)] + Sqrt[2]*(Sqrt[c]*d + I*Sqrt[a]*e)*g*Sqrt[(d - (I*Sqrt[a]*e)/Sqrt[c] + 
 (I*Sqrt[c]*d*x)/Sqrt[a] + e*x)/(d + e*x)]*EllipticF[ArcSin[Sqrt[((e*f - d 
*g)*(I*Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]], -((( 
I*Sqrt[c]*d*f)/Sqrt[a] - e*f + d*g + (I*Sqrt[a]*e*g)/Sqrt[c])/(2*e*f - 2*d 
*g))] + 2*Sqrt[c]*(-(e*f) + d*g)*Sqrt[((e*f - d*g)*((-I)*Sqrt[a] + Sqrt[c] 
*x))/((Sqrt[c]*f + I*Sqrt[a]*g)*(d + e*x))]*EllipticF[ArcSin[Sqrt[((Sqrt[c 
]*d - I*Sqrt[a]*e)*(f + g*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]], ((S 
qrt[c]*d + I*Sqrt[a]*e)*(Sqrt[c]*f - I*Sqrt[a]*g))/((Sqrt[c]*d - I*Sqrt[a] 
*e)*(Sqrt[c]*f + I*Sqrt[a]*g))]))/((c*d^2*e + a*e^3)*Sqrt[((e*f - d*g)*(I* 
Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - I*Sqrt[a]*g)*(d + e*x))]*Sqrt[f + g*x] 
*Sqrt[a + c*x^2])
 

Rubi [F]

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.

\(\displaystyle \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2} (d+e x)^{3/2}} \, dx\)

\(\Big \downarrow \) 744

\(\displaystyle \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2} (d+e x)^{3/2}}dx\)

Input:

Int[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + c*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 744

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2090\) vs. \(2(887)=1774\).

Time = 10.74 (sec) , antiderivative size = 2091, normalized size of antiderivative = 2.12

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2091\)
default	\(\text {Expression too large to display}\)	\(3816\)

Input:

int((g*x+f)^(1/2)/(e*x+d)^(3/2)/(c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

integrate((g*x+f)^(1/2)/(e*x+d)^(3/2)/(c*x^2+a)^(1/2),x, algorithm="fricas 
")
 

Output:

integral(sqrt(c*x^2 + a)*sqrt(e*x + d)*sqrt(g*x + f)/(c*e^2*x^4 + 2*c*d*e* 
x^3 + 2*a*d*e*x + a*d^2 + (c*d^2 + a*e^2)*x^2), x)
 

Sympy [F]

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

integrate((g*x+f)**(1/2)/(e*x+d)**(3/2)/(c*x**2+a)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((g*x+f)^(1/2)/(e*x+d)^(3/2)/(c*x^2+a)^(1/2),x, algorithm="maxima 
")
 

Output:

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

Giac [F]

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

integrate((g*x+f)^(1/2)/(e*x+d)^(3/2)/(c*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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