Integrand size = 20, antiderivative size = 164 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^7} \, dx=\frac {2 c^4 (a c-b d) \left (c+\frac {d}{x}\right )^{5/2}}{5 d^6}-\frac {2 c^3 (5 a c-4 b d) \left (c+\frac {d}{x}\right )^{7/2}}{7 d^6}+\frac {4 c^2 (5 a c-3 b d) \left (c+\frac {d}{x}\right )^{9/2}}{9 d^6}-\frac {4 c (5 a c-2 b d) \left (c+\frac {d}{x}\right )^{11/2}}{11 d^6}+\frac {2 (5 a c-b d) \left (c+\frac {d}{x}\right )^{13/2}}{13 d^6}-\frac {2 a \left (c+\frac {d}{x}\right )^{15/2}}{15 d^6} \] Output:
2/5*c^4*(a*c-b*d)*(c+d/x)^(5/2)/d^6-2/7*c^3*(5*a*c-4*b*d)*(c+d/x)^(7/2)/d^ 6+4/9*c^2*(5*a*c-3*b*d)*(c+d/x)^(9/2)/d^6-4/11*c*(5*a*c-2*b*d)*(c+d/x)^(11 /2)/d^6+2/13*(5*a*c-b*d)*(c+d/x)^(13/2)/d^6-2/15*a*(c+d/x)^(15/2)/d^6
Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.79 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^7} \, dx=-\frac {2 \sqrt {c+\frac {d}{x}} (d+c x)^2 \left (3 b d x \left (1155 d^4-840 c d^3 x+560 c^2 d^2 x^2-320 c^3 d x^3+128 c^4 x^4\right )+a \left (3003 d^5-2310 c d^4 x+1680 c^2 d^3 x^2-1120 c^3 d^2 x^3+640 c^4 d x^4-256 c^5 x^5\right )\right )}{45045 d^6 x^7} \] Input:
Integrate[((c + d/x)^(3/2)*(a + b*x))/x^7,x]
Output:
(-2*Sqrt[c + d/x]*(d + c*x)^2*(3*b*d*x*(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) + a*(3003*d^5 - 2310*c*d^4*x + 1680* c^2*d^3*x^2 - 1120*c^3*d^2*x^3 + 640*c^4*d*x^4 - 256*c^5*x^5)))/(45045*d^6 *x^7)
Time = 0.55 (sec) , antiderivative size = 164, 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.
| \(\displaystyle \int \frac {(a+b x) \left (c+\frac {d}{x}\right )^{3/2}}{x^7} \, dx\) |
| \(\Big \downarrow \) 1016 |
| \(\displaystyle \int \frac {\left (\frac {a}{x}+b\right ) \left (c+\frac {d}{x}\right )^{3/2}}{x^6}dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle -\int \frac {\left (\frac {a}{x}+b\right ) \left (c+\frac {d}{x}\right )^{3/2}}{x^4}d\frac {1}{x}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle -\int \left (\frac {a \left (c+\frac {d}{x}\right )^{13/2}}{d^5}+\frac {(b d-5 a c) \left (c+\frac {d}{x}\right )^{11/2}}{d^5}+\frac {2 c (5 a c-2 b d) \left (c+\frac {d}{x}\right )^{9/2}}{d^5}-\frac {2 c^2 (5 a c-3 b d) \left (c+\frac {d}{x}\right )^{7/2}}{d^5}+\frac {c^3 (5 a c-4 b d) \left (c+\frac {d}{x}\right )^{5/2}}{d^5}-\frac {c^4 (a c-b d) \left (c+\frac {d}{x}\right )^{3/2}}{d^5}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 c^4 \left (c+\frac {d}{x}\right )^{5/2} (a c-b d)}{5 d^6}-\frac {2 c^3 \left (c+\frac {d}{x}\right )^{7/2} (5 a c-4 b d)}{7 d^6}+\frac {4 c^2 \left (c+\frac {d}{x}\right )^{9/2} (5 a c-3 b d)}{9 d^6}+\frac {2 \left (c+\frac {d}{x}\right )^{13/2} (5 a c-b d)}{13 d^6}-\frac {4 c \left (c+\frac {d}{x}\right )^{11/2} (5 a c-2 b d)}{11 d^6}-\frac {2 a \left (c+\frac {d}{x}\right )^{15/2}}{15 d^6}\) |
Input:
Int[((c + d/x)^(3/2)*(a + b*x))/x^7,x]
Output:
(2*c^4*(a*c - b*d)*(c + d/x)^(5/2))/(5*d^6) - (2*c^3*(5*a*c - 4*b*d)*(c + d/x)^(7/2))/(7*d^6) + (4*c^2*(5*a*c - 3*b*d)*(c + d/x)^(9/2))/(9*d^6) - (4 *c*(5*a*c - 2*b*d)*(c + d/x)^(11/2))/(11*d^6) + (2*(5*a*c - b*d)*(c + d/x) ^(13/2))/(13*d^6) - (2*a*(c + d/x)^(15/2))/(15*d^6)
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]))))
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]]
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])
Time = 0.08 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.80
| method | result | size |
| orering | \(\frac {2 \left (256 a \,c^{5} x^{5}-384 b \,c^{4} d \,x^{5}-640 a \,c^{4} d \,x^{4}+960 b \,c^{3} d^{2} x^{4}+1120 a \,c^{3} d^{2} x^{3}-1680 b \,c^{2} d^{3} x^{3}-1680 a \,c^{2} d^{3} x^{2}+2520 b c \,d^{4} x^{2}+2310 a c \,d^{4} x -3465 b \,d^{5} x -3003 a \,d^{5}\right ) \left (c x +d \right ) \left (c +\frac {d}{x}\right )^{\frac {3}{2}}}{45045 d^{6} x^{6}}\) | \(132\) |
| gosper | \(\frac {2 \left (c x +d \right ) \left (256 a \,c^{5} x^{5}-384 b \,c^{4} d \,x^{5}-640 a \,c^{4} d \,x^{4}+960 b \,c^{3} d^{2} x^{4}+1120 a \,c^{3} d^{2} x^{3}-1680 b \,c^{2} d^{3} x^{3}-1680 a \,c^{2} d^{3} x^{2}+2520 b c \,d^{4} x^{2}+2310 a c \,d^{4} x -3465 b \,d^{5} x -3003 a \,d^{5}\right ) \left (\frac {c x +d}{x}\right )^{\frac {3}{2}}}{45045 x^{6} d^{6}}\) | \(134\) |
| default | \(\frac {2 \sqrt {\frac {c x +d}{x}}\, \left (c \,x^{2}+d x \right )^{\frac {3}{2}} \left (256 a \,c^{6} x^{6}-384 b \,c^{5} d \,x^{6}-384 a \,c^{5} d \,x^{5}+576 b \,c^{4} d^{2} x^{5}+480 a \,c^{4} d^{2} x^{4}-720 b \,c^{3} d^{3} x^{4}-560 a \,c^{3} d^{3} x^{3}+840 b \,c^{2} d^{4} x^{3}+630 a \,c^{2} d^{4} x^{2}-945 b c \,d^{5} x^{2}-693 a c \,d^{5} x -3465 b \,d^{6} x -3003 a \,d^{6}\right )}{45045 x^{8} d^{6} \sqrt {\left (c x +d \right ) x}}\) | \(173\) |
| risch | \(\frac {2 \sqrt {\frac {c x +d}{x}}\, \left (256 a \,c^{7} x^{7}-384 b \,c^{6} d \,x^{7}-128 a \,c^{6} d \,x^{6}+192 b \,c^{5} d^{2} x^{6}+96 a \,c^{5} d^{2} x^{5}-144 b \,c^{4} d^{3} x^{5}-80 a \,c^{4} d^{3} x^{4}+120 b \,c^{3} d^{4} x^{4}+70 a \,c^{3} d^{4} x^{3}-105 b \,c^{2} d^{5} x^{3}-63 a \,c^{2} d^{5} x^{2}-4410 b c \,d^{6} x^{2}-3696 a c \,d^{6} x -3465 b \,d^{7} x -3003 a \,d^{7}\right )}{45045 x^{7} d^{6}}\) | \(177\) |
| trager | \(\frac {2 \left (256 a \,c^{7} x^{7}-384 b \,c^{6} d \,x^{7}-128 a \,c^{6} d \,x^{6}+192 b \,c^{5} d^{2} x^{6}+96 a \,c^{5} d^{2} x^{5}-144 b \,c^{4} d^{3} x^{5}-80 a \,c^{4} d^{3} x^{4}+120 b \,c^{3} d^{4} x^{4}+70 a \,c^{3} d^{4} x^{3}-105 b \,c^{2} d^{5} x^{3}-63 a \,c^{2} d^{5} x^{2}-4410 b c \,d^{6} x^{2}-3696 a c \,d^{6} x -3465 b \,d^{7} x -3003 a \,d^{7}\right ) \sqrt {-\frac {-c x -d}{x}}}{45045 x^{7} d^{6}}\) | \(181\) |
Input:
int((c+d/x)^(3/2)*(b*x+a)/x^7,x,method=_RETURNVERBOSE)
Output:
2/45045*(256*a*c^5*x^5-384*b*c^4*d*x^5-640*a*c^4*d*x^4+960*b*c^3*d^2*x^4+1 120*a*c^3*d^2*x^3-1680*b*c^2*d^3*x^3-1680*a*c^2*d^3*x^2+2520*b*c*d^4*x^2+2 310*a*c*d^4*x-3465*b*d^5*x-3003*a*d^5)/d^6/x^6*(c*x+d)*(c+d/x)^(3/2)
Time = 0.07 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.08 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^7} \, dx=-\frac {2 \, {\left (3003 \, a d^{7} - 128 \, {\left (2 \, a c^{7} - 3 \, b c^{6} d\right )} x^{7} + 64 \, {\left (2 \, a c^{6} d - 3 \, b c^{5} d^{2}\right )} x^{6} - 48 \, {\left (2 \, a c^{5} d^{2} - 3 \, b c^{4} d^{3}\right )} x^{5} + 40 \, {\left (2 \, a c^{4} d^{3} - 3 \, b c^{3} d^{4}\right )} x^{4} - 35 \, {\left (2 \, a c^{3} d^{4} - 3 \, b c^{2} d^{5}\right )} x^{3} + 63 \, {\left (a c^{2} d^{5} + 70 \, b c d^{6}\right )} x^{2} + 231 \, {\left (16 \, a c d^{6} + 15 \, b d^{7}\right )} x\right )} \sqrt {\frac {c x + d}{x}}}{45045 \, d^{6} x^{7}} \] Input:
integrate((c+d/x)^(3/2)*(b*x+a)/x^7,x, algorithm="fricas")
Output:
-2/45045*(3003*a*d^7 - 128*(2*a*c^7 - 3*b*c^6*d)*x^7 + 64*(2*a*c^6*d - 3*b *c^5*d^2)*x^6 - 48*(2*a*c^5*d^2 - 3*b*c^4*d^3)*x^5 + 40*(2*a*c^4*d^3 - 3*b *c^3*d^4)*x^4 - 35*(2*a*c^3*d^4 - 3*b*c^2*d^5)*x^3 + 63*(a*c^2*d^5 + 70*b* c*d^6)*x^2 + 231*(16*a*c*d^6 + 15*b*d^7)*x)*sqrt((c*x + d)/x)/(d^6*x^7)
Time = 2.29 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.51 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^7} \, dx =\text {Too large to display} \] Input:
integrate((c+d/x)**(3/2)*(b*x+a)/x**7,x)
Output:
-a*c*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)) - a*d*Piecewise((2*(c**6*(c + d/x)**(3/2)/3 - 6*c**5*(c + d/x)**(5/2)/5 + 15 *c**4*(c + d/x)**(7/2)/7 - 20*c**3*(c + d/x)**(9/2)/9 + 15*c**2*(c + d/x)* *(11/2)/11 - 6*c*(c + d/x)**(13/2)/13 + (c + d/x)**(15/2)/15)/d**7, Ne(d, 0)), (sqrt(c)/(7*x**7), True)) - b*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)) - b*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))
Time = 0.04 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.13 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^7} \, dx=-\frac {2}{45045} \, {\left (\frac {3003 \, {\left (c + \frac {d}{x}\right )}^{\frac {15}{2}}}{d^{6}} - \frac {17325 \, {\left (c + \frac {d}{x}\right )}^{\frac {13}{2}} c}{d^{6}} + \frac {40950 \, {\left (c + \frac {d}{x}\right )}^{\frac {11}{2}} c^{2}}{d^{6}} - \frac {50050 \, {\left (c + \frac {d}{x}\right )}^{\frac {9}{2}} c^{3}}{d^{6}} + \frac {32175 \, {\left (c + \frac {d}{x}\right )}^{\frac {7}{2}} c^{4}}{d^{6}} - \frac {9009 \, {\left (c + \frac {d}{x}\right )}^{\frac {5}{2}} c^{5}}{d^{6}}\right )} a - \frac {2}{15015} \, {\left (\frac {1155 \, {\left (c + \frac {d}{x}\right )}^{\frac {13}{2}}}{d^{5}} - \frac {5460 \, {\left (c + \frac {d}{x}\right )}^{\frac {11}{2}} c}{d^{5}} + \frac {10010 \, {\left (c + \frac {d}{x}\right )}^{\frac {9}{2}} c^{2}}{d^{5}} - \frac {8580 \, {\left (c + \frac {d}{x}\right )}^{\frac {7}{2}} c^{3}}{d^{5}} + \frac {3003 \, {\left (c + \frac {d}{x}\right )}^{\frac {5}{2}} c^{4}}{d^{5}}\right )} b \] Input:
integrate((c+d/x)^(3/2)*(b*x+a)/x^7,x, algorithm="maxima")
Output:
-2/45045*(3003*(c + d/x)^(15/2)/d^6 - 17325*(c + d/x)^(13/2)*c/d^6 + 40950 *(c + d/x)^(11/2)*c^2/d^6 - 50050*(c + d/x)^(9/2)*c^3/d^6 + 32175*(c + d/x )^(7/2)*c^4/d^6 - 9009*(c + d/x)^(5/2)*c^5/d^6)*a - 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)*b
Leaf count of result is larger than twice the leaf count of optimal. 589 vs. \(2 (140) = 280\).
Time = 1.04 (sec) , antiderivative size = 589, normalized size of antiderivative = 3.59 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^7} \, dx =\text {Too large to display} \] Input:
integrate((c+d/x)^(3/2)*(b*x+a)/x^7,x, algorithm="giac")
Output:
2/45045*(144144*(sqrt(c)*x - sqrt(c*x^2 + d*x))^10*b*c^4*sgn(x) + 240240*( sqrt(c)*x - sqrt(c*x^2 + d*x))^9*a*c^(9/2)*sgn(x) + 720720*(sqrt(c)*x - sq rt(c*x^2 + d*x))^9*b*c^(7/2)*d*sgn(x) + 1338480*(sqrt(c)*x - sqrt(c*x^2 + d*x))^8*a*c^4*d*sgn(x) + 1595880*(sqrt(c)*x - sqrt(c*x^2 + d*x))^8*b*c^3*d ^2*sgn(x) + 3333330*(sqrt(c)*x - sqrt(c*x^2 + d*x))^7*a*c^(7/2)*d^2*sgn(x) + 2027025*(sqrt(c)*x - sqrt(c*x^2 + d*x))^7*b*c^(5/2)*d^3*sgn(x) + 484484 0*(sqrt(c)*x - sqrt(c*x^2 + d*x))^6*a*c^3*d^3*sgn(x) + 1606605*(sqrt(c)*x - sqrt(c*x^2 + d*x))^6*b*c^2*d^4*sgn(x) + 4513509*(sqrt(c)*x - sqrt(c*x^2 + d*x))^5*a*c^(5/2)*d^4*sgn(x) + 810810*(sqrt(c)*x - sqrt(c*x^2 + d*x))^5* b*c^(3/2)*d^5*sgn(x) + 2788695*(sqrt(c)*x - sqrt(c*x^2 + d*x))^4*a*c^2*d^5 *sgn(x) + 253890*(sqrt(c)*x - sqrt(c*x^2 + d*x))^4*b*c*d^6*sgn(x) + 114114 0*(sqrt(c)*x - sqrt(c*x^2 + d*x))^3*a*c^(3/2)*d^6*sgn(x) + 45045*(sqrt(c)* x - sqrt(c*x^2 + d*x))^3*b*sqrt(c)*d^7*sgn(x) + 297990*(sqrt(c)*x - sqrt(c *x^2 + d*x))^2*a*c*d^7*sgn(x) + 3465*(sqrt(c)*x - sqrt(c*x^2 + d*x))^2*b*d ^8*sgn(x) + 45045*(sqrt(c)*x - sqrt(c*x^2 + d*x))*a*sqrt(c)*d^8*sgn(x) + 3 003*a*d^9*sgn(x))/(sqrt(c)*x - sqrt(c*x^2 + d*x))^15
Time = 9.27 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.77 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^7} \, dx=\frac {512\,a\,c^7\,\sqrt {c+\frac {d}{x}}}{45045\,d^6}-\frac {256\,b\,c^6\,\sqrt {c+\frac {d}{x}}}{15015\,d^5}-\frac {32\,a\,c\,\sqrt {c+\frac {d}{x}}}{195\,x^6}-\frac {28\,b\,c\,\sqrt {c+\frac {d}{x}}}{143\,x^5}-\frac {2\,a\,d\,\sqrt {c+\frac {d}{x}}}{15\,x^7}-\frac {2\,b\,d\,\sqrt {c+\frac {d}{x}}}{13\,x^6}-\frac {2\,a\,c^2\,\sqrt {c+\frac {d}{x}}}{715\,d\,x^5}+\frac {4\,a\,c^3\,\sqrt {c+\frac {d}{x}}}{1287\,d^2\,x^4}-\frac {32\,a\,c^4\,\sqrt {c+\frac {d}{x}}}{9009\,d^3\,x^3}+\frac {64\,a\,c^5\,\sqrt {c+\frac {d}{x}}}{15015\,d^4\,x^2}-\frac {256\,a\,c^6\,\sqrt {c+\frac {d}{x}}}{45045\,d^5\,x}-\frac {2\,b\,c^2\,\sqrt {c+\frac {d}{x}}}{429\,d\,x^4}+\frac {16\,b\,c^3\,\sqrt {c+\frac {d}{x}}}{3003\,d^2\,x^3}-\frac {32\,b\,c^4\,\sqrt {c+\frac {d}{x}}}{5005\,d^3\,x^2}+\frac {128\,b\,c^5\,\sqrt {c+\frac {d}{x}}}{15015\,d^4\,x} \] Input:
int(((c + d/x)^(3/2)*(a + b*x))/x^7,x)
Output:
(512*a*c^7*(c + d/x)^(1/2))/(45045*d^6) - (256*b*c^6*(c + d/x)^(1/2))/(150 15*d^5) - (32*a*c*(c + d/x)^(1/2))/(195*x^6) - (28*b*c*(c + d/x)^(1/2))/(1 43*x^5) - (2*a*d*(c + d/x)^(1/2))/(15*x^7) - (2*b*d*(c + d/x)^(1/2))/(13*x ^6) - (2*a*c^2*(c + d/x)^(1/2))/(715*d*x^5) + (4*a*c^3*(c + d/x)^(1/2))/(1 287*d^2*x^4) - (32*a*c^4*(c + d/x)^(1/2))/(9009*d^3*x^3) + (64*a*c^5*(c + d/x)^(1/2))/(15015*d^4*x^2) - (256*a*c^6*(c + d/x)^(1/2))/(45045*d^5*x) - (2*b*c^2*(c + d/x)^(1/2))/(429*d*x^4) + (16*b*c^3*(c + d/x)^(1/2))/(3003*d ^2*x^3) - (32*b*c^4*(c + d/x)^(1/2))/(5005*d^3*x^2) + (128*b*c^5*(c + d/x) ^(1/2))/(15015*d^4*x)
Time = 0.21 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.88 \[ \int \frac {\left (c+\frac {d}{x}\right )^{3/2} (a+b x)}{x^7} \, dx=\frac {\frac {512 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{7} x^{7}}{45045}-\frac {256 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{6} d \,x^{6}}{45045}+\frac {64 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{5} d^{2} x^{5}}{15015}-\frac {32 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{4} d^{3} x^{4}}{9009}+\frac {4 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{3} d^{4} x^{3}}{1287}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, a \,c^{2} d^{5} x^{2}}{715}-\frac {32 \sqrt {x}\, \sqrt {c x +d}\, a c \,d^{6} x}{195}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, a \,d^{7}}{15}-\frac {256 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{6} d \,x^{7}}{15015}+\frac {128 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{5} d^{2} x^{6}}{15015}-\frac {32 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{4} d^{3} x^{5}}{5005}+\frac {16 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{3} d^{4} x^{4}}{3003}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, b \,c^{2} d^{5} x^{3}}{429}-\frac {28 \sqrt {x}\, \sqrt {c x +d}\, b c \,d^{6} x^{2}}{143}-\frac {2 \sqrt {x}\, \sqrt {c x +d}\, b \,d^{7} x}{13}-\frac {512 \sqrt {c}\, a \,c^{7} x^{8}}{45045}+\frac {256 \sqrt {c}\, b \,c^{6} d \,x^{8}}{15015}}{d^{6} x^{8}} \] Input:
int((c+d/x)^(3/2)*(b*x+a)/x^7,x)
Output:
(2*(256*sqrt(x)*sqrt(c*x + d)*a*c**7*x**7 - 128*sqrt(x)*sqrt(c*x + d)*a*c* *6*d*x**6 + 96*sqrt(x)*sqrt(c*x + d)*a*c**5*d**2*x**5 - 80*sqrt(x)*sqrt(c* x + d)*a*c**4*d**3*x**4 + 70*sqrt(x)*sqrt(c*x + d)*a*c**3*d**4*x**3 - 63*s qrt(x)*sqrt(c*x + d)*a*c**2*d**5*x**2 - 3696*sqrt(x)*sqrt(c*x + d)*a*c*d** 6*x - 3003*sqrt(x)*sqrt(c*x + d)*a*d**7 - 384*sqrt(x)*sqrt(c*x + d)*b*c**6 *d*x**7 + 192*sqrt(x)*sqrt(c*x + d)*b*c**5*d**2*x**6 - 144*sqrt(x)*sqrt(c* x + d)*b*c**4*d**3*x**5 + 120*sqrt(x)*sqrt(c*x + d)*b*c**3*d**4*x**4 - 105 *sqrt(x)*sqrt(c*x + d)*b*c**2*d**5*x**3 - 4410*sqrt(x)*sqrt(c*x + d)*b*c*d **6*x**2 - 3465*sqrt(x)*sqrt(c*x + d)*b*d**7*x - 256*sqrt(c)*a*c**7*x**8 + 384*sqrt(c)*b*c**6*d*x**8))/(45045*d**6*x**8)