\(\int \frac {(c+\frac {d}{x})^{3/2} (a+b x)}{x^5} \, dx\) [10]

Optimal result

Integrand size = 20, antiderivative size = 104 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^5} \, dx=\frac {2 c^2 (a c-b d) \left (c+\frac {d}{x}\right )^{5/2}}{5 d^4}-\frac {2 c (3 a c-2 b d) \left (c+\frac {d}{x}\right )^{7/2}}{7 d^4}+\frac {2 (3 a c-b d) \left (c+\frac {d}{x}\right )^{9/2}}{9 d^4}-\frac {2 a \left (c+\frac {d}{x}\right )^{11/2}}{11 d^4} \] Output:

2/5*c^2*(a*c-b*d)*(c+d/x)^(5/2)/d^4-2/7*c*(3*a*c-2*b*d)*(c+d/x)^(7/2)/d^4+ 
2/9*(3*a*c-b*d)*(c+d/x)^(9/2)/d^4-2/11*a*(c+d/x)^(11/2)/d^4
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^5} \, dx=-\frac {2 \sqrt {c+\frac {d}{x}} (d+c x)^2 \left (11 b d x \left (35 d^2-20 c d x+8 c^2 x^2\right )+3 a \left (105 d^3-70 c d^2 x+40 c^2 d x^2-16 c^3 x^3\right )\right )}{3465 d^4 x^5} \] Input:

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

Output:

(-2*Sqrt[c + d/x]*(d + c*x)^2*(11*b*d*x*(35*d^2 - 20*c*d*x + 8*c^2*x^2) + 
3*a*(105*d^3 - 70*c*d^2*x + 40*c^2*d*x^2 - 16*c^3*x^3)))/(3465*d^4*x^5)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1016, 948, 86, 2009}

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.

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

\(\Big \downarrow \) 1016

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 86

\(\displaystyle -\int \left (\frac {a \left (c+\frac {d}{x}\right )^{9/2}}{d^3}+\frac {(b d-3 a c) \left (c+\frac {d}{x}\right )^{7/2}}{d^3}+\frac {c (3 a c-2 b d) \left (c+\frac {d}{x}\right )^{5/2}}{d^3}-\frac {c^2 (a c-b d) \left (c+\frac {d}{x}\right )^{3/2}}{d^3}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 c^2 \left (c+\frac {d}{x}\right )^{5/2} (a c-b d)}{5 d^4}+\frac {2 \left (c+\frac {d}{x}\right )^{9/2} (3 a c-b d)}{9 d^4}-\frac {2 c \left (c+\frac {d}{x}\right )^{7/2} (3 a c-2 b d)}{7 d^4}-\frac {2 a \left (c+\frac {d}{x}\right )^{11/2}}{11 d^4}\)

Input:

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

Output:

(2*c^2*(a*c - b*d)*(c + d/x)^(5/2))/(5*d^4) - (2*c*(3*a*c - 2*b*d)*(c + d/ 
x)^(7/2))/(7*d^4) + (2*(3*a*c - b*d)*(c + d/x)^(9/2))/(9*d^4) - (2*a*(c + 
d/x)^(11/2))/(11*d^4)
 

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 948

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. 
), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ 
p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ 
[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1016

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^( 
p_.), x_Symbol] :> Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ 
[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] ||  !I 
ntegerQ[p])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.81

method	result	size
orering	\(\frac {2 \left (48 a \,c^{3} x^{3}-88 b \,c^{2} d \,x^{3}-120 a \,c^{2} d \,x^{2}+220 b c \,d^{2} x^{2}+210 a c \,d^{2} x -385 b \,d^{3} x -315 a \,d^{3}\right ) \left (c x +d \right ) \left (c +\frac {d}{x}\right )^{\frac {3}{2}}}{3465 d^{4} x^{4}}\)	\(84\)
gosper	\(\frac {2 \left (c x +d \right ) \left (48 a \,c^{3} x^{3}-88 b \,c^{2} d \,x^{3}-120 a \,c^{2} d \,x^{2}+220 b c \,d^{2} x^{2}+210 a c \,d^{2} x -385 b \,d^{3} x -315 a \,d^{3}\right ) \left (\frac {c x +d}{x}\right )^{\frac {3}{2}}}{3465 x^{4} d^{4}}\)	\(86\)
default	\(\frac {2 \sqrt {\frac {c x +d}{x}}\, \left (c \,x^{2}+d x \right )^{\frac {3}{2}} \left (48 a \,c^{4} x^{4}-88 b \,c^{3} d \,x^{4}-72 a \,c^{3} d \,x^{3}+132 b \,c^{2} d^{2} x^{3}+90 a \,c^{2} d^{2} x^{2}-165 b c \,d^{3} x^{2}-105 a c \,d^{3} x -385 b \,d^{4} x -315 a \,d^{4}\right )}{3465 x^{6} d^{4} \sqrt {\left (c x +d \right ) x}}\)	\(125\)
risch	\(\frac {2 \sqrt {\frac {c x +d}{x}}\, \left (48 a \,c^{5} x^{5}-88 b \,c^{4} d \,x^{5}-24 a \,c^{4} d \,x^{4}+44 b \,c^{3} d^{2} x^{4}+18 a \,c^{3} d^{2} x^{3}-33 b \,c^{2} d^{3} x^{3}-15 a \,c^{2} d^{3} x^{2}-550 b c \,d^{4} x^{2}-420 a c \,d^{4} x -385 b \,d^{5} x -315 a \,d^{5}\right )}{3465 x^{5} d^{4}}\)	\(129\)
trager	\(\frac {2 \left (48 a \,c^{5} x^{5}-88 b \,c^{4} d \,x^{5}-24 a \,c^{4} d \,x^{4}+44 b \,c^{3} d^{2} x^{4}+18 a \,c^{3} d^{2} x^{3}-33 b \,c^{2} d^{3} x^{3}-15 a \,c^{2} d^{3} x^{2}-550 b c \,d^{4} x^{2}-420 a c \,d^{4} x -385 b \,d^{5} x -315 a \,d^{5}\right ) \sqrt {-\frac {-c x -d}{x}}}{3465 x^{5} d^{4}}\)	\(133\)

Input:

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

Output:

2/3465*(48*a*c^3*x^3-88*b*c^2*d*x^3-120*a*c^2*d*x^2+220*b*c*d^2*x^2+210*a* 
c*d^2*x-385*b*d^3*x-315*a*d^3)/d^4/x^4*(c*x+d)*(c+d/x)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.25 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^5} \, dx=-\frac {2 \, {\left (315 \, a d^{5} - 8 \, {\left (6 \, a c^{5} - 11 \, b c^{4} d\right )} x^{5} + 4 \, {\left (6 \, a c^{4} d - 11 \, b c^{3} d^{2}\right )} x^{4} - 3 \, {\left (6 \, a c^{3} d^{2} - 11 \, b c^{2} d^{3}\right )} x^{3} + 5 \, {\left (3 \, a c^{2} d^{3} + 110 \, b c d^{4}\right )} x^{2} + 35 \, {\left (12 \, a c d^{4} + 11 \, b d^{5}\right )} x\right )} \sqrt {\frac {c x + d}{x}}}{3465 \, d^{4} x^{5}} \] Input:

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

Output:

-2/3465*(315*a*d^5 - 8*(6*a*c^5 - 11*b*c^4*d)*x^5 + 4*(6*a*c^4*d - 11*b*c^ 
3*d^2)*x^4 - 3*(6*a*c^3*d^2 - 11*b*c^2*d^3)*x^3 + 5*(3*a*c^2*d^3 + 110*b*c 
*d^4)*x^2 + 35*(12*a*c*d^4 + 11*b*d^5)*x)*sqrt((c*x + d)/x)/(d^4*x^5)
 

Sympy [A] (verification not implemented)

Time = 2.08 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.81 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^5} \, dx=- a c \left (\begin {cases} \frac {2 \left (- \frac {c^{3} \left (c + \frac {d}{x}\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + \frac {d}{x}\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + \frac {d}{x}\right )^{\frac {7}{2}}}{7} + \frac {\left (c + \frac {d}{x}\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{4 x^{4}} & \text {otherwise} \end {cases}\right ) - a d \left (\begin {cases} \frac {2 \left (\frac {c^{4} \left (c + \frac {d}{x}\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + \frac {d}{x}\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + \frac {d}{x}\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + \frac {d}{x}\right )^{\frac {9}{2}}}{9} + \frac {\left (c + \frac {d}{x}\right )^{\frac {11}{2}}}{11}\right )}{d^{5}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{5 x^{5}} & \text {otherwise} \end {cases}\right ) - b c \left (\begin {cases} \frac {2 \left (\frac {c^{2} \left (c + \frac {d}{x}\right )^{\frac {3}{2}}}{3} - \frac {2 c \left (c + \frac {d}{x}\right )^{\frac {5}{2}}}{5} + \frac {\left (c + \frac {d}{x}\right )^{\frac {7}{2}}}{7}\right )}{d^{3}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{3 x^{3}} & \text {otherwise} \end {cases}\right ) - b d \left (\begin {cases} \frac {2 \left (- \frac {c^{3} \left (c + \frac {d}{x}\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + \frac {d}{x}\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + \frac {d}{x}\right )^{\frac {7}{2}}}{7} + \frac {\left (c + \frac {d}{x}\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{4 x^{4}} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

-a*c*Piecewise((2*(-c**3*(c + d/x)**(3/2)/3 + 3*c**2*(c + d/x)**(5/2)/5 - 
3*c*(c + d/x)**(7/2)/7 + (c + d/x)**(9/2)/9)/d**4, Ne(d, 0)), (sqrt(c)/(4* 
x**4), True)) - a*d*Piecewise((2*(c**4*(c + d/x)**(3/2)/3 - 4*c**3*(c + d/ 
x)**(5/2)/5 + 6*c**2*(c + d/x)**(7/2)/7 - 4*c*(c + d/x)**(9/2)/9 + (c + d/ 
x)**(11/2)/11)/d**5, Ne(d, 0)), (sqrt(c)/(5*x**5), True)) - b*c*Piecewise( 
(2*(c**2*(c + d/x)**(3/2)/3 - 2*c*(c + d/x)**(5/2)/5 + (c + d/x)**(7/2)/7) 
/d**3, Ne(d, 0)), (sqrt(c)/(3*x**3), True)) - b*d*Piecewise((2*(-c**3*(c + 
 d/x)**(3/2)/3 + 3*c**2*(c + d/x)**(5/2)/5 - 3*c*(c + d/x)**(7/2)/7 + (c + 
 d/x)**(9/2)/9)/d**4, Ne(d, 0)), (sqrt(c)/(4*x**4), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.13 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^5} \, dx=-\frac {2}{1155} \, {\left (\frac {105 \, {\left (c + \frac {d}{x}\right )}^{\frac {11}{2}}}{d^{4}} - \frac {385 \, {\left (c + \frac {d}{x}\right )}^{\frac {9}{2}} c}{d^{4}} + \frac {495 \, {\left (c + \frac {d}{x}\right )}^{\frac {7}{2}} c^{2}}{d^{4}} - \frac {231 \, {\left (c + \frac {d}{x}\right )}^{\frac {5}{2}} c^{3}}{d^{4}}\right )} a - \frac {2}{315} \, {\left (\frac {35 \, {\left (c + \frac {d}{x}\right )}^{\frac {9}{2}}}{d^{3}} - \frac {90 \, {\left (c + \frac {d}{x}\right )}^{\frac {7}{2}} c}{d^{3}} + \frac {63 \, {\left (c + \frac {d}{x}\right )}^{\frac {5}{2}} c^{2}}{d^{3}}\right )} b \] Input:

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

Output:

-2/1155*(105*(c + d/x)^(11/2)/d^4 - 385*(c + d/x)^(9/2)*c/d^4 + 495*(c + d 
/x)^(7/2)*c^2/d^4 - 231*(c + d/x)^(5/2)*c^3/d^4)*a - 2/315*(35*(c + d/x)^( 
9/2)/d^3 - 90*(c + d/x)^(7/2)*c/d^3 + 63*(c + d/x)^(5/2)*c^2/d^3)*b
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (88) = 176\).

Time = 0.64 (sec) , antiderivative size = 461, normalized size of antiderivative = 4.43 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^5} \, dx=\frac {2 \, {\left (4620 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{8} b c^{3} \mathrm {sgn}\left (x\right ) + 6930 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{7} a c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 17325 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{7} b c^{\frac {5}{2}} d \mathrm {sgn}\left (x\right ) + 30492 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{6} a c^{3} d \mathrm {sgn}\left (x\right ) + 28413 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{6} b c^{2} d^{2} \mathrm {sgn}\left (x\right ) + 58905 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{5} a c^{\frac {5}{2}} d^{2} \mathrm {sgn}\left (x\right ) + 25410 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{5} b c^{\frac {3}{2}} d^{3} \mathrm {sgn}\left (x\right ) + 63855 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{4} a c^{2} d^{3} \mathrm {sgn}\left (x\right ) + 12870 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{4} b c d^{4} \mathrm {sgn}\left (x\right ) + 41580 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{3} a c^{\frac {3}{2}} d^{4} \mathrm {sgn}\left (x\right ) + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{3} b \sqrt {c} d^{5} \mathrm {sgn}\left (x\right ) + 16170 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{2} a c d^{5} \mathrm {sgn}\left (x\right ) + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{2} b d^{6} \mathrm {sgn}\left (x\right ) + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )} a \sqrt {c} d^{6} \mathrm {sgn}\left (x\right ) + 315 \, a d^{7} \mathrm {sgn}\left (x\right )\right )}}{3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{11}} \] Input:

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

Output:

2/3465*(4620*(sqrt(c)*x - sqrt(c*x^2 + d*x))^8*b*c^3*sgn(x) + 6930*(sqrt(c 
)*x - sqrt(c*x^2 + d*x))^7*a*c^(7/2)*sgn(x) + 17325*(sqrt(c)*x - sqrt(c*x^ 
2 + d*x))^7*b*c^(5/2)*d*sgn(x) + 30492*(sqrt(c)*x - sqrt(c*x^2 + d*x))^6*a 
*c^3*d*sgn(x) + 28413*(sqrt(c)*x - sqrt(c*x^2 + d*x))^6*b*c^2*d^2*sgn(x) + 
 58905*(sqrt(c)*x - sqrt(c*x^2 + d*x))^5*a*c^(5/2)*d^2*sgn(x) + 25410*(sqr 
t(c)*x - sqrt(c*x^2 + d*x))^5*b*c^(3/2)*d^3*sgn(x) + 63855*(sqrt(c)*x - sq 
rt(c*x^2 + d*x))^4*a*c^2*d^3*sgn(x) + 12870*(sqrt(c)*x - sqrt(c*x^2 + d*x) 
)^4*b*c*d^4*sgn(x) + 41580*(sqrt(c)*x - sqrt(c*x^2 + d*x))^3*a*c^(3/2)*d^4 
*sgn(x) + 3465*(sqrt(c)*x - sqrt(c*x^2 + d*x))^3*b*sqrt(c)*d^5*sgn(x) + 16 
170*(sqrt(c)*x - sqrt(c*x^2 + d*x))^2*a*c*d^5*sgn(x) + 385*(sqrt(c)*x - sq 
rt(c*x^2 + d*x))^2*b*d^6*sgn(x) + 3465*(sqrt(c)*x - sqrt(c*x^2 + d*x))*a*s 
qrt(c)*d^6*sgn(x) + 315*a*d^7*sgn(x))/(sqrt(c)*x - sqrt(c*x^2 + d*x))^11
 

Mupad [B] (verification not implemented)

Time = 8.23 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.98 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^5} \, dx=\frac {32\,a\,c^5\,\sqrt {c+\frac {d}{x}}}{1155\,d^4}-\frac {16\,b\,c^4\,\sqrt {c+\frac {d}{x}}}{315\,d^3}-\frac {8\,a\,c\,\sqrt {c+\frac {d}{x}}}{33\,x^4}-\frac {20\,b\,c\,\sqrt {c+\frac {d}{x}}}{63\,x^3}-\frac {2\,a\,d\,\sqrt {c+\frac {d}{x}}}{11\,x^5}-\frac {2\,b\,d\,\sqrt {c+\frac {d}{x}}}{9\,x^4}-\frac {2\,a\,c^2\,\sqrt {c+\frac {d}{x}}}{231\,d\,x^3}+\frac {4\,a\,c^3\,\sqrt {c+\frac {d}{x}}}{385\,d^2\,x^2}-\frac {16\,a\,c^4\,\sqrt {c+\frac {d}{x}}}{1155\,d^3\,x}-\frac {2\,b\,c^2\,\sqrt {c+\frac {d}{x}}}{105\,d\,x^2}+\frac {8\,b\,c^3\,\sqrt {c+\frac {d}{x}}}{315\,d^2\,x} \] Input:

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

Output:

(32*a*c^5*(c + d/x)^(1/2))/(1155*d^4) - (16*b*c^4*(c + d/x)^(1/2))/(315*d^ 
3) - (8*a*c*(c + d/x)^(1/2))/(33*x^4) - (20*b*c*(c + d/x)^(1/2))/(63*x^3) 
- (2*a*d*(c + d/x)^(1/2))/(11*x^5) - (2*b*d*(c + d/x)^(1/2))/(9*x^4) - (2* 
a*c^2*(c + d/x)^(1/2))/(231*d*x^3) + (4*a*c^3*(c + d/x)^(1/2))/(385*d^2*x^ 
2) - (16*a*c^4*(c + d/x)^(1/2))/(1155*d^3*x) - (2*b*c^2*(c + d/x)^(1/2))/( 
105*d*x^2) + (8*b*c^3*(c + d/x)^(1/2))/(315*d^2*x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.19 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^5} \, dx=\frac {\frac {32 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{5} x^{5}}{1155}-\frac {16 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{4} d \,x^{4}}{1155}+\frac {4 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{3} d^{2} x^{3}}{385}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{2} d^{3} x^{2}}{231}-\frac {8 \sqrt {x}\, \sqrt {c x +d}\, a c \,d^{4} x}{33}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, a \,d^{5}}{11}-\frac {16 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{4} d \,x^{5}}{315}+\frac {8 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{3} d^{2} x^{4}}{315}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{2} d^{3} x^{3}}{105}-\frac {20 \sqrt {x}\, \sqrt {c x +d}\, b c \,d^{4} x^{2}}{63}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, b \,d^{5} x}{9}-\frac {32 \sqrt {c}\, a \,c^{5} x^{6}}{1155}+\frac {16 \sqrt {c}\, b \,c^{4} d \,x^{6}}{315}}{d^{4} x^{6}} \] Input:

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

Output:

(2*(48*sqrt(x)*sqrt(c*x + d)*a*c**5*x**5 - 24*sqrt(x)*sqrt(c*x + d)*a*c**4 
*d*x**4 + 18*sqrt(x)*sqrt(c*x + d)*a*c**3*d**2*x**3 - 15*sqrt(x)*sqrt(c*x 
+ d)*a*c**2*d**3*x**2 - 420*sqrt(x)*sqrt(c*x + d)*a*c*d**4*x - 315*sqrt(x) 
*sqrt(c*x + d)*a*d**5 - 88*sqrt(x)*sqrt(c*x + d)*b*c**4*d*x**5 + 44*sqrt(x 
)*sqrt(c*x + d)*b*c**3*d**2*x**4 - 33*sqrt(x)*sqrt(c*x + d)*b*c**2*d**3*x* 
*3 - 550*sqrt(x)*sqrt(c*x + d)*b*c*d**4*x**2 - 385*sqrt(x)*sqrt(c*x + d)*b 
*d**5*x - 48*sqrt(c)*a*c**5*x**6 + 88*sqrt(c)*b*c**4*d*x**6))/(3465*d**4*x 
**6)