\(\int \frac {1}{x^2 (a+\frac {b}{(c+d x)^2})^{3/2}} \, dx\) [92]

Optimal result

Integrand size = 19, antiderivative size = 169 \[ \int \frac {1}{x^2 \left (a+\frac {b}{(c+d x)^2}\right )^{3/2}} \, dx=\frac {c \left (b-2 a c^2\right ) d \sqrt {a+\frac {b}{(c+d x)^2}}}{a \left (b+a c^2\right )^2 \left (1-\frac {c}{c+d x}\right )}+\frac {d \left (a c-\frac {b}{c+d x}\right )}{a \left (b+a c^2\right ) \sqrt {a+\frac {b}{(c+d x)^2}} \left (1-\frac {c}{c+d x}\right )}-\frac {3 b c^2 d \text {arctanh}\left (\frac {a c+\frac {b}{c+d x}}{\sqrt {b+a c^2} \sqrt {a+\frac {b}{(c+d x)^2}}}\right )}{\left (b+a c^2\right )^{5/2}} \] Output:

c*(-2*a*c^2+b)*d*(a+b/(d*x+c)^2)^(1/2)/a/(a*c^2+b)^2/(1-c/(d*x+c))+d*(a*c- 
b/(d*x+c))/a/(a*c^2+b)/(a+b/(d*x+c)^2)^(1/2)/(1-c/(d*x+c))-3*b*c^2*d*arcta 
nh((a*c+b/(d*x+c))/(a*c^2+b)^(1/2)/(a+b/(d*x+c)^2)^(1/2))/(a*c^2+b)^(5/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.41 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \left (a+\frac {b}{(c+d x)^2}\right )^{3/2}} \, dx=-\frac {b^2 d x+a^2 c^3 (c+d x)^2+a b c \left (c^2-3 c d x-2 d^2 x^2\right )}{a \left (b+a c^2\right )^2 x (c+d x) \sqrt {\frac {b+a (c+d x)^2}{(c+d x)^2}}}+\frac {3 b c^2 d \log (x)}{\left (b+a c^2\right )^{5/2}}-\frac {3 b c^2 d \log \left (b+(c+d x) \left (a c+\sqrt {b+a c^2} \sqrt {\frac {b+a (c+d x)^2}{(c+d x)^2}}\right )\right )}{\left (b+a c^2\right )^{5/2}} \] Input:

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

Output:

-((b^2*d*x + a^2*c^3*(c + d*x)^2 + a*b*c*(c^2 - 3*c*d*x - 2*d^2*x^2))/(a*( 
b + a*c^2)^2*x*(c + d*x)*Sqrt[(b + a*(c + d*x)^2)/(c + d*x)^2])) + (3*b*c^ 
2*d*Log[x])/(b + a*c^2)^(5/2) - (3*b*c^2*d*Log[b + (c + d*x)*(a*c + Sqrt[b 
 + a*c^2]*Sqrt[(b + a*(c + d*x)^2)/(c + d*x)^2])])/(b + a*c^2)^(5/2)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {896, 1774, 1799, 496, 25, 27, 679, 488, 219}

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.

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

\(\Big \downarrow \) 896

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

\(\Big \downarrow \) 1774

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

\(\Big \downarrow \) 1799

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

\(\Big \downarrow \) 496

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 679

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 496

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 
 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^n*(a 
 + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 
*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad 
raticQ[a, 0, b, c, d, n, p, x]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 896

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff 
icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1)   Subst[Int[Si 
mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; 
 FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
 

rule 1774

Int[((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.), x_Sy 
mbol] :> Int[x^(mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, 
e, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n2] ||  !IntegerQ[p 
])
 

rule 1799

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^ 
n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Simplif 
y[m - n + 1], 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(372\) vs. \(2(159)=318\).

Time = 0.12 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.21

method	result	size
default	\(-\frac {\left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b \right ) \left (\left (a \,c^{2}+b \right )^{\frac {3}{2}} a^{2} c^{3} d^{2} x^{2}+3 \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\, \ln \left (\frac {2 a \,c^{2}+2 b +2 a d x c +2 \sqrt {a \,c^{2}+b}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}{x}\right ) a^{2} b \,c^{4} d x +2 \left (a \,c^{2}+b \right )^{\frac {3}{2}} a^{2} c^{4} d x +\left (a \,c^{2}+b \right )^{\frac {3}{2}} a^{2} c^{5}-2 \left (a \,c^{2}+b \right )^{\frac {3}{2}} a b c \,d^{2} x^{2}+3 \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\, \ln \left (\frac {2 a \,c^{2}+2 b +2 a d x c +2 \sqrt {a \,c^{2}+b}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}{x}\right ) a \,b^{2} c^{2} d x -3 \left (a \,c^{2}+b \right )^{\frac {3}{2}} a b \,c^{2} d x +\left (a \,c^{2}+b \right )^{\frac {3}{2}} a b \,c^{3}+\left (a \,c^{2}+b \right )^{\frac {3}{2}} b^{2} d x \right )}{\left (\frac {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}{\left (d x +c \right )^{2}}\right )^{\frac {3}{2}} \left (d x +c \right )^{3} \left (a \,c^{2}+b \right )^{\frac {7}{2}} x a}\)	\(373\)
risch	\(-\frac {c^{3} \left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b \right )}{\left (a \,c^{2}+b \right )^{2} x \sqrt {\frac {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}{\left (d x +c \right )^{2}}}\, \left (d x +c \right )}+\frac {\left (-\frac {b^{2} d}{\left (a \,c^{2}+b \right )^{2} a \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}+\frac {2 b \,d^{2} c x}{\left (a \,c^{2}+b \right )^{2} \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}+\frac {3 d^{2} a \,c^{3} x}{\left (a \,c^{2}+b \right )^{2} \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}+\frac {3 d a \,c^{4}}{\left (a \,c^{2}+b \right )^{2} \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}+\frac {3 b^{2} d \,c^{2}}{\left (a \,c^{2}+b \right )^{3} \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}-\frac {3 d^{2} c^{5} x \,a^{2}}{\left (a \,c^{2}+b \right )^{3} \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}-\frac {3 b \,d^{2} c^{3} x a}{\left (a \,c^{2}+b \right )^{3} \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}-\frac {3 d \,c^{6} a^{2}}{\left (a \,c^{2}+b \right )^{3} \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}-\frac {3 b d \,c^{4} \ln \left (\frac {2 a \,c^{2}+2 b +2 a d x c +2 \sqrt {a \,c^{2}+b}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}{x}\right ) a}{\left (a \,c^{2}+b \right )^{\frac {7}{2}}}-\frac {3 b^{2} d \,c^{2} \ln \left (\frac {2 a \,c^{2}+2 b +2 a d x c +2 \sqrt {a \,c^{2}+b}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}{x}\right )}{\left (a \,c^{2}+b \right )^{\frac {7}{2}}}\right ) \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}{\sqrt {\frac {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}{\left (d x +c \right )^{2}}}\, \left (d x +c \right )}\)	\(617\)

Input:

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

Output:

-(a*d^2*x^2+2*a*c*d*x+a*c^2+b)*((a*c^2+b)^(3/2)*a^2*c^3*d^2*x^2+3*(a*d^2*x 
^2+2*a*c*d*x+a*c^2+b)^(1/2)*ln(2*(a*d*x*c+a*c^2+(a*c^2+b)^(1/2)*(a*d^2*x^2 
+2*a*c*d*x+a*c^2+b)^(1/2)+b)/x)*a^2*b*c^4*d*x+2*(a*c^2+b)^(3/2)*a^2*c^4*d* 
x+(a*c^2+b)^(3/2)*a^2*c^5-2*(a*c^2+b)^(3/2)*a*b*c*d^2*x^2+3*(a*d^2*x^2+2*a 
*c*d*x+a*c^2+b)^(1/2)*ln(2*(a*d*x*c+a*c^2+(a*c^2+b)^(1/2)*(a*d^2*x^2+2*a*c 
*d*x+a*c^2+b)^(1/2)+b)/x)*a*b^2*c^2*d*x-3*(a*c^2+b)^(3/2)*a*b*c^2*d*x+(a*c 
^2+b)^(3/2)*a*b*c^3+(a*c^2+b)^(3/2)*b^2*d*x)/((a*d^2*x^2+2*a*c*d*x+a*c^2+b 
)/(d*x+c)^2)^(3/2)/(d*x+c)^3/(a*c^2+b)^(7/2)/x/a
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (160) = 320\).

Time = 0.92 (sec) , antiderivative size = 1022, normalized size of antiderivative = 6.05 \[ \int \frac {1}{x^2 \left (a+\frac {b}{(c+d x)^2}\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/2*(3*(a^2*b*c^2*d^3*x^3 + 2*a^2*b*c^3*d^2*x^2 + (a^2*b*c^4 + a*b^2*c^2) 
*d*x)*sqrt(a*c^2 + b)*log(-(2*a^2*c^4 + (2*a^2*c^2 + a*b)*d^2*x^2 + 4*a*b* 
c^2 + 4*(a^2*c^3 + a*b*c)*d*x + 2*b^2 - 2*(a*c*d^2*x^2 + a*c^3 + (2*a*c^2 
+ b)*d*x + b*c)*sqrt(a*c^2 + b)*sqrt((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/( 
d^2*x^2 + 2*c*d*x + c^2)))/x^2) - 2*(a^3*c^8 + 2*a^2*b*c^6 + a*b^2*c^4 + ( 
a^3*c^5 - a^2*b*c^3 - 2*a*b^2*c)*d^3*x^3 + (3*a^3*c^6 - 2*a^2*b*c^4 - 4*a* 
b^2*c^2 + b^3)*d^2*x^2 + (3*a^3*c^7 + a^2*b*c^5 - a*b^2*c^3 + b^3*c)*d*x)* 
sqrt((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2)))/((a^5 
*c^6 + 3*a^4*b*c^4 + 3*a^3*b^2*c^2 + a^2*b^3)*d^2*x^3 + 2*(a^5*c^7 + 3*a^4 
*b*c^5 + 3*a^3*b^2*c^3 + a^2*b^3*c)*d*x^2 + (a^5*c^8 + 4*a^4*b*c^6 + 6*a^3 
*b^2*c^4 + 4*a^2*b^3*c^2 + a*b^4)*x), (3*(a^2*b*c^2*d^3*x^3 + 2*a^2*b*c^3* 
d^2*x^2 + (a^2*b*c^4 + a*b^2*c^2)*d*x)*sqrt(-a*c^2 - b)*arctan((a*c*d^2*x^ 
2 + a*c^3 + (2*a*c^2 + b)*d*x + b*c)*sqrt(-a*c^2 - b)*sqrt((a*d^2*x^2 + 2* 
a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2*c*d*x + c^2))/(a^2*c^4 + (a^2*c^2 + a*b) 
*d^2*x^2 + 2*a*b*c^2 + 2*(a^2*c^3 + a*b*c)*d*x + b^2)) - (a^3*c^8 + 2*a^2* 
b*c^6 + a*b^2*c^4 + (a^3*c^5 - a^2*b*c^3 - 2*a*b^2*c)*d^3*x^3 + (3*a^3*c^6 
 - 2*a^2*b*c^4 - 4*a*b^2*c^2 + b^3)*d^2*x^2 + (3*a^3*c^7 + a^2*b*c^5 - a*b 
^2*c^3 + b^3*c)*d*x)*sqrt((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 + 2 
*c*d*x + c^2)))/((a^5*c^6 + 3*a^4*b*c^4 + 3*a^3*b^2*c^2 + a^2*b^3)*d^2*x^3 
 + 2*(a^5*c^7 + 3*a^4*b*c^5 + 3*a^3*b^2*c^3 + a^2*b^3*c)*d*x^2 + (a^5*c...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (160) = 320\).

Time = 0.21 (sec) , antiderivative size = 523, normalized size of antiderivative = 3.09 \[ \int \frac {1}{x^2 \left (a+\frac {b}{(c+d x)^2}\right )^{3/2}} \, dx=\frac {6 \, b c^{2} d \arctan \left (-\frac {\sqrt {a d^{2}} x - \sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}}{\sqrt {-a c^{2} - b}}\right )}{{\left (a^{2} c^{4} \mathrm {sgn}\left (d x + c\right ) + 2 \, a b c^{2} \mathrm {sgn}\left (d x + c\right ) + b^{2} \mathrm {sgn}\left (d x + c\right )\right )} \sqrt {-a c^{2} - b}} + \frac {\frac {2 \, {\left (a^{3} b^{2} c^{5} d^{3} \mathrm {sgn}\left (d x + c\right ) + 2 \, a^{2} b^{3} c^{3} d^{3} \mathrm {sgn}\left (d x + c\right ) + a b^{4} c d^{3} \mathrm {sgn}\left (d x + c\right )\right )} x}{a^{5} b c^{8} d + 4 \, a^{4} b^{2} c^{6} d + 6 \, a^{3} b^{3} c^{4} d + 4 \, a^{2} b^{4} c^{2} d + a b^{5} d} + \frac {3 \, a^{3} b^{2} c^{6} d^{2} \mathrm {sgn}\left (d x + c\right ) + 5 \, a^{2} b^{3} c^{4} d^{2} \mathrm {sgn}\left (d x + c\right ) + a b^{4} c^{2} d^{2} \mathrm {sgn}\left (d x + c\right ) - b^{5} d^{2} \mathrm {sgn}\left (d x + c\right )}{a^{5} b c^{8} d + 4 \, a^{4} b^{2} c^{6} d + 6 \, a^{3} b^{3} c^{4} d + 4 \, a^{2} b^{4} c^{2} d + a b^{5} d}}{\sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}} - \frac {2 \, {\left (a^{\frac {3}{2}} c^{5} {\left | d \right |} + {\left (\sqrt {a d^{2}} x - \sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}\right )} a c^{4} d + \sqrt {a} b c^{3} {\left | d \right |}\right )}}{{\left (a^{2} c^{4} \mathrm {sgn}\left (d x + c\right ) + 2 \, a b c^{2} \mathrm {sgn}\left (d x + c\right ) + b^{2} \mathrm {sgn}\left (d x + c\right )\right )} {\left (a c^{2} - {\left (\sqrt {a d^{2}} x - \sqrt {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}\right )}^{2} + b\right )}} \] Input:

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

Output:

6*b*c^2*d*arctan(-(sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b) 
)/sqrt(-a*c^2 - b))/((a^2*c^4*sgn(d*x + c) + 2*a*b*c^2*sgn(d*x + c) + b^2* 
sgn(d*x + c))*sqrt(-a*c^2 - b)) + (2*(a^3*b^2*c^5*d^3*sgn(d*x + c) + 2*a^2 
*b^3*c^3*d^3*sgn(d*x + c) + a*b^4*c*d^3*sgn(d*x + c))*x/(a^5*b*c^8*d + 4*a 
^4*b^2*c^6*d + 6*a^3*b^3*c^4*d + 4*a^2*b^4*c^2*d + a*b^5*d) + (3*a^3*b^2*c 
^6*d^2*sgn(d*x + c) + 5*a^2*b^3*c^4*d^2*sgn(d*x + c) + a*b^4*c^2*d^2*sgn(d 
*x + c) - b^5*d^2*sgn(d*x + c))/(a^5*b*c^8*d + 4*a^4*b^2*c^6*d + 6*a^3*b^3 
*c^4*d + 4*a^2*b^4*c^2*d + a*b^5*d))/sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + 
b) - 2*(a^(3/2)*c^5*abs(d) + (sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + 
 a*c^2 + b))*a*c^4*d + sqrt(a)*b*c^3*abs(d))/((a^2*c^4*sgn(d*x + c) + 2*a* 
b*c^2*sgn(d*x + c) + b^2*sgn(d*x + c))*(a*c^2 - (sqrt(a*d^2)*x - sqrt(a*d^ 
2*x^2 + 2*a*c*d*x + a*c^2 + b))^2 + b))
 

Mupad [F(-1)]

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

Reduce [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 842, normalized size of antiderivative = 4.98 \[ \int \frac {1}{x^2 \left (a+\frac {b}{(c+d x)^2}\right )^{3/2}} \, dx=\frac {-\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a^{3} c^{7}-2 \sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a^{3} c^{6} d x -\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a^{3} c^{5} d^{2} x^{2}-2 \sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a^{2} b \,c^{5}+\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a^{2} b \,c^{4} d x +\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a^{2} b \,c^{3} d^{2} x^{2}-\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a \,b^{2} c^{3}+2 \sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a \,b^{2} c^{2} d x +2 \sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a \,b^{2} c \,d^{2} x^{2}-\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, b^{3} d x +3 \sqrt {a \,c^{2}+b}\, \mathrm {log}\left (\sqrt {a \,c^{2}+b}\, \sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}-a \,c^{2}-a c d x -b \right ) a^{2} b \,c^{4} d x +6 \sqrt {a \,c^{2}+b}\, \mathrm {log}\left (\sqrt {a \,c^{2}+b}\, \sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}-a \,c^{2}-a c d x -b \right ) a^{2} b \,c^{3} d^{2} x^{2}+3 \sqrt {a \,c^{2}+b}\, \mathrm {log}\left (\sqrt {a \,c^{2}+b}\, \sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}-a \,c^{2}-a c d x -b \right ) a^{2} b \,c^{2} d^{3} x^{3}+3 \sqrt {a \,c^{2}+b}\, \mathrm {log}\left (\sqrt {a \,c^{2}+b}\, \sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}-a \,c^{2}-a c d x -b \right ) a \,b^{2} c^{2} d x -3 \sqrt {a \,c^{2}+b}\, \mathrm {log}\left (x \right ) a^{2} b \,c^{4} d x -6 \sqrt {a \,c^{2}+b}\, \mathrm {log}\left (x \right ) a^{2} b \,c^{3} d^{2} x^{2}-3 \sqrt {a \,c^{2}+b}\, \mathrm {log}\left (x \right ) a^{2} b \,c^{2} d^{3} x^{3}-3 \sqrt {a \,c^{2}+b}\, \mathrm {log}\left (x \right ) a \,b^{2} c^{2} d x}{a x \left (a^{4} c^{6} d^{2} x^{2}+2 a^{4} c^{7} d x +a^{4} c^{8}+3 a^{3} b \,c^{4} d^{2} x^{2}+6 a^{3} b \,c^{5} d x +4 a^{3} b \,c^{6}+3 a^{2} b^{2} c^{2} d^{2} x^{2}+6 a^{2} b^{2} c^{3} d x +6 a^{2} b^{2} c^{4}+a \,b^{3} d^{2} x^{2}+2 a \,b^{3} c d x +4 a \,b^{3} c^{2}+b^{4}\right )} \] Input:

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

Output:

( - sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b)*a**3*c**7 - 2*sqrt(a*c**2 + 
 2*a*c*d*x + a*d**2*x**2 + b)*a**3*c**6*d*x - sqrt(a*c**2 + 2*a*c*d*x + a* 
d**2*x**2 + b)*a**3*c**5*d**2*x**2 - 2*sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x* 
*2 + b)*a**2*b*c**5 + sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b)*a**2*b*c* 
*4*d*x + sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b)*a**2*b*c**3*d**2*x**2 
- sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b)*a*b**2*c**3 + 2*sqrt(a*c**2 + 
 2*a*c*d*x + a*d**2*x**2 + b)*a*b**2*c**2*d*x + 2*sqrt(a*c**2 + 2*a*c*d*x 
+ a*d**2*x**2 + b)*a*b**2*c*d**2*x**2 - sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x 
**2 + b)*b**3*d*x + 3*sqrt(a*c**2 + b)*log(sqrt(a*c**2 + b)*sqrt(a*c**2 + 
2*a*c*d*x + a*d**2*x**2 + b) - a*c**2 - a*c*d*x - b)*a**2*b*c**4*d*x + 6*s 
qrt(a*c**2 + b)*log(sqrt(a*c**2 + b)*sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 
 + b) - a*c**2 - a*c*d*x - b)*a**2*b*c**3*d**2*x**2 + 3*sqrt(a*c**2 + b)*l 
og(sqrt(a*c**2 + b)*sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b) - a*c**2 - 
a*c*d*x - b)*a**2*b*c**2*d**3*x**3 + 3*sqrt(a*c**2 + b)*log(sqrt(a*c**2 + 
b)*sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b) - a*c**2 - a*c*d*x - b)*a*b* 
*2*c**2*d*x - 3*sqrt(a*c**2 + b)*log(x)*a**2*b*c**4*d*x - 6*sqrt(a*c**2 + 
b)*log(x)*a**2*b*c**3*d**2*x**2 - 3*sqrt(a*c**2 + b)*log(x)*a**2*b*c**2*d* 
*3*x**3 - 3*sqrt(a*c**2 + b)*log(x)*a*b**2*c**2*d*x)/(a*x*(a**4*c**8 + 2*a 
**4*c**7*d*x + a**4*c**6*d**2*x**2 + 4*a**3*b*c**6 + 6*a**3*b*c**5*d*x + 3 
*a**3*b*c**4*d**2*x**2 + 6*a**2*b**2*c**4 + 6*a**2*b**2*c**3*d*x + 3*a*...