\(\int \frac {(a+\frac {b}{c+d x^2})^{3/2}}{x} \, dx\) [179]

Optimal result

Integrand size = 21, antiderivative size = 102 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=\frac {b \sqrt {a+\frac {b}{c+d x^2}}}{c}+a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )-\frac {(b+a c)^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {b+a c}}\right )}{c^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=\frac {b \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c}-\frac {(-b-a c)^{3/2} \arctan \left (\frac {\sqrt {c} \sqrt {-b-a c} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{b+a c}\right )}{c^{3/2}}+a^{3/2} \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.37, 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, 381, 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.

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

\(\Big \downarrow \) 2057

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

\(\Big \downarrow \) 2053

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

\(\Big \downarrow \) 2052

\(\displaystyle -b d \int -\frac {x^8}{d \left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 25

\(\displaystyle b d \int \frac {x^8}{d \left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 27

\(\displaystyle b \int \frac {x^8}{\left (a-x^4\right ) \left (-c x^4+b+a c\right )}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}\)

\(\Big \downarrow \) 381

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

\(\Big \downarrow \) 397

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

\(\Big \downarrow \) 219

\(\displaystyle b \left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{c}-\frac {\frac {(a c+b)^2 \int \frac {1}{-c x^4+b+a c}d\sqrt {\frac {a d x^2+b+a c}{d x^2+c}}}{b}-\frac {a^{3/2} c \text {arctanh}\left (\frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{\sqrt {a}}\right )}{b}}{c}\right )\)

\(\Big \downarrow \) 221

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

Input:

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

Output:

b*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]/c - (-((a^(3/2)*c*ArcTanh[Sqrt[(b 
 + a*c + a*d*x^2)/(c + d*x^2)]/Sqrt[a]])/b) + ((b + a*c)^(3/2)*ArcTanh[(Sq 
rt[c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)])/Sqrt[b + a*c]])/(b*Sqrt[c]))/ 
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 381

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
+ 1)/(b*d*(m + 2*(p + q) + 1))), x] - Simp[e^4/(b*d*(m + 2*(p + q) + 1)) 
Int[(e*x)^(m - 4)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 
2*q - 1) + b*c*(m + 2*p - 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p, q 
}, x] && NeQ[b*c - a*d, 0] && GtQ[m, 3] && 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. \(651\) vs. \(2(84)=168\).

Time = 0.13 (sec) , antiderivative size = 652, normalized size of antiderivative = 6.39

method	result	size
default	\(\frac {\left (\ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{2} c^{2} d^{2} x^{2}-\sqrt {a \,c^{2}+b c}\, \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) \sqrt {a \,d^{2}}\, a c d \,x^{2}+\ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a^{2} c^{3} d -\sqrt {a \,c^{2}+b c}\, \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) \sqrt {a \,d^{2}}\, b d \,x^{2}-\sqrt {a \,c^{2}+b c}\, \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) \sqrt {a \,d^{2}}\, a \,c^{2}-\sqrt {a \,c^{2}+b c}\, \ln \left (\frac {2 a d \,x^{2} c +b d \,x^{2}+2 a \,c^{2}+2 \sqrt {a \,c^{2}+b c}\, \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}+2 b c}{x^{2}}\right ) \sqrt {a \,d^{2}}\, b c +2 \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}\, b c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{2 \sqrt {a \,d^{2}}\, c^{2} \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}\)	\(652\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (84) = 168\).

Time = 0.20 (sec) , antiderivative size = 1073, normalized size of antiderivative = 10.52 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 201 vs. \(2 (84) = 168\).

Time = 0.15 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.97 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=-\frac {1}{2} \, a^{\frac {3}{2}} \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right ) + \frac {b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{c} + \frac {{\left (a^{2} c^{2} + 2 \, a b c + b^{2}\right )} \log \left (\frac {c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} - \sqrt {{\left (a c + b\right )} c}}{c \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}} + \sqrt {{\left (a c + b\right )} c}}\right )}{2 \, \sqrt {{\left (a c + b\right )} c} c} \] Input:

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 427, normalized size of antiderivative = 4.19 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b c +\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) a \,c^{2}+\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) a c d \,x^{2}+\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) b c +\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (\sqrt {a c +b}\, \sqrt {a d \,x^{2}+a c +b}\, c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, a c -\sqrt {c}\, \sqrt {d \,x^{2}+c}\, b \right ) b d \,x^{2}-\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) a \,c^{2}-\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) a c d \,x^{2}-\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) b c -\sqrt {c}\, \sqrt {a c +b}\, \mathrm {log}\left (x \right ) b d \,x^{2}+\sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}-\sqrt {d \,x^{2}+c}\, a \right ) a \,c^{3}+\sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}-\sqrt {d \,x^{2}+c}\, a \right ) a \,c^{2} d \,x^{2}}{c^{2} \left (d \,x^{2}+c \right )} \] Input:

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

Output:

(sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*b*c + 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*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)*sq 
rt(c + d*x**2)*b)*a*c*d*x**2 + sqrt(c)*sqrt(a*c + b)*log(sqrt(a*c + b)*sqr 
t(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)*b*c + 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)*b*d 
*x**2 - sqrt(c)*sqrt(a*c + b)*log(x)*a*c**2 - sqrt(c)*sqrt(a*c + b)*log(x) 
*a*c*d*x**2 - sqrt(c)*sqrt(a*c + b)*log(x)*b*c - sqrt(c)*sqrt(a*c + b)*log 
(x)*b*d*x**2 + sqrt(a)*log( - sqrt(a)*sqrt(a*c + a*d*x**2 + b) - sqrt(c + 
d*x**2)*a)*a*c**3 + sqrt(a)*log( - sqrt(a)*sqrt(a*c + a*d*x**2 + b) - sqrt 
(c + d*x**2)*a)*a*c**2*d*x**2)/(c**2*(c + d*x**2))