\(\int \frac {(c+\frac {d}{x})^{3/2} (a+b x)}{x^6} \, dx\) [11]

Optimal result

Integrand size = 20, antiderivative size = 134 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^6} \, dx=-\frac {2 c^3 (a c-b d) \left (c+\frac {d}{x}\right )^{5/2}}{5 d^5}+\frac {2 c^2 (4 a c-3 b d) \left (c+\frac {d}{x}\right )^{7/2}}{7 d^5}-\frac {2 c (2 a c-b d) \left (c+\frac {d}{x}\right )^{9/2}}{3 d^5}+\frac {2 (4 a c-b d) \left (c+\frac {d}{x}\right )^{11/2}}{11 d^5}-\frac {2 a \left (c+\frac {d}{x}\right )^{13/2}}{13 d^5} \] Output:

-2/5*c^3*(a*c-b*d)*(c+d/x)^(5/2)/d^5+2/7*c^2*(4*a*c-3*b*d)*(c+d/x)^(7/2)/d 
^5-2/3*c*(2*a*c-b*d)*(c+d/x)^(9/2)/d^5+2/11*(4*a*c-b*d)*(c+d/x)^(11/2)/d^5 
-2/13*a*(c+d/x)^(13/2)/d^5
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.80 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^6} \, dx=\frac {2 \sqrt {c+\frac {d}{x}} (d+c x)^2 \left (13 b d x \left (-105 d^3+70 c d^2 x-40 c^2 d x^2+16 c^3 x^3\right )+a \left (-1155 d^4+840 c d^3 x-560 c^2 d^2 x^2+320 c^3 d x^3-128 c^4 x^4\right )\right )}{15015 d^5 x^6} \] Input:

Integrate[((c + d/x)^(3/2)*(a + b*x))/x^6,x]
 

Output:

(2*Sqrt[c + d/x]*(d + c*x)^2*(13*b*d*x*(-105*d^3 + 70*c*d^2*x - 40*c^2*d*x 
^2 + 16*c^3*x^3) + a*(-1155*d^4 + 840*c*d^3*x - 560*c^2*d^2*x^2 + 320*c^3* 
d*x^3 - 128*c^4*x^4)))/(15015*d^5*x^6)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 134, 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.

\(\displaystyle \int \frac {(a+b x) \left (c+\frac {d}{x}\right )^{3/2}}{x^6} \, dx\)

\(\Big \downarrow \) 1016

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

\(\Big \downarrow \) 948

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

\(\Big \downarrow \) 86

\(\displaystyle -\int \left (\frac {a \left (c+\frac {d}{x}\right )^{11/2}}{d^4}+\frac {(b d-4 a c) \left (c+\frac {d}{x}\right )^{9/2}}{d^4}+\frac {3 c (2 a c-b d) \left (c+\frac {d}{x}\right )^{7/2}}{d^4}-\frac {c^2 (4 a c-3 b d) \left (c+\frac {d}{x}\right )^{5/2}}{d^4}+\frac {c^3 (a c-b d) \left (c+\frac {d}{x}\right )^{3/2}}{d^4}\right )d\frac {1}{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 c^3 \left (c+\frac {d}{x}\right )^{5/2} (a c-b d)}{5 d^5}+\frac {2 c^2 \left (c+\frac {d}{x}\right )^{7/2} (4 a c-3 b d)}{7 d^5}+\frac {2 \left (c+\frac {d}{x}\right )^{11/2} (4 a c-b d)}{11 d^5}-\frac {2 c \left (c+\frac {d}{x}\right )^{9/2} (2 a c-b d)}{3 d^5}-\frac {2 a \left (c+\frac {d}{x}\right )^{13/2}}{13 d^5}\)

Input:

Int[((c + d/x)^(3/2)*(a + b*x))/x^6,x]
 

Output:

(-2*c^3*(a*c - b*d)*(c + d/x)^(5/2))/(5*d^5) + (2*c^2*(4*a*c - 3*b*d)*(c + 
 d/x)^(7/2))/(7*d^5) - (2*c*(2*a*c - b*d)*(c + d/x)^(9/2))/(3*d^5) + (2*(4 
*a*c - b*d)*(c + d/x)^(11/2))/(11*d^5) - (2*a*(c + d/x)^(13/2))/(13*d^5)
 

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 = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81

method	result	size
orering	\(-\frac {2 \left (128 a \,c^{4} x^{4}-208 b \,c^{3} d \,x^{4}-320 a \,c^{3} d \,x^{3}+520 b \,c^{2} d^{2} x^{3}+560 a \,c^{2} d^{2} x^{2}-910 b c \,d^{3} x^{2}-840 a c \,d^{3} x +1365 b \,d^{4} x +1155 a \,d^{4}\right ) \left (c x +d \right ) \left (c +\frac {d}{x}\right )^{\frac {3}{2}}}{15015 d^{5} x^{5}}\)	\(108\)
gosper	\(-\frac {2 \left (c x +d \right ) \left (128 a \,c^{4} x^{4}-208 b \,c^{3} d \,x^{4}-320 a \,c^{3} d \,x^{3}+520 b \,c^{2} d^{2} x^{3}+560 a \,c^{2} d^{2} x^{2}-910 b c \,d^{3} x^{2}-840 a c \,d^{3} x +1365 b \,d^{4} x +1155 a \,d^{4}\right ) \left (\frac {c x +d}{x}\right )^{\frac {3}{2}}}{15015 x^{5} d^{5}}\)	\(110\)
default	\(-\frac {2 \sqrt {\frac {c x +d}{x}}\, \left (c \,x^{2}+d x \right )^{\frac {3}{2}} \left (128 a \,c^{5} x^{5}-208 b \,c^{4} d \,x^{5}-192 a \,c^{4} d \,x^{4}+312 b \,c^{3} d^{2} x^{4}+240 a \,c^{3} d^{2} x^{3}-390 b \,c^{2} d^{3} x^{3}-280 a \,c^{2} d^{3} x^{2}+455 b c \,d^{4} x^{2}+315 a c \,d^{4} x +1365 b \,d^{5} x +1155 a \,d^{5}\right )}{15015 x^{7} d^{5} \sqrt {\left (c x +d \right ) x}}\)	\(149\)
risch	\(-\frac {2 \sqrt {\frac {c x +d}{x}}\, \left (128 a \,c^{6} x^{6}-208 b \,c^{5} d \,x^{6}-64 a \,c^{5} d \,x^{5}+104 b \,c^{4} d^{2} x^{5}+48 a \,c^{4} d^{2} x^{4}-78 b \,c^{3} d^{3} x^{4}-40 a \,c^{3} d^{3} x^{3}+65 b \,c^{2} d^{4} x^{3}+35 a \,c^{2} d^{4} x^{2}+1820 b c \,d^{5} x^{2}+1470 a c \,d^{5} x +1365 b \,d^{6} x +1155 a \,d^{6}\right )}{15015 x^{6} d^{5}}\)	\(153\)
trager	\(-\frac {2 \left (128 a \,c^{6} x^{6}-208 b \,c^{5} d \,x^{6}-64 a \,c^{5} d \,x^{5}+104 b \,c^{4} d^{2} x^{5}+48 a \,c^{4} d^{2} x^{4}-78 b \,c^{3} d^{3} x^{4}-40 a \,c^{3} d^{3} x^{3}+65 b \,c^{2} d^{4} x^{3}+35 a \,c^{2} d^{4} x^{2}+1820 b c \,d^{5} x^{2}+1470 a c \,d^{5} x +1365 b \,d^{6} x +1155 a \,d^{6}\right ) \sqrt {-\frac {-c x -d}{x}}}{15015 x^{6} d^{5}}\)	\(157\)

Input:

int((c+d/x)^(3/2)*(b*x+a)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-2/15015*(128*a*c^4*x^4-208*b*c^3*d*x^4-320*a*c^3*d*x^3+520*b*c^2*d^2*x^3+ 
560*a*c^2*d^2*x^2-910*b*c*d^3*x^2-840*a*c*d^3*x+1365*b*d^4*x+1155*a*d^4)/d 
^5/x^5*(c*x+d)*(c+d/x)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.14 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^6} \, dx=-\frac {2 \, {\left (1155 \, a d^{6} + 16 \, {\left (8 \, a c^{6} - 13 \, b c^{5} d\right )} x^{6} - 8 \, {\left (8 \, a c^{5} d - 13 \, b c^{4} d^{2}\right )} x^{5} + 6 \, {\left (8 \, a c^{4} d^{2} - 13 \, b c^{3} d^{3}\right )} x^{4} - 5 \, {\left (8 \, a c^{3} d^{3} - 13 \, b c^{2} d^{4}\right )} x^{3} + 35 \, {\left (a c^{2} d^{4} + 52 \, b c d^{5}\right )} x^{2} + 105 \, {\left (14 \, a c d^{5} + 13 \, b d^{6}\right )} x\right )} \sqrt {\frac {c x + d}{x}}}{15015 \, d^{5} x^{6}} \] Input:

integrate((c+d/x)^(3/2)*(b*x+a)/x^6,x, algorithm="fricas")
 

Output:

-2/15015*(1155*a*d^6 + 16*(8*a*c^6 - 13*b*c^5*d)*x^6 - 8*(8*a*c^5*d - 13*b 
*c^4*d^2)*x^5 + 6*(8*a*c^4*d^2 - 13*b*c^3*d^3)*x^4 - 5*(8*a*c^3*d^3 - 13*b 
*c^2*d^4)*x^3 + 35*(a*c^2*d^4 + 52*b*c*d^5)*x^2 + 105*(14*a*c*d^5 + 13*b*d 
^6)*x)*sqrt((c*x + d)/x)/(d^5*x^6)
 

Sympy [A] (verification not implemented)

Time = 2.24 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.63 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^6} \, dx=- a c \left (\begin {cases} \frac {2 \left (\frac {c^{4} \left (c + \frac {d}{x}\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + \frac {d}{x}\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + \frac {d}{x}\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + \frac {d}{x}\right )^{\frac {9}{2}}}{9} + \frac {\left (c + \frac {d}{x}\right )^{\frac {11}{2}}}{11}\right )}{d^{5}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{5 x^{5}} & \text {otherwise} \end {cases}\right ) - a d \left (\begin {cases} \frac {2 \left (- \frac {c^{5} \left (c + \frac {d}{x}\right )^{\frac {3}{2}}}{3} + c^{4} \left (c + \frac {d}{x}\right )^{\frac {5}{2}} - \frac {10 c^{3} \left (c + \frac {d}{x}\right )^{\frac {7}{2}}}{7} + \frac {10 c^{2} \left (c + \frac {d}{x}\right )^{\frac {9}{2}}}{9} - \frac {5 c \left (c + \frac {d}{x}\right )^{\frac {11}{2}}}{11} + \frac {\left (c + \frac {d}{x}\right )^{\frac {13}{2}}}{13}\right )}{d^{6}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{6 x^{6}} & \text {otherwise} \end {cases}\right ) - b c \left (\begin {cases} \frac {2 \left (- \frac {c^{3} \left (c + \frac {d}{x}\right )^{\frac {3}{2}}}{3} + \frac {3 c^{2} \left (c + \frac {d}{x}\right )^{\frac {5}{2}}}{5} - \frac {3 c \left (c + \frac {d}{x}\right )^{\frac {7}{2}}}{7} + \frac {\left (c + \frac {d}{x}\right )^{\frac {9}{2}}}{9}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{4 x^{4}} & \text {otherwise} \end {cases}\right ) - b d \left (\begin {cases} \frac {2 \left (\frac {c^{4} \left (c + \frac {d}{x}\right )^{\frac {3}{2}}}{3} - \frac {4 c^{3} \left (c + \frac {d}{x}\right )^{\frac {5}{2}}}{5} + \frac {6 c^{2} \left (c + \frac {d}{x}\right )^{\frac {7}{2}}}{7} - \frac {4 c \left (c + \frac {d}{x}\right )^{\frac {9}{2}}}{9} + \frac {\left (c + \frac {d}{x}\right )^{\frac {11}{2}}}{11}\right )}{d^{5}} & \text {for}\: d \neq 0 \\\frac {\sqrt {c}}{5 x^{5}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((c+d/x)**(3/2)*(b*x+a)/x**6,x)
 

Output:

-a*c*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)) - a*d*Piecewise((2*(-c**5*(c + 
d/x)**(3/2)/3 + c**4*(c + d/x)**(5/2) - 10*c**3*(c + d/x)**(7/2)/7 + 10*c* 
*2*(c + d/x)**(9/2)/9 - 5*c*(c + d/x)**(11/2)/11 + (c + d/x)**(13/2)/13)/d 
**6, Ne(d, 0)), (sqrt(c)/(6*x**6), True)) - b*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)) - b*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))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.13 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^6} \, dx=-\frac {2}{15015} \, {\left (\frac {1155 \, {\left (c + \frac {d}{x}\right )}^{\frac {13}{2}}}{d^{5}} - \frac {5460 \, {\left (c + \frac {d}{x}\right )}^{\frac {11}{2}} c}{d^{5}} + \frac {10010 \, {\left (c + \frac {d}{x}\right )}^{\frac {9}{2}} c^{2}}{d^{5}} - \frac {8580 \, {\left (c + \frac {d}{x}\right )}^{\frac {7}{2}} c^{3}}{d^{5}} + \frac {3003 \, {\left (c + \frac {d}{x}\right )}^{\frac {5}{2}} c^{4}}{d^{5}}\right )} a - \frac {2}{1155} \, {\left (\frac {105 \, {\left (c + \frac {d}{x}\right )}^{\frac {11}{2}}}{d^{4}} - \frac {385 \, {\left (c + \frac {d}{x}\right )}^{\frac {9}{2}} c}{d^{4}} + \frac {495 \, {\left (c + \frac {d}{x}\right )}^{\frac {7}{2}} c^{2}}{d^{4}} - \frac {231 \, {\left (c + \frac {d}{x}\right )}^{\frac {5}{2}} c^{3}}{d^{4}}\right )} b \] Input:

integrate((c+d/x)^(3/2)*(b*x+a)/x^6,x, algorithm="maxima")
 

Output:

-2/15015*(1155*(c + d/x)^(13/2)/d^5 - 5460*(c + d/x)^(11/2)*c/d^5 + 10010* 
(c + d/x)^(9/2)*c^2/d^5 - 8580*(c + d/x)^(7/2)*c^3/d^5 + 3003*(c + d/x)^(5 
/2)*c^4/d^5)*a - 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)*b
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (114) = 228\).

Time = 0.85 (sec) , antiderivative size = 525, normalized size of antiderivative = 3.92 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^6} \, dx=\frac {2 \, {\left (30030 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{9} b c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 48048 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{8} a c^{4} \mathrm {sgn}\left (x\right ) + 132132 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{8} b c^{3} d \mathrm {sgn}\left (x\right ) + 240240 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{7} a c^{\frac {7}{2}} d \mathrm {sgn}\left (x\right ) + 255255 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{7} b c^{\frac {5}{2}} d^{2} \mathrm {sgn}\left (x\right ) + 531960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{6} a c^{3} d^{2} \mathrm {sgn}\left (x\right ) + 276705 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{6} b c^{2} d^{3} \mathrm {sgn}\left (x\right ) + 675675 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{5} a c^{\frac {5}{2}} d^{3} \mathrm {sgn}\left (x\right ) + 180180 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{5} b c^{\frac {3}{2}} d^{4} \mathrm {sgn}\left (x\right ) + 535535 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{4} a c^{2} d^{4} \mathrm {sgn}\left (x\right ) + 70070 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{4} b c d^{5} \mathrm {sgn}\left (x\right ) + 270270 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{3} a c^{\frac {3}{2}} d^{5} \mathrm {sgn}\left (x\right ) + 15015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{3} b \sqrt {c} d^{6} \mathrm {sgn}\left (x\right ) + 84630 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{2} a c d^{6} \mathrm {sgn}\left (x\right ) + 1365 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{2} b d^{7} \mathrm {sgn}\left (x\right ) + 15015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )} a \sqrt {c} d^{7} \mathrm {sgn}\left (x\right ) + 1155 \, a d^{8} \mathrm {sgn}\left (x\right )\right )}}{15015 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d x}\right )}^{13}} \] Input:

integrate((c+d/x)^(3/2)*(b*x+a)/x^6,x, algorithm="giac")
 

Output:

2/15015*(30030*(sqrt(c)*x - sqrt(c*x^2 + d*x))^9*b*c^(7/2)*sgn(x) + 48048* 
(sqrt(c)*x - sqrt(c*x^2 + d*x))^8*a*c^4*sgn(x) + 132132*(sqrt(c)*x - sqrt( 
c*x^2 + d*x))^8*b*c^3*d*sgn(x) + 240240*(sqrt(c)*x - sqrt(c*x^2 + d*x))^7* 
a*c^(7/2)*d*sgn(x) + 255255*(sqrt(c)*x - sqrt(c*x^2 + d*x))^7*b*c^(5/2)*d^ 
2*sgn(x) + 531960*(sqrt(c)*x - sqrt(c*x^2 + d*x))^6*a*c^3*d^2*sgn(x) + 276 
705*(sqrt(c)*x - sqrt(c*x^2 + d*x))^6*b*c^2*d^3*sgn(x) + 675675*(sqrt(c)*x 
 - sqrt(c*x^2 + d*x))^5*a*c^(5/2)*d^3*sgn(x) + 180180*(sqrt(c)*x - sqrt(c* 
x^2 + d*x))^5*b*c^(3/2)*d^4*sgn(x) + 535535*(sqrt(c)*x - sqrt(c*x^2 + d*x) 
)^4*a*c^2*d^4*sgn(x) + 70070*(sqrt(c)*x - sqrt(c*x^2 + d*x))^4*b*c*d^5*sgn 
(x) + 270270*(sqrt(c)*x - sqrt(c*x^2 + d*x))^3*a*c^(3/2)*d^5*sgn(x) + 1501 
5*(sqrt(c)*x - sqrt(c*x^2 + d*x))^3*b*sqrt(c)*d^6*sgn(x) + 84630*(sqrt(c)* 
x - sqrt(c*x^2 + d*x))^2*a*c*d^6*sgn(x) + 1365*(sqrt(c)*x - sqrt(c*x^2 + d 
*x))^2*b*d^7*sgn(x) + 15015*(sqrt(c)*x - sqrt(c*x^2 + d*x))*a*sqrt(c)*d^7* 
sgn(x) + 1155*a*d^8*sgn(x))/(sqrt(c)*x - sqrt(c*x^2 + d*x))^13
 

Mupad [B] (verification not implemented)

Time = 8.84 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.85 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^6} \, dx=\frac {32\,b\,c^5\,\sqrt {c+\frac {d}{x}}}{1155\,d^4}-\frac {256\,a\,c^6\,\sqrt {c+\frac {d}{x}}}{15015\,d^5}-\frac {28\,a\,c\,\sqrt {c+\frac {d}{x}}}{143\,x^5}-\frac {8\,b\,c\,\sqrt {c+\frac {d}{x}}}{33\,x^4}-\frac {2\,a\,d\,\sqrt {c+\frac {d}{x}}}{13\,x^6}-\frac {2\,b\,d\,\sqrt {c+\frac {d}{x}}}{11\,x^5}-\frac {2\,a\,c^2\,\sqrt {c+\frac {d}{x}}}{429\,d\,x^4}+\frac {16\,a\,c^3\,\sqrt {c+\frac {d}{x}}}{3003\,d^2\,x^3}-\frac {32\,a\,c^4\,\sqrt {c+\frac {d}{x}}}{5005\,d^3\,x^2}+\frac {128\,a\,c^5\,\sqrt {c+\frac {d}{x}}}{15015\,d^4\,x}-\frac {2\,b\,c^2\,\sqrt {c+\frac {d}{x}}}{231\,d\,x^3}+\frac {4\,b\,c^3\,\sqrt {c+\frac {d}{x}}}{385\,d^2\,x^2}-\frac {16\,b\,c^4\,\sqrt {c+\frac {d}{x}}}{1155\,d^3\,x} \] Input:

int(((c + d/x)^(3/2)*(a + b*x))/x^6,x)
 

Output:

(32*b*c^5*(c + d/x)^(1/2))/(1155*d^4) - (256*a*c^6*(c + d/x)^(1/2))/(15015 
*d^5) - (28*a*c*(c + d/x)^(1/2))/(143*x^5) - (8*b*c*(c + d/x)^(1/2))/(33*x 
^4) - (2*a*d*(c + d/x)^(1/2))/(13*x^6) - (2*b*d*(c + d/x)^(1/2))/(11*x^5) 
- (2*a*c^2*(c + d/x)^(1/2))/(429*d*x^4) + (16*a*c^3*(c + d/x)^(1/2))/(3003 
*d^2*x^3) - (32*a*c^4*(c + d/x)^(1/2))/(5005*d^3*x^2) + (128*a*c^5*(c + d/ 
x)^(1/2))/(15015*d^4*x) - (2*b*c^2*(c + d/x)^(1/2))/(231*d*x^3) + (4*b*c^3 
*(c + d/x)^(1/2))/(385*d^2*x^2) - (16*b*c^4*(c + d/x)^(1/2))/(1155*d^3*x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.00 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^6} \, dx=\frac {-\frac {256 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{6} x^{6}}{15015}+\frac {128 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{5} d \,x^{5}}{15015}-\frac {32 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{4} d^{2} x^{4}}{5005}+\frac {16 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{3} d^{3} x^{3}}{3003}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{2} d^{4} x^{2}}{429}-\frac {28 \sqrt {x}\, \sqrt {c x +d}\, a c \,d^{5} x}{143}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, a \,d^{6}}{13}+\frac {32 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{5} d \,x^{6}}{1155}-\frac {16 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{4} d^{2} x^{5}}{1155}+\frac {4 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{3} d^{3} x^{4}}{385}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{2} d^{4} x^{3}}{231}-\frac {8 \sqrt {x}\, \sqrt {c x +d}\, b c \,d^{5} x^{2}}{33}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, b \,d^{6} x}{11}+\frac {256 \sqrt {c}\, a \,c^{6} x^{7}}{15015}-\frac {32 \sqrt {c}\, b \,c^{5} d \,x^{7}}{1155}}{d^{5} x^{7}} \] Input:

int((c+d/x)^(3/2)*(b*x+a)/x^6,x)
 

Output:

(2*( - 128*sqrt(x)*sqrt(c*x + d)*a*c**6*x**6 + 64*sqrt(x)*sqrt(c*x + d)*a* 
c**5*d*x**5 - 48*sqrt(x)*sqrt(c*x + d)*a*c**4*d**2*x**4 + 40*sqrt(x)*sqrt( 
c*x + d)*a*c**3*d**3*x**3 - 35*sqrt(x)*sqrt(c*x + d)*a*c**2*d**4*x**2 - 14 
70*sqrt(x)*sqrt(c*x + d)*a*c*d**5*x - 1155*sqrt(x)*sqrt(c*x + d)*a*d**6 + 
208*sqrt(x)*sqrt(c*x + d)*b*c**5*d*x**6 - 104*sqrt(x)*sqrt(c*x + d)*b*c**4 
*d**2*x**5 + 78*sqrt(x)*sqrt(c*x + d)*b*c**3*d**3*x**4 - 65*sqrt(x)*sqrt(c 
*x + d)*b*c**2*d**4*x**3 - 1820*sqrt(x)*sqrt(c*x + d)*b*c*d**5*x**2 - 1365 
*sqrt(x)*sqrt(c*x + d)*b*d**6*x + 128*sqrt(c)*a*c**6*x**7 - 208*sqrt(c)*b* 
c**5*d*x**7))/(15015*d**5*x**7)