Integrand size = 15, antiderivative size = 241 \[ \int \frac {(c+d x)^{10}}{a+b x} \, dx=\frac {d (b c-a d)^9 x}{b^{10}}+\frac {(b c-a d)^8 (c+d x)^2}{2 b^9}+\frac {(b c-a d)^7 (c+d x)^3}{3 b^8}+\frac {(b c-a d)^6 (c+d x)^4}{4 b^7}+\frac {(b c-a d)^5 (c+d x)^5}{5 b^6}+\frac {(b c-a d)^4 (c+d x)^6}{6 b^5}+\frac {(b c-a d)^3 (c+d x)^7}{7 b^4}+\frac {(b c-a d)^2 (c+d x)^8}{8 b^3}+\frac {(b c-a d) (c+d x)^9}{9 b^2}+\frac {(c+d x)^{10}}{10 b}+\frac {(b c-a d)^{10} \log (a+b x)}{b^{11}} \]
d*(-a*d+b*c)^9*x/b^10+1/2*(-a*d+b*c)^8*(d*x+c)^2/b^9+1/3*(-a*d+b*c)^7*(d*x +c)^3/b^8+1/4*(-a*d+b*c)^6*(d*x+c)^4/b^7+1/5*(-a*d+b*c)^5*(d*x+c)^5/b^6+1/ 6*(-a*d+b*c)^4*(d*x+c)^6/b^5+1/7*(-a*d+b*c)^3*(d*x+c)^7/b^4+1/8*(-a*d+b*c) ^2*(d*x+c)^8/b^3+1/9*(-a*d+b*c)*(d*x+c)^9/b^2+1/10*(d*x+c)^10/b+(-a*d+b*c) ^10*ln(b*x+a)/b^11
Leaf count is larger than twice the leaf count of optimal. \(591\) vs. \(2(241)=482\).
Time = 0.16 (sec) , antiderivative size = 591, normalized size of antiderivative = 2.45 \[ \int \frac {(c+d x)^{10}}{a+b x} \, dx=\frac {d x \left (-2520 a^9 d^9+1260 a^8 b d^8 (20 c+d x)-840 a^7 b^2 d^7 \left (135 c^2+15 c d x+d^2 x^2\right )+210 a^6 b^3 d^6 \left (1440 c^3+270 c^2 d x+40 c d^2 x^2+3 d^3 x^3\right )-252 a^5 b^4 d^5 \left (2100 c^4+600 c^3 d x+150 c^2 d^2 x^2+25 c d^3 x^3+2 d^4 x^4\right )+210 a^4 b^5 d^4 \left (3024 c^5+1260 c^4 d x+480 c^3 d^2 x^2+135 c^2 d^3 x^3+24 c d^4 x^4+2 d^5 x^5\right )-120 a^3 b^6 d^3 \left (4410 c^6+2646 c^5 d x+1470 c^4 d^2 x^2+630 c^3 d^3 x^3+189 c^2 d^4 x^4+35 c d^5 x^5+3 d^6 x^6\right )+45 a^2 b^7 d^2 \left (6720 c^7+5880 c^6 d x+4704 c^5 d^2 x^2+2940 c^4 d^3 x^3+1344 c^3 d^4 x^4+420 c^2 d^5 x^5+80 c d^6 x^6+7 d^7 x^7\right )-10 a b^8 d \left (11340 c^8+15120 c^7 d x+17640 c^6 d^2 x^2+15876 c^5 d^3 x^3+10584 c^4 d^4 x^4+5040 c^3 d^5 x^5+1620 c^2 d^6 x^6+315 c d^7 x^7+28 d^8 x^8\right )+b^9 \left (25200 c^9+56700 c^8 d x+100800 c^7 d^2 x^2+132300 c^6 d^3 x^3+127008 c^5 d^4 x^4+88200 c^4 d^5 x^5+43200 c^3 d^6 x^6+14175 c^2 d^7 x^7+2800 c d^8 x^8+252 d^9 x^9\right )\right )}{2520 b^{10}}+\frac {(b c-a d)^{10} \log (a+b x)}{b^{11}} \]
(d*x*(-2520*a^9*d^9 + 1260*a^8*b*d^8*(20*c + d*x) - 840*a^7*b^2*d^7*(135*c ^2 + 15*c*d*x + d^2*x^2) + 210*a^6*b^3*d^6*(1440*c^3 + 270*c^2*d*x + 40*c* d^2*x^2 + 3*d^3*x^3) - 252*a^5*b^4*d^5*(2100*c^4 + 600*c^3*d*x + 150*c^2*d ^2*x^2 + 25*c*d^3*x^3 + 2*d^4*x^4) + 210*a^4*b^5*d^4*(3024*c^5 + 1260*c^4* d*x + 480*c^3*d^2*x^2 + 135*c^2*d^3*x^3 + 24*c*d^4*x^4 + 2*d^5*x^5) - 120* a^3*b^6*d^3*(4410*c^6 + 2646*c^5*d*x + 1470*c^4*d^2*x^2 + 630*c^3*d^3*x^3 + 189*c^2*d^4*x^4 + 35*c*d^5*x^5 + 3*d^6*x^6) + 45*a^2*b^7*d^2*(6720*c^7 + 5880*c^6*d*x + 4704*c^5*d^2*x^2 + 2940*c^4*d^3*x^3 + 1344*c^3*d^4*x^4 + 4 20*c^2*d^5*x^5 + 80*c*d^6*x^6 + 7*d^7*x^7) - 10*a*b^8*d*(11340*c^8 + 15120 *c^7*d*x + 17640*c^6*d^2*x^2 + 15876*c^5*d^3*x^3 + 10584*c^4*d^4*x^4 + 504 0*c^3*d^5*x^5 + 1620*c^2*d^6*x^6 + 315*c*d^7*x^7 + 28*d^8*x^8) + b^9*(2520 0*c^9 + 56700*c^8*d*x + 100800*c^7*d^2*x^2 + 132300*c^6*d^3*x^3 + 127008*c ^5*d^4*x^4 + 88200*c^4*d^5*x^5 + 43200*c^3*d^6*x^6 + 14175*c^2*d^7*x^7 + 2 800*c*d^8*x^8 + 252*d^9*x^9)))/(2520*b^10) + ((b*c - a*d)^10*Log[a + b*x]) /b^11
Time = 0.37 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {49, 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.
| \(\displaystyle \int \frac {(c+d x)^{10}}{a+b x} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {(b c-a d)^{10}}{b^{10} (a+b x)}+\frac {d (b c-a d)^9}{b^{10}}+\frac {d (c+d x) (b c-a d)^8}{b^9}+\frac {d (c+d x)^2 (b c-a d)^7}{b^8}+\frac {d (c+d x)^3 (b c-a d)^6}{b^7}+\frac {d (c+d x)^4 (b c-a d)^5}{b^6}+\frac {d (c+d x)^5 (b c-a d)^4}{b^5}+\frac {d (c+d x)^6 (b c-a d)^3}{b^4}+\frac {d (c+d x)^7 (b c-a d)^2}{b^3}+\frac {d (c+d x)^8 (b c-a d)}{b^2}+\frac {d (c+d x)^9}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(b c-a d)^{10} \log (a+b x)}{b^{11}}+\frac {d x (b c-a d)^9}{b^{10}}+\frac {(c+d x)^2 (b c-a d)^8}{2 b^9}+\frac {(c+d x)^3 (b c-a d)^7}{3 b^8}+\frac {(c+d x)^4 (b c-a d)^6}{4 b^7}+\frac {(c+d x)^5 (b c-a d)^5}{5 b^6}+\frac {(c+d x)^6 (b c-a d)^4}{6 b^5}+\frac {(c+d x)^7 (b c-a d)^3}{7 b^4}+\frac {(c+d x)^8 (b c-a d)^2}{8 b^3}+\frac {(c+d x)^9 (b c-a d)}{9 b^2}+\frac {(c+d x)^{10}}{10 b}\) |
(d*(b*c - a*d)^9*x)/b^10 + ((b*c - a*d)^8*(c + d*x)^2)/(2*b^9) + ((b*c - a *d)^7*(c + d*x)^3)/(3*b^8) + ((b*c - a*d)^6*(c + d*x)^4)/(4*b^7) + ((b*c - a*d)^5*(c + d*x)^5)/(5*b^6) + ((b*c - a*d)^4*(c + d*x)^6)/(6*b^5) + ((b*c - a*d)^3*(c + d*x)^7)/(7*b^4) + ((b*c - a*d)^2*(c + d*x)^8)/(8*b^3) + ((b *c - a*d)*(c + d*x)^9)/(9*b^2) + (c + d*x)^10/(10*b) + ((b*c - a*d)^10*Log [a + b*x])/b^11
3.14.12.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(826\) vs. \(2(223)=446\).
Time = 0.28 (sec) , antiderivative size = 827, normalized size of antiderivative = 3.43
| method | result | size |
| norman | \(\frac {d^{10} x^{10}}{10 b}+\frac {d^{2} \left (a^{8} d^{8}-10 a^{7} b c \,d^{7}+45 a^{6} b^{2} c^{2} d^{6}-120 a^{5} b^{3} c^{3} d^{5}+210 a^{4} b^{4} c^{4} d^{4}-252 a^{3} b^{5} c^{5} d^{3}+210 a^{2} b^{6} c^{6} d^{2}-120 a \,b^{7} c^{7} d +45 b^{8} c^{8}\right ) x^{2}}{2 b^{9}}-\frac {d^{3} \left (a^{7} d^{7}-10 a^{6} b c \,d^{6}+45 a^{5} b^{2} c^{2} d^{5}-120 a^{4} b^{3} c^{3} d^{4}+210 a^{3} b^{4} c^{4} d^{3}-252 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d -120 b^{7} c^{7}\right ) x^{3}}{3 b^{8}}+\frac {d^{4} \left (a^{6} d^{6}-10 a^{5} b c \,d^{5}+45 a^{4} b^{2} c^{2} d^{4}-120 a^{3} b^{3} c^{3} d^{3}+210 a^{2} b^{4} c^{4} d^{2}-252 a \,b^{5} c^{5} d +210 b^{6} c^{6}\right ) x^{4}}{4 b^{7}}-\frac {d^{5} \left (a^{5} d^{5}-10 a^{4} b c \,d^{4}+45 a^{3} b^{2} c^{2} d^{3}-120 a^{2} b^{3} c^{3} d^{2}+210 a \,b^{4} c^{4} d -252 b^{5} c^{5}\right ) x^{5}}{5 b^{6}}+\frac {d^{6} \left (a^{4} d^{4}-10 a^{3} b c \,d^{3}+45 a^{2} b^{2} c^{2} d^{2}-120 a \,b^{3} c^{3} d +210 b^{4} c^{4}\right ) x^{6}}{6 b^{5}}-\frac {d^{7} \left (a^{3} d^{3}-10 a^{2} b c \,d^{2}+45 a \,b^{2} c^{2} d -120 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}+\frac {d^{8} \left (a^{2} d^{2}-10 a b c d +45 b^{2} c^{2}\right ) x^{8}}{8 b^{3}}-\frac {d^{9} \left (a d -10 b c \right ) x^{9}}{9 b^{2}}-\frac {d \left (a^{9} d^{9}-10 a^{8} b c \,d^{8}+45 a^{7} b^{2} c^{2} d^{7}-120 a^{6} b^{3} c^{3} d^{6}+210 a^{5} b^{4} c^{4} d^{5}-252 a^{4} b^{5} c^{5} d^{4}+210 a^{3} b^{6} c^{6} d^{3}-120 a^{2} b^{7} c^{7} d^{2}+45 a \,b^{8} c^{8} d -10 b^{9} c^{9}\right ) x}{b^{10}}+\frac {\left (a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}\right ) \ln \left (b x +a \right )}{b^{11}}\) | \(827\) |
| risch | \(\text {Expression too large to display}\) | \(1022\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1022\) |
| default | \(\text {Expression too large to display}\) | \(2929\) |
1/10/b*d^10*x^10+1/2/b^9*d^2*(a^8*d^8-10*a^7*b*c*d^7+45*a^6*b^2*c^2*d^6-12 0*a^5*b^3*c^3*d^5+210*a^4*b^4*c^4*d^4-252*a^3*b^5*c^5*d^3+210*a^2*b^6*c^6* d^2-120*a*b^7*c^7*d+45*b^8*c^8)*x^2-1/3/b^8*d^3*(a^7*d^7-10*a^6*b*c*d^6+45 *a^5*b^2*c^2*d^5-120*a^4*b^3*c^3*d^4+210*a^3*b^4*c^4*d^3-252*a^2*b^5*c^5*d ^2+210*a*b^6*c^6*d-120*b^7*c^7)*x^3+1/4/b^7*d^4*(a^6*d^6-10*a^5*b*c*d^5+45 *a^4*b^2*c^2*d^4-120*a^3*b^3*c^3*d^3+210*a^2*b^4*c^4*d^2-252*a*b^5*c^5*d+2 10*b^6*c^6)*x^4-1/5/b^6*d^5*(a^5*d^5-10*a^4*b*c*d^4+45*a^3*b^2*c^2*d^3-120 *a^2*b^3*c^3*d^2+210*a*b^4*c^4*d-252*b^5*c^5)*x^5+1/6/b^5*d^6*(a^4*d^4-10* a^3*b*c*d^3+45*a^2*b^2*c^2*d^2-120*a*b^3*c^3*d+210*b^4*c^4)*x^6-1/7/b^4*d^ 7*(a^3*d^3-10*a^2*b*c*d^2+45*a*b^2*c^2*d-120*b^3*c^3)*x^7+1/8/b^3*d^8*(a^2 *d^2-10*a*b*c*d+45*b^2*c^2)*x^8-1/9/b^2*d^9*(a*d-10*b*c)*x^9-d*(a^9*d^9-10 *a^8*b*c*d^8+45*a^7*b^2*c^2*d^7-120*a^6*b^3*c^3*d^6+210*a^5*b^4*c^4*d^5-25 2*a^4*b^5*c^5*d^4+210*a^3*b^6*c^6*d^3-120*a^2*b^7*c^7*d^2+45*a*b^8*c^8*d-1 0*b^9*c^9)/b^10*x+(a^10*d^10-10*a^9*b*c*d^9+45*a^8*b^2*c^2*d^8-120*a^7*b^3 *c^3*d^7+210*a^6*b^4*c^4*d^6-252*a^5*b^5*c^5*d^5+210*a^4*b^6*c^6*d^4-120*a ^3*b^7*c^7*d^3+45*a^2*b^8*c^8*d^2-10*a*b^9*c^9*d+b^10*c^10)/b^11*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 868 vs. \(2 (223) = 446\).
Time = 0.23 (sec) , antiderivative size = 868, normalized size of antiderivative = 3.60 \[ \int \frac {(c+d x)^{10}}{a+b x} \, dx =\text {Too large to display} \]
1/2520*(252*b^10*d^10*x^10 + 280*(10*b^10*c*d^9 - a*b^9*d^10)*x^9 + 315*(4 5*b^10*c^2*d^8 - 10*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 360*(120*b^10*c^3*d^ 7 - 45*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 420*(210*b^1 0*c^4*d^6 - 120*a*b^9*c^3*d^7 + 45*a^2*b^8*c^2*d^8 - 10*a^3*b^7*c*d^9 + a^ 4*b^6*d^10)*x^6 + 504*(252*b^10*c^5*d^5 - 210*a*b^9*c^4*d^6 + 120*a^2*b^8* c^3*d^7 - 45*a^3*b^7*c^2*d^8 + 10*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 630* (210*b^10*c^6*d^4 - 252*a*b^9*c^5*d^5 + 210*a^2*b^8*c^4*d^6 - 120*a^3*b^7* c^3*d^7 + 45*a^4*b^6*c^2*d^8 - 10*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 840* (120*b^10*c^7*d^3 - 210*a*b^9*c^6*d^4 + 252*a^2*b^8*c^5*d^5 - 210*a^3*b^7* c^4*d^6 + 120*a^4*b^6*c^3*d^7 - 45*a^5*b^5*c^2*d^8 + 10*a^6*b^4*c*d^9 - a^ 7*b^3*d^10)*x^3 + 1260*(45*b^10*c^8*d^2 - 120*a*b^9*c^7*d^3 + 210*a^2*b^8* c^6*d^4 - 252*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 120*a^5*b^5*c^3*d^7 + 45*a^6*b^4*c^2*d^8 - 10*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 2520*(10*b^1 0*c^9*d - 45*a*b^9*c^8*d^2 + 120*a^2*b^8*c^7*d^3 - 210*a^3*b^7*c^6*d^4 + 2 52*a^4*b^6*c^5*d^5 - 210*a^5*b^5*c^4*d^6 + 120*a^6*b^4*c^3*d^7 - 45*a^7*b^ 3*c^2*d^8 + 10*a^8*b^2*c*d^9 - a^9*b*d^10)*x + 2520*(b^10*c^10 - 10*a*b^9* c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 - 2 52*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^ 2*c^2*d^8 - 10*a^9*b*c*d^9 + a^10*d^10)*log(b*x + a))/b^11
Leaf count of result is larger than twice the leaf count of optimal. 799 vs. \(2 (206) = 412\).
Time = 0.82 (sec) , antiderivative size = 799, normalized size of antiderivative = 3.32 \[ \int \frac {(c+d x)^{10}}{a+b x} \, dx =\text {Too large to display} \]
x**9*(-a*d**10/(9*b**2) + 10*c*d**9/(9*b)) + x**8*(a**2*d**10/(8*b**3) - 5 *a*c*d**9/(4*b**2) + 45*c**2*d**8/(8*b)) + x**7*(-a**3*d**10/(7*b**4) + 10 *a**2*c*d**9/(7*b**3) - 45*a*c**2*d**8/(7*b**2) + 120*c**3*d**7/(7*b)) + x **6*(a**4*d**10/(6*b**5) - 5*a**3*c*d**9/(3*b**4) + 15*a**2*c**2*d**8/(2*b **3) - 20*a*c**3*d**7/b**2 + 35*c**4*d**6/b) + x**5*(-a**5*d**10/(5*b**6) + 2*a**4*c*d**9/b**5 - 9*a**3*c**2*d**8/b**4 + 24*a**2*c**3*d**7/b**3 - 42 *a*c**4*d**6/b**2 + 252*c**5*d**5/(5*b)) + x**4*(a**6*d**10/(4*b**7) - 5*a **5*c*d**9/(2*b**6) + 45*a**4*c**2*d**8/(4*b**5) - 30*a**3*c**3*d**7/b**4 + 105*a**2*c**4*d**6/(2*b**3) - 63*a*c**5*d**5/b**2 + 105*c**6*d**4/(2*b)) + x**3*(-a**7*d**10/(3*b**8) + 10*a**6*c*d**9/(3*b**7) - 15*a**5*c**2*d** 8/b**6 + 40*a**4*c**3*d**7/b**5 - 70*a**3*c**4*d**6/b**4 + 84*a**2*c**5*d* *5/b**3 - 70*a*c**6*d**4/b**2 + 40*c**7*d**3/b) + x**2*(a**8*d**10/(2*b**9 ) - 5*a**7*c*d**9/b**8 + 45*a**6*c**2*d**8/(2*b**7) - 60*a**5*c**3*d**7/b* *6 + 105*a**4*c**4*d**6/b**5 - 126*a**3*c**5*d**5/b**4 + 105*a**2*c**6*d** 4/b**3 - 60*a*c**7*d**3/b**2 + 45*c**8*d**2/(2*b)) + x*(-a**9*d**10/b**10 + 10*a**8*c*d**9/b**9 - 45*a**7*c**2*d**8/b**8 + 120*a**6*c**3*d**7/b**7 - 210*a**5*c**4*d**6/b**6 + 252*a**4*c**5*d**5/b**5 - 210*a**3*c**6*d**4/b* *4 + 120*a**2*c**7*d**3/b**3 - 45*a*c**8*d**2/b**2 + 10*c**9*d/b) + d**10* x**10/(10*b) + (a*d - b*c)**10*log(a + b*x)/b**11
Leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (223) = 446\).
Time = 0.22 (sec) , antiderivative size = 866, normalized size of antiderivative = 3.59 \[ \int \frac {(c+d x)^{10}}{a+b x} \, dx =\text {Too large to display} \]
1/2520*(252*b^9*d^10*x^10 + 280*(10*b^9*c*d^9 - a*b^8*d^10)*x^9 + 315*(45* b^9*c^2*d^8 - 10*a*b^8*c*d^9 + a^2*b^7*d^10)*x^8 + 360*(120*b^9*c^3*d^7 - 45*a*b^8*c^2*d^8 + 10*a^2*b^7*c*d^9 - a^3*b^6*d^10)*x^7 + 420*(210*b^9*c^4 *d^6 - 120*a*b^8*c^3*d^7 + 45*a^2*b^7*c^2*d^8 - 10*a^3*b^6*c*d^9 + a^4*b^5 *d^10)*x^6 + 504*(252*b^9*c^5*d^5 - 210*a*b^8*c^4*d^6 + 120*a^2*b^7*c^3*d^ 7 - 45*a^3*b^6*c^2*d^8 + 10*a^4*b^5*c*d^9 - a^5*b^4*d^10)*x^5 + 630*(210*b ^9*c^6*d^4 - 252*a*b^8*c^5*d^5 + 210*a^2*b^7*c^4*d^6 - 120*a^3*b^6*c^3*d^7 + 45*a^4*b^5*c^2*d^8 - 10*a^5*b^4*c*d^9 + a^6*b^3*d^10)*x^4 + 840*(120*b^ 9*c^7*d^3 - 210*a*b^8*c^6*d^4 + 252*a^2*b^7*c^5*d^5 - 210*a^3*b^6*c^4*d^6 + 120*a^4*b^5*c^3*d^7 - 45*a^5*b^4*c^2*d^8 + 10*a^6*b^3*c*d^9 - a^7*b^2*d^ 10)*x^3 + 1260*(45*b^9*c^8*d^2 - 120*a*b^8*c^7*d^3 + 210*a^2*b^7*c^6*d^4 - 252*a^3*b^6*c^5*d^5 + 210*a^4*b^5*c^4*d^6 - 120*a^5*b^4*c^3*d^7 + 45*a^6* b^3*c^2*d^8 - 10*a^7*b^2*c*d^9 + a^8*b*d^10)*x^2 + 2520*(10*b^9*c^9*d - 45 *a*b^8*c^8*d^2 + 120*a^2*b^7*c^7*d^3 - 210*a^3*b^6*c^6*d^4 + 252*a^4*b^5*c ^5*d^5 - 210*a^5*b^4*c^4*d^6 + 120*a^6*b^3*c^3*d^7 - 45*a^7*b^2*c^2*d^8 + 10*a^8*b*c*d^9 - a^9*d^10)*x)/b^10 + (b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2* b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 210*a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5* d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 - 10* a^9*b*c*d^9 + a^10*d^10)*log(b*x + a)/b^11
Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (223) = 446\).
Time = 0.40 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.99 \[ \int \frac {(c+d x)^{10}}{a+b x} \, dx =\text {Too large to display} \]
1/2520*(252*b^9*d^10*x^10 + 2800*b^9*c*d^9*x^9 - 280*a*b^8*d^10*x^9 + 1417 5*b^9*c^2*d^8*x^8 - 3150*a*b^8*c*d^9*x^8 + 315*a^2*b^7*d^10*x^8 + 43200*b^ 9*c^3*d^7*x^7 - 16200*a*b^8*c^2*d^8*x^7 + 3600*a^2*b^7*c*d^9*x^7 - 360*a^3 *b^6*d^10*x^7 + 88200*b^9*c^4*d^6*x^6 - 50400*a*b^8*c^3*d^7*x^6 + 18900*a^ 2*b^7*c^2*d^8*x^6 - 4200*a^3*b^6*c*d^9*x^6 + 420*a^4*b^5*d^10*x^6 + 127008 *b^9*c^5*d^5*x^5 - 105840*a*b^8*c^4*d^6*x^5 + 60480*a^2*b^7*c^3*d^7*x^5 - 22680*a^3*b^6*c^2*d^8*x^5 + 5040*a^4*b^5*c*d^9*x^5 - 504*a^5*b^4*d^10*x^5 + 132300*b^9*c^6*d^4*x^4 - 158760*a*b^8*c^5*d^5*x^4 + 132300*a^2*b^7*c^4*d ^6*x^4 - 75600*a^3*b^6*c^3*d^7*x^4 + 28350*a^4*b^5*c^2*d^8*x^4 - 6300*a^5* b^4*c*d^9*x^4 + 630*a^6*b^3*d^10*x^4 + 100800*b^9*c^7*d^3*x^3 - 176400*a*b ^8*c^6*d^4*x^3 + 211680*a^2*b^7*c^5*d^5*x^3 - 176400*a^3*b^6*c^4*d^6*x^3 + 100800*a^4*b^5*c^3*d^7*x^3 - 37800*a^5*b^4*c^2*d^8*x^3 + 8400*a^6*b^3*c*d ^9*x^3 - 840*a^7*b^2*d^10*x^3 + 56700*b^9*c^8*d^2*x^2 - 151200*a*b^8*c^7*d ^3*x^2 + 264600*a^2*b^7*c^6*d^4*x^2 - 317520*a^3*b^6*c^5*d^5*x^2 + 264600* a^4*b^5*c^4*d^6*x^2 - 151200*a^5*b^4*c^3*d^7*x^2 + 56700*a^6*b^3*c^2*d^8*x ^2 - 12600*a^7*b^2*c*d^9*x^2 + 1260*a^8*b*d^10*x^2 + 25200*b^9*c^9*d*x - 1 13400*a*b^8*c^8*d^2*x + 302400*a^2*b^7*c^7*d^3*x - 529200*a^3*b^6*c^6*d^4* x + 635040*a^4*b^5*c^5*d^5*x - 529200*a^5*b^4*c^4*d^6*x + 302400*a^6*b^3*c ^3*d^7*x - 113400*a^7*b^2*c^2*d^8*x + 25200*a^8*b*c*d^9*x - 2520*a^9*d^10* x)/b^10 + (b^10*c^10 - 10*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^...
Time = 0.13 (sec) , antiderivative size = 979, normalized size of antiderivative = 4.06 \[ \int \frac {(c+d x)^{10}}{a+b x} \, dx =\text {Too large to display} \]
x^7*((120*c^3*d^7)/(7*b) - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c ^2*d^8)/b))/(7*b)) - x^9*((a*d^10)/(9*b^2) - (10*c*d^9)/(9*b)) + x^5*((a*( (a*((120*c^3*d^7)/b - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*d^ 8)/b))/b))/b - (210*c^4*d^6)/b))/(5*b) + (252*c^5*d^5)/(5*b)) + x^3*((a*(( a*((a*((a*((120*c^3*d^7)/b - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45 *c^2*d^8)/b))/b))/b - (210*c^4*d^6)/b))/b + (252*c^5*d^5)/b))/b - (210*c^6 *d^4)/b))/(3*b) + (40*c^7*d^3)/b) + x*((a*((a*((a*((a*((a*((a*((120*c^3*d^ 7)/b - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*d^8)/b))/b))/b - (210*c^4*d^6)/b))/b + (252*c^5*d^5)/b))/b - (210*c^6*d^4)/b))/b + (120*c^7 *d^3)/b))/b - (45*c^8*d^2)/b))/b + (10*c^9*d)/b) + x^8*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/(8*b) + (45*c^2*d^8)/(8*b)) - x^6*((a*((120*c^3*d^7)/b - ( a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*d^8)/b))/b))/(6*b) - (35* c^4*d^6)/b) - x^4*((a*((a*((a*((120*c^3*d^7)/b - (a*((a*((a*d^10)/b^2 - (1 0*c*d^9)/b))/b + (45*c^2*d^8)/b))/b))/b - (210*c^4*d^6)/b))/b + (252*c^5*d ^5)/b))/(4*b) - (105*c^6*d^4)/(2*b)) - x^2*((a*((a*((a*((a*((a*((120*c^3*d ^7)/b - (a*((a*((a*d^10)/b^2 - (10*c*d^9)/b))/b + (45*c^2*d^8)/b))/b))/b - (210*c^4*d^6)/b))/b + (252*c^5*d^5)/b))/b - (210*c^6*d^4)/b))/b + (120*c^ 7*d^3)/b))/(2*b) - (45*c^8*d^2)/(2*b)) + (d^10*x^10)/(10*b) + (log(a + b*x )*(a^10*d^10 + b^10*c^10 + 45*a^2*b^8*c^8*d^2 - 120*a^3*b^7*c^7*d^3 + 210* a^4*b^6*c^6*d^4 - 252*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 - 120*a^7*b...
Time = 0.00 (sec) , antiderivative size = 1021, normalized size of antiderivative = 4.24 \[ \int \frac {(c+d x)^{10}}{a+b x} \, dx =\text {Too large to display} \]
int((c**10 + 10*c**9*d*x + 45*c**8*d**2*x**2 + 120*c**7*d**3*x**3 + 210*c* *6*d**4*x**4 + 252*c**5*d**5*x**5 + 210*c**4*d**6*x**6 + 120*c**3*d**7*x** 7 + 45*c**2*d**8*x**8 + 10*c*d**9*x**9 + d**10*x**10)/(a + b*x),x)
(2520*log(a + b*x)*a**10*d**10 - 25200*log(a + b*x)*a**9*b*c*d**9 + 113400 *log(a + b*x)*a**8*b**2*c**2*d**8 - 302400*log(a + b*x)*a**7*b**3*c**3*d** 7 + 529200*log(a + b*x)*a**6*b**4*c**4*d**6 - 635040*log(a + b*x)*a**5*b** 5*c**5*d**5 + 529200*log(a + b*x)*a**4*b**6*c**6*d**4 - 302400*log(a + b*x )*a**3*b**7*c**7*d**3 + 113400*log(a + b*x)*a**2*b**8*c**8*d**2 - 25200*lo g(a + b*x)*a*b**9*c**9*d + 2520*log(a + b*x)*b**10*c**10 - 2520*a**9*b*d** 10*x + 25200*a**8*b**2*c*d**9*x + 1260*a**8*b**2*d**10*x**2 - 113400*a**7* b**3*c**2*d**8*x - 12600*a**7*b**3*c*d**9*x**2 - 840*a**7*b**3*d**10*x**3 + 302400*a**6*b**4*c**3*d**7*x + 56700*a**6*b**4*c**2*d**8*x**2 + 8400*a** 6*b**4*c*d**9*x**3 + 630*a**6*b**4*d**10*x**4 - 529200*a**5*b**5*c**4*d**6 *x - 151200*a**5*b**5*c**3*d**7*x**2 - 37800*a**5*b**5*c**2*d**8*x**3 - 63 00*a**5*b**5*c*d**9*x**4 - 504*a**5*b**5*d**10*x**5 + 635040*a**4*b**6*c** 5*d**5*x + 264600*a**4*b**6*c**4*d**6*x**2 + 100800*a**4*b**6*c**3*d**7*x* *3 + 28350*a**4*b**6*c**2*d**8*x**4 + 5040*a**4*b**6*c*d**9*x**5 + 420*a** 4*b**6*d**10*x**6 - 529200*a**3*b**7*c**6*d**4*x - 317520*a**3*b**7*c**5*d **5*x**2 - 176400*a**3*b**7*c**4*d**6*x**3 - 75600*a**3*b**7*c**3*d**7*x** 4 - 22680*a**3*b**7*c**2*d**8*x**5 - 4200*a**3*b**7*c*d**9*x**6 - 360*a**3 *b**7*d**10*x**7 + 302400*a**2*b**8*c**7*d**3*x + 264600*a**2*b**8*c**6*d* *4*x**2 + 211680*a**2*b**8*c**5*d**5*x**3 + 132300*a**2*b**8*c**4*d**6*x** 4 + 60480*a**2*b**8*c**3*d**7*x**5 + 18900*a**2*b**8*c**2*d**8*x**6 + 3...