\(\int \frac {(c+\frac {d}{x^2})^{3/2} (a+b x)}{x^2} \, dx\) [22]

Optimal result

Integrand size = 20, antiderivative size = 111 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^2} \, dx=-\frac {1}{12} \left (c+\frac {d}{x^2}\right )^{3/2} \left (4 b+\frac {3 a}{x}\right )-\frac {1}{8} c \sqrt {c+\frac {d}{x^2}} \left (8 b+\frac {3 a}{x}\right )+b c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {3 a c^2 \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{8 \sqrt {d}} \] Output:

-1/12*(c+d/x^2)^(3/2)*(4*b+3*a/x)-1/8*c*(c+d/x^2)^(1/2)*(8*b+3*a/x)+b*c^(3 
/2)*arctanh((c+d/x^2)^(1/2)/c^(1/2))-3/8*a*c^2*arctanh(d^(1/2)/(c+d/x^2)^( 
1/2)/x)/d^(1/2)
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.33 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {\sqrt {c+\frac {d}{x^2}} \left (18 a c^2 x^4 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {d+c x^2}}{\sqrt {d}}\right )-\sqrt {d} \left (\sqrt {d+c x^2} \left (6 a d+8 b d x+15 a c x^2+32 b c x^3\right )+24 b c^{3/2} x^4 \log \left (-\sqrt {c} x+\sqrt {d+c x^2}\right )\right )\right )}{24 \sqrt {d} x^3 \sqrt {d+c x^2}} \] Input:

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

Output:

(Sqrt[c + d/x^2]*(18*a*c^2*x^4*ArcTanh[(Sqrt[c]*x - Sqrt[d + c*x^2])/Sqrt[ 
d]] - Sqrt[d]*(Sqrt[d + c*x^2]*(6*a*d + 8*b*d*x + 15*a*c*x^2 + 32*b*c*x^3) 
 + 24*b*c^(3/2)*x^4*Log[-(Sqrt[c]*x) + Sqrt[d + c*x^2]])))/(24*Sqrt[d]*x^3 
*Sqrt[d + c*x^2])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1892, 1803, 535, 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.

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

\(\Big \downarrow \) 1892

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

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 535

\(\displaystyle -\frac {1}{4} c \int \sqrt {c+\frac {d}{x^2}} \left (\frac {3 a}{x}+4 b\right ) xd\frac {1}{x}-\frac {1}{12} \left (\frac {3 a}{x}+4 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

\(\Big \downarrow \) 535

\(\displaystyle -\frac {1}{4} c \left (\frac {1}{2} c \int \frac {\left (\frac {3 a}{x}+8 b\right ) x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+\frac {1}{2} \left (\frac {3 a}{x}+8 b\right ) \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{12} \left (\frac {3 a}{x}+4 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

\(\Big \downarrow \) 538

\(\displaystyle -\frac {1}{4} c \left (\frac {1}{2} c \left (3 a \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+8 b \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}\right )+\frac {1}{2} \left (\frac {3 a}{x}+8 b\right ) \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{12} \left (\frac {3 a}{x}+4 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {1}{4} c \left (\frac {1}{2} c \left (3 a \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}+8 b \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}\right )+\frac {1}{2} \left (\frac {3 a}{x}+8 b\right ) \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{12} \left (\frac {3 a}{x}+4 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {1}{4} c \left (\frac {1}{2} c \left (8 b \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+\frac {3 a \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}}\right )+\frac {1}{2} \left (\frac {3 a}{x}+8 b\right ) \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{12} \left (\frac {3 a}{x}+4 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

\(\Big \downarrow \) 243

\(\displaystyle -\frac {1}{4} c \left (\frac {1}{2} c \left (4 b \int \frac {x}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x^2}+\frac {3 a \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}}\right )+\frac {1}{2} \left (\frac {3 a}{x}+8 b\right ) \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{12} \left (\frac {3 a}{x}+4 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {1}{4} c \left (\frac {1}{2} c \left (\frac {8 b \int \frac {1}{\frac {\sqrt {c+\frac {d}{x^2}}}{d}-\frac {c}{d}}d\sqrt {c+\frac {d}{x^2}}}{d}+\frac {3 a \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}}\right )+\frac {1}{2} \left (\frac {3 a}{x}+8 b\right ) \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{12} \left (\frac {3 a}{x}+4 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {1}{4} c \left (\frac {1}{2} c \left (\frac {3 a \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}}-\frac {8 b \text {arctanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}\right )+\frac {1}{2} \left (\frac {3 a}{x}+8 b\right ) \sqrt {c+\frac {d}{x^2}}\right )-\frac {1}{12} \left (\frac {3 a}{x}+4 b\right ) \left (c+\frac {d}{x^2}\right )^{3/2}\)

Input:

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

Output:

-1/12*((c + d/x^2)^(3/2)*(4*b + (3*a)/x)) - (c*((Sqrt[c + d/x^2]*(8*b + (3 
*a)/x))/2 + (c*((-8*b*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]])/Sqrt[c] + (3*a*Arc 
Tanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/Sqrt[d]))/2))/4
 

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 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 = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.10

method	result	size
risch	\(-\frac {\left (32 b c \,x^{3}+15 a c \,x^{2}+8 b d x +6 a d \right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{24 x^{3}}+\frac {\left (c^{\frac {3}{2}} b \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )-\frac {3 c^{2} a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right )}{8 \sqrt {d}}\right ) x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{\sqrt {c \,x^{2}+d}}\)	\(122\)
default	\(-\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (-16 c^{\frac {5}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,x^{5}+9 d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) c^{\frac {5}{2}} a \,x^{4}+16 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,x^{3}-3 c^{\frac {5}{2}} \left (c \,x^{2}+d \right )^{\frac {3}{2}} a \,x^{4}-24 c^{\frac {5}{2}} \sqrt {c \,x^{2}+d}\, b d \,x^{5}+3 c^{\frac {3}{2}} \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \,x^{2}-9 c^{\frac {5}{2}} \sqrt {c \,x^{2}+d}\, a d \,x^{4}-24 \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right ) b \,c^{2} d^{2} x^{4}+8 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} b d x +6 \sqrt {c}\, \left (c \,x^{2}+d \right )^{\frac {5}{2}} a d \right )}{24 x \left (c \,x^{2}+d \right )^{\frac {3}{2}} d^{2} \sqrt {c}}\)	\(239\)

Input:

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

Output:

-1/24*(32*b*c*x^3+15*a*c*x^2+8*b*d*x+6*a*d)/x^3*((c*x^2+d)/x^2)^(1/2)+(c^( 
3/2)*b*ln(c^(1/2)*x+(c*x^2+d)^(1/2))-3/8*c^2*a/d^(1/2)*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 = 0.10 (sec) , antiderivative size = 558, normalized size of antiderivative = 5.03 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^2} \, dx=\left [\frac {24 \, b c^{\frac {3}{2}} d x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) + 9 \, a c^{2} \sqrt {d} x^{3} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) - 2 \, {\left (32 \, b c d x^{3} + 15 \, a c d x^{2} + 8 \, b d^{2} x + 6 \, a d^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{48 \, d x^{3}}, -\frac {48 \, b \sqrt {-c} c d x^{3} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - 9 \, a c^{2} \sqrt {d} x^{3} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (32 \, b c d x^{3} + 15 \, a c d x^{2} + 8 \, b d^{2} x + 6 \, a d^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{48 \, d x^{3}}, \frac {9 \, a c^{2} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + 12 \, b c^{\frac {3}{2}} d x^{3} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - {\left (32 \, b c d x^{3} + 15 \, a c d x^{2} + 8 \, b d^{2} x + 6 \, a d^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{24 \, d x^{3}}, -\frac {24 \, b \sqrt {-c} c d x^{3} \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) - 9 \, a c^{2} \sqrt {-d} x^{3} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) + {\left (32 \, b c d x^{3} + 15 \, a c d x^{2} + 8 \, b d^{2} x + 6 \, a d^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{24 \, d x^{3}}\right ] \] Input:

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

Output:

[1/48*(24*b*c^(3/2)*d*x^3*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^ 
2) - d) + 9*a*c^2*sqrt(d)*x^3*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x 
^2) + 2*d)/x^2) - 2*(32*b*c*d*x^3 + 15*a*c*d*x^2 + 8*b*d^2*x + 6*a*d^2)*sq 
rt((c*x^2 + d)/x^2))/(d*x^3), -1/48*(48*b*sqrt(-c)*c*d*x^3*arctan(sqrt(-c) 
*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) - 9*a*c^2*sqrt(d)*x^3*log(-(c*x^2 
- 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 2*d)/x^2) + 2*(32*b*c*d*x^3 + 15*a*c 
*d*x^2 + 8*b*d^2*x + 6*a*d^2)*sqrt((c*x^2 + d)/x^2))/(d*x^3), 1/24*(9*a*c^ 
2*sqrt(-d)*x^3*arctan(sqrt(-d)*x*sqrt((c*x^2 + d)/x^2)/d) + 12*b*c^(3/2)*d 
*x^3*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d) - (32*b*c*d*x 
^3 + 15*a*c*d*x^2 + 8*b*d^2*x + 6*a*d^2)*sqrt((c*x^2 + d)/x^2))/(d*x^3), - 
1/24*(24*b*sqrt(-c)*c*d*x^3*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x 
^2 + d)) - 9*a*c^2*sqrt(-d)*x^3*arctan(sqrt(-d)*x*sqrt((c*x^2 + d)/x^2)/d) 
 + (32*b*c*d*x^3 + 15*a*c*d*x^2 + 8*b*d^2*x + 6*a*d^2)*sqrt((c*x^2 + d)/x^ 
2))/(d*x^3)]
 

Sympy [A] (verification not implemented)

Time = 5.98 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.99 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^2} \, dx=- \frac {a c^{\frac {3}{2}} \sqrt {1 + \frac {d}{c x^{2}}}}{2 x} - \frac {a c^{\frac {3}{2}}}{8 x \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 a \sqrt {c} d}{8 x^{3} \sqrt {1 + \frac {d}{c x^{2}}}} - \frac {3 a c^{2} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{8 \sqrt {d}} - \frac {a d^{2}}{4 \sqrt {c} x^{5} \sqrt {1 + \frac {d}{c x^{2}}}} + b c^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {d}} \right )} - \frac {b c^{2} x}{\sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}} - \frac {b c \sqrt {d}}{x \sqrt {\frac {c x^{2}}{d} + 1}} + b d \left (\begin {cases} - \frac {\sqrt {c}}{2 x^{2}} & \text {for}\: d = 0 \\- \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

-a*c**(3/2)*sqrt(1 + d/(c*x**2))/(2*x) - a*c**(3/2)/(8*x*sqrt(1 + d/(c*x** 
2))) - 3*a*sqrt(c)*d/(8*x**3*sqrt(1 + d/(c*x**2))) - 3*a*c**2*asinh(sqrt(d 
)/(sqrt(c)*x))/(8*sqrt(d)) - a*d**2/(4*sqrt(c)*x**5*sqrt(1 + d/(c*x**2))) 
+ b*c**(3/2)*asinh(sqrt(c)*x/sqrt(d)) - b*c**2*x/(sqrt(d)*sqrt(c*x**2/d + 
1)) - b*c*sqrt(d)/(x*sqrt(c*x**2/d + 1)) + b*d*Piecewise((-sqrt(c)/(2*x**2 
), Eq(d, 0)), (-(c + d/x**2)**(3/2)/(3*d), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (89) = 178\).

Time = 0.11 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.63 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {1}{16} \, {\left (\frac {3 \, c^{2} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{\sqrt {d}} - \frac {2 \, {\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{2} x^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{2} d x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{2} x^{4} - 2 \, {\left (c + \frac {d}{x^{2}}\right )} d x^{2} + d^{2}}\right )} a - \frac {1}{6} \, {\left (3 \, c^{\frac {3}{2}} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right ) + 2 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} + 6 \, \sqrt {c + \frac {d}{x^{2}}} c\right )} b \] Input:

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

Output:

1/16*(3*c^2*log((sqrt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d) 
))/sqrt(d) - 2*(5*(c + d/x^2)^(3/2)*c^2*x^3 - 3*sqrt(c + d/x^2)*c^2*d*x)/( 
(c + d/x^2)^2*x^4 - 2*(c + d/x^2)*d*x^2 + d^2))*a - 1/6*(3*c^(3/2)*log((sq 
rt(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)*b
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (89) = 178\).

Time = 0.21 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.75 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {3 \, a c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + d}}{\sqrt {-d}}\right ) \mathrm {sgn}\left (x\right )}{4 \, \sqrt {-d}} - b c^{\frac {3}{2}} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + d} \right |}\right ) \mathrm {sgn}\left (x\right ) + \frac {15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{7} a c^{2} \mathrm {sgn}\left (x\right ) + 48 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {3}{2}} d \mathrm {sgn}\left (x\right ) + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{5} a c^{2} d \mathrm {sgn}\left (x\right ) - 96 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {3}{2}} d^{2} \mathrm {sgn}\left (x\right ) + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{3} a c^{2} d^{2} \mathrm {sgn}\left (x\right ) + 80 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {3}{2}} d^{3} \mathrm {sgn}\left (x\right ) + 15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )} a c^{2} d^{3} \mathrm {sgn}\left (x\right ) - 32 \, b c^{\frac {3}{2}} d^{4} \mathrm {sgn}\left (x\right )}{12 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{4}} \] Input:

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

Output:

3/4*a*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + d))/sqrt(-d))*sgn(x)/sqrt(-d) 
- b*c^(3/2)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + d)))*sgn(x) + 1/12*(15*(sqrt 
(c)*x - sqrt(c*x^2 + d))^7*a*c^2*sgn(x) + 48*(sqrt(c)*x - sqrt(c*x^2 + d)) 
^6*b*c^(3/2)*d*sgn(x) + 9*(sqrt(c)*x - sqrt(c*x^2 + d))^5*a*c^2*d*sgn(x) - 
 96*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(3/2)*d^2*sgn(x) + 9*(sqrt(c)*x - 
sqrt(c*x^2 + d))^3*a*c^2*d^2*sgn(x) + 80*(sqrt(c)*x - sqrt(c*x^2 + d))^2*b 
*c^(3/2)*d^3*sgn(x) + 15*(sqrt(c)*x - sqrt(c*x^2 + d))*a*c^2*d^3*sgn(x) - 
32*b*c^(3/2)*d^4*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^4
 

Mupad [B] (verification not implemented)

Time = 9.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.77 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^2} \, dx=b\,c^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )-\frac {b\,{\left (c+\frac {d}{x^2}\right )}^{3/2}}{3}-b\,c\,\sqrt {c+\frac {d}{x^2}}-\frac {a\,{\left (c\,x^2+d\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {d}{c\,x^2}\right )}{x\,{\left (\frac {d}{c}+x^2\right )}^{3/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.58 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^2} \, dx=\frac {-15 \sqrt {c \,x^{2}+d}\, a c d \,x^{2}-6 \sqrt {c \,x^{2}+d}\, a \,d^{2}-32 \sqrt {c \,x^{2}+d}\, b c d \,x^{3}-8 \sqrt {c \,x^{2}+d}\, b \,d^{2} x +24 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x}{\sqrt {d}}\right ) b c d \,x^{4}+8 \sqrt {c}\, b c d \,x^{4}+9 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) a \,c^{2} x^{4}-9 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) a \,c^{2} x^{4}}{24 d \,x^{4}} \] Input:

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

Output:

( - 15*sqrt(c*x**2 + d)*a*c*d*x**2 - 6*sqrt(c*x**2 + d)*a*d**2 - 32*sqrt(c 
*x**2 + d)*b*c*d*x**3 - 8*sqrt(c*x**2 + d)*b*d**2*x + 24*sqrt(c)*log((sqrt 
(c*x**2 + d) + sqrt(c)*x)/sqrt(d))*b*c*d*x**4 + 8*sqrt(c)*b*c*d*x**4 + 9*s 
qrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x - sqrt(d))/sqrt(d))*a*c**2*x**4 - 
 9*sqrt(d)*log((sqrt(c*x**2 + d) + sqrt(c)*x + sqrt(d))/sqrt(d))*a*c**2*x* 
*4)/(24*d*x**4)