\(\int \frac {\sqrt {a+b x^2}}{(c+d x^2)^{3/2} (e+f x^2)} \, dx\) [151]

Optimal result

Integrand size = 32, antiderivative size = 209 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {\sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {a^{3/2} f \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e (d e-c f) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\frac {-\sqrt {\frac {b}{a}} d e x \left (a+b x^2\right )-i b c e \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b e+a f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {a f}{b e},i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c e (-d e+c f) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(-(Sqrt[b/a]*d*e*x*(a + b*x^2)) - I*b*c*e*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d* 
x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*e) + a*f 
)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticPi[(a*f)/(b*e), I*ArcSin 
h[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*c*e*(-(d*e) + c*f)*Sqrt[a + b*x^2 
]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {416, 313, 414}

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.

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

\(\Big \downarrow \) 416

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 414

\(\displaystyle \frac {\sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a^{3/2} f \sqrt {c+d x^2} \operatorname {EllipticPi}\left (1-\frac {a f}{b e},\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c e \sqrt {a+b x^2} (d e-c f) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}\)

Input:

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

Output:

(Sqrt[d]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/ 
(a*d)])/(Sqrt[c]*(d*e - c*f)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c 
+ d*x^2]) - (a^(3/2)*f*Sqrt[c + d*x^2]*EllipticPi[1 - (a*f)/(b*e), ArcTan[ 
(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[b]*c*e*(d*e - c*f)*Sqrt[a + 
b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))])
 

Defintions of rubi rules used

rule 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 414

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

rule 416

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

Maple [A] (verified)

Time = 6.59 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {\left (-\sqrt {-\frac {b}{a}}\, b d e \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c e +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a c f -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b c e -\sqrt {-\frac {b}{a}}\, a d e x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{c e \sqrt {-\frac {b}{a}}\, \left (c f -d e \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(286\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+a d \right ) x}{c \left (c f -d e \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\left (c f -d e \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) a}{\left (c f -d e \right ) e \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {b}{a}}, \frac {a f}{b e}, \frac {\sqrt {-\frac {d}{c}}}{\sqrt {-\frac {b}{a}}}\right ) b}{\left (c f -d e \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(398\)

Input:

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

Output:

(-(-b/a)^(1/2)*b*d*e*x^3+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE 
(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c*e+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^( 
1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e,(-1/c*d)^(1/2)/(-b/a)^(1/2))*a*c*f- 
((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticPi(x*(-b/a)^(1/2),a*f/b/e, 
(-1/c*d)^(1/2)/(-b/a)^(1/2))*b*c*e-(-b/a)^(1/2)*a*d*e*x)*(d*x^2+c)^(1/2)*( 
b*x^2+a)^(1/2)/c/e/(-b/a)^(1/2)/(c*f-d*e)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d^{2} f \,x^{6}+2 c d f \,x^{4}+d^{2} e \,x^{4}+c^{2} f \,x^{2}+2 c d e \,x^{2}+c^{2} e}d x \] Input:

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

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c**2*e + c**2*f*x**2 + 2*c*d*e*x* 
*2 + 2*c*d*f*x**4 + d**2*e*x**4 + d**2*f*x**6),x)