\(\int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx\) [44]

Optimal result

Integrand size = 31, antiderivative size = 659 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\frac {2 e \sqrt {\sqrt {c} f+\sqrt {a} g} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {-\frac {(e f-d g) \left (\sqrt {a}-\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} E\left (\arcsin \left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {a} g} \sqrt {d+e x}}\right )|\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\sqrt {c} d+\sqrt {a} e} (e f-d g) \sqrt {\frac {(e f-d g) \left (\sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} \sqrt {a-c x^2}}-\frac {2 \sqrt {c} \sqrt {\sqrt {c} f+\sqrt {a} g} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {-\frac {(e f-d g) \left (\sqrt {a}-\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {a} g} \sqrt {d+e x}}\right ),\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\sqrt {c} d+\sqrt {a} e} \left (\sqrt {c} f-\sqrt {a} g\right ) \sqrt {\frac {(e f-d g) \left (\sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} \sqrt {a-c x^2}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 30.28 (sec) , antiderivative size = 519, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=-\frac {2 \sqrt {a-c x^2} \left (\sqrt {a} \sqrt {c} e (e f-d g) \sqrt {-\frac {\left (-c f^2+a g^2\right ) \left (a-c x^2\right )}{a c (f+g x)^2}}-e \left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right ) \sqrt {\frac {\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}{\left (\sqrt {c} d+\sqrt {a} e\right ) (f+g x)}} E\left (\arcsin \left (\sqrt {\frac {f-\frac {\sqrt {a} g}{\sqrt {c}}-\frac {\sqrt {c} f x}{\sqrt {a}}+g x}{2 f+2 g x}}\right )|\frac {2 \sqrt {a} \sqrt {c} (e f-d g)}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )+\left (c d^2-a e^2\right ) g \sqrt {\frac {\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}{\left (\sqrt {c} d+\sqrt {a} e\right ) (f+g x)}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {f-\frac {\sqrt {a} g}{\sqrt {c}}-\frac {\sqrt {c} f x}{\sqrt {a}}+g x}{2 f+2 g x}}\right ),\frac {2 \sqrt {a} \sqrt {c} (e f-d g)}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )\right )}{\sqrt {a} \sqrt {c} \left (c d^2-a e^2\right ) (-e f+d g) \sqrt {d+e x} \sqrt {f+g x} \sqrt {-\frac {\left (-c f^2+a g^2\right ) \left (a-c x^2\right )}{a c (f+g x)^2}}} \] Input:

Integrate[1/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a - c*x^2]),x]
 

Output:

(-2*Sqrt[a - c*x^2]*(Sqrt[a]*Sqrt[c]*e*(e*f - d*g)*Sqrt[-(((-(c*f^2) + a*g 
^2)*(a - c*x^2))/(a*c*(f + g*x)^2))] - e*(Sqrt[c]*d + Sqrt[a]*e)*(Sqrt[c]* 
f - Sqrt[a]*g)*Sqrt[((Sqrt[c]*f + Sqrt[a]*g)*(d + e*x))/((Sqrt[c]*d + Sqrt 
[a]*e)*(f + g*x))]*EllipticE[ArcSin[Sqrt[(f - (Sqrt[a]*g)/Sqrt[c] - (Sqrt[ 
c]*f*x)/Sqrt[a] + g*x)/(2*f + 2*g*x)]], (2*Sqrt[a]*Sqrt[c]*(e*f - d*g))/(( 
Sqrt[c]*d + Sqrt[a]*e)*(Sqrt[c]*f - Sqrt[a]*g))] + (c*d^2 - a*e^2)*g*Sqrt[ 
((Sqrt[c]*f + Sqrt[a]*g)*(d + e*x))/((Sqrt[c]*d + Sqrt[a]*e)*(f + g*x))]*E 
llipticF[ArcSin[Sqrt[(f - (Sqrt[a]*g)/Sqrt[c] - (Sqrt[c]*f*x)/Sqrt[a] + g* 
x)/(2*f + 2*g*x)]], (2*Sqrt[a]*Sqrt[c]*(e*f - d*g))/((Sqrt[c]*d + Sqrt[a]* 
e)*(Sqrt[c]*f - Sqrt[a]*g))]))/(Sqrt[a]*Sqrt[c]*(c*d^2 - a*e^2)*(-(e*f) + 
d*g)*Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[-(((-(c*f^2) + a*g^2)*(a - c*x^2))/( 
a*c*(f + g*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.

\(\displaystyle \int \frac {1}{\sqrt {a-c x^2} (d+e x)^{3/2} \sqrt {f+g x}} \, dx\)

\(\Big \downarrow \) 733

\(\displaystyle \frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}-\frac {g \int \frac {1}{\sqrt {d+e x} \sqrt {f+g x} \sqrt {a-c x^2}}dx}{e f-d g}\)

\(\Big \downarrow \) 732

\(\displaystyle \frac {2 g (d+e x) \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a-c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\)

\(\Big \downarrow \) 744

\(\displaystyle \frac {2 g (d+e x) \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \int \frac {1}{\sqrt {\frac {\left (c d^2-a e^2\right ) (f+g x)^2}{\left (c f^2-a g^2\right ) (d+e x)^2}-\frac {2 (c d f-a e g) (f+g x)}{\left (c f^2-a g^2\right ) (d+e x)}+1}}d\frac {\sqrt {f+g x}}{\sqrt {d+e x}}}{\sqrt {a-c x^2} (e f-d g)^2}+\frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {a-c x^2}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-a g^2} \sqrt {-\frac {\left (a-c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (c f^2-a g^2\right )}} \left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}{\left (\frac {(f+g x) \sqrt {c d^2-a e^2}}{(d+e x) \sqrt {c f^2-a g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-a g^2} \sqrt {d+e x}}\right ),\frac {1}{2} \left (\frac {c d f-a e g}{\sqrt {c d^2-a e^2} \sqrt {c f^2-a g^2}}+1\right )\right )}{\sqrt {a-c x^2} \sqrt [4]{c d^2-a e^2} (e f-d g)^2 \sqrt {\frac {(f+g x)^2 \left (c d^2-a e^2\right )}{(d+e x)^2 \left (c f^2-a g^2\right )}-\frac {2 (f+g x) (c d f-a e g)}{(d+e x) \left (c f^2-a g^2\right )}+1}}\)

Input:

Int[1/((d + e*x)^(3/2)*Sqrt[f + g*x]*Sqrt[a - c*x^2]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 732

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

rule 733

Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(3/2)*Sqrt[(a_) + (b_. 
)*(x_)^2]), x_Symbol] :> Simp[d/(d*e - c*f)   Int[1/(Sqrt[c + d*x]*Sqrt[e + 
 f*x]*Sqrt[a + b*x^2]), x], x] - Simp[f/(d*e - c*f)   Int[Sqrt[c + d*x]/((e 
 + f*x)^(3/2)*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
 

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]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2132\) vs. \(2(507)=1014\).

Time = 11.76 (sec) , antiderivative size = 2133, normalized size of antiderivative = 3.24

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2133\)
default	\(\text {Expression too large to display}\)	\(7582\)

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int { \frac {1}{\sqrt {-c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:

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

Output:

integral(-sqrt(-c*x^2 + a)*sqrt(e*x + d)*sqrt(g*x + f)/(c*e^2*g*x^5 + (c*e 
^2*f + 2*c*d*e*g)*x^4 - a*d^2*f + (2*c*d*e*f + (c*d^2 - a*e^2)*g)*x^3 - (2 
*a*d*e*g - (c*d^2 - a*e^2)*f)*x^2 - (2*a*d*e*f + a*d^2*g)*x), x)
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int \frac {1}{\sqrt {a - c x^{2}} \left (d + e x\right )^{\frac {3}{2}} \sqrt {f + g x}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int { \frac {1}{\sqrt {-c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int { \frac {1}{\sqrt {-c x^{2} + a} {\left (e x + d\right )}^{\frac {3}{2}} \sqrt {g x + f}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {a-c\,x^2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\int \frac {1}{\left (e x +d \right )^{\frac {3}{2}} \sqrt {g x +f}\, \sqrt {-c \,x^{2}+a}}d x \] Input:

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

Output:

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