\(\int (c+\frac {d}{x^2})^{3/2} (a+b x) \, dx\) [20]

Optimal result

Integrand size = 17, antiderivative size = 111 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} (a+b x) \, dx=-\frac {3}{2} d \sqrt {c+\frac {d}{x^2}} \left (b+\frac {a}{x}\right )+\frac {1}{2} \left (c+\frac {d}{x^2}\right )^{3/2} \left (b+\frac {2 a}{x}\right ) x^2+\frac {3}{2} b \sqrt {c} d \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {3}{2} a c \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.24 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} (a+b x) \, dx=\frac {\sqrt {c+\frac {d}{x^2}} \left (\sqrt {d+c x^2} \left (-a d-2 b d x+2 a c x^2+b c x^3\right )+6 a c \sqrt {d} x^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {d+c x^2}}{\sqrt {d}}\right )-3 b \sqrt {c} d x^2 \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )\right )}{2 x \sqrt {d+c x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {1774, 1803, 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.

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

\(\Big \downarrow \) 1774

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

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 537

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.18

method	result	size
risch	\(-\frac {d \left (2 b x +a \right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{2 x}+\frac {\left (\frac {3 \sqrt {c}\, b d \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )}{2}+a \sqrt {c \,x^{2}+d}\, c -\frac {3 c a \sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right )}{2}+\frac {b c x \sqrt {c \,x^{2}+d}}{2}\right ) x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{\sqrt {c \,x^{2}+d}}\)	\(131\)
default	\(-\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} x \left (-2 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,x^{3}+3 d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) c^{\frac {3}{2}} a \,x^{2}+2 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} b x -c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,x^{2}-3 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, b d \,x^{3}+a \left (c \,x^{2}+d \right )^{\frac {5}{2}} \sqrt {c}-3 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, a d \,x^{2}-3 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) b c \,d^{2} x^{2}\right )}{2 \left (c \,x^{2}+d \right )^{\frac {3}{2}} d \sqrt {c}}\)	\(196\)

Input:

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

Output:

-1/2*d*(2*b*x+a)/x*((c*x^2+d)/x^2)^(1/2)+(3/2*c^(1/2)*b*d*ln(c^(1/2)*x+(c* 
x^2+d)^(1/2))+a*(c*x^2+d)^(1/2)*c-3/2*c*a*d^(1/2)*ln((2*d+2*d^(1/2)*(c*x^2 
+d)^(1/2))/x)+1/2*b*c*x*(c*x^2+d)^(1/2))/(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 = 493, normalized size of antiderivative = 4.44 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} (a+b x) \, dx=\left [\frac {3 \, b \sqrt {c} d x \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 3 \, a c \sqrt {d} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (b c x^{3} + 2 \, a c x^{2} - 2 \, b d x - a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{4 \, x}, -\frac {6 \, b \sqrt {-c} d x \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - 3 \, a c \sqrt {d} x \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (b c x^{3} + 2 \, a c x^{2} - 2 \, b d x - a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{4 \, x}, \frac {6 \, a c \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + 3 \, b \sqrt {c} d x \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 2 \, {\left (b c x^{3} + 2 \, a c x^{2} - 2 \, b d x - a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{4 \, x}, -\frac {3 \, b \sqrt {-c} d x \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - 3 \, a c \sqrt {-d} x \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) - {\left (b c x^{3} + 2 \, a c x^{2} - 2 \, b d x - a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{2 \, x}\right ] \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 3.49 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.64 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} (a+b x) \, dx=\frac {a c^{\frac {3}{2}} x}{\sqrt {1 + \frac {d}{c x^{2}}}} - \frac {a \sqrt {c} d \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} + \frac {a \sqrt {c} d}{x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 a c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2} + \frac {3 b \sqrt {c} d \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )}}{2} + \frac {b c \sqrt {d} x \sqrt {\frac {c x^{2}}{d} + 1}}{2} - \frac {b c \sqrt {d} x}{\sqrt {\frac {c x^{2}}{d} + 1}} - \frac {b d^{\frac {3}{2}}}{x \sqrt {\frac {c x^{2}}{d} + 1}} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.44 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} (a+b x) \, dx=\frac {1}{4} \, {\left (4 \, \sqrt {c + \frac {d}{x^{2}}} c x - \frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c d x}{{\left (c + \frac {d}{x^{2}}\right )} x^{2} - d} + 3 \, c \sqrt {d} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} a + \frac {1}{4} \, {\left (2 \, \sqrt {c + \frac {d}{x^{2}}} c x^{2} - 3 \, \sqrt {c} d \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right ) - 4 \, \sqrt {c + \frac {d}{x^{2}}} d\right )} b \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (87) = 174\).

Time = 45.66 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.86 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} (a+b x) \, dx=\frac {3 \, a c d \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + d}}{\sqrt {-d}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-d}} - \frac {3}{2} \, b \sqrt {c} d \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right ) \mathrm {sgn}\left (x\right ) + \frac {1}{2} \, {\left (b c x \mathrm {sgn}\left (x\right ) + 2 \, a c \mathrm {sgn}\left (x\right )\right )} \sqrt {c x^{2} + d} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{3} a c d \mathrm {sgn}\left (x\right ) + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b \sqrt {c} d^{2} \mathrm {sgn}\left (x\right ) + {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )} a c d^{2} \mathrm {sgn}\left (x\right ) - 2 \, b \sqrt {c} d^{3} \mathrm {sgn}\left (x\right )}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.79 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} (a+b x) \, dx=\frac {3\,b\,\sqrt {c}\,d\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{2}-b\,d\,\sqrt {c+\frac {d}{x^2}}+\frac {b\,c\,x^2\,\sqrt {c+\frac {d}{x^2}}}{2}+\frac {a\,x\,{\left (c\,x^2+d\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {d}{c\,x^2}\right )}{{\left (\frac {d}{c}+x^2\right )}^{3/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.35 \[ \int \left (c+\frac {d}{x^2}\right )^{3/2} (a+b x) \, dx=\frac {2 \sqrt {c \,x^{2}+d}\, a c \,x^{2}-\sqrt {c \,x^{2}+d}\, a d +\sqrt {c \,x^{2}+d}\, b c \,x^{3}-2 \sqrt {c \,x^{2}+d}\, b d x +3 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) b d \,x^{2}+3 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) a c \,x^{2}-3 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) a c \,x^{2}}{2 x^{2}} \] Input:

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

Output:

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