3.4.63 \(\int \frac {1}{(a+\frac {b}{c+d x^2})^{3/2}} \, dx\) [363]

3.4.63.1 Optimal result

Integrand size = 17, antiderivative size = 356 \[ \int \frac {1}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {b x}{a (b+a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}+\frac {(2 b+a c) x \left (b+a c+a d x^2\right )}{a^2 (b+a c) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {\sqrt {c} (2 b+a c) \left (b+a c+a d x^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a^2 (b+a c) \sqrt {d} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {c^{3/2} \left (b+a c+a d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{a (b+a c) \sqrt {d} \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]

output

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

3.4.63.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.43 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {i \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (-i a b \sqrt {\frac {d}{c}} x \left (c+d x^2\right )+\left (2 b^2+3 a b c+a^2 c^2\right ) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-2 b (b+a c) \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{a^2 (b+a c) \sqrt {\frac {d}{c}} \left (b+a \left (c+d x^2\right )\right )} \]

input

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

output

((-I)*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*((-I)*a*b*Sqrt[d/c]*x*(c + d*x 
^2) + (2*b^2 + 3*a*b*c + a^2*c^2)*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)] - 2*b*( 
b + a*c)*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^2*(b + a*c)*Sqrt[d/c]*(b + 
a*(c + d*x^2)))
 

3.4.63.3 Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.95, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2057, 2058, 315, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2058

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

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

output

(Sqrt[b + a*c + a*d*x^2]*(-((b*x*Sqrt[c + d*x^2])/(a*(b + a*c)*Sqrt[b + a* 
c + a*d*x^2])) + ((2*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)/S 
qrt[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]*Ellipti 
cF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(Sqrt[d]*Sqrt[c + d*x^2]*Sqr 
t[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]))/(a*(b + a*c))))/(Sqrt 
[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])
 

3.4.63.3.1 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 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

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 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]
 

3.4.63.4 Maple [A] (verified)

Time = 3.38 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.31

method	result	size
default	\(-\frac {\left (-\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a \,c^{2}+\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, b d \,x^{3}+\sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c -2 \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b c +\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, b c x \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{a \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \left (a c +b \right ) \left (a d \,x^{2}+a c +b \right )}\)	\(466\)

input

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

3.4.63.5 Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {{\left ({\left (a^{2} c^{2} + 2 \, a b c\right )} d x^{3} + {\left (a^{2} c^{3} + 3 \, a b c^{2} + 2 \, b^{2} c\right )} x\right )} \sqrt {a} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left ({\left (a^{2} c + a b\right )} d^{2} + {\left (a^{2} c^{2} + 2 \, a b c\right )} d\right )} x^{3} + {\left (a^{2} c^{3} + 3 \, a b c^{2} + 2 \, b^{2} c + {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} d\right )} x\right )} \sqrt {a} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{2} c + a b\right )} d^{2} x^{4} + a^{2} c^{3} + 3 \, a b c^{2} + 2 \, {\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} d x^{2} + 2 \, b^{2} c\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{{\left (a^{4} c + a^{3} b\right )} d^{2} x^{3} + {\left (a^{4} c^{2} + 2 \, a^{3} b c + a^{2} b^{2}\right )} d x} \]

input

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

output

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

3.4.63.6 Sympy [F]

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

input

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

output

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

3.4.63.7 Maxima [F]

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

input

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

output

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

3.4.63.8 Giac [F]

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

input

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

output

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

3.4.63.9 Mupad [F(-1)]

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

input

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

output

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