\(\int \frac {(c+\frac {d}{x^2})^{3/2} (a+b x)}{x^5} \, dx\) [25]

Optimal result

Integrand size = 20, antiderivative size = 138 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^5} \, dx=\frac {a c \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^2}-\frac {a \left (c+\frac {d}{x^2}\right )^{7/2}}{7 d^2}-\frac {b c \sqrt {c+\frac {d}{x^2}}}{8 x^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{6 x^3}-\frac {b c^2 \sqrt {c+\frac {d}{x^2}}}{16 d x}+\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {d}}{\sqrt {c+\frac {d}{x^2}} x}\right )}{16 d^{3/2}} \] Output:

1/5*a*c*(c+d/x^2)^(5/2)/d^2-1/7*a*(c+d/x^2)^(7/2)/d^2-1/8*b*c*(c+d/x^2)^(1 
/2)/x^3-1/6*b*(c+d/x^2)^(3/2)/x^3-1/16*b*c^2*(c+d/x^2)^(1/2)/d/x+1/16*b*c^ 
3*arctanh(d^(1/2)/(c+d/x^2)^(1/2)/x)/d^(3/2)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^5} \, dx=-\frac {\sqrt {c+\frac {d}{x^2}} \left (\sqrt {d+c x^2} \left (48 a \left (5 d-2 c x^2\right ) \left (d+c x^2\right )^2+35 b d x \left (8 d^2+14 c d x^2+3 c^2 x^4\right )\right )+210 b c^3 \sqrt {d} x^7 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {d+c x^2}}{\sqrt {d}}\right )\right )}{1680 d^2 x^6 \sqrt {d+c x^2}} \] Input:

Integrate[((c + d/x^2)^(3/2)*(a + b*x))/x^5,x]
 

Output:

-1/1680*(Sqrt[c + d/x^2]*(Sqrt[d + c*x^2]*(48*a*(5*d - 2*c*x^2)*(d + c*x^2 
)^2 + 35*b*d*x*(8*d^2 + 14*c*d*x^2 + 3*c^2*x^4)) + 210*b*c^3*Sqrt[d]*x^7*A 
rcTanh[(Sqrt[c]*x - Sqrt[d + c*x^2])/Sqrt[d]]))/(d^2*x^6*Sqrt[d + c*x^2])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {1892, 1803, 533, 533, 25, 27, 455, 211, 211, 224, 219}

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.

\(\displaystyle \int \frac {(a+b x) \left (c+\frac {d}{x^2}\right )^{3/2}}{x^5} \, dx\)

\(\Big \downarrow \) 1892

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

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 533

\(\displaystyle \frac {\int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} \left (2 a c-\frac {7 b d}{x}\right )}{x}d\frac {1}{x}}{7 d}-\frac {a \left (c+\frac {d}{x^2}\right )^{5/2}}{7 d x^2}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {-\frac {\int -c d \left (c+\frac {d}{x^2}\right )^{3/2} \left (\frac {12 a}{x}+7 b\right )d\frac {1}{x}}{6 d}-\frac {7 b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 x}}{7 d}-\frac {a \left (c+\frac {d}{x^2}\right )^{5/2}}{7 d x^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int c d \left (c+\frac {d}{x^2}\right )^{3/2} \left (\frac {12 a}{x}+7 b\right )d\frac {1}{x}}{6 d}-\frac {7 b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 x}}{7 d}-\frac {a \left (c+\frac {d}{x^2}\right )^{5/2}}{7 d x^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} c \int \left (c+\frac {d}{x^2}\right )^{3/2} \left (\frac {12 a}{x}+7 b\right )d\frac {1}{x}-\frac {7 b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 x}}{7 d}-\frac {a \left (c+\frac {d}{x^2}\right )^{5/2}}{7 d x^2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {1}{6} c \left (7 b \int \left (c+\frac {d}{x^2}\right )^{3/2}d\frac {1}{x}+\frac {12 a \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}\right )-\frac {7 b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 x}}{7 d}-\frac {a \left (c+\frac {d}{x^2}\right )^{5/2}}{7 d x^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{6} c \left (7 b \left (\frac {3}{4} c \int \sqrt {c+\frac {d}{x^2}}d\frac {1}{x}+\frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{4 x}\right )+\frac {12 a \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}\right )-\frac {7 b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 x}}{7 d}-\frac {a \left (c+\frac {d}{x^2}\right )^{5/2}}{7 d x^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{6} c \left (7 b \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {c+\frac {d}{x^2}}}d\frac {1}{x}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )+\frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{4 x}\right )+\frac {12 a \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}\right )-\frac {7 b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 x}}{7 d}-\frac {a \left (c+\frac {d}{x^2}\right )^{5/2}}{7 d x^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{6} c \left (7 b \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{1-\frac {d}{x^2}}d\frac {1}{\sqrt {c+\frac {d}{x^2}} x}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )+\frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{4 x}\right )+\frac {12 a \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}\right )-\frac {7 b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 x}}{7 d}-\frac {a \left (c+\frac {d}{x^2}\right )^{5/2}}{7 d x^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{6} c \left (\frac {12 a \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d}+7 b \left (\frac {3}{4} c \left (\frac {c \text {arctanh}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{2 \sqrt {d}}+\frac {\sqrt {c+\frac {d}{x^2}}}{2 x}\right )+\frac {\left (c+\frac {d}{x^2}\right )^{3/2}}{4 x}\right )\right )-\frac {7 b \left (c+\frac {d}{x^2}\right )^{5/2}}{6 x}}{7 d}-\frac {a \left (c+\frac {d}{x^2}\right )^{5/2}}{7 d x^2}\)

Input:

Int[((c + d/x^2)^(3/2)*(a + b*x))/x^5,x]
 

Output:

-1/7*(a*(c + d/x^2)^(5/2))/(d*x^2) + ((-7*b*(c + d/x^2)^(5/2))/(6*x) + (c* 
((12*a*(c + d/x^2)^(5/2))/(5*d) + 7*b*((c + d/x^2)^(3/2)/(4*x) + (3*c*(Sqr 
t[c + d/x^2]/(2*x) + (c*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(2*Sqrt[d])) 
)/4)))/6)/(7*d)
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 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 455

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

rule 533

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

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 = 141, normalized size of antiderivative = 1.02

method	result	size
risch	\(\frac {\left (96 a \,c^{3} x^{6}-105 b \,c^{2} d \,x^{5}-48 a \,c^{2} d \,x^{4}-490 x^{3} b c \,d^{2}-384 a \,d^{2} x^{2} c -280 b \,d^{3} x -240 a \,d^{3}\right ) \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{1680 x^{6} d^{2}}+\frac {c^{3} b \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}}{16 d^{\frac {3}{2}} \sqrt {c \,x^{2}+d}}\)	\(141\)
default	\(\frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (-35 \left (c \,x^{2}+d \right )^{\frac {3}{2}} b \,c^{3} x^{7}+105 d^{\frac {3}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c \,x^{2}+d}}{x}\right ) b \,c^{3} x^{7}+35 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b \,c^{2} x^{5}-105 \sqrt {c \,x^{2}+d}\, b \,c^{3} d \,x^{7}+70 \left (c \,x^{2}+d \right )^{\frac {5}{2}} b c d \,x^{3}+96 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a c d \,x^{2}-280 b \left (c \,x^{2}+d \right )^{\frac {5}{2}} d^{2} x -240 \left (c \,x^{2}+d \right )^{\frac {5}{2}} a \,d^{2}\right )}{1680 x^{4} \left (c \,x^{2}+d \right )^{\frac {3}{2}} d^{3}}\)	\(185\)

Input:

int((c+d/x^2)^(3/2)*(b*x+a)/x^5,x,method=_RETURNVERBOSE)
 

Output:

1/1680*(96*a*c^3*x^6-105*b*c^2*d*x^5-48*a*c^2*d*x^4-490*b*c*d^2*x^3-384*a* 
c*d^2*x^2-280*b*d^3*x-240*a*d^3)/x^6/d^2*((c*x^2+d)/x^2)^(1/2)+1/16*c^3*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 = 259, normalized size of antiderivative = 1.88 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^5} \, dx=\left [\frac {105 \, b c^{3} \sqrt {d} x^{6} \log \left (-\frac {c x^{2} + 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \, {\left (96 \, a c^{3} x^{6} - 105 \, b c^{2} d x^{5} - 48 \, a c^{2} d x^{4} - 490 \, b c d^{2} x^{3} - 384 \, a c d^{2} x^{2} - 280 \, b d^{3} x - 240 \, a d^{3}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3360 \, d^{2} x^{6}}, -\frac {105 \, b c^{3} \sqrt {-d} x^{6} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{d}\right ) - {\left (96 \, a c^{3} x^{6} - 105 \, b c^{2} d x^{5} - 48 \, a c^{2} d x^{4} - 490 \, b c d^{2} x^{3} - 384 \, a c d^{2} x^{2} - 280 \, b d^{3} x - 240 \, a d^{3}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{1680 \, d^{2} x^{6}}\right ] \] Input:

integrate((c+d/x^2)^(3/2)*(b*x+a)/x^5,x, algorithm="fricas")
 

Output:

[1/3360*(105*b*c^3*sqrt(d)*x^6*log(-(c*x^2 + 2*sqrt(d)*x*sqrt((c*x^2 + d)/ 
x^2) + 2*d)/x^2) + 2*(96*a*c^3*x^6 - 105*b*c^2*d*x^5 - 48*a*c^2*d*x^4 - 49 
0*b*c*d^2*x^3 - 384*a*c*d^2*x^2 - 280*b*d^3*x - 240*a*d^3)*sqrt((c*x^2 + d 
)/x^2))/(d^2*x^6), -1/1680*(105*b*c^3*sqrt(-d)*x^6*arctan(sqrt(-d)*x*sqrt( 
(c*x^2 + d)/x^2)/d) - (96*a*c^3*x^6 - 105*b*c^2*d*x^5 - 48*a*c^2*d*x^4 - 4 
90*b*c*d^2*x^3 - 384*a*c*d^2*x^2 - 280*b*d^3*x - 240*a*d^3)*sqrt((c*x^2 + 
d)/x^2))/(d^2*x^6)]
 

Sympy [A] (verification not implemented)

Time = 1.75 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.20 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^5} \, dx=- a c \left (\begin {cases} \sqrt {c + \frac {d}{x^{2}}} \left (- \frac {2 c^{2}}{15 d^{2}} + \frac {c}{15 d x^{2}} + \frac {1}{5 x^{4}}\right ) & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{4 x^{4}} & \text {otherwise} \end {cases}\right ) - a d \left (\begin {cases} \sqrt {c + \frac {d}{x^{2}}} \cdot \left (\frac {8 c^{3}}{105 d^{3}} - \frac {4 c^{2}}{105 d^{2} x^{2}} + \frac {c}{35 d x^{4}} + \frac {1}{7 x^{6}}\right ) & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{6 x^{6}} & \text {otherwise} \end {cases}\right ) - b c \left (\begin {cases} - \frac {c^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {d} \sqrt {c + \frac {d}{x^{2}}} + \frac {2 d}{x} \right )}}{\sqrt {d}} & \text {for}\: c \neq 0 \\- \frac {\log {\left (x \right )}}{x \sqrt {\frac {d}{x^{2}}}} & \text {otherwise} \end {cases}\right )}{8 d} + \sqrt {c + \frac {d}{x^{2}}} \left (\frac {c}{8 d x} + \frac {1}{4 x^{3}}\right ) & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) - b d \left (\begin {cases} \frac {c^{3} \left (\begin {cases} \frac {\log {\left (2 \sqrt {d} \sqrt {c + \frac {d}{x^{2}}} + \frac {2 d}{x} \right )}}{\sqrt {d}} & \text {for}\: c \neq 0 \\- \frac {\log {\left (x \right )}}{x \sqrt {\frac {d}{x^{2}}}} & \text {otherwise} \end {cases}\right )}{16 d^{2}} + \sqrt {c + \frac {d}{x^{2}}} \left (- \frac {c^{2}}{16 d^{2} x} + \frac {c}{24 d x^{3}} + \frac {1}{6 x^{5}}\right ) & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{5 x^{5}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((c+d/x**2)**(3/2)*(b*x+a)/x**5,x)
 

Output:

-a*c*Piecewise((sqrt(c + d/x**2)*(-2*c**2/(15*d**2) + c/(15*d*x**2) + 1/(5 
*x**4)), Ne(d, 0)), (sqrt(c)/(4*x**4), True)) - a*d*Piecewise((sqrt(c + d/ 
x**2)*(8*c**3/(105*d**3) - 4*c**2/(105*d**2*x**2) + c/(35*d*x**4) + 1/(7*x 
**6)), Ne(d, 0)), (sqrt(c)/(6*x**6), True)) - b*c*Piecewise((-c**2*Piecewi 
se((log(2*sqrt(d)*sqrt(c + d/x**2) + 2*d/x)/sqrt(d), Ne(c, 0)), (-log(x)/( 
x*sqrt(d/x**2)), True))/(8*d) + sqrt(c + d/x**2)*(c/(8*d*x) + 1/(4*x**3)), 
 Ne(d, 0)), (sqrt(c)/(3*x**3), True)) - b*d*Piecewise((c**3*Piecewise((log 
(2*sqrt(d)*sqrt(c + d/x**2) + 2*d/x)/sqrt(d), Ne(c, 0)), (-log(x)/(x*sqrt( 
d/x**2)), True))/(16*d**2) + sqrt(c + d/x**2)*(-c**2/(16*d**2*x) + c/(24*d 
*x**3) + 1/(6*x**5)), Ne(d, 0)), (sqrt(c)/(5*x**5), True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.39 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^5} \, dx=-\frac {1}{35} \, {\left (\frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}}}{d^{2}} - \frac {7 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c}{d^{2}}\right )} a - \frac {1}{96} \, {\left (\frac {3 \, c^{3} \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, {\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} c^{3} x^{5} + 8 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c^{3} d x^{3} - 3 \, \sqrt {c + \frac {d}{x^{2}}} c^{3} d^{2} x\right )}}{{\left (c + \frac {d}{x^{2}}\right )}^{3} d x^{6} - 3 \, {\left (c + \frac {d}{x^{2}}\right )}^{2} d^{2} x^{4} + 3 \, {\left (c + \frac {d}{x^{2}}\right )} d^{3} x^{2} - d^{4}}\right )} b \] Input:

integrate((c+d/x^2)^(3/2)*(b*x+a)/x^5,x, algorithm="maxima")
 

Output:

-1/35*(5*(c + d/x^2)^(7/2)/d^2 - 7*(c + d/x^2)^(5/2)*c/d^2)*a - 1/96*(3*c^ 
3*log((sqrt(c + d/x^2)*x - sqrt(d))/(sqrt(c + d/x^2)*x + sqrt(d)))/d^(3/2) 
 + 2*(3*(c + d/x^2)^(5/2)*c^3*x^5 + 8*(c + d/x^2)^(3/2)*c^3*d*x^3 - 3*sqrt 
(c + d/x^2)*c^3*d^2*x)/((c + d/x^2)^3*d*x^6 - 3*(c + d/x^2)^2*d^2*x^4 + 3* 
(c + d/x^2)*d^3*x^2 - d^4))*b
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (110) = 220\).

Time = 171.45 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.93 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^5} \, dx=-\frac {b c^{3} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + d}}{\sqrt {-d}}\right ) \mathrm {sgn}\left (x\right )}{8 \, \sqrt {-d} d} + \frac {105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{13} b c^{3} \mathrm {sgn}\left (x\right ) + 1540 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{11} b c^{3} d \mathrm {sgn}\left (x\right ) + 3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{10} a c^{\frac {7}{2}} d \mathrm {sgn}\left (x\right ) + 1085 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{9} b c^{3} d^{2} \mathrm {sgn}\left (x\right ) + 3360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {7}{2}} d^{2} \mathrm {sgn}\left (x\right ) + 6720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {7}{2}} d^{3} \mathrm {sgn}\left (x\right ) - 1085 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{5} b c^{3} d^{4} \mathrm {sgn}\left (x\right ) + 1344 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {7}{2}} d^{4} \mathrm {sgn}\left (x\right ) - 1540 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{3} b c^{3} d^{5} \mathrm {sgn}\left (x\right ) + 672 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {7}{2}} d^{5} \mathrm {sgn}\left (x\right ) - 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )} b c^{3} d^{6} \mathrm {sgn}\left (x\right ) - 96 \, a c^{\frac {7}{2}} d^{6} \mathrm {sgn}\left (x\right )}{840 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{7} d} \] Input:

integrate((c+d/x^2)^(3/2)*(b*x+a)/x^5,x, algorithm="giac")
 

Output:

-1/8*b*c^3*arctan(-(sqrt(c)*x - sqrt(c*x^2 + d))/sqrt(-d))*sgn(x)/(sqrt(-d 
)*d) + 1/840*(105*(sqrt(c)*x - sqrt(c*x^2 + d))^13*b*c^3*sgn(x) + 1540*(sq 
rt(c)*x - sqrt(c*x^2 + d))^11*b*c^3*d*sgn(x) + 3360*(sqrt(c)*x - sqrt(c*x^ 
2 + d))^10*a*c^(7/2)*d*sgn(x) + 1085*(sqrt(c)*x - sqrt(c*x^2 + d))^9*b*c^3 
*d^2*sgn(x) + 3360*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(7/2)*d^2*sgn(x) + 
6720*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(7/2)*d^3*sgn(x) - 1085*(sqrt(c)* 
x - sqrt(c*x^2 + d))^5*b*c^3*d^4*sgn(x) + 1344*(sqrt(c)*x - sqrt(c*x^2 + d 
))^4*a*c^(7/2)*d^4*sgn(x) - 1540*(sqrt(c)*x - sqrt(c*x^2 + d))^3*b*c^3*d^5 
*sgn(x) + 672*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(7/2)*d^5*sgn(x) - 105*( 
sqrt(c)*x - sqrt(c*x^2 + d))*b*c^3*d^6*sgn(x) - 96*a*c^(7/2)*d^6*sgn(x))/( 
((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)^7*d)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^5} \, dx=\int \frac {{\left (c+\frac {d}{x^2}\right )}^{3/2}\,\left (a+b\,x\right )}{x^5} \,d x \] Input:

int(((c + d/x^2)^(3/2)*(a + b*x))/x^5,x)
 

Output:

int(((c + d/x^2)^(3/2)*(a + b*x))/x^5, x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.48 \[ \int \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (a+b x)}{x^5} \, dx=\frac {96 \sqrt {c \,x^{2}+d}\, a \,c^{3} x^{6}-48 \sqrt {c \,x^{2}+d}\, a \,c^{2} d \,x^{4}-384 \sqrt {c \,x^{2}+d}\, a c \,d^{2} x^{2}-240 \sqrt {c \,x^{2}+d}\, a \,d^{3}-105 \sqrt {c \,x^{2}+d}\, b \,c^{2} d \,x^{5}-490 \sqrt {c \,x^{2}+d}\, b c \,d^{2} x^{3}-280 \sqrt {c \,x^{2}+d}\, b \,d^{3} x -96 \sqrt {c}\, a \,c^{3} x^{7}-105 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x -\sqrt {d}}{\sqrt {d}}\right ) b \,c^{3} x^{7}+105 \sqrt {d}\, \mathrm {log}\left (\frac {\sqrt {c \,x^{2}+d}+\sqrt {c}\, x +\sqrt {d}}{\sqrt {d}}\right ) b \,c^{3} x^{7}}{1680 d^{2} x^{7}} \] Input:

int((c+d/x^2)^(3/2)*(b*x+a)/x^5,x)
 

Output:

(96*sqrt(c*x**2 + d)*a*c**3*x**6 - 48*sqrt(c*x**2 + d)*a*c**2*d*x**4 - 384 
*sqrt(c*x**2 + d)*a*c*d**2*x**2 - 240*sqrt(c*x**2 + d)*a*d**3 - 105*sqrt(c 
*x**2 + d)*b*c**2*d*x**5 - 490*sqrt(c*x**2 + d)*b*c*d**2*x**3 - 280*sqrt(c 
*x**2 + d)*b*d**3*x - 96*sqrt(c)*a*c**3*x**7 - 105*sqrt(d)*log((sqrt(c*x** 
2 + d) + sqrt(c)*x - sqrt(d))/sqrt(d))*b*c**3*x**7 + 105*sqrt(d)*log((sqrt 
(c*x**2 + d) + sqrt(c)*x + sqrt(d))/sqrt(d))*b*c**3*x**7)/(1680*d**2*x**7)