\(\int \frac {(\frac {e (a+b x^2)}{c+d x^2})^{3/2}}{x^2} \, dx\) [65]

Optimal result

Integrand size = 26, antiderivative size = 311 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=\frac {(b c-2 a d) e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{c d x}-\frac {(b c-a d) e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{c d x}+\frac {(b c-2 a d) e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{c^{3/2} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b e \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \] Output:

(-2*a*d+b*c)*e*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/c/d/x-(-a*d+b*c)*e*( 
b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/c/d/x+(-2*a*d+b*c)*e*(b*e/d-(-a*d+b* 
c)*e/d/(d*x^2+c))^(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^(3/2)/d^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)+b*e*(b*e/d- 
(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)), 
(1-b*c/a/d)^(1/2))/c^(1/2)/d^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.36 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.73 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=-\frac {e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} d \left (a+b x^2\right ) \left (a c-b c x^2+2 a d x^2\right )+i b c (-b c+2 a d) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c (-b c+a d) x \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}} c^2 d x \left (a+b x^2\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {2058, 370, 27, 445, 27, 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.

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 370

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

Input:

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

Output:

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

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 370

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + 
 d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(e*x) 
^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a 
*d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], 
 x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] 
&& GtQ[q, 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 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 445

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 + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(669\) vs. \(2(298)=596\).

Time = 10.64 (sec) , antiderivative size = 670, normalized size of antiderivative = 2.15

method	result	size
default	\(-\frac {{\left (\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}\right )}^{\frac {3}{2}} \left (d \,x^{2}+c \right ) \left (\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{4}+\sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{4}-\sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{4}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d x -\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x -2 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d x +\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x +\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{2}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a b c d \,x^{2}+\sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{2}-\sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {-\frac {b}{a}}\, a b c d \,x^{2}+\sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, \sqrt {-\frac {b}{a}}\, a^{2} c d \right )}{\left (b \,x^{2}+a \right )^{2} c^{2} x \sqrt {-\frac {b}{a}}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}\, d}\)	\(670\)
risch	\(-\frac {a \left (d \,x^{2}+c \right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{c^{2} x}+\frac {\left (\frac {b^{2} c^{2} \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{d \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^{2} b c e \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{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 (a d e +b c e +e \left (a d -b c \right )\right )}-\frac {c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\left (b d \,x^{2} e +a d e \right ) x}{c \left (a d -b c \right ) e \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2} e +a d e \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{c \left (a d -b c \right )}\right ) \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{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 b d a e \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )\right )}{\left (a d -b c \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 (a d e +b c e +e \left (a d -b c \right )\right )}\right )}{d}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{c^{2} \left (b \,x^{2}+a \right )}\)	\(723\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.55 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^2} \, dx=-\frac {{\left (b^{2} c - 2 \, a b d\right )} \sqrt {\frac {a c e}{d^{2}}} e x \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (b^{2} c - {\left (a^{2} + 2 \, a b\right )} d\right )} \sqrt {\frac {a c e}{d^{2}}} e x \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left (a^{2} c e - {\left (a b c - 2 \, a^{2} d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{a c^{2} x} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e)*e*( - 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c - sqrt(c + d*x**2)* 
sqrt(a + b*x**2)*a*d*x**2 + sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*x**2 + i 
nt((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d** 
2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a*b*c*d**2*x + int(( 
sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x* 
*4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a*b*d**3*x**3 - int((sqr 
t(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 
+ b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*b**2*c**2*d*x - int((sqrt(c 
 + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b 
*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*b**2*c*d**2*x**3 - 3*int((sqrt 
(c + d*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c* 
*2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*c**2*d*x - 3*int((sqrt(c + d 
*x**2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x** 
2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*c*d**2*x**3 + 3*int((sqrt(c + d*x* 
*2)*sqrt(a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 
 2*b*c*d*x**4 + b*d**2*x**6),x)*a*b*c**3*x + 3*int((sqrt(c + d*x**2)*sqrt( 
a + b*x**2))/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d* 
x**4 + b*d**2*x**6),x)*a*b*c**2*d*x**3))/(2*c**2*x*(c + d*x**2))