\(\int (c+\frac {d}{x^2})^{3/2} x (a+b x) \, dx\) [19]

Optimal result

Integrand size = 18, antiderivative size = 114 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x (a+b x) \, dx=\frac {1}{2} d \sqrt {c+\frac {d}{x^2}} \left (2 b-\frac {3 a}{x}\right ) x+\frac {1}{6} \left (c+\frac {d}{x^2}\right )^{3/2} \left (2 b+\frac {3 a}{x}\right ) x^3+\frac {3}{2} a \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-b d^{3/2} \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right ) \] Output:

1/2*d*(c+d/x^2)^(1/2)*(2*b-3*a/x)*x+1/6*(c+d/x^2)^(3/2)*(2*b+3*a/x)*x^3+3/ 
2*a*c^(1/2)*d*arctanh((c+d/x^2)^(1/2)/c^(1/2))-b*d^(3/2)*arctanh(d^(1/2)/( 
c+d/x^2)^(1/2)/x)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.15 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x (a+b x) \, dx=\frac {\sqrt {c+\frac {d}{x^2}} \left (\sqrt {d+c x^2} \left (-6 a d+8 b d x+3 a c x^2+2 b c x^3\right )+12 b d^{3/2} x \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {d+c x^2}}{\sqrt {d}}\right )-9 a \sqrt {c} d x \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )\right )}{6 \sqrt {d+c x^2}} \] Input:

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

Output:

(Sqrt[c + d/x^2]*(Sqrt[d + c*x^2]*(-6*a*d + 8*b*d*x + 3*a*c*x^2 + 2*b*c*x^ 
3) + 12*b*d^(3/2)*x*ArcTanh[(Sqrt[c]*x - Sqrt[d + c*x^2])/Sqrt[d]] - 9*a*S 
qrt[c]*d*x*Log[-(Sqrt[c]*x) + Sqrt[d + c*x^2]]))/(6*Sqrt[d + c*x^2])
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {1892, 1803, 537, 25, 536, 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.

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

\(\Big \downarrow \) 1892

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

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 537

\(\displaystyle \frac {1}{2} d \int -\sqrt {c+\frac {d}{x^2}} \left (\frac {3 a}{x}+2 b\right ) x^2d\frac {1}{x}+\frac {1}{6} x^3 \left (\frac {3 a}{x}+2 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} x^3 \left (\frac {3 a}{x}+2 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{2} d \int \sqrt {c+\frac {d}{x^2}} \left (\frac {3 a}{x}+2 b\right ) x^2d\frac {1}{x}\)

\(\Big \downarrow \) 536

\(\displaystyle \frac {1}{6} x^3 \left (\frac {3 a}{x}+2 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{2} d \left (\int \frac {\left (3 a c+\frac {2 b d}{x}\right ) x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}-x \left (2 b-\frac {3 a}{x}\right ) \sqrt {c+\frac {d}{x^2}}\right )\)

\(\Big \downarrow \) 538

\(\displaystyle \frac {1}{6} x^3 \left (\frac {3 a}{x}+2 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{2} d \left (3 a c \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+2 b d \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+x \left (2 b-\frac {3 a}{x}\right ) \left (-\sqrt {c+\frac {d}{x^2}}\right )\right )\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {1}{6} x^3 \left (\frac {3 a}{x}+2 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{2} d \left (3 a c \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+2 b d \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}+x \left (2 b-\frac {3 a}{x}\right ) \left (-\sqrt {c+\frac {d}{x^2}}\right )\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{6} x^3 \left (\frac {3 a}{x}+2 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{2} d \left (3 a c \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+x \left (2 b-\frac {3 a}{x}\right ) \left (-\sqrt {c+\frac {d}{x^2}}\right )+2 b \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )\right )\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{6} x^3 \left (\frac {3 a}{x}+2 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{2} d \left (\frac {3}{2} a c \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}+x \left (2 b-\frac {3 a}{x}\right ) \left (-\sqrt {c+\frac {d}{x^2}}\right )+2 b \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{6} x^3 \left (\frac {3 a}{x}+2 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{2} d \left (\frac {3 a c \int \frac {1}{\frac {\sqrt {c+\frac {d}{x^2}}}{d}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}}{d}+x \left (2 b-\frac {3 a}{x}\right ) \left (-\sqrt {c+\frac {d}{x^2}}\right )+2 b \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{6} x^3 \left (\frac {3 a}{x}+2 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}-\frac {1}{2} d \left (-3 a \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )+x \left (2 b-\frac {3 a}{x}\right ) \left (-\sqrt {c+\frac {d}{x^2}}\right )+2 b \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )\right )\)

Input:

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

Output:

((c + d/x^2)^(3/2)*(2*b + (3*a)/x)*x^3)/6 - (d*(-(Sqrt[c + d/x^2]*(2*b - ( 
3*a)/x)*x) - 3*a*Sqrt[c]*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]] + 2*b*Sqrt[d]*Ar 
cTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)]))/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 536

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

rule 1892

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

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.22

method	result	size
risch	\(-a d \sqrt {\frac {c \,x^{2}+d}{x^{2}}}+\frac {\left (\frac {a \sqrt {c \,x^{2}+d}\, c x}{2}+\frac {3 a \sqrt {c}\, d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )}{2}+\frac {b c \,x^{2} \sqrt {c \,x^{2}+d}}{3}+\frac {4 d b \sqrt {c \,x^{2}+d}}{3}-b \,d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right )\right ) x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{\sqrt {c \,x^{2}+d}}\)	\(139\)
default	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x^{2} \left (6 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,x^{2}-6 \sqrt {c}\, d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b x +9 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, a d \,x^{2}-6 a \left (c \,x^{2}+d \right )^{\frac {5}{2}} \sqrt {c}+2 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {3}{2}} b d x +6 \sqrt {c}\, \sqrt {c \,x^{2}+d}\, b \,d^{2} x +9 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a c \,d^{2} x \right )}{6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} d \sqrt {c}}\)	\(178\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 465, normalized size of antiderivative = 4.08 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x (a+b x) \, dx=\left [\frac {3}{4} \, a \sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + \frac {1}{2} \, b d^{\frac {3}{2}} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + \frac {1}{6} \, {\left (2 \, b c x^{3} + 3 \, a c x^{2} + 8 \, b d x - 6 \, a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}, -\frac {3}{2} \, a \sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + \frac {1}{2} \, b d^{\frac {3}{2}} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + \frac {1}{6} \, {\left (2 \, b c x^{3} + 3 \, a c x^{2} + 8 \, b d x - 6 \, a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}, b \sqrt {-d} d \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + \frac {3}{4} \, a \sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + \frac {1}{6} \, {\left (2 \, b c x^{3} + 3 \, a c x^{2} + 8 \, b d x - 6 \, a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}, -\frac {3}{2} \, a \sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + b \sqrt {-d} d \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + \frac {1}{6} \, {\left (2 \, b c x^{3} + 3 \, a c x^{2} + 8 \, b d x - 6 \, a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}\right ] \] Input:

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

Output:

[3/4*a*sqrt(c)*d*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d) + 
 1/2*b*d^(3/2)*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 2*d)/x^2) 
 + 1/6*(2*b*c*x^3 + 3*a*c*x^2 + 8*b*d*x - 6*a*d)*sqrt((c*x^2 + d)/x^2), -3 
/2*a*sqrt(-c)*d*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) + 1 
/2*b*d^(3/2)*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 2*d)/x^2) + 
 1/6*(2*b*c*x^3 + 3*a*c*x^2 + 8*b*d*x - 6*a*d)*sqrt((c*x^2 + d)/x^2), b*sq 
rt(-d)*d*arctan(sqrt(-d)*x*sqrt((c*x^2 + d)/x^2)/d) + 3/4*a*sqrt(c)*d*log( 
-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d) + 1/6*(2*b*c*x^3 + 3*a 
*c*x^2 + 8*b*d*x - 6*a*d)*sqrt((c*x^2 + d)/x^2), -3/2*a*sqrt(-c)*d*arctan( 
sqrt(-c)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) + b*sqrt(-d)*d*arctan(sqrt 
(-d)*x*sqrt((c*x^2 + d)/x^2)/d) + 1/6*(2*b*c*x^3 + 3*a*c*x^2 + 8*b*d*x - 6 
*a*d)*sqrt((c*x^2 + d)/x^2)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (100) = 200\).

Time = 3.12 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.77 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x (a+b x) \, dx=\frac {3 a \sqrt {c} d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2} + \frac {a c \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} - \frac {a c \sqrt {d} x}{\sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a d^{\frac {3}{2}}}{x \sqrt {\frac {c x^{2}}{d} + 1}} + \frac {b \sqrt {c} d x}{\sqrt {1 + \frac {d}{c x^{2}}}} + \frac {b c \sqrt {d} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3} + \frac {b d^{\frac {3}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3} - b d^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )} + \frac {b d^{2}}{\sqrt {c} x \sqrt {1 + \frac {d}{c x^{2}}}} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.25 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x (a+b x) \, dx=\frac {1}{4} \, {\left (2 \, \sqrt {c + \frac {d}{x^{2}}} c x^{2} - 3 \, \sqrt {c} d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right ) - 4 \, \sqrt {c + \frac {d}{x^{2}}} d\right )} a + \frac {1}{6} \, {\left (2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3} + 6 \, \sqrt {c + \frac {d}{x^{2}}} d x + 3 \, d^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} b \] Input:

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

Output:

1/4*(2*sqrt(c + d/x^2)*c*x^2 - 3*sqrt(c)*d*log((sqrt(c + d/x^2) - sqrt(c)) 
/(sqrt(c + d/x^2) + sqrt(c))) - 4*sqrt(c + d/x^2)*d)*a + 1/6*(2*(c + d/x^2 
)^(3/2)*x^3 + 6*sqrt(c + d/x^2)*d*x + 3*d^(3/2)*log((sqrt(c + d/x^2)*x - s 
qrt(d))/(sqrt(c + d/x^2)*x + sqrt(d))))*b
 

Giac [A] (verification not implemented)

Time = 15.58 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.19 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x (a+b x) \, dx=\frac {2 \, b d^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + d}}{\sqrt {-d}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-d}} - \frac {3}{2} \, a \sqrt {c} d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right ) \mathrm {sgn}\left (x\right ) + \frac {2 \, a \sqrt {c} d^{2} \mathrm {sgn}\left (x\right )}{{\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d} + \frac {1}{6} \, \sqrt {c x^{2} + d} {\left (8 \, b d \mathrm {sgn}\left (x\right ) + {\left (2 \, b c x \mathrm {sgn}\left (x\right ) + 3 \, a c \mathrm {sgn}\left (x\right )\right )} x\right )} \] Input:

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

Output:

2*b*d^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + d))/sqrt(-d))*sgn(x)/sqrt(-d) - 
3/2*a*sqrt(c)*d*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))*sgn(x) + 2*a*sqrt(c 
)*d^2*sgn(x)/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d) + 1/6*sqrt(c*x^2 + d)*( 
8*b*d*sgn(x) + (2*b*c*x*sgn(x) + 3*a*c*sgn(x))*x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.33 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} x (a+b x) \, dx=\frac {12 \sqrt {c \,x^{2}+d}\, a c \,x^{2}-24 \sqrt {c \,x^{2}+d}\, a d +8 \sqrt {c \,x^{2}+d}\, b c \,x^{3}+32 \sqrt {c \,x^{2}+d}\, b d x +36 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) a d x -27 \sqrt {c}\, a d x +24 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) b d x -24 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) b d x}{24 x} \] Input:

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

Output:

(12*sqrt(c*x**2 + d)*a*c*x**2 - 24*sqrt(c*x**2 + d)*a*d + 8*sqrt(c*x**2 + 
d)*b*c*x**3 + 32*sqrt(c*x**2 + d)*b*d*x + 36*sqrt(c)*log((sqrt(c*x**2 + d) 
 + sqrt(c)*x)/sqrt(d))*a*d*x - 27*sqrt(c)*a*d*x + 24*sqrt(d)*log((sqrt(c*x 
**2 + d) + sqrt(c)*x - sqrt(d))/sqrt(d))*b*d*x - 24*sqrt(d)*log((sqrt(c*x* 
*2 + d) + sqrt(c)*x + sqrt(d))/sqrt(d))*b*d*x)/(24*x)