3.4.64 \(\int \frac {1}{x^2 (a+\frac {b}{c+d x^2})^{3/2}} \, dx\) [364]

3.4.64.1 Optimal result

Integrand size = 21, antiderivative size = 410 \[ \int \frac {1}{x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {b}{a (b+a c) x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(b-a c) \left (b+a c+a d x^2\right )}{a (b+a c)^2 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {(b-a c) d x \left (b+a c+a d x^2\right )}{a (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {\sqrt {c} (b-a c) \sqrt {d} \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a (b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {c^{3/2} \sqrt {d} \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{(b+a c)^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

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

3.4.64.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.51 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (a \sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (b \left (c-d x^2\right )+a c \left (c+d x^2\right )\right )+i \left (-b^2+a^2 c^2\right ) d x \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )+i b (b+a c) d x \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{a (b+a c)^2 \sqrt {\frac {d}{c}} x \left (b+a \left (c+d x^2\right )\right )} \]

input

Integrate[1/(x^2*(a + b/(c + d*x^2))^(3/2)),x]
 

output

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

3.4.64.3 Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2057, 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.

\(\displaystyle \int \frac {1}{x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2057

\(\displaystyle \int \frac {1}{x^2 \left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}}dx\)

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 370

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

Int[1/(x^2*(a + b/(c + d*x^2))^(3/2)),x]
 

output

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

3.4.64.3.1 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 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 2057

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

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]
 

3.4.64.4 Maple [A] (verified)

Time = 11.04 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.67

method	result	size
default	\(-\frac {\left (\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a c \,d^{2} x^{4}-a \,c^{2} d \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) x \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}-\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, b \,d^{2} x^{4}+2 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a \,c^{2} d \,x^{2}-2 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c d x +\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c d x +\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, b c d \,x^{2}-\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, b c d \,x^{2}+\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, a \,c^{3}+\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {-\frac {a d}{a c +b}}\, b \,c^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{\sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, x \left (a c +b \right )^{2} \left (a d \,x^{2}+a c +b \right )}\)	\(685\)
risch	\(\text {Expression too large to display}\)	\(1061\)

input

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

output

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

3.4.64.5 Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {{\left ({\left (a^{3} c - a^{2} b\right )} d^{3} x^{3} + {\left (a^{3} c^{2} - a b^{2}\right )} d^{2} x\right )} \sqrt {-\frac {a d}{a c + b}} \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left ({\left (a^{3} c - a^{2} b\right )} d^{3} + {\left (a^{3} c^{2} + 2 \, a^{2} b c + a b^{2}\right )} d^{2}\right )} x^{3} + {\left ({\left (a^{3} c^{2} - a b^{2}\right )} d^{2} + {\left (a^{3} c^{3} + 3 \, a^{2} b c^{2} + 3 \, a b^{2} c + b^{3}\right )} d\right )} x\right )} \sqrt {-\frac {a d}{a c + b}} \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left (a^{3} c^{4} + {\left (a^{3} c^{2} - a b^{2}\right )} d^{2} x^{4} + 2 \, a^{2} b c^{3} + a b^{2} c^{2} + 2 \, {\left (a^{3} c^{3} + a^{2} b c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{{\left (a^{5} c^{3} + 3 \, a^{4} b c^{2} + 3 \, a^{3} b^{2} c + a^{2} b^{3}\right )} d x^{3} + {\left (a^{5} c^{4} + 4 \, a^{4} b c^{3} + 6 \, a^{3} b^{2} c^{2} + 4 \, a^{2} b^{3} c + a b^{4}\right )} x} \]

input

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

output

(((a^3*c - a^2*b)*d^3*x^3 + (a^3*c^2 - a*b^2)*d^2*x)*sqrt(-a*d/(a*c + b))* 
sqrt((a*c^2 + b*c)/d^2)*elliptic_e(arcsin(sqrt(-a*d/(a*c + b))*x), (a*c + 
b)/(a*c)) - (((a^3*c - a^2*b)*d^3 + (a^3*c^2 + 2*a^2*b*c + a*b^2)*d^2)*x^3 
 + ((a^3*c^2 - a*b^2)*d^2 + (a^3*c^3 + 3*a^2*b*c^2 + 3*a*b^2*c + b^3)*d)*x 
)*sqrt(-a*d/(a*c + b))*sqrt((a*c^2 + b*c)/d^2)*elliptic_f(arcsin(sqrt(-a*d 
/(a*c + b))*x), (a*c + b)/(a*c)) - (a^3*c^4 + (a^3*c^2 - a*b^2)*d^2*x^4 + 
2*a^2*b*c^3 + a*b^2*c^2 + 2*(a^3*c^3 + a^2*b*c^2)*d*x^2)*sqrt((a*d*x^2 + a 
*c + b)/(d*x^2 + c)))/((a^5*c^3 + 3*a^4*b*c^2 + 3*a^3*b^2*c + a^2*b^3)*d*x 
^3 + (a^5*c^4 + 4*a^4*b*c^3 + 6*a^3*b^2*c^2 + 4*a^2*b^3*c + a*b^4)*x)
 

3.4.64.6 Sympy [F]

\[ \int \frac {1}{x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}\, dx \]

input

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

output

Integral(1/(x**2*((a*c + a*d*x**2 + b)/(c + d*x**2))**(3/2)), x)
 

3.4.64.7 Maxima [F]

\[ \int \frac {1}{x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

input

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

output

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

3.4.64.8 Giac [F]

\[ \int \frac {1}{x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

input

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

output

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

3.4.64.9 Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}} \,d x \]

input

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

output

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