Integrand size = 15, antiderivative size = 257 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\frac {d^{10} x}{b^{10}}-\frac {(b c-a d)^{10}}{9 b^{11} (a+b x)^9}-\frac {5 d (b c-a d)^9}{4 b^{11} (a+b x)^8}-\frac {45 d^2 (b c-a d)^8}{7 b^{11} (a+b x)^7}-\frac {20 d^3 (b c-a d)^7}{b^{11} (a+b x)^6}-\frac {42 d^4 (b c-a d)^6}{b^{11} (a+b x)^5}-\frac {63 d^5 (b c-a d)^5}{b^{11} (a+b x)^4}-\frac {70 d^6 (b c-a d)^4}{b^{11} (a+b x)^3}-\frac {60 d^7 (b c-a d)^3}{b^{11} (a+b x)^2}-\frac {45 d^8 (b c-a d)^2}{b^{11} (a+b x)}+\frac {10 d^9 (b c-a d) \log (a+b x)}{b^{11}} \]
d^10*x/b^10-1/9*(-a*d+b*c)^10/b^11/(b*x+a)^9-5/4*d*(-a*d+b*c)^9/b^11/(b*x+ a)^8-45/7*d^2*(-a*d+b*c)^8/b^11/(b*x+a)^7-20*d^3*(-a*d+b*c)^7/b^11/(b*x+a) ^6-42*d^4*(-a*d+b*c)^6/b^11/(b*x+a)^5-63*d^5*(-a*d+b*c)^5/b^11/(b*x+a)^4-7 0*d^6*(-a*d+b*c)^4/b^11/(b*x+a)^3-60*d^7*(-a*d+b*c)^3/b^11/(b*x+a)^2-45*d^ 8*(-a*d+b*c)^2/b^11/(b*x+a)+10*d^9*(-a*d+b*c)*ln(b*x+a)/b^11
Leaf count is larger than twice the leaf count of optimal. \(708\) vs. \(2(257)=514\).
Time = 0.25 (sec) , antiderivative size = 708, normalized size of antiderivative = 2.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=-\frac {4861 a^{10} d^{10}+a^9 b d^9 (-7129 c+41229 d x)+9 a^8 b^2 d^8 \left (140 c^2-6849 c d x+17064 d^2 x^2\right )+12 a^7 b^3 d^7 \left (35 c^3+945 c^2 d x-19602 c d^2 x^2+27342 d^3 x^3\right )+42 a^6 b^4 d^6 \left (5 c^4+90 c^3 d x+1080 c^2 d^2 x^2-12348 c d^3 x^3+10458 d^4 x^4\right )+126 a^5 b^5 d^5 \left (c^5+15 c^4 d x+120 c^3 d^2 x^2+840 c^2 d^3 x^3-5754 c d^4 x^4+2982 d^5 x^5\right )+42 a^4 b^6 d^4 \left (2 c^6+27 c^5 d x+180 c^4 d^2 x^2+840 c^3 d^3 x^3+3780 c^2 d^4 x^4-15750 c d^5 x^5+4704 d^6 x^6\right )+12 a^3 b^7 d^3 \left (5 c^7+63 c^6 d x+378 c^5 d^2 x^2+1470 c^4 d^3 x^3+4410 c^3 d^4 x^4+13230 c^2 d^5 x^5-32340 c d^6 x^6+4536 d^7 x^7\right )+9 a^2 b^8 d^2 \left (5 c^8+60 c^7 d x+336 c^6 d^2 x^2+1176 c^5 d^3 x^3+2940 c^4 d^4 x^4+5880 c^3 d^5 x^5+11760 c^2 d^6 x^6-15120 c d^7 x^7+252 d^8 x^8\right )+a b^9 d \left (35 c^9+405 c^8 d x+2160 c^7 d^2 x^2+7056 c^6 d^3 x^3+15876 c^5 d^4 x^4+26460 c^4 d^5 x^5+35280 c^3 d^6 x^6+45360 c^2 d^7 x^7-22680 c d^8 x^8-2268 d^9 x^9\right )+b^{10} \left (28 c^{10}+315 c^9 d x+1620 c^8 d^2 x^2+5040 c^7 d^3 x^3+10584 c^6 d^4 x^4+15876 c^5 d^5 x^5+17640 c^4 d^6 x^6+15120 c^3 d^7 x^7+11340 c^2 d^8 x^8-252 d^{10} x^{10}\right )+2520 d^9 (-b c+a d) (a+b x)^9 \log (a+b x)}{252 b^{11} (a+b x)^9} \]
-1/252*(4861*a^10*d^10 + a^9*b*d^9*(-7129*c + 41229*d*x) + 9*a^8*b^2*d^8*( 140*c^2 - 6849*c*d*x + 17064*d^2*x^2) + 12*a^7*b^3*d^7*(35*c^3 + 945*c^2*d *x - 19602*c*d^2*x^2 + 27342*d^3*x^3) + 42*a^6*b^4*d^6*(5*c^4 + 90*c^3*d*x + 1080*c^2*d^2*x^2 - 12348*c*d^3*x^3 + 10458*d^4*x^4) + 126*a^5*b^5*d^5*( c^5 + 15*c^4*d*x + 120*c^3*d^2*x^2 + 840*c^2*d^3*x^3 - 5754*c*d^4*x^4 + 29 82*d^5*x^5) + 42*a^4*b^6*d^4*(2*c^6 + 27*c^5*d*x + 180*c^4*d^2*x^2 + 840*c ^3*d^3*x^3 + 3780*c^2*d^4*x^4 - 15750*c*d^5*x^5 + 4704*d^6*x^6) + 12*a^3*b ^7*d^3*(5*c^7 + 63*c^6*d*x + 378*c^5*d^2*x^2 + 1470*c^4*d^3*x^3 + 4410*c^3 *d^4*x^4 + 13230*c^2*d^5*x^5 - 32340*c*d^6*x^6 + 4536*d^7*x^7) + 9*a^2*b^8 *d^2*(5*c^8 + 60*c^7*d*x + 336*c^6*d^2*x^2 + 1176*c^5*d^3*x^3 + 2940*c^4*d ^4*x^4 + 5880*c^3*d^5*x^5 + 11760*c^2*d^6*x^6 - 15120*c*d^7*x^7 + 252*d^8* x^8) + a*b^9*d*(35*c^9 + 405*c^8*d*x + 2160*c^7*d^2*x^2 + 7056*c^6*d^3*x^3 + 15876*c^5*d^4*x^4 + 26460*c^4*d^5*x^5 + 35280*c^3*d^6*x^6 + 45360*c^2*d ^7*x^7 - 22680*c*d^8*x^8 - 2268*d^9*x^9) + b^10*(28*c^10 + 315*c^9*d*x + 1 620*c^8*d^2*x^2 + 5040*c^7*d^3*x^3 + 10584*c^6*d^4*x^4 + 15876*c^5*d^5*x^5 + 17640*c^4*d^6*x^6 + 15120*c^3*d^7*x^7 + 11340*c^2*d^8*x^8 - 252*d^10*x^ 10) + 2520*d^9*(-(b*c) + a*d)*(a + b*x)^9*Log[a + b*x])/(b^11*(a + b*x)^9)
Time = 0.55 (sec) , antiderivative size = 257, 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.
| \(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {10 d^9 (b c-a d)}{b^{10} (a+b x)}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)^2}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^3}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^4}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^5}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^6}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^7}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^8}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^9}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^{10}}+\frac {d^{10}}{b^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {10 d^9 (b c-a d) \log (a+b x)}{b^{11}}-\frac {45 d^8 (b c-a d)^2}{b^{11} (a+b x)}-\frac {60 d^7 (b c-a d)^3}{b^{11} (a+b x)^2}-\frac {70 d^6 (b c-a d)^4}{b^{11} (a+b x)^3}-\frac {63 d^5 (b c-a d)^5}{b^{11} (a+b x)^4}-\frac {42 d^4 (b c-a d)^6}{b^{11} (a+b x)^5}-\frac {20 d^3 (b c-a d)^7}{b^{11} (a+b x)^6}-\frac {45 d^2 (b c-a d)^8}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{4 b^{11} (a+b x)^8}-\frac {(b c-a d)^{10}}{9 b^{11} (a+b x)^9}+\frac {d^{10} x}{b^{10}}\) |
(d^10*x)/b^10 - (b*c - a*d)^10/(9*b^11*(a + b*x)^9) - (5*d*(b*c - a*d)^9)/ (4*b^11*(a + b*x)^8) - (45*d^2*(b*c - a*d)^8)/(7*b^11*(a + b*x)^7) - (20*d ^3*(b*c - a*d)^7)/(b^11*(a + b*x)^6) - (42*d^4*(b*c - a*d)^6)/(b^11*(a + b *x)^5) - (63*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^4) - (70*d^6*(b*c - a*d)^4 )/(b^11*(a + b*x)^3) - (60*d^7*(b*c - a*d)^3)/(b^11*(a + b*x)^2) - (45*d^8 *(b*c - a*d)^2)/(b^11*(a + b*x)) + (10*d^9*(b*c - a*d)*Log[a + b*x])/b^11
3.14.21.3.1 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. \(847\) vs. \(2(251)=502\).
Time = 0.23 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.30
| method | result | size |
| risch | \(\frac {d^{10} x}{b^{10}}+\frac {\left (-45 a^{2} b^{7} d^{10}+90 a \,b^{8} c \,d^{9}-45 b^{9} c^{2} d^{8}\right ) x^{8}-60 b^{6} d^{7} \left (5 a^{3} d^{3}-9 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{7}-70 b^{5} d^{6} \left (13 a^{4} d^{4}-22 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) x^{6}-21 b^{4} d^{5} \left (77 a^{5} d^{5}-125 a^{4} b c \,d^{4}+30 a^{3} b^{2} c^{2} d^{3}+10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right ) x^{5}-21 b^{3} d^{4} \left (87 a^{6} d^{6}-137 a^{5} b c \,d^{5}+30 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+5 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +2 b^{6} c^{6}\right ) x^{4}-2 b^{2} d^{3} \left (669 a^{7} d^{7}-1029 a^{6} b c \,d^{6}+210 a^{5} b^{2} c^{2} d^{5}+70 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+14 a \,b^{6} c^{6} d +10 b^{7} c^{7}\right ) x^{3}-\frac {3 b \,d^{2} \left (1443 a^{8} d^{8}-2178 a^{7} b c \,d^{7}+420 a^{6} b^{2} c^{2} d^{6}+140 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}+42 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}+20 a \,b^{7} c^{7} d +15 b^{8} c^{8}\right ) x^{2}}{7}-\frac {d \left (4609 a^{9} d^{9}-6849 a^{8} b c \,d^{8}+1260 a^{7} b^{2} c^{2} d^{7}+420 a^{6} b^{3} c^{3} d^{6}+210 a^{5} b^{4} c^{4} d^{5}+126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}+60 a^{2} b^{7} c^{7} d^{2}+45 a \,b^{8} c^{8} d +35 b^{9} c^{9}\right ) x}{28}-\frac {4861 a^{10} d^{10}-7129 a^{9} b c \,d^{9}+1260 a^{8} b^{2} c^{2} d^{8}+420 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+84 a^{4} b^{6} c^{6} d^{4}+60 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+35 a \,b^{9} c^{9} d +28 b^{10} c^{10}}{252 b}}{b^{10} \left (b x +a \right )^{9}}-\frac {10 d^{10} \ln \left (b x +a \right ) a}{b^{11}}+\frac {10 d^{9} \ln \left (b x +a \right ) c}{b^{10}}\) | \(848\) |
| default | \(\frac {d^{10} x}{b^{10}}-\frac {70 d^{6} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{b^{11} \left (b x +a \right )^{3}}-\frac {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}}{9 b^{11} \left (b x +a \right )^{9}}-\frac {10 d^{9} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{11}}+\frac {20 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{6}}+\frac {5 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{4 b^{11} \left (b x +a \right )^{8}}+\frac {63 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{11} \left (b x +a \right )^{4}}-\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{7 b^{11} \left (b x +a \right )^{7}}+\frac {60 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{11} \left (b x +a \right )^{2}}-\frac {42 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{b^{11} \left (b x +a \right )^{5}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{11} \left (b x +a \right )}\) | \(859\) |
| norman | \(\frac {\frac {d^{10} x^{10}}{b}-\frac {7129 a^{10} d^{10}-7129 a^{9} b c \,d^{9}+1260 a^{8} b^{2} c^{2} d^{8}+420 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+84 a^{4} b^{6} c^{6} d^{4}+60 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+35 a \,b^{9} c^{9} d +28 b^{10} c^{10}}{252 b^{11}}-\frac {9 \left (10 a^{2} d^{10}-10 a b c \,d^{9}+5 b^{2} c^{2} d^{8}\right ) x^{8}}{b^{3}}-\frac {12 \left (45 a^{3} d^{10}-45 a^{2} b c \,d^{9}+15 a \,b^{2} c^{2} d^{8}+5 b^{3} c^{3} d^{7}\right ) x^{7}}{b^{4}}-\frac {14 \left (110 a^{4} d^{10}-110 a^{3} b c \,d^{9}+30 a^{2} b^{2} c^{2} d^{8}+10 a \,b^{3} c^{3} d^{7}+5 b^{4} c^{4} d^{6}\right ) x^{6}}{b^{5}}-\frac {21 \left (125 a^{5} d^{10}-125 a^{4} b c \,d^{9}+30 a^{3} b^{2} c^{2} d^{8}+10 a^{2} b^{3} c^{3} d^{7}+5 a \,b^{4} c^{4} d^{6}+3 b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}-\frac {21 \left (137 a^{6} d^{10}-137 a^{5} b c \,d^{9}+30 a^{4} b^{2} c^{2} d^{8}+10 a^{3} b^{3} c^{3} d^{7}+5 a^{2} b^{4} c^{4} d^{6}+3 a \,b^{5} c^{5} d^{5}+2 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}-\frac {2 \left (1029 a^{7} d^{10}-1029 a^{6} b c \,d^{9}+210 a^{5} b^{2} c^{2} d^{8}+70 a^{4} b^{3} c^{3} d^{7}+35 a^{3} b^{4} c^{4} d^{6}+21 a^{2} b^{5} c^{5} d^{5}+14 a \,b^{6} c^{6} d^{4}+10 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}-\frac {3 \left (2178 a^{8} d^{10}-2178 a^{7} b c \,d^{9}+420 a^{6} b^{2} c^{2} d^{8}+140 a^{5} b^{3} c^{3} d^{7}+70 a^{4} b^{4} c^{4} d^{6}+42 a^{3} b^{5} c^{5} d^{5}+28 a^{2} b^{6} c^{6} d^{4}+20 a \,b^{7} c^{7} d^{3}+15 b^{8} c^{8} d^{2}\right ) x^{2}}{7 b^{9}}-\frac {\left (6849 a^{9} d^{10}-6849 a^{8} b c \,d^{9}+1260 a^{7} b^{2} c^{2} d^{8}+420 a^{6} b^{3} c^{3} d^{7}+210 a^{5} b^{4} c^{4} d^{6}+126 a^{4} b^{5} c^{5} d^{5}+84 a^{3} b^{6} c^{6} d^{4}+60 a^{2} b^{7} c^{7} d^{3}+45 a \,b^{8} c^{8} d^{2}+35 b^{9} c^{9} d \right ) x}{28 b^{10}}}{\left (b x +a \right )^{9}}-\frac {10 d^{9} \left (a d -b c \right ) \ln \left (b x +a \right )}{b^{11}}\) | \(859\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1325\) |
d^10*x/b^10+((-45*a^2*b^7*d^10+90*a*b^8*c*d^9-45*b^9*c^2*d^8)*x^8-60*b^6*d ^7*(5*a^3*d^3-9*a^2*b*c*d^2+3*a*b^2*c^2*d+b^3*c^3)*x^7-70*b^5*d^6*(13*a^4* d^4-22*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2+2*a*b^3*c^3*d+b^4*c^4)*x^6-21*b^4*d^5 *(77*a^5*d^5-125*a^4*b*c*d^4+30*a^3*b^2*c^2*d^3+10*a^2*b^3*c^3*d^2+5*a*b^4 *c^4*d+3*b^5*c^5)*x^5-21*b^3*d^4*(87*a^6*d^6-137*a^5*b*c*d^5+30*a^4*b^2*c^ 2*d^4+10*a^3*b^3*c^3*d^3+5*a^2*b^4*c^4*d^2+3*a*b^5*c^5*d+2*b^6*c^6)*x^4-2* b^2*d^3*(669*a^7*d^7-1029*a^6*b*c*d^6+210*a^5*b^2*c^2*d^5+70*a^4*b^3*c^3*d ^4+35*a^3*b^4*c^4*d^3+21*a^2*b^5*c^5*d^2+14*a*b^6*c^6*d+10*b^7*c^7)*x^3-3/ 7*b*d^2*(1443*a^8*d^8-2178*a^7*b*c*d^7+420*a^6*b^2*c^2*d^6+140*a^5*b^3*c^3 *d^5+70*a^4*b^4*c^4*d^4+42*a^3*b^5*c^5*d^3+28*a^2*b^6*c^6*d^2+20*a*b^7*c^7 *d+15*b^8*c^8)*x^2-1/28*d*(4609*a^9*d^9-6849*a^8*b*c*d^8+1260*a^7*b^2*c^2* d^7+420*a^6*b^3*c^3*d^6+210*a^5*b^4*c^4*d^5+126*a^4*b^5*c^5*d^4+84*a^3*b^6 *c^6*d^3+60*a^2*b^7*c^7*d^2+45*a*b^8*c^8*d+35*b^9*c^9)*x-1/252/b*(4861*a^1 0*d^10-7129*a^9*b*c*d^9+1260*a^8*b^2*c^2*d^8+420*a^7*b^3*c^3*d^7+210*a^6*b ^4*c^4*d^6+126*a^5*b^5*c^5*d^5+84*a^4*b^6*c^6*d^4+60*a^3*b^7*c^7*d^3+45*a^ 2*b^8*c^8*d^2+35*a*b^9*c^9*d+28*b^10*c^10))/b^10/(b*x+a)^9-10/b^11*d^10*ln (b*x+a)*a+10/b^10*d^9*ln(b*x+a)*c
Leaf count of result is larger than twice the leaf count of optimal. 1216 vs. \(2 (251) = 502\).
Time = 0.24 (sec) , antiderivative size = 1216, normalized size of antiderivative = 4.73 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\text {Too large to display} \]
1/252*(252*b^10*d^10*x^10 + 2268*a*b^9*d^10*x^9 - 28*b^10*c^10 - 35*a*b^9* c^9*d - 45*a^2*b^8*c^8*d^2 - 60*a^3*b^7*c^7*d^3 - 84*a^4*b^6*c^6*d^4 - 126 *a^5*b^5*c^5*d^5 - 210*a^6*b^4*c^4*d^6 - 420*a^7*b^3*c^3*d^7 - 1260*a^8*b^ 2*c^2*d^8 + 7129*a^9*b*c*d^9 - 4861*a^10*d^10 - 2268*(5*b^10*c^2*d^8 - 10* a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 - 3024*(5*b^10*c^3*d^7 + 15*a*b^9*c^2*d^8 - 45*a^2*b^8*c*d^9 + 18*a^3*b^7*d^10)*x^7 - 3528*(5*b^10*c^4*d^6 + 10*a*b^ 9*c^3*d^7 + 30*a^2*b^8*c^2*d^8 - 110*a^3*b^7*c*d^9 + 56*a^4*b^6*d^10)*x^6 - 5292*(3*b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 30*a^3*b^7 *c^2*d^8 - 125*a^4*b^6*c*d^9 + 71*a^5*b^5*d^10)*x^5 - 5292*(2*b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 30*a^4*b^6*c ^2*d^8 - 137*a^5*b^5*c*d^9 + 83*a^6*b^4*d^10)*x^4 - 504*(10*b^10*c^7*d^3 + 14*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 70*a^4*b^6*c ^3*d^7 + 210*a^5*b^5*c^2*d^8 - 1029*a^6*b^4*c*d^9 + 651*a^7*b^3*d^10)*x^3 - 108*(15*b^10*c^8*d^2 + 20*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 42*a^3*b^ 7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 140*a^5*b^5*c^3*d^7 + 420*a^6*b^4*c^2*d^8 - 2178*a^7*b^3*c*d^9 + 1422*a^8*b^2*d^10)*x^2 - 9*(35*b^10*c^9*d + 45*a*b ^9*c^8*d^2 + 60*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 420*a^6*b^4*c^3*d^7 + 1260*a^7*b^3*c^2*d^8 - 6849 *a^8*b^2*c*d^9 + 4581*a^9*b*d^10)*x + 2520*(a^9*b*c*d^9 - a^10*d^10 + (b^1 0*c*d^9 - a*b^9*d^10)*x^9 + 9*(a*b^9*c*d^9 - a^2*b^8*d^10)*x^8 + 36*(a^...
Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 957 vs. \(2 (251) = 502\).
Time = 0.27 (sec) , antiderivative size = 957, normalized size of antiderivative = 3.72 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx =\text {Too large to display} \]
d^10*x/b^10 - 1/252*(28*b^10*c^10 + 35*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 210*a^6*b^ 4*c^4*d^6 + 420*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 - 7129*a^9*b*c*d^9 + 4861*a^10*d^10 + 11340*(b^10*c^2*d^8 - 2*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 15120*(b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 - 9*a^2*b^8*c*d^9 + 5*a^3*b^7*d^1 0)*x^7 + 17640*(b^10*c^4*d^6 + 2*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 22*a^ 3*b^7*c*d^9 + 13*a^4*b^6*d^10)*x^6 + 5292*(3*b^10*c^5*d^5 + 5*a*b^9*c^4*d^ 6 + 10*a^2*b^8*c^3*d^7 + 30*a^3*b^7*c^2*d^8 - 125*a^4*b^6*c*d^9 + 77*a^5*b ^5*d^10)*x^5 + 5292*(2*b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 30*a^4*b^6*c^2*d^8 - 137*a^5*b^5*c*d^9 + 87*a^6*b^4 *d^10)*x^4 + 504*(10*b^10*c^7*d^3 + 14*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 70*a^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 - 1029*a^6 *b^4*c*d^9 + 669*a^7*b^3*d^10)*x^3 + 108*(15*b^10*c^8*d^2 + 20*a*b^9*c^7*d ^3 + 28*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 140*a^ 5*b^5*c^3*d^7 + 420*a^6*b^4*c^2*d^8 - 2178*a^7*b^3*c*d^9 + 1443*a^8*b^2*d^ 10)*x^2 + 9*(35*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 60*a^2*b^8*c^7*d^3 + 84*a^ 3*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 420*a^6*b^4*c^ 3*d^7 + 1260*a^7*b^3*c^2*d^8 - 6849*a^8*b^2*c*d^9 + 4609*a^9*b*d^10)*x)/(b ^20*x^9 + 9*a*b^19*x^8 + 36*a^2*b^18*x^7 + 84*a^3*b^17*x^6 + 126*a^4*b^16* x^5 + 126*a^5*b^15*x^4 + 84*a^6*b^14*x^3 + 36*a^7*b^13*x^2 + 9*a^8*b^12...
Leaf count of result is larger than twice the leaf count of optimal. 867 vs. \(2 (251) = 502\).
Time = 0.30 (sec) , antiderivative size = 867, normalized size of antiderivative = 3.37 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx =\text {Too large to display} \]
d^10*x/b^10 + 10*(b*c*d^9 - a*d^10)*log(abs(b*x + a))/b^11 - 1/252*(28*b^1 0*c^10 + 35*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 84*a^4 *b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 210*a^6*b^4*c^4*d^6 + 420*a^7*b^3*c^3 *d^7 + 1260*a^8*b^2*c^2*d^8 - 7129*a^9*b*c*d^9 + 4861*a^10*d^10 + 11340*(b ^10*c^2*d^8 - 2*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 15120*(b^10*c^3*d^7 + 3* a*b^9*c^2*d^8 - 9*a^2*b^8*c*d^9 + 5*a^3*b^7*d^10)*x^7 + 17640*(b^10*c^4*d^ 6 + 2*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 22*a^3*b^7*c*d^9 + 13*a^4*b^6*d^ 10)*x^6 + 5292*(3*b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 30 *a^3*b^7*c^2*d^8 - 125*a^4*b^6*c*d^9 + 77*a^5*b^5*d^10)*x^5 + 5292*(2*b^10 *c^6*d^4 + 3*a*b^9*c^5*d^5 + 5*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 30*a ^4*b^6*c^2*d^8 - 137*a^5*b^5*c*d^9 + 87*a^6*b^4*d^10)*x^4 + 504*(10*b^10*c ^7*d^3 + 14*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 35*a^3*b^7*c^4*d^6 + 70*a ^4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 - 1029*a^6*b^4*c*d^9 + 669*a^7*b^3*d^ 10)*x^3 + 108*(15*b^10*c^8*d^2 + 20*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 4 2*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 + 140*a^5*b^5*c^3*d^7 + 420*a^6*b^4 *c^2*d^8 - 2178*a^7*b^3*c*d^9 + 1443*a^8*b^2*d^10)*x^2 + 9*(35*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 60*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 126*a^4*b^6 *c^5*d^5 + 210*a^5*b^5*c^4*d^6 + 420*a^6*b^4*c^3*d^7 + 1260*a^7*b^3*c^2*d^ 8 - 6849*a^8*b^2*c*d^9 + 4609*a^9*b*d^10)*x)/((b*x + a)^9*b^11)
Time = 0.54 (sec) , antiderivative size = 955, normalized size of antiderivative = 3.72 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx=\frac {d^{10}\,x}{b^{10}}-\frac {\ln \left (a+b\,x\right )\,\left (10\,a\,d^{10}-10\,b\,c\,d^9\right )}{b^{11}}-\frac {x^4\,\left (1827\,a^6\,b^3\,d^{10}-2877\,a^5\,b^4\,c\,d^9+630\,a^4\,b^5\,c^2\,d^8+210\,a^3\,b^6\,c^3\,d^7+105\,a^2\,b^7\,c^4\,d^6+63\,a\,b^8\,c^5\,d^5+42\,b^9\,c^6\,d^4\right )+x^6\,\left (910\,a^4\,b^5\,d^{10}-1540\,a^3\,b^6\,c\,d^9+420\,a^2\,b^7\,c^2\,d^8+140\,a\,b^8\,c^3\,d^7+70\,b^9\,c^4\,d^6\right )+\frac {4861\,a^{10}\,d^{10}-7129\,a^9\,b\,c\,d^9+1260\,a^8\,b^2\,c^2\,d^8+420\,a^7\,b^3\,c^3\,d^7+210\,a^6\,b^4\,c^4\,d^6+126\,a^5\,b^5\,c^5\,d^5+84\,a^4\,b^6\,c^6\,d^4+60\,a^3\,b^7\,c^7\,d^3+45\,a^2\,b^8\,c^8\,d^2+35\,a\,b^9\,c^9\,d+28\,b^{10}\,c^{10}}{252\,b}+x\,\left (\frac {4609\,a^9\,d^{10}}{28}-\frac {6849\,a^8\,b\,c\,d^9}{28}+45\,a^7\,b^2\,c^2\,d^8+15\,a^6\,b^3\,c^3\,d^7+\frac {15\,a^5\,b^4\,c^4\,d^6}{2}+\frac {9\,a^4\,b^5\,c^5\,d^5}{2}+3\,a^3\,b^6\,c^6\,d^4+\frac {15\,a^2\,b^7\,c^7\,d^3}{7}+\frac {45\,a\,b^8\,c^8\,d^2}{28}+\frac {5\,b^9\,c^9\,d}{4}\right )+x^8\,\left (45\,a^2\,b^7\,d^{10}-90\,a\,b^8\,c\,d^9+45\,b^9\,c^2\,d^8\right )+x^3\,\left (1338\,a^7\,b^2\,d^{10}-2058\,a^6\,b^3\,c\,d^9+420\,a^5\,b^4\,c^2\,d^8+140\,a^4\,b^5\,c^3\,d^7+70\,a^3\,b^6\,c^4\,d^6+42\,a^2\,b^7\,c^5\,d^5+28\,a\,b^8\,c^6\,d^4+20\,b^9\,c^7\,d^3\right )+x^2\,\left (\frac {4329\,a^8\,b\,d^{10}}{7}-\frac {6534\,a^7\,b^2\,c\,d^9}{7}+180\,a^6\,b^3\,c^2\,d^8+60\,a^5\,b^4\,c^3\,d^7+30\,a^4\,b^5\,c^4\,d^6+18\,a^3\,b^6\,c^5\,d^5+12\,a^2\,b^7\,c^6\,d^4+\frac {60\,a\,b^8\,c^7\,d^3}{7}+\frac {45\,b^9\,c^8\,d^2}{7}\right )+x^5\,\left (1617\,a^5\,b^4\,d^{10}-2625\,a^4\,b^5\,c\,d^9+630\,a^3\,b^6\,c^2\,d^8+210\,a^2\,b^7\,c^3\,d^7+105\,a\,b^8\,c^4\,d^6+63\,b^9\,c^5\,d^5\right )+x^7\,\left (300\,a^3\,b^6\,d^{10}-540\,a^2\,b^7\,c\,d^9+180\,a\,b^8\,c^2\,d^8+60\,b^9\,c^3\,d^7\right )}{a^9\,b^{10}+9\,a^8\,b^{11}\,x+36\,a^7\,b^{12}\,x^2+84\,a^6\,b^{13}\,x^3+126\,a^5\,b^{14}\,x^4+126\,a^4\,b^{15}\,x^5+84\,a^3\,b^{16}\,x^6+36\,a^2\,b^{17}\,x^7+9\,a\,b^{18}\,x^8+b^{19}\,x^9} \]
(d^10*x)/b^10 - (log(a + b*x)*(10*a*d^10 - 10*b*c*d^9))/b^11 - (x^4*(1827* a^6*b^3*d^10 + 42*b^9*c^6*d^4 + 63*a*b^8*c^5*d^5 - 2877*a^5*b^4*c*d^9 + 10 5*a^2*b^7*c^4*d^6 + 210*a^3*b^6*c^3*d^7 + 630*a^4*b^5*c^2*d^8) + x^6*(910* a^4*b^5*d^10 + 70*b^9*c^4*d^6 + 140*a*b^8*c^3*d^7 - 1540*a^3*b^6*c*d^9 + 4 20*a^2*b^7*c^2*d^8) + (4861*a^10*d^10 + 28*b^10*c^10 + 45*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 210*a^6* b^4*c^4*d^6 + 420*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 35*a*b^9*c^9*d - 7129*a^9*b*c*d^9)/(252*b) + x*((4609*a^9*d^10)/28 + (5*b^9*c^9*d)/4 + (4 5*a*b^8*c^8*d^2)/28 + (15*a^2*b^7*c^7*d^3)/7 + 3*a^3*b^6*c^6*d^4 + (9*a^4* b^5*c^5*d^5)/2 + (15*a^5*b^4*c^4*d^6)/2 + 15*a^6*b^3*c^3*d^7 + 45*a^7*b^2* c^2*d^8 - (6849*a^8*b*c*d^9)/28) + x^8*(45*a^2*b^7*d^10 + 45*b^9*c^2*d^8 - 90*a*b^8*c*d^9) + x^3*(1338*a^7*b^2*d^10 + 20*b^9*c^7*d^3 + 28*a*b^8*c^6* d^4 - 2058*a^6*b^3*c*d^9 + 42*a^2*b^7*c^5*d^5 + 70*a^3*b^6*c^4*d^6 + 140*a ^4*b^5*c^3*d^7 + 420*a^5*b^4*c^2*d^8) + x^2*((4329*a^8*b*d^10)/7 + (45*b^9 *c^8*d^2)/7 + (60*a*b^8*c^7*d^3)/7 - (6534*a^7*b^2*c*d^9)/7 + 12*a^2*b^7*c ^6*d^4 + 18*a^3*b^6*c^5*d^5 + 30*a^4*b^5*c^4*d^6 + 60*a^5*b^4*c^3*d^7 + 18 0*a^6*b^3*c^2*d^8) + x^5*(1617*a^5*b^4*d^10 + 63*b^9*c^5*d^5 + 105*a*b^8*c ^4*d^6 - 2625*a^4*b^5*c*d^9 + 210*a^2*b^7*c^3*d^7 + 630*a^3*b^6*c^2*d^8) + x^7*(300*a^3*b^6*d^10 + 60*b^9*c^3*d^7 + 180*a*b^8*c^2*d^8 - 540*a^2*b^7* c*d^9))/(a^9*b^10 + b^19*x^9 + 9*a^8*b^11*x + 9*a*b^18*x^8 + 36*a^7*b^1...
Time = 0.00 (sec) , antiderivative size = 1314, normalized size of antiderivative = 5.11 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{10}} \, dx =\text {Too large to display} \]
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**10 + 10*a**9*b*x + 45*a**8*b**2*x**2 + 120*a**7*b**3*x**3 + 210*a**6*b**4*x**4 + 252*a**5* b**5*x**5 + 210*a**4*b**6*x**6 + 120*a**3*b**7*x**7 + 45*a**2*b**8*x**8 + 10*a*b**9*x**9 + b**10*x**10),x)
( - 2520*log(a + b*x)*a**11*d**10 + 2520*log(a + b*x)*a**10*b*c*d**9 - 226 80*log(a + b*x)*a**10*b*d**10*x + 22680*log(a + b*x)*a**9*b**2*c*d**9*x - 90720*log(a + b*x)*a**9*b**2*d**10*x**2 + 90720*log(a + b*x)*a**8*b**3*c*d **9*x**2 - 211680*log(a + b*x)*a**8*b**3*d**10*x**3 + 211680*log(a + b*x)* a**7*b**4*c*d**9*x**3 - 317520*log(a + b*x)*a**7*b**4*d**10*x**4 + 317520* log(a + b*x)*a**6*b**5*c*d**9*x**4 - 317520*log(a + b*x)*a**6*b**5*d**10*x **5 + 317520*log(a + b*x)*a**5*b**6*c*d**9*x**5 - 211680*log(a + b*x)*a**5 *b**6*d**10*x**6 + 211680*log(a + b*x)*a**4*b**7*c*d**9*x**6 - 90720*log(a + b*x)*a**4*b**7*d**10*x**7 + 90720*log(a + b*x)*a**3*b**8*c*d**9*x**7 - 22680*log(a + b*x)*a**3*b**8*d**10*x**8 + 22680*log(a + b*x)*a**2*b**9*c*d **9*x**8 - 2520*log(a + b*x)*a**2*b**9*d**10*x**9 + 2520*log(a + b*x)*a*b* *10*c*d**9*x**9 - 4609*a**11*d**10 + 4609*a**10*b*c*d**9 - 38961*a**10*b*d **10*x + 38961*a**9*b**2*c*d**9*x - 144504*a**9*b**2*d**10*x**2 - 420*a**8 *b**3*c**3*d**7 + 144504*a**8*b**3*c*d**9*x**2 - 306936*a**8*b**3*d**10*x* *3 - 210*a**7*b**4*c**4*d**6 - 3780*a**7*b**4*c**3*d**7*x + 306936*a**7*b* *4*c*d**9*x**3 - 407484*a**7*b**4*d**10*x**4 - 126*a**6*b**5*c**5*d**5 - 1 890*a**6*b**5*c**4*d**6*x - 15120*a**6*b**5*c**3*d**7*x**2 + 407484*a**6*b **5*c*d**9*x**4 - 343980*a**6*b**5*d**10*x**5 - 84*a**5*b**6*c**6*d**4 - 1 134*a**5*b**6*c**5*d**5*x - 7560*a**5*b**6*c**4*d**6*x**2 - 35280*a**5*b** 6*c**3*d**7*x**3 + 343980*a**5*b**6*c*d**9*x**5 - 176400*a**5*b**6*d**1...