\(\int \frac {(c+\frac {d}{x^2})^{3/2} (a+b x)}{x} \, dx\) [21]

Optimal result

Integrand size = 20, antiderivative size = 111 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x} \, dx=-\frac {1}{2} \sqrt {c+\frac {d}{x^2}} \left (2 a c+\frac {3 b d}{x}\right )+\frac {1}{3} \left (c+\frac {d}{x^2}\right )^{3/2} \left (3 b-\frac {a}{x}\right ) x+a c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {3}{2} b c \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.24 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x} \, dx=-\frac {\sqrt {c+\frac {d}{x^2}} \left (\sqrt {d+c x^2} \left (2 a d+3 b d x+8 a c x^2-6 b c x^3\right )-18 b c \sqrt {d} x^3 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {d+c x^2}}{\sqrt {d}}\right )+6 a c^{3/2} x^3 \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )\right )}{6 x^2 \sqrt {d+c x^2}} \] Input:

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

Output:

-1/6*(Sqrt[c + d/x^2]*(Sqrt[d + c*x^2]*(2*a*d + 3*b*d*x + 8*a*c*x^2 - 6*b* 
c*x^3) - 18*b*c*Sqrt[d]*x^3*ArcTanh[(Sqrt[c]*x - Sqrt[d + c*x^2])/Sqrt[d]] 
 + 6*a*c^(3/2)*x^3*Log[-(Sqrt[c]*x) + Sqrt[d + c*x^2]]))/(x^2*Sqrt[d + c*x 
^2])
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1892, 1730, 536, 535, 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.

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

\(\Big \downarrow \) 1892

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

\(\Big \downarrow \) 1730

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

\(\Big \downarrow \) 536

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

\(\Big \downarrow \) 535

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

\(\Big \downarrow \) 538

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 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 1730

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

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 = 125, normalized size of antiderivative = 1.13

method	result	size
risch	\(-\frac {\left (8 a c \,x^{2}+3 b d x +2 a d \right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{6 x^{2}}+\frac {\left (-\frac {3 \sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b c}{2}+a \,c^{\frac {3}{2}} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+\sqrt {c \,x^{2}+d}\, b c \right ) x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{\sqrt {c \,x^{2}+d}}\)	\(125\)
default	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (4 c^{\frac {5}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,x^{4}-9 d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) c^{\frac {3}{2}} b \,x^{3}-4 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \,x^{2}+3 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} b d \,x^{3}+6 c^{\frac {5}{2}} \sqrt {c \,x^{2}+d}\, a d \,x^{4}+6 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) a \,c^{2} d^{2} x^{3}-3 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} b d x +9 c^{\frac {3}{2}} \sqrt {c \,x^{2}+d}\, b \,d^{2} x^{3}-2 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} a d \right )}{6 \left (c \,x^{2}+d \right )^{\frac {3}{2}} d^{2} \sqrt {c}}\)	\(221\)

Input:

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

Output:

-1/6*(8*a*c*x^2+3*b*d*x+2*a*d)/x^2*((c*x^2+d)/x^2)^(1/2)+(-3/2*d^(1/2)*ln( 
(2*d+2*d^(1/2)*(c*x^2+d)^(1/2))/x)*b*c+a*c^(3/2)*ln(c^(1/2)*x+(c*x^2+d)^(1 
/2))+(c*x^2+d)^(1/2)*b*c)/(c*x^2+d)^(1/2)*x*((c*x^2+d)/x^2)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 510, normalized size of antiderivative = 4.59 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x} \, dx=\left [\frac {6 \, a c^{\frac {3}{2}} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 9 \, b c \sqrt {d} x^{2} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (6 \, b c x^{3} - 8 \, a c x^{2} - 3 \, b d x - 2 \, a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{12 \, x^{2}}, -\frac {12 \, a \sqrt {-c} c x^{2} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - 9 \, b c \sqrt {d} x^{2} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (6 \, b c x^{3} - 8 \, a c x^{2} - 3 \, b d x - 2 \, a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{12 \, x^{2}}, \frac {9 \, b c \sqrt {-d} x^{2} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + 3 \, a c^{\frac {3}{2}} x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + {\left (6 \, b c x^{3} - 8 \, a c x^{2} - 3 \, b d x - 2 \, a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{6 \, x^{2}}, -\frac {6 \, a \sqrt {-c} c x^{2} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - 9 \, b c \sqrt {-d} x^{2} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) - {\left (6 \, b c x^{3} - 8 \, a c x^{2} - 3 \, b d x - 2 \, a d\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{6 \, x^{2}}\right ] \] Input:

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

Output:

[1/12*(6*a*c^(3/2)*x^2*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) 
- d) + 9*b*c*sqrt(d)*x^2*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 
 2*d)/x^2) + 2*(6*b*c*x^3 - 8*a*c*x^2 - 3*b*d*x - 2*a*d)*sqrt((c*x^2 + d)/ 
x^2))/x^2, -1/12*(12*a*sqrt(-c)*c*x^2*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d) 
/x^2)/(c*x^2 + d)) - 9*b*c*sqrt(d)*x^2*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x 
^2 + d)/x^2) + 2*d)/x^2) - 2*(6*b*c*x^3 - 8*a*c*x^2 - 3*b*d*x - 2*a*d)*sqr 
t((c*x^2 + d)/x^2))/x^2, 1/6*(9*b*c*sqrt(-d)*x^2*arctan(sqrt(-d)*x*sqrt((c 
*x^2 + d)/x^2)/d) + 3*a*c^(3/2)*x^2*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x 
^2 + d)/x^2) - d) + (6*b*c*x^3 - 8*a*c*x^2 - 3*b*d*x - 2*a*d)*sqrt((c*x^2 
+ d)/x^2))/x^2, -1/6*(6*a*sqrt(-c)*c*x^2*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + 
 d)/x^2)/(c*x^2 + d)) - 9*b*c*sqrt(-d)*x^2*arctan(sqrt(-d)*x*sqrt((c*x^2 + 
 d)/x^2)/d) - (6*b*c*x^3 - 8*a*c*x^2 - 3*b*d*x - 2*a*d)*sqrt((c*x^2 + d)/x 
^2))/x^2]
 

Sympy [A] (verification not implemented)

Time = 4.69 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.68 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x} \, dx=a c^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )} - \frac {a c^{2} x}{\sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {a c \sqrt {d}}{x \sqrt {\frac {c x^{2}}{d} + 1}} + a d \left (\begin {cases} - \frac {\sqrt {c}}{2 x^{2}} & \text {for}\: d = 0 \\- \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) + \frac {b c^{\frac {3}{2}} x}{\sqrt {1 + \frac {d}{c x^{2}}}} - \frac {b \sqrt {c} d \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} + \frac {b \sqrt {c} d}{x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 b c \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{2} \] Input:

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

Output:

a*c**(3/2)*asinh(sqrt(c)*x/sqrt(d)) - a*c**2*x/(sqrt(d)*sqrt(c*x**2/d + 1) 
) - a*c*sqrt(d)/(x*sqrt(c*x**2/d + 1)) + a*d*Piecewise((-sqrt(c)/(2*x**2), 
 Eq(d, 0)), (-(c + d/x**2)**(3/2)/(3*d), True)) + b*c**(3/2)*x/sqrt(1 + d/ 
(c*x**2)) - b*sqrt(c)*d*sqrt(1 + d/(c*x**2))/(2*x) + b*sqrt(c)*d/(x*sqrt(1 
 + d/(c*x**2))) - 3*b*c*sqrt(d)*asinh(sqrt(d)/(sqrt(c)*x))/2
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.40 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x} \, dx=-\frac {1}{6} \, {\left (3 \, c^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right ) + 2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} + 6 \, \sqrt {c + \frac {d}{x^{2}}} c\right )} a + \frac {1}{4} \, {\left (4 \, \sqrt {c + \frac {d}{x^{2}}} c x - \frac {2 \, \sqrt {c + \frac {d}{x^{2}}} c d x}{{\left (c + \frac {d}{x^{2}}\right )} x^{2} - d} + 3 \, c \sqrt {d} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )\right )} b \] Input:

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

Output:

-1/6*(3*c^(3/2)*log((sqrt(c + d/x^2) - sqrt(c))/(sqrt(c + d/x^2) + sqrt(c) 
)) + 2*(c + d/x^2)^(3/2) + 6*sqrt(c + d/x^2)*c)*a + 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((sq 
rt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d))))*b
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (89) = 178\).

Time = 67.55 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.05 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x} \, dx=\frac {3 \, b c d \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + d}}{\sqrt {-d}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-d}} - a c^{\frac {3}{2}} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right ) \mathrm {sgn}\left (x\right ) + \sqrt {c x^{2} + d} b c \mathrm {sgn}\left (x\right ) + \frac {3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{5} b c d \mathrm {sgn}\left (x\right ) + 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {3}{2}} d \mathrm {sgn}\left (x\right ) - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {3}{2}} d^{2} \mathrm {sgn}\left (x\right ) - 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )} b c d^{3} \mathrm {sgn}\left (x\right ) + 8 \, a c^{\frac {3}{2}} d^{3} \mathrm {sgn}\left (x\right )}{3 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{3}} \] Input:

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

Output:

3*b*c*d*arctan(-(sqrt(c)*x - sqrt(c*x^2 + d))/sqrt(-d))*sgn(x)/sqrt(-d) - 
a*c^(3/2)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))*sgn(x) + sqrt(c*x^2 + d)* 
b*c*sgn(x) + 1/3*(3*(sqrt(c)*x - sqrt(c*x^2 + d))^5*b*c*d*sgn(x) + 12*(sqr 
t(c)*x - sqrt(c*x^2 + d))^4*a*c^(3/2)*d*sgn(x) - 12*(sqrt(c)*x - sqrt(c*x^ 
2 + d))^2*a*c^(3/2)*d^2*sgn(x) - 3*(sqrt(c)*x - sqrt(c*x^2 + d))*b*c*d^3*s 
gn(x) + 8*a*c^(3/2)*d^3*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^3
 

Mupad [B] (verification not implemented)

Time = 8.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.74 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x} \, dx=a\,c^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {a\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{3}-a\,c\,\sqrt {c+\frac {d}{x^2}}+\frac {b\,x\,{\left (c\,x^2+d\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {d}{c\,x^2}\right )}{{\left (\frac {d}{c}+x^2\right )}^{3/2}} \] Input:

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

Output:

a*c^(3/2)*atanh((c + d/x^2)^(1/2)/c^(1/2)) - (a*(c + d/x^2)^(3/2))/3 - a*c 
*(c + d/x^2)^(1/2) + (b*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 = 0.18 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.03 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x} \, dx=\frac {-16 \sqrt {c \,x^{2}+d}\, a c \,x^{2}-4 \sqrt {c \,x^{2}+d}\, a d +12 \sqrt {c \,x^{2}+d}\, b c \,x^{3}-6 \sqrt {c \,x^{2}+d}\, b d x +12 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) a c \,x^{3}+9 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {c \,x^{2}+d}\, x -\sqrt {d}\, \sqrt {c \,x^{2}+d}-\sqrt {d}\, \sqrt {c}\, x +c \,x^{2}+d}{\sqrt {d}\, \sqrt {c \,x^{2}+d}+\sqrt {d}\, \sqrt {c}\, x}\right ) b c \,x^{3}-9 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c}\, \sqrt {c \,x^{2}+d}\, x +\sqrt {d}\, \sqrt {c \,x^{2}+d}+\sqrt {d}\, \sqrt {c}\, x +c \,x^{2}+d}{\sqrt {d}\, \sqrt {c \,x^{2}+d}+\sqrt {d}\, \sqrt {c}\, x}\right ) b c \,x^{3}}{12 x^{3}} \] Input:

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

Output:

( - 16*sqrt(c*x**2 + d)*a*c*x**2 - 4*sqrt(c*x**2 + d)*a*d + 12*sqrt(c*x**2 
 + d)*b*c*x**3 - 6*sqrt(c*x**2 + d)*b*d*x + 12*sqrt(c)*log((sqrt(c*x**2 + 
d) + sqrt(c)*x)/sqrt(d))*a*c*x**3 + 9*sqrt(d)*log((sqrt(c)*sqrt(c*x**2 + d 
)*x - sqrt(d)*sqrt(c*x**2 + d) - sqrt(d)*sqrt(c)*x + c*x**2 + d)/(sqrt(d)* 
sqrt(c*x**2 + d) + sqrt(d)*sqrt(c)*x))*b*c*x**3 - 9*sqrt(d)*log((sqrt(c)*s 
qrt(c*x**2 + d)*x + sqrt(d)*sqrt(c*x**2 + d) + sqrt(d)*sqrt(c)*x + c*x**2 
+ d)/(sqrt(d)*sqrt(c*x**2 + d) + sqrt(d)*sqrt(c)*x))*b*c*x**3)/(12*x**3)