\(\int \frac {1}{x (a+\frac {b}{c+d x^2})^{3/2}} \, dx\) [205]

Optimal result

Integrand size = 21, antiderivative size = 110 \[ \int \frac {1}{x \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {b}{a (b+a c) \sqrt {a+\frac {b}{c+d x^2}}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {b+a c}}\right )}{(b+a c)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.27 \[ \int \frac {1}{x \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=-\frac {b}{a (b+a c) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}-\frac {c^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {-b-a c}}\right )}{(-b-a c)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{a^{3/2}} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.71 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.25, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2057, 2053, 2052, 25, 27, 382, 397, 219, 221}

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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2053

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

\(\Big \downarrow \) 2052

\(\displaystyle -b d \int -\frac {1}{d x^4 \left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 25

\(\displaystyle b d \int \frac {1}{d x^4 \left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 27

\(\displaystyle b \int \frac {1}{x^4 \left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 382

\(\displaystyle b \left (\frac {\int \frac {-c x^4+b+2 a c}{\left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{a (a c+b)}-\frac {1}{a x^2 (a c+b)}\right )\)

\(\Big \downarrow \) 397

\(\displaystyle b \left (\frac {\frac {(a c+b) \int \frac {1}{a-x^4}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}-\frac {a c^2 \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}}{a (a c+b)}-\frac {1}{a x^2 (a c+b)}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle b \left (\frac {\frac {(a c+b) \text {arctanh}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{\sqrt {a} b}-\frac {a c^2 \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}}{a (a c+b)}-\frac {1}{a x^2 (a c+b)}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle b \left (\frac {\frac {(a c+b) \text {arctanh}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{\sqrt {a} b}-\frac {a c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a c+b}}\right )}{b \sqrt {a c+b}}}{a (a c+b)}-\frac {1}{a x^2 (a c+b)}\right )\)

Input:

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

Output:

b*(-(1/(a*(b + a*c)*x^2)) + (((b + a*c)*ArcTanh[Sqrt[(b + a*c + a*d*x^2)/( 
c + d*x^2)]/Sqrt[a]])/(Sqrt[a]*b) - (a*c^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[(b + 
a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + a*c]])/(b*Sqrt[b + a*c]))/(a*(b + a* 
c)))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 382

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 + 1)/ 
(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b* 
x^2)^p*(c + d*x^2)^q*Simp[(b*c + a*d)*(m + 3) + 2*(b*c*p + a*d*q) + b*d*(m 
+ 2*p + 2*q + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[ 
b*c - a*d, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
 

rule 397

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ 
Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^2), x], x] - Simp[ 
(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e 
, f}, x]
 

rule 2052

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)))/((c_) + (d_.)*(x_)))^(p_), x_S 
ymbol] :> With[{q = Denominator[p]}, Simp[q*e*(b*c - a*d)   Subst[Int[x^(q* 
(p + 1) - 1)*(((-a)*e + c*x^q)^m/(b*e - d*x^q)^(m + 2)), x], x, (e*((a + b* 
x)/(c + d*x)))^(1/q)], x]] /; FreeQ[{a, b, c, d, e, m}, x] && FractionQ[p] 
&& IntegerQ[m]
 

rule 2053

Int[(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.) 
))^(p_), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(e*( 
(a + b*x)/(c + d*x)))^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, 
 x] && IntegerQ[Simplify[(m + 1)/n]]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1014\) vs. \(2(92)=184\).

Time = 0.13 (sec) , antiderivative size = 1015, normalized size of antiderivative = 9.23

method	result	size
default	\(\text {Expression too large to display}\)	\(1015\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (92) = 184\).

Time = 0.21 (sec) , antiderivative size = 1477, normalized size of antiderivative = 13.43 \[ \int \frac {1}{x \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*((a^2*c^2 + (a^2*c + a*b)*d*x^2 + 2*a*b*c + b^2)*sqrt(a)*log(8*a^2*d^ 
2*x^4 + 8*a^2*c^2 + 8*(2*a^2*c + a*b)*d*x^2 + 8*a*b*c + b^2 + 4*(2*a*d^2*x 
^4 + (4*a*c + b)*d*x^2 + 2*a*c^2 + b*c)*sqrt(a)*sqrt((a*d*x^2 + a*c + b)/( 
d*x^2 + c))) + (a^3*c*d*x^2 + a^3*c^2 + a^2*b*c)*sqrt(c/(a*c + b))*log(((8 
*a^2*c^2 + 8*a*b*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8 
*(2*a^2*c^3 + 3*a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a^2*c^2 + 3*a*b*c + b^2)*d^ 
2*x^4 + 2*a^2*c^4 + 4*a*b*c^3 + 2*b^2*c^2 + (4*a^2*c^3 + 7*a*b*c^2 + 3*b^2 
*c)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt(c/(a*c + b)))/x^4) - 
 4*(a*b*d*x^2 + a*b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^4*c^2 + 2 
*a^3*b*c + a^2*b^2 + (a^4*c + a^3*b)*d*x^2), -1/4*(2*(a^2*c^2 + (a^2*c + a 
*b)*d*x^2 + 2*a*b*c + b^2)*sqrt(-a)*arctan(1/2*(2*a*d*x^2 + 2*a*c + b)*sqr 
t(-a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*d*x^2 + a^2*c + a*b)) - ( 
a^3*c*d*x^2 + a^3*c^2 + a^2*b*c)*sqrt(c/(a*c + b))*log(((8*a^2*c^2 + 8*a*b 
*c + b^2)*d^2*x^4 + 8*a^2*c^4 + 16*a*b*c^3 + 8*b^2*c^2 + 8*(2*a^2*c^3 + 3* 
a*b*c^2 + b^2*c)*d*x^2 - 4*((2*a^2*c^2 + 3*a*b*c + b^2)*d^2*x^4 + 2*a^2*c^ 
4 + 4*a*b*c^3 + 2*b^2*c^2 + (4*a^2*c^3 + 7*a*b*c^2 + 3*b^2*c)*d*x^2)*sqrt( 
(a*d*x^2 + a*c + b)/(d*x^2 + c))*sqrt(c/(a*c + b)))/x^4) + 4*(a*b*d*x^2 + 
a*b*c)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c)))/(a^4*c^2 + 2*a^3*b*c + a^2*b 
^2 + (a^4*c + a^3*b)*d*x^2), 1/4*(2*(a^3*c*d*x^2 + a^3*c^2 + a^2*b*c)*sqrt 
(-c/(a*c + b))*arctan(1/2*((2*a*c + b)*d*x^2 + 2*a*c^2 + 2*b*c)*sqrt((a...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (92) = 184\).

Time = 0.15 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.83 \[ \int \frac {1}{x \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {c^{2} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{2 \, \sqrt {{\left (a c + b\right )} c} {\left (a c + b\right )}} - \frac {b}{{\left (a^{2} c + a b\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}} - \frac {\log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{2 \, a^{\frac {3}{2}}} \] Input:

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 627, normalized size of antiderivative = 5.70 \[ \int \frac {1}{x \left (a+\frac {b}{c+d x^2}\right )^{3/2}} \, dx=\frac {-\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a^{2} b c -\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a \,b^{2}+\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{3} c^{2}+\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{3} c d \,x^{2}+\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) a^{2} b c -\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) a^{3} c^{2}-\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) a^{3} c d \,x^{2}-\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) a^{2} b c +\sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) a^{3} c^{3}+\sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) a^{3} c^{2} d \,x^{2}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) a^{2} b \,c^{2}+2 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) a^{2} b c d \,x^{2}+3 \sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) a \,b^{2} c +\sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) a \,b^{2} d \,x^{2}+\sqrt {a}\, \mathrm {log}\left (\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}+\sqrt {d \,x^{2}+c}\, a \right ) b^{3}}{a^{2} \left (a^{3} c^{2} d \,x^{2}+2 a^{2} b c d \,x^{2}+a^{3} c^{3}+a \,b^{2} d \,x^{2}+3 a^{2} b \,c^{2}+3 a \,b^{2} c +b^{3}\right )} \] Input:

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

Output:

( - sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*a**2*b*c - sqrt(c + d*x**2)* 
sqrt(a*c + a*d*x**2 + b)*a*b**2 + sqrt(c)*sqrt(a*c + b)*log(sqrt(a*c + b)* 
sqrt(a*c + a*d*x**2 + b)*c - sqrt(c)*sqrt(c + d*x**2)*a*c - sqrt(c)*sqrt(c 
 + d*x**2)*b)*a**3*c**2 + sqrt(c)*sqrt(a*c + b)*log(sqrt(a*c + b)*sqrt(a*c 
 + a*d*x**2 + b)*c - sqrt(c)*sqrt(c + d*x**2)*a*c - sqrt(c)*sqrt(c + d*x** 
2)*b)*a**3*c*d*x**2 + sqrt(c)*sqrt(a*c + b)*log(sqrt(a*c + b)*sqrt(a*c + a 
*d*x**2 + b)*c - sqrt(c)*sqrt(c + d*x**2)*a*c - sqrt(c)*sqrt(c + d*x**2)*b 
)*a**2*b*c - sqrt(c)*sqrt(a*c + b)*log(x)*a**3*c**2 - sqrt(c)*sqrt(a*c + b 
)*log(x)*a**3*c*d*x**2 - sqrt(c)*sqrt(a*c + b)*log(x)*a**2*b*c + sqrt(a)*l 
og(sqrt(a)*sqrt(a*c + a*d*x**2 + b) + sqrt(c + d*x**2)*a)*a**3*c**3 + sqrt 
(a)*log(sqrt(a)*sqrt(a*c + a*d*x**2 + b) + sqrt(c + d*x**2)*a)*a**3*c**2*d 
*x**2 + 3*sqrt(a)*log(sqrt(a)*sqrt(a*c + a*d*x**2 + b) + sqrt(c + d*x**2)* 
a)*a**2*b*c**2 + 2*sqrt(a)*log(sqrt(a)*sqrt(a*c + a*d*x**2 + b) + sqrt(c + 
 d*x**2)*a)*a**2*b*c*d*x**2 + 3*sqrt(a)*log(sqrt(a)*sqrt(a*c + a*d*x**2 + 
b) + sqrt(c + d*x**2)*a)*a*b**2*c + sqrt(a)*log(sqrt(a)*sqrt(a*c + a*d*x** 
2 + b) + sqrt(c + d*x**2)*a)*a*b**2*d*x**2 + sqrt(a)*log(sqrt(a)*sqrt(a*c 
+ a*d*x**2 + b) + sqrt(c + d*x**2)*a)*b**3)/(a**2*(a**3*c**3 + a**3*c**2*d 
*x**2 + 3*a**2*b*c**2 + 2*a**2*b*c*d*x**2 + 3*a*b**2*c + a*b**2*d*x**2 + b 
**3))