\(\int \frac {\sqrt {a+b x^2} (e+f x^2)}{x^2 \sqrt {c+d x^2}} \, dx\) [119]

Optimal result

Integrand size = 33, antiderivative size = 262 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^2 \sqrt {c+d x^2}} \, dx=\frac {b (d e+c f) x \sqrt {c+d x^2}}{c d \sqrt {a+b x^2}}-\frac {e \sqrt {a+b x^2} \sqrt {c+d x^2}}{c x}-\frac {\sqrt {a} \sqrt {b} (d e+c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{c d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} (b e+a f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {b} c \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

b*(c*f+d*e)*x*(d*x^2+c)^(1/2)/c/d/(b*x^2+a)^(1/2)-e*(b*x^2+a)^(1/2)*(d*x^2 
+c)^(1/2)/c/x-a^(1/2)*b^(1/2)*(c*f+d*e)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)* 
x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/c/d/(b*x^2+a)^(1/2)/(a*(d*x 
^2+c)/c/(b*x^2+a))^(1/2)+a^(1/2)*(a*f+b*e)*(d*x^2+c)^(1/2)*InverseJacobiAM 
(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(1/2)/c/(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 = 1.41 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{x^2 \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d e \left (a+b x^2\right ) \left (c+d x^2\right )-i b c (d e+c f) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) f x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c d x \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(-(Sqrt[b/a]*d*e*(a + b*x^2)*(c + d*x^2)) - I*b*c*(d*e + c*f)*x*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*c) + a*d)*f*x*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipt 
icF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(Sqrt[b/a]*c*d*x*Sqrt[a + b*x^2] 
*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {442, 406, 320, 388, 313}

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.

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

\(\Big \downarrow \) 442

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

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

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 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[e   Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim 
p[f   Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, 
f, p, q}, x]
 

rule 442

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^q/(a*g*(m + 1))), x] - Simp[1/(a*g^2*(m + 1))   Int[(g*x) 
^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f)*(m + 1) + e*2 
*(b*c*(p + 1) + a*d*q) + d*((b*e - a*f)*(m + 1) + b*e*2*(p + q + 1))*x^2, x 
], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && GtQ[q, 0] && LtQ[m, -1] 
&&  !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^2])
 

Maple [A] (verified)

Time = 5.13 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.14

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {e \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{c x}+\frac {\left (a f +b e \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (f b +\frac {d b e}{c}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(298\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (-\sqrt {-\frac {b}{a}}\, b \,d^{2} e \,x^{4}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d f x -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f x +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e x -\sqrt {-\frac {b}{a}}\, a \,d^{2} e \,x^{2}-\sqrt {-\frac {b}{a}}\, b c d e \,x^{2}-\sqrt {-\frac {b}{a}}\, a c d e \right )}{\left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) c x \sqrt {-\frac {b}{a}}\, d}\)	\(348\)
risch	\(-\frac {e \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{c x}+\frac {\left (-\frac {b \left (c f +d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a c f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {b c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{c \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(378\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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