\(\int x (a+\frac {b}{(c+d x)^2})^{3/2} \, dx\) [73]

Optimal result

Integrand size = 17, antiderivative size = 155 \[ \int x \left (a+\frac {b}{(c+d x)^2}\right )^{3/2} \, dx=\frac {(c+d x)^2 \left (a+\frac {b}{(c+d x)^2}\right )^{3/2} \left (1-\frac {2 c}{c+d x}\right )}{2 d^2}-\frac {3 b \sqrt {a+\frac {b}{(c+d x)^2}} \left (1-\frac {c}{c+d x}\right )}{2 d^2}+\frac {3 a \sqrt {b} c \text {arctanh}\left (\frac {\sqrt {b}}{(c+d x) \sqrt {a+\frac {b}{(c+d x)^2}}}\right )}{2 d^2}+\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{(c+d x)^2}}}{\sqrt {a}}\right )}{2 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 10.79 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.17 \[ \int x \left (a+\frac {b}{(c+d x)^2}\right )^{3/2} \, dx=-\frac {\sqrt {a+\frac {b}{(c+d x)^2}} \left (\sqrt {b+a (c+d x)^2} \left (a (c-d x) (c+d x)^2+b (c+2 d x)\right )+6 a \sqrt {b} c (c+d x)^2 \text {arctanh}\left (\frac {\sqrt {a} (c+d x)-\sqrt {b+a (c+d x)^2}}{\sqrt {b}}\right )+3 \sqrt {a} b (c+d x)^2 \log \left (-\sqrt {a} (c+d x)+\sqrt {b+a (c+d x)^2}\right )\right )}{2 d^2 (c+d x) \sqrt {b+a (c+d x)^2}} \] Input:

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

Output:

-1/2*(Sqrt[a + b/(c + d*x)^2]*(Sqrt[b + a*(c + d*x)^2]*(a*(c - d*x)*(c + d 
*x)^2 + b*(c + 2*d*x)) + 6*a*Sqrt[b]*c*(c + d*x)^2*ArcTanh[(Sqrt[a]*(c + d 
*x) - Sqrt[b + a*(c + d*x)^2])/Sqrt[b]] + 3*Sqrt[a]*b*(c + d*x)^2*Log[-(Sq 
rt[a]*(c + d*x)) + Sqrt[b + a*(c + d*x)^2]]))/(d^2*(c + d*x)*Sqrt[b + a*(c 
 + d*x)^2])
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.882, Rules used = {896, 25, 1774, 1803, 25, 537, 25, 535, 27, 538, 224, 219, 243, 73, 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.

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

\(\Big \downarrow \) 896

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1774

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

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(((c + d*x)^2*(a + b/(c + d*x)^2)^(3/2)*(1 - (2*c)/(c + d*x)))/2 - (3*b*(S 
qrt[a + b/(c + d*x)^2]*(1 - c/(c + d*x)) + a*(-((c*ArcTanh[Sqrt[b]/((c + d 
*x)*Sqrt[a + b/(c + d*x)^2])])/Sqrt[b]) - ArcTanh[Sqrt[a + b/(c + d*x)^2]/ 
Sqrt[a]]/Sqrt[a])))/2)/d^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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, 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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 535

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim 
p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p 
 + 1)   Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free 
Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 537

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), 
 x] - Simp[2*b*(p/((m + 1)*(m + 2)))   Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) 
*x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && 
 GtQ[p, 0] &&  !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
 

rule 538

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

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 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(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] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(283\) vs. \(2(131)=262\).

Time = 0.17 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.83

method	result	size
risch	\(-\frac {\left (-d x +c \right ) a \sqrt {\frac {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}{\left (d x +c \right )^{2}}}\, \left (d x +c \right )}{2 d^{2}}+\frac {\left (\frac {3 b a \ln \left (\frac {a \,d^{2} x +a c d}{\sqrt {a \,d^{2}}}+\sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\right )}{2 d \sqrt {a \,d^{2}}}-\frac {b \sqrt {a \,d^{2} \left (x +\frac {c}{d}\right )^{2}+b}}{d^{3} \left (x +\frac {c}{d}\right )}+\frac {3 \sqrt {b}\, a c \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,d^{2} \left (x +\frac {c}{d}\right )^{2}+b}}{x +\frac {c}{d}}\right )}{2 d^{2}}+\frac {b c \sqrt {a \,d^{2} \left (x +\frac {c}{d}\right )^{2}+b}}{2 d^{4} \left (x +\frac {c}{d}\right )^{2}}\right ) \sqrt {\frac {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}{\left (d x +c \right )^{2}}}\, \left (d x +c \right )}{\sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}}\)	\(284\)
default	\(-\frac {\left (-2 \left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a \,d^{3} x^{3}-5 \left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a c \,d^{2} x^{2}-3 \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\, \sqrt {a \,d^{2}}\, a b \,d^{3} x^{3}-3 \sqrt {a \,d^{2}}\, b^{\frac {3}{2}} \ln \left (\frac {2 \left (\sqrt {b}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}+b \right ) d}{d x +c}\right ) a c \,d^{2} x^{2}-4 \left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a \,c^{2} d x -6 \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\, \sqrt {a \,d^{2}}\, a b c \,d^{2} x^{2}-3 \ln \left (\frac {a \,d^{2} x +a c d +\sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\, \sqrt {a \,d^{2}}}{\sqrt {a \,d^{2}}}\right ) a \,b^{2} d^{3} x^{2}-6 \sqrt {a \,d^{2}}\, b^{\frac {3}{2}} \ln \left (\frac {2 \left (\sqrt {b}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}+b \right ) d}{d x +c}\right ) a \,c^{2} d x +2 \left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b \right )^{\frac {5}{2}} \sqrt {a \,d^{2}}\, d x -\left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b \right )^{\frac {3}{2}} \sqrt {a \,d^{2}}\, a \,c^{3}-3 \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\, \sqrt {a \,d^{2}}\, a b \,c^{2} d x -6 \ln \left (\frac {a \,d^{2} x +a c d +\sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\, \sqrt {a \,d^{2}}}{\sqrt {a \,d^{2}}}\right ) a \,b^{2} c \,d^{2} x -3 \sqrt {a \,d^{2}}\, b^{\frac {3}{2}} \ln \left (\frac {2 \left (\sqrt {b}\, \sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}+b \right ) d}{d x +c}\right ) a \,c^{3}+\left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b \right )^{\frac {5}{2}} \sqrt {a \,d^{2}}\, c -3 \ln \left (\frac {a \,d^{2} x +a c d +\sqrt {a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b}\, \sqrt {a \,d^{2}}}{\sqrt {a \,d^{2}}}\right ) a \,b^{2} c^{2} d \right ) \left (d x +c \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}}}{2 d^{2} \sqrt {a \,d^{2}}\, b \left (a \,d^{2} x^{2}+2 a d x c +a \,c^{2}+b \right )^{\frac {3}{2}}}\)	\(775\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (130) = 260\).

Time = 0.61 (sec) , antiderivative size = 1189, normalized size of antiderivative = 7.67 \[ \int x \left (a+\frac {b}{(c+d x)^2}\right )^{3/2} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

Integral(x*((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]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 832 vs. \(2 (130) = 260\).

Time = 1.58 (sec) , antiderivative size = 832, normalized size of antiderivative = 5.37 \[ \int x \left (a+\frac {b}{(c+d x)^2}\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)*(a*x*sgn(d*x + c)/d - a*c*sgn( 
d*x + c)/d^2) - 3/14*sqrt(a)*b*log(abs(-a^(7/2)*c^7*d - 7*(sqrt(a*d^2)*x - 
 sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b))*a^3*c^6*abs(d) - 21*(sqrt(a*d^2) 
*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b))^2*a^(5/2)*c^5*d - 35*(sqrt(a 
*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b))^3*a^2*c^4*abs(d) - 35*( 
sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b))^4*a^(3/2)*c^3*d + 
 3*a^(5/2)*b*c^5*d - 21*(sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^ 
2 + b))^5*a*c^2*abs(d) + 15*(sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + 
a*c^2 + b))*a^2*b*c^4*abs(d) - 7*(sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d 
*x + a*c^2 + b))^6*sqrt(a)*c*d + 30*(sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a* 
c*d*x + a*c^2 + b))^2*a^(3/2)*b*c^3*d - (sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 
2*a*c*d*x + a*c^2 + b))^7*abs(d) + 30*(sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2* 
a*c*d*x + a*c^2 + b))^3*a*b*c^2*abs(d) + 15*(sqrt(a*d^2)*x - sqrt(a*d^2*x^ 
2 + 2*a*c*d*x + a*c^2 + b))^4*sqrt(a)*b*c*d - 3*a^(3/2)*b^2*c^3*d + 3*(sqr 
t(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b))^5*b*abs(d) - 9*(sqrt 
(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b))*a*b^2*c^2*abs(d) - 9* 
(sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b))^2*sqrt(a)*b^2*c* 
d - 3*(sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b))^3*b^2*abs( 
d) + sqrt(a)*b^3*c*d + (sqrt(a*d^2)*x - sqrt(a*d^2*x^2 + 2*a*c*d*x + a*c^2 
 + b))*b^3*abs(d)))*sgn(d*x + c)/(d*abs(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 643, normalized size of antiderivative = 4.15 \[ \int x \left (a+\frac {b}{(c+d x)^2}\right )^{3/2} \, dx=\frac {-\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a \,c^{3}-\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a \,c^{2} d x +\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a c \,d^{2} x^{2}+\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, a \,d^{3} x^{3}-\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, b c -2 \sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}\, b d x +3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x}{\sqrt {b}}\right ) b \,c^{2}+6 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x}{\sqrt {b}}\right ) b c d x +3 \sqrt {a}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x}{\sqrt {b}}\right ) b \,d^{2} x^{2}-3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x -\sqrt {b}}{\sqrt {b}}\right ) a \,c^{3}-6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x -\sqrt {b}}{\sqrt {b}}\right ) a \,c^{2} d x -3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x -\sqrt {b}}{\sqrt {b}}\right ) a c \,d^{2} x^{2}+3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x +\sqrt {b}}{\sqrt {b}}\right ) a \,c^{3}+6 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x +\sqrt {b}}{\sqrt {b}}\right ) a \,c^{2} d x +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}+\sqrt {a}\, c +\sqrt {a}\, d x +\sqrt {b}}{\sqrt {b}}\right ) a c \,d^{2} x^{2}}{2 d^{2} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:

int(x*(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*c**3 - sqrt(a*c**2 + 2*a* 
c*d*x + a*d**2*x**2 + b)*a*c**2*d*x + sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x** 
2 + b)*a*c*d**2*x**2 + sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b)*a*d**3*x 
**3 - sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b)*b*c - 2*sqrt(a*c**2 + 2*a 
*c*d*x + a*d**2*x**2 + b)*b*d*x + 3*sqrt(a)*log((sqrt(a*c**2 + 2*a*c*d*x + 
 a*d**2*x**2 + b) + sqrt(a)*c + sqrt(a)*d*x)/sqrt(b))*b*c**2 + 6*sqrt(a)*l 
og((sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b) + sqrt(a)*c + sqrt(a)*d*x)/ 
sqrt(b))*b*c*d*x + 3*sqrt(a)*log((sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + 
b) + sqrt(a)*c + sqrt(a)*d*x)/sqrt(b))*b*d**2*x**2 - 3*sqrt(b)*log((sqrt(a 
*c**2 + 2*a*c*d*x + a*d**2*x**2 + b) + sqrt(a)*c + sqrt(a)*d*x - sqrt(b))/ 
sqrt(b))*a*c**3 - 6*sqrt(b)*log((sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b 
) + sqrt(a)*c + sqrt(a)*d*x - sqrt(b))/sqrt(b))*a*c**2*d*x - 3*sqrt(b)*log 
((sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b) + sqrt(a)*c + sqrt(a)*d*x - s 
qrt(b))/sqrt(b))*a*c*d**2*x**2 + 3*sqrt(b)*log((sqrt(a*c**2 + 2*a*c*d*x + 
a*d**2*x**2 + b) + sqrt(a)*c + sqrt(a)*d*x + sqrt(b))/sqrt(b))*a*c**3 + 6* 
sqrt(b)*log((sqrt(a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b) + sqrt(a)*c + sqrt 
(a)*d*x + sqrt(b))/sqrt(b))*a*c**2*d*x + 3*sqrt(b)*log((sqrt(a*c**2 + 2*a* 
c*d*x + a*d**2*x**2 + b) + sqrt(a)*c + sqrt(a)*d*x + sqrt(b))/sqrt(b))*a*c 
*d**2*x**2)/(2*d**2*(c**2 + 2*c*d*x + d**2*x**2))