3.3.73 \(\int \frac {\sqrt {\frac {e (a+b x^2)}{c+d x^2}}}{x^2} \, dx\) [273]

3.3.73.1 Optimal result

Integrand size = 26, antiderivative size = 239 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^2} \, dx=\frac {d x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c}-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{c x}-\frac {\sqrt {d} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b \sqrt {c} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

output

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

3.3.73.2 Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.46 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^2} \, dx=\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right ) \left (-\frac {1}{x}+\frac {b \sqrt {1+\frac {b x^2}{a}} E\left (\arcsin \left (\sqrt {-\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \left (a+b x^2\right ) \sqrt {1+\frac {d x^2}{c}}}\right )}{c} \]

input

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

output

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

3.3.73.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.20, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {2058, 377, 27, 324, 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.

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 324

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

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

output

(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(-((Sqrt[a + b*x^2]*Sqr 
t[c + d*x^2])/(c*x)) + (b*(d*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (S 
qrt[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)*Sqrt[a + b*x^2]*EllipticF[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))/Sqrt[a + b*x^2]
 

3.3.73.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 324

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b   Int[x^2/(Sqr 
t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c 
] && PosQ[b/a]
 

rule 377

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

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 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + 
 b*x^n)^(p*q)*(c + d*x^n)^(p*r))]   Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* 
r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
 

3.3.73.4 Maple [A] (verified)

Time = 4.66 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (\sqrt {-\frac {b}{a}}\, b d \,x^{4}-b c \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, x E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right )+\sqrt {-\frac {b}{a}}\, a d \,x^{2}+\sqrt {-\frac {b}{a}}\, b c \,x^{2}+\sqrt {-\frac {b}{a}}\, a c \right )}{\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, c x \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\)	\(192\)
risch	\(-\frac {\left (d \,x^{2}+c \right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{c x}+\frac {b \left (\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 d a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{c \left (b \,x^{2}+a \right )}\)	\(344\)

input

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

output

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

3.3.73.5 Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^2} \, dx=\frac {b d \sqrt {\frac {a c e}{d^{2}}} x \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a + b\right )} d \sqrt {\frac {a c e}{d^{2}}} x \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a d x^{2} + a c\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{a c x} \]

input

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

output

(b*d*sqrt(a*c*e/d^2)*x*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b* 
c)) - (a + b)*d*sqrt(a*c*e/d^2)*x*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a 
)), a*d/(b*c)) - (a*d*x^2 + a*c)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/(a*c*x 
)
 

3.3.73.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.3.73.7 Maxima [F]

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

input

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

output

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

3.3.73.8 Giac [F]

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

input

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

output

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

3.3.73.9 Mupad [F(-1)]

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

input

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

output

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