\(\int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx\) [1331]

Optimal result

Integrand size = 15, antiderivative size = 273 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx=-\frac {(b c-a d)^{10}}{19 b^{11} (a+b x)^{19}}-\frac {5 d (b c-a d)^9}{9 b^{11} (a+b x)^{18}}-\frac {45 d^2 (b c-a d)^8}{17 b^{11} (a+b x)^{17}}-\frac {15 d^3 (b c-a d)^7}{2 b^{11} (a+b x)^{16}}-\frac {14 d^4 (b c-a d)^6}{b^{11} (a+b x)^{15}}-\frac {18 d^5 (b c-a d)^5}{b^{11} (a+b x)^{14}}-\frac {210 d^6 (b c-a d)^4}{13 b^{11} (a+b x)^{13}}-\frac {10 d^7 (b c-a d)^3}{b^{11} (a+b x)^{12}}-\frac {45 d^8 (b c-a d)^2}{11 b^{11} (a+b x)^{11}}-\frac {d^9 (b c-a d)}{b^{11} (a+b x)^{10}}-\frac {d^{10}}{9 b^{11} (a+b x)^9} \]

[Out]

-1/19*(-a*d+b*c)^10/b^11/(b*x+a)^19-5/9*d*(-a*d+b*c)^9/b^11/(b*x+a)^18-45/17*d^2*(-a*d+b*c)^8/b^11/(b*x+a)^17-
15/2*d^3*(-a*d+b*c)^7/b^11/(b*x+a)^16-14*d^4*(-a*d+b*c)^6/b^11/(b*x+a)^15-18*d^5*(-a*d+b*c)^5/b^11/(b*x+a)^14-
210/13*d^6*(-a*d+b*c)^4/b^11/(b*x+a)^13-10*d^7*(-a*d+b*c)^3/b^11/(b*x+a)^12-45/11*d^8*(-a*d+b*c)^2/b^11/(b*x+a
)^11-d^9*(-a*d+b*c)/b^11/(b*x+a)^10-1/9*d^10/b^11/(b*x+a)^9

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx=-\frac {d^9 (b c-a d)}{b^{11} (a+b x)^{10}}-\frac {45 d^8 (b c-a d)^2}{11 b^{11} (a+b x)^{11}}-\frac {10 d^7 (b c-a d)^3}{b^{11} (a+b x)^{12}}-\frac {210 d^6 (b c-a d)^4}{13 b^{11} (a+b x)^{13}}-\frac {18 d^5 (b c-a d)^5}{b^{11} (a+b x)^{14}}-\frac {14 d^4 (b c-a d)^6}{b^{11} (a+b x)^{15}}-\frac {15 d^3 (b c-a d)^7}{2 b^{11} (a+b x)^{16}}-\frac {45 d^2 (b c-a d)^8}{17 b^{11} (a+b x)^{17}}-\frac {5 d (b c-a d)^9}{9 b^{11} (a+b x)^{18}}-\frac {(b c-a d)^{10}}{19 b^{11} (a+b x)^{19}}-\frac {d^{10}}{9 b^{11} (a+b x)^9} \]

[In]

Int[(c + d*x)^10/(a + b*x)^20,x]

[Out]

-1/19*(b*c - a*d)^10/(b^11*(a + b*x)^19) - (5*d*(b*c - a*d)^9)/(9*b^11*(a + b*x)^18) - (45*d^2*(b*c - a*d)^8)/
(17*b^11*(a + b*x)^17) - (15*d^3*(b*c - a*d)^7)/(2*b^11*(a + b*x)^16) - (14*d^4*(b*c - a*d)^6)/(b^11*(a + b*x)
^15) - (18*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^14) - (210*d^6*(b*c - a*d)^4)/(13*b^11*(a + b*x)^13) - (10*d^7*(
b*c - a*d)^3)/(b^11*(a + b*x)^12) - (45*d^8*(b*c - a*d)^2)/(11*b^11*(a + b*x)^11) - (d^9*(b*c - a*d))/(b^11*(a
+ b*x)^10) - d^10/(9*b^11*(a + b*x)^9)

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^{10}}{b^{10} (a+b x)^{20}}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^{19}}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^{18}}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^{17}}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^{16}}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^{15}}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^{14}}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^{13}}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)^{12}}+\frac {10 d^9 (b c-a d)}{b^{10} (a+b x)^{11}}+\frac {d^{10}}{b^{10} (a+b x)^{10}}\right ) \, dx \\ & = -\frac {(b c-a d)^{10}}{19 b^{11} (a+b x)^{19}}-\frac {5 d (b c-a d)^9}{9 b^{11} (a+b x)^{18}}-\frac {45 d^2 (b c-a d)^8}{17 b^{11} (a+b x)^{17}}-\frac {15 d^3 (b c-a d)^7}{2 b^{11} (a+b x)^{16}}-\frac {14 d^4 (b c-a d)^6}{b^{11} (a+b x)^{15}}-\frac {18 d^5 (b c-a d)^5}{b^{11} (a+b x)^{14}}-\frac {210 d^6 (b c-a d)^4}{13 b^{11} (a+b x)^{13}}-\frac {10 d^7 (b c-a d)^3}{b^{11} (a+b x)^{12}}-\frac {45 d^8 (b c-a d)^2}{11 b^{11} (a+b x)^{11}}-\frac {d^9 (b c-a d)}{b^{11} (a+b x)^{10}}-\frac {d^{10}}{9 b^{11} (a+b x)^9} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(692\) vs. \(2(273)=546\).

Time = 0.17 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.53 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx=-\frac {a^{10} d^{10}+a^9 b d^9 (9 c+19 d x)+9 a^8 b^2 d^8 \left (5 c^2+19 c d x+19 d^2 x^2\right )+3 a^7 b^3 d^7 \left (55 c^3+285 c^2 d x+513 c d^2 x^2+323 d^3 x^3\right )+3 a^6 b^4 d^6 \left (165 c^4+1045 c^3 d x+2565 c^2 d^2 x^2+2907 c d^3 x^3+1292 d^4 x^4\right )+9 a^5 b^5 d^5 \left (143 c^5+1045 c^4 d x+3135 c^3 d^2 x^2+4845 c^2 d^3 x^3+3876 c d^4 x^4+1292 d^5 x^5\right )+3 a^4 b^6 d^4 \left (1001 c^6+8151 c^5 d x+28215 c^4 d^2 x^2+53295 c^3 d^3 x^3+58140 c^2 d^4 x^4+34884 c d^5 x^5+9044 d^6 x^6\right )+3 a^3 b^7 d^3 \left (2145 c^7+19019 c^6 d x+73359 c^5 d^2 x^2+159885 c^4 d^3 x^3+213180 c^3 d^4 x^4+174420 c^2 d^5 x^5+81396 c d^6 x^6+16796 d^7 x^7\right )+9 a^2 b^8 d^2 \left (1430 c^8+13585 c^7 d x+57057 c^6 d^2 x^2+138567 c^5 d^3 x^3+213180 c^4 d^4 x^4+213180 c^3 d^5 x^5+135660 c^2 d^6 x^6+50388 c d^7 x^7+8398 d^8 x^8\right )+a b^9 d \left (24310 c^9+244530 c^8 d x+1100385 c^7 d^2 x^2+2909907 c^6 d^3 x^3+4988412 c^5 d^4 x^4+5755860 c^4 d^5 x^5+4476780 c^3 d^6 x^6+2267460 c^2 d^7 x^7+680238 c d^8 x^8+92378 d^9 x^9\right )+b^{10} \left (43758 c^{10}+461890 c^9 d x+2200770 c^8 d^2 x^2+6235515 c^7 d^3 x^3+11639628 c^6 d^4 x^4+14965236 c^5 d^5 x^5+13430340 c^4 d^6 x^6+8314020 c^3 d^7 x^7+3401190 c^2 d^8 x^8+831402 c d^9 x^9+92378 d^{10} x^{10}\right )}{831402 b^{11} (a+b x)^{19}} \]

[In]

Integrate[(c + d*x)^10/(a + b*x)^20,x]

[Out]

-1/831402*(a^10*d^10 + a^9*b*d^9*(9*c + 19*d*x) + 9*a^8*b^2*d^8*(5*c^2 + 19*c*d*x + 19*d^2*x^2) + 3*a^7*b^3*d^
7*(55*c^3 + 285*c^2*d*x + 513*c*d^2*x^2 + 323*d^3*x^3) + 3*a^6*b^4*d^6*(165*c^4 + 1045*c^3*d*x + 2565*c^2*d^2*
x^2 + 2907*c*d^3*x^3 + 1292*d^4*x^4) + 9*a^5*b^5*d^5*(143*c^5 + 1045*c^4*d*x + 3135*c^3*d^2*x^2 + 4845*c^2*d^3
*x^3 + 3876*c*d^4*x^4 + 1292*d^5*x^5) + 3*a^4*b^6*d^4*(1001*c^6 + 8151*c^5*d*x + 28215*c^4*d^2*x^2 + 53295*c^3
*d^3*x^3 + 58140*c^2*d^4*x^4 + 34884*c*d^5*x^5 + 9044*d^6*x^6) + 3*a^3*b^7*d^3*(2145*c^7 + 19019*c^6*d*x + 733
59*c^5*d^2*x^2 + 159885*c^4*d^3*x^3 + 213180*c^3*d^4*x^4 + 174420*c^2*d^5*x^5 + 81396*c*d^6*x^6 + 16796*d^7*x^
7) + 9*a^2*b^8*d^2*(1430*c^8 + 13585*c^7*d*x + 57057*c^6*d^2*x^2 + 138567*c^5*d^3*x^3 + 213180*c^4*d^4*x^4 + 2
13180*c^3*d^5*x^5 + 135660*c^2*d^6*x^6 + 50388*c*d^7*x^7 + 8398*d^8*x^8) + a*b^9*d*(24310*c^9 + 244530*c^8*d*x
+ 1100385*c^7*d^2*x^2 + 2909907*c^6*d^3*x^3 + 4988412*c^5*d^4*x^4 + 5755860*c^4*d^5*x^5 + 4476780*c^3*d^6*x^6
+ 2267460*c^2*d^7*x^7 + 680238*c*d^8*x^8 + 92378*d^9*x^9) + b^10*(43758*c^10 + 461890*c^9*d*x + 2200770*c^8*d
^2*x^2 + 6235515*c^7*d^3*x^3 + 11639628*c^6*d^4*x^4 + 14965236*c^5*d^5*x^5 + 13430340*c^4*d^6*x^6 + 8314020*c^
3*d^7*x^7 + 3401190*c^2*d^8*x^8 + 831402*c*d^9*x^9 + 92378*d^10*x^10))/(b^11*(a + b*x)^19)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(259)=518\).

Time = 0.25 (sec) , antiderivative size = 831, normalized size of antiderivative = 3.04

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+9 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}+165 a^{7} b^{3} c^{3} d^{7}+495 a^{6} b^{4} c^{4} d^{6}+1287 a^{5} b^{5} c^{5} d^{5}+3003 a^{4} b^{6} c^{6} d^{4}+6435 a^{3} b^{7} c^{7} d^{3}+12870 a^{2} b^{8} c^{8} d^{2}+24310 a \,b^{9} c^{9} d +43758 b^{10} c^{10}}{831402 b^{11}}-\frac {d \left (a^{9} d^{9}+9 a^{8} b c \,d^{8}+45 a^{7} b^{2} c^{2} d^{7}+165 a^{6} b^{3} c^{3} d^{6}+495 a^{5} b^{4} c^{4} d^{5}+1287 a^{4} b^{5} c^{5} d^{4}+3003 a^{3} b^{6} c^{6} d^{3}+6435 a^{2} b^{7} c^{7} d^{2}+12870 a \,b^{8} c^{8} d +24310 b^{9} c^{9}\right ) x}{43758 b^{10}}-\frac {d^{2} \left (a^{8} d^{8}+9 a^{7} b c \,d^{7}+45 a^{6} b^{2} c^{2} d^{6}+165 a^{5} b^{3} c^{3} d^{5}+495 a^{4} b^{4} c^{4} d^{4}+1287 a^{3} b^{5} c^{5} d^{3}+3003 a^{2} b^{6} c^{6} d^{2}+6435 a \,b^{7} c^{7} d +12870 b^{8} c^{8}\right ) x^{2}}{4862 b^{9}}-\frac {d^{3} \left (a^{7} d^{7}+9 a^{6} b c \,d^{6}+45 a^{5} b^{2} c^{2} d^{5}+165 a^{4} b^{3} c^{3} d^{4}+495 a^{3} b^{4} c^{4} d^{3}+1287 a^{2} b^{5} c^{5} d^{2}+3003 a \,b^{6} c^{6} d +6435 b^{7} c^{7}\right ) x^{3}}{858 b^{8}}-\frac {2 d^{4} \left (a^{6} d^{6}+9 a^{5} b c \,d^{5}+45 a^{4} b^{2} c^{2} d^{4}+165 a^{3} b^{3} c^{3} d^{3}+495 a^{2} b^{4} c^{4} d^{2}+1287 a \,b^{5} c^{5} d +3003 b^{6} c^{6}\right ) x^{4}}{429 b^{7}}-\frac {2 d^{5} \left (a^{5} d^{5}+9 a^{4} b c \,d^{4}+45 a^{3} b^{2} c^{2} d^{3}+165 a^{2} b^{3} c^{3} d^{2}+495 a \,b^{4} c^{4} d +1287 b^{5} c^{5}\right ) x^{5}}{143 b^{6}}-\frac {14 d^{6} \left (a^{4} d^{4}+9 a^{3} b c \,d^{3}+45 a^{2} b^{2} c^{2} d^{2}+165 a \,b^{3} c^{3} d +495 b^{4} c^{4}\right ) x^{6}}{429 b^{5}}-\frac {2 d^{7} \left (a^{3} d^{3}+9 a^{2} b c \,d^{2}+45 a \,b^{2} c^{2} d +165 b^{3} c^{3}\right ) x^{7}}{33 b^{4}}-\frac {d^{8} \left (a^{2} d^{2}+9 a b c d +45 b^{2} c^{2}\right ) x^{8}}{11 b^{3}}-\frac {d^{9} \left (a d +9 b c \right ) x^{9}}{9 b^{2}}-\frac {d^{10} x^{10}}{9 b}}{\left (b x +a \right )^{19}}\)	\(831\)
default	\(-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{11 b^{11} \left (b x +a \right )^{11}}-\frac {210 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 )}{13 b^{11} \left (b x +a \right )^{13}}-\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}}{19 b^{11} \left (b x +a \right )^{19}}-\frac {d^{10}}{9 b^{11} \left (b x +a \right )^{9}}+\frac {18 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 )^{14}}+\frac {10 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 )^{12}}+\frac {d^{9} \left (a d -b c \right )}{b^{11} \left (b x +a \right )^{10}}+\frac {15 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 )}{2 b^{11} \left (b x +a \right )^{16}}+\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 )}{9 b^{11} \left (b x +a \right )^{18}}-\frac {14 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 )^{15}}-\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 )}{17 b^{11} \left (b x +a \right )^{17}}\)	\(866\)
norman	\(\frac {\frac {-a^{10} b^{8} d^{10}-9 a^{9} b^{9} c \,d^{9}-45 a^{8} b^{10} c^{2} d^{8}-165 a^{7} b^{11} c^{3} d^{7}-495 a^{6} b^{12} c^{4} d^{6}-1287 a^{5} b^{13} c^{5} d^{5}-3003 a^{4} b^{14} c^{6} d^{4}-6435 a^{3} b^{15} c^{7} d^{3}-12870 a^{2} b^{16} c^{8} d^{2}-24310 a \,c^{9} d \,b^{17}-43758 b^{18} c^{10}}{831402 b^{19}}+\frac {\left (-a^{9} b^{8} d^{10}-9 a^{8} b^{9} c \,d^{9}-45 a^{7} b^{10} c^{2} d^{8}-165 a^{6} b^{11} c^{3} d^{7}-495 a^{5} b^{12} c^{4} d^{6}-1287 a^{4} b^{13} c^{5} d^{5}-3003 a^{3} b^{14} c^{6} d^{4}-6435 a^{2} b^{15} c^{7} d^{3}-12870 a \,b^{16} c^{8} d^{2}-24310 b^{17} c^{9} d \right ) x}{43758 b^{18}}+\frac {\left (-a^{8} b^{8} d^{10}-9 a^{7} b^{9} c \,d^{9}-45 a^{6} b^{10} c^{2} d^{8}-165 a^{5} b^{11} c^{3} d^{7}-495 a^{4} b^{12} c^{4} d^{6}-1287 a^{3} b^{13} c^{5} d^{5}-3003 a^{2} b^{14} c^{6} d^{4}-6435 a \,b^{15} c^{7} d^{3}-12870 b^{16} c^{8} d^{2}\right ) x^{2}}{4862 b^{17}}+\frac {\left (-a^{7} b^{8} d^{10}-9 a^{6} b^{9} c \,d^{9}-45 a^{5} b^{10} c^{2} d^{8}-165 a^{4} b^{11} c^{3} d^{7}-495 a^{3} b^{12} c^{4} d^{6}-1287 a^{2} b^{13} c^{5} d^{5}-3003 a \,b^{14} c^{6} d^{4}-6435 c^{7} d^{3} b^{15}\right ) x^{3}}{858 b^{16}}+\frac {2 \left (-a^{6} b^{8} d^{10}-9 a^{5} b^{9} c \,d^{9}-45 a^{4} b^{10} c^{2} d^{8}-165 a^{3} b^{11} c^{3} d^{7}-495 a^{2} b^{12} c^{4} d^{6}-1287 a \,b^{13} c^{5} d^{5}-3003 c^{6} d^{4} b^{14}\right ) x^{4}}{429 b^{15}}+\frac {2 \left (-a^{5} b^{8} d^{10}-9 a^{4} b^{9} c \,d^{9}-45 a^{3} b^{10} c^{2} d^{8}-165 a^{2} b^{11} c^{3} d^{7}-495 a \,b^{12} c^{4} d^{6}-1287 c^{5} d^{5} b^{13}\right ) x^{5}}{143 b^{14}}+\frac {14 \left (-a^{4} b^{8} d^{10}-9 a^{3} b^{9} c \,d^{9}-45 a^{2} b^{10} c^{2} d^{8}-165 a \,b^{11} c^{3} d^{7}-495 b^{12} c^{4} d^{6}\right ) x^{6}}{429 b^{13}}+\frac {2 \left (-a^{3} b^{8} d^{10}-9 a^{2} b^{9} c \,d^{9}-45 a \,b^{10} c^{2} d^{8}-165 b^{11} c^{3} d^{7}\right ) x^{7}}{33 b^{12}}+\frac {\left (-a^{2} b^{8} d^{10}-9 a \,b^{9} c \,d^{9}-45 b^{10} c^{2} d^{8}\right ) x^{8}}{11 b^{11}}+\frac {\left (-a \,b^{8} d^{10}-9 b^{9} c \,d^{9}\right ) x^{9}}{9 b^{10}}-\frac {d^{10} x^{10}}{9 b}}{\left (b x +a \right )^{19}}\)	\(909\)
gosper	\(-\frac {92378 x^{10} d^{10} b^{10}+92378 x^{9} a \,b^{9} d^{10}+831402 x^{9} b^{10} c \,d^{9}+75582 x^{8} a^{2} b^{8} d^{10}+680238 x^{8} a \,b^{9} c \,d^{9}+3401190 x^{8} b^{10} c^{2} d^{8}+50388 x^{7} a^{3} b^{7} d^{10}+453492 x^{7} a^{2} b^{8} c \,d^{9}+2267460 x^{7} a \,b^{9} c^{2} d^{8}+8314020 x^{7} b^{10} c^{3} d^{7}+27132 x^{6} a^{4} b^{6} d^{10}+244188 x^{6} a^{3} b^{7} c \,d^{9}+1220940 x^{6} a^{2} b^{8} c^{2} d^{8}+4476780 x^{6} a \,b^{9} c^{3} d^{7}[phone] x^{6} b^{10} c^{4} d^{6}+11628 x^{5} a^{5} b^{5} d^{10}+104652 x^{5} a^{4} b^{6} c \,d^{9}+523260 x^{5} a^{3} b^{7} c^{2} d^{8}+1918620 x^{5} a^{2} b^{8} c^{3} d^{7}+5755860 x^{5} a \,b^{9} c^{4} d^{6}[phone] x^{5} b^{10} c^{5} d^{5}+3876 x^{4} a^{6} b^{4} d^{10}+34884 x^{4} a^{5} b^{5} c \,d^{9}+174420 x^{4} a^{4} b^{6} c^{2} d^{8}+639540 x^{4} a^{3} b^{7} c^{3} d^{7}+1918620 x^{4} a^{2} b^{8} c^{4} d^{6}+4988412 x^{4} a \,b^{9} c^{5} d^{5}[phone] x^{4} b^{10} c^{6} d^{4}+969 x^{3} a^{7} b^{3} d^{10}+8721 x^{3} a^{6} b^{4} c \,d^{9}+43605 x^{3} a^{5} b^{5} c^{2} d^{8}+159885 x^{3} a^{4} b^{6} c^{3} d^{7}+479655 x^{3} a^{3} b^{7} c^{4} d^{6}+1247103 x^{3} a^{2} b^{8} c^{5} d^{5}+2909907 x^{3} a \,b^{9} c^{6} d^{4}+6235515 x^{3} b^{10} c^{7} d^{3}+171 x^{2} a^{8} b^{2} d^{10}+1539 x^{2} a^{7} b^{3} c \,d^{9}+7695 x^{2} a^{6} b^{4} c^{2} d^{8}+28215 x^{2} a^{5} b^{5} c^{3} d^{7}+84645 x^{2} a^{4} b^{6} c^{4} d^{6}+220077 x^{2} a^{3} b^{7} c^{5} d^{5}+513513 x^{2} a^{2} b^{8} c^{6} d^{4}+1100385 x^{2} a \,b^{9} c^{7} d^{3}+2200770 x^{2} b^{10} c^{8} d^{2}+19 x \,a^{9} b \,d^{10}+171 x \,a^{8} b^{2} c \,d^{9}+855 x \,a^{7} b^{3} c^{2} d^{8}+3135 x \,a^{6} b^{4} c^{3} d^{7}+9405 x \,a^{5} b^{5} c^{4} d^{6}+24453 x \,a^{4} b^{6} c^{5} d^{5}+57057 x \,a^{3} b^{7} c^{6} d^{4}+122265 x \,a^{2} b^{8} c^{7} d^{3}+244530 x a \,b^{9} c^{8} d^{2}+461890 x \,b^{10} c^{9} d +a^{10} d^{10}+9 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}+165 a^{7} b^{3} c^{3} d^{7}+495 a^{6} b^{4} c^{4} d^{6}+1287 a^{5} b^{5} c^{5} d^{5}+3003 a^{4} b^{6} c^{6} d^{4}+6435 a^{3} b^{7} c^{7} d^{3}+12870 a^{2} b^{8} c^{8} d^{2}+24310 a \,b^{9} c^{9} d +43758 b^{10} c^{10}}{831402 b^{11} \left (b x +a \right )^{19}}\)	\(962\)
parallelrisch	\(\frac {-92378 d^{10} x^{10} b^{18}-92378 a \,b^{17} d^{10} x^{9}-831402 b^{18} c \,d^{9} x^{9}-75582 a^{2} b^{16} d^{10} x^{8}-680238 a \,b^{17} c \,d^{9} x^{8}-3401190 b^{18} c^{2} d^{8} x^{8}-50388 a^{3} b^{15} d^{10} x^{7}-453492 a^{2} b^{16} c \,d^{9} x^{7}-2267460 a \,b^{17} c^{2} d^{8} x^{7}-8314020 b^{18} c^{3} d^{7} x^{7}-27132 a^{4} b^{14} d^{10} x^{6}-244188 a^{3} b^{15} c \,d^{9} x^{6}-1220940 a^{2} b^{16} c^{2} d^{8} x^{6}-4476780 a \,b^{17} c^{3} d^{7} x^{6}-13430340 b^{18} c^{4} d^{6} x^{6}-11628 a^{5} b^{13} d^{10} x^{5}-104652 a^{4} b^{14} c \,d^{9} x^{5}-523260 a^{3} b^{15} c^{2} d^{8} x^{5}-1918620 a^{2} b^{16} c^{3} d^{7} x^{5}-5755860 a \,b^{17} c^{4} d^{6} x^{5}-14965236 b^{18} c^{5} d^{5} x^{5}-3876 a^{6} b^{12} d^{10} x^{4}-34884 a^{5} b^{13} c \,d^{9} x^{4}-174420 a^{4} b^{14} c^{2} d^{8} x^{4}-639540 a^{3} b^{15} c^{3} d^{7} x^{4}-1918620 a^{2} b^{16} c^{4} d^{6} x^{4}-4988412 a \,b^{17} c^{5} d^{5} x^{4}-11639628 b^{18} c^{6} d^{4} x^{4}-969 a^{7} b^{11} d^{10} x^{3}-8721 a^{6} b^{12} c \,d^{9} x^{3}-43605 a^{5} b^{13} c^{2} d^{8} x^{3}-159885 a^{4} b^{14} c^{3} d^{7} x^{3}-479655 a^{3} b^{15} c^{4} d^{6} x^{3}-1247103 a^{2} b^{16} c^{5} d^{5} x^{3}-2909907 a \,b^{17} c^{6} d^{4} x^{3}-6235515 b^{18} c^{7} d^{3} x^{3}-171 a^{8} b^{10} d^{10} x^{2}-1539 a^{7} b^{11} c \,d^{9} x^{2}-7695 a^{6} b^{12} c^{2} d^{8} x^{2}-28215 a^{5} b^{13} c^{3} d^{7} x^{2}-84645 a^{4} b^{14} c^{4} d^{6} x^{2}-220077 a^{3} b^{15} c^{5} d^{5} x^{2}-513513 a^{2} b^{16} c^{6} d^{4} x^{2}-1100385 a \,b^{17} c^{7} d^{3} x^{2}-2200770 b^{18} c^{8} d^{2} x^{2}-19 a^{9} b^{9} d^{10} x -171 a^{8} b^{10} c \,d^{9} x -855 a^{7} b^{11} c^{2} d^{8} x -3135 a^{6} b^{12} c^{3} d^{7} x -9405 a^{5} b^{13} c^{4} d^{6} x -24453 a^{4} b^{14} c^{5} d^{5} x -57057 a^{3} b^{15} c^{6} d^{4} x -122265 a^{2} b^{16} c^{7} d^{3} x -244530 a \,b^{17} c^{8} d^{2} x -461890 b^{18} c^{9} d x -a^{10} b^{8} d^{10}-9 a^{9} b^{9} c \,d^{9}-45 a^{8} b^{10} c^{2} d^{8}-165 a^{7} b^{11} c^{3} d^{7}-495 a^{6} b^{12} c^{4} d^{6}-1287 a^{5} b^{13} c^{5} d^{5}-3003 a^{4} b^{14} c^{6} d^{4}-6435 a^{3} b^{15} c^{7} d^{3}-12870 a^{2} b^{16} c^{8} d^{2}-24310 a \,c^{9} d \,b^{17}-43758 b^{18} c^{10}}{831402 b^{19} \left (b x +a \right )^{19}}\)	\(970\)

[In]

int((d*x+c)^10/(b*x+a)^20,x,method=_RETURNVERBOSE)

[Out]

(-1/831402/b^11*(a^10*d^10+9*a^9*b*c*d^9+45*a^8*b^2*c^2*d^8+165*a^7*b^3*c^3*d^7+495*a^6*b^4*c^4*d^6+1287*a^5*b
^5*c^5*d^5+3003*a^4*b^6*c^6*d^4+6435*a^3*b^7*c^7*d^3+12870*a^2*b^8*c^8*d^2+24310*a*b^9*c^9*d+43758*b^10*c^10)-
1/43758/b^10*d*(a^9*d^9+9*a^8*b*c*d^8+45*a^7*b^2*c^2*d^7+165*a^6*b^3*c^3*d^6+495*a^5*b^4*c^4*d^5+1287*a^4*b^5*
c^5*d^4+3003*a^3*b^6*c^6*d^3+6435*a^2*b^7*c^7*d^2+12870*a*b^8*c^8*d+24310*b^9*c^9)*x-1/4862/b^9*d^2*(a^8*d^8+9
*a^7*b*c*d^7+45*a^6*b^2*c^2*d^6+165*a^5*b^3*c^3*d^5+495*a^4*b^4*c^4*d^4+1287*a^3*b^5*c^5*d^3+3003*a^2*b^6*c^6*
d^2+6435*a*b^7*c^7*d+12870*b^8*c^8)*x^2-1/858/b^8*d^3*(a^7*d^7+9*a^6*b*c*d^6+45*a^5*b^2*c^2*d^5+165*a^4*b^3*c^
3*d^4+495*a^3*b^4*c^4*d^3+1287*a^2*b^5*c^5*d^2+3003*a*b^6*c^6*d+6435*b^7*c^7)*x^3-2/429/b^7*d^4*(a^6*d^6+9*a^5
*b*c*d^5+45*a^4*b^2*c^2*d^4+165*a^3*b^3*c^3*d^3+495*a^2*b^4*c^4*d^2+1287*a*b^5*c^5*d+3003*b^6*c^6)*x^4-2/143/b
^6*d^5*(a^5*d^5+9*a^4*b*c*d^4+45*a^3*b^2*c^2*d^3+165*a^2*b^3*c^3*d^2+495*a*b^4*c^4*d+1287*b^5*c^5)*x^5-14/429/
b^5*d^6*(a^4*d^4+9*a^3*b*c*d^3+45*a^2*b^2*c^2*d^2+165*a*b^3*c^3*d+495*b^4*c^4)*x^6-2/33/b^4*d^7*(a^3*d^3+9*a^2
*b*c*d^2+45*a*b^2*c^2*d+165*b^3*c^3)*x^7-1/11/b^3*d^8*(a^2*d^2+9*a*b*c*d+45*b^2*c^2)*x^8-1/9/b^2*d^9*(a*d+9*b*
c)*x^9-1/9/b*d^10*x^10)/(b*x+a)^19

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1063 vs. \(2 (259) = 518\).

Time = 0.23 (sec) , antiderivative size = 1063, normalized size of antiderivative = 3.89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx=-\frac {92378 \, b^{10} d^{10} x^{10} + 43758 \, b^{10} c^{10} + 24310 \, a b^{9} c^{9} d + 12870 \, a^{2} b^{8} c^{8} d^{2} + 6435 \, a^{3} b^{7} c^{7} d^{3} + 3003 \, a^{4} b^{6} c^{6} d^{4} + 1287 \, a^{5} b^{5} c^{5} d^{5} + 495 \, a^{6} b^{4} c^{4} d^{6} + 165 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} + 9 \, a^{9} b c d^{9} + a^{10} d^{10} + 92378 \, {\left (9 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 75582 \, {\left (45 \, b^{10} c^{2} d^{8} + 9 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 50388 \, {\left (165 \, b^{10} c^{3} d^{7} + 45 \, a b^{9} c^{2} d^{8} + 9 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 27132 \, {\left (495 \, b^{10} c^{4} d^{6} + 165 \, a b^{9} c^{3} d^{7} + 45 \, a^{2} b^{8} c^{2} d^{8} + 9 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 11628 \, {\left (1287 \, b^{10} c^{5} d^{5} + 495 \, a b^{9} c^{4} d^{6} + 165 \, a^{2} b^{8} c^{3} d^{7} + 45 \, a^{3} b^{7} c^{2} d^{8} + 9 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 3876 \, {\left (3003 \, b^{10} c^{6} d^{4} + 1287 \, a b^{9} c^{5} d^{5} + 495 \, a^{2} b^{8} c^{4} d^{6} + 165 \, a^{3} b^{7} c^{3} d^{7} + 45 \, a^{4} b^{6} c^{2} d^{8} + 9 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 969 \, {\left (6435 \, b^{10} c^{7} d^{3} + 3003 \, a b^{9} c^{6} d^{4} + 1287 \, a^{2} b^{8} c^{5} d^{5} + 495 \, a^{3} b^{7} c^{4} d^{6} + 165 \, a^{4} b^{6} c^{3} d^{7} + 45 \, a^{5} b^{5} c^{2} d^{8} + 9 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 171 \, {\left (12870 \, b^{10} c^{8} d^{2} + 6435 \, a b^{9} c^{7} d^{3} + 3003 \, a^{2} b^{8} c^{6} d^{4} + 1287 \, a^{3} b^{7} c^{5} d^{5} + 495 \, a^{4} b^{6} c^{4} d^{6} + 165 \, a^{5} b^{5} c^{3} d^{7} + 45 \, a^{6} b^{4} c^{2} d^{8} + 9 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 19 \, {\left (24310 \, b^{10} c^{9} d + 12870 \, a b^{9} c^{8} d^{2} + 6435 \, a^{2} b^{8} c^{7} d^{3} + 3003 \, a^{3} b^{7} c^{6} d^{4} + 1287 \, a^{4} b^{6} c^{5} d^{5} + 495 \, a^{5} b^{5} c^{4} d^{6} + 165 \, a^{6} b^{4} c^{3} d^{7} + 45 \, a^{7} b^{3} c^{2} d^{8} + 9 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{831402 \, {\left (b^{30} x^{19} + 19 \, a b^{29} x^{18} + 171 \, a^{2} b^{28} x^{17} + 969 \, a^{3} b^{27} x^{16} + 3876 \, a^{4} b^{26} x^{15} + 11628 \, a^{5} b^{25} x^{14} + 27132 \, a^{6} b^{24} x^{13} + 50388 \, a^{7} b^{23} x^{12} + 75582 \, a^{8} b^{22} x^{11} + 92378 \, a^{9} b^{21} x^{10} + 92378 \, a^{10} b^{20} x^{9} + 75582 \, a^{11} b^{19} x^{8} + 50388 \, a^{12} b^{18} x^{7} + 27132 \, a^{13} b^{17} x^{6} + 11628 \, a^{14} b^{16} x^{5} + 3876 \, a^{15} b^{15} x^{4} + 969 \, a^{16} b^{14} x^{3} + 171 \, a^{17} b^{13} x^{2} + 19 \, a^{18} b^{12} x + a^{19} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^20,x, algorithm="fricas")

[Out]

-1/831402*(92378*b^10*d^10*x^10 + 43758*b^10*c^10 + 24310*a*b^9*c^9*d + 12870*a^2*b^8*c^8*d^2 + 6435*a^3*b^7*c
^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165*a^7*b^3*c^3*d^7 + 45*a^8*b^2*
c^2*d^8 + 9*a^9*b*c*d^9 + a^10*d^10 + 92378*(9*b^10*c*d^9 + a*b^9*d^10)*x^9 + 75582*(45*b^10*c^2*d^8 + 9*a*b^9
*c*d^9 + a^2*b^8*d^10)*x^8 + 50388*(165*b^10*c^3*d^7 + 45*a*b^9*c^2*d^8 + 9*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7
+ 27132*(495*b^10*c^4*d^6 + 165*a*b^9*c^3*d^7 + 45*a^2*b^8*c^2*d^8 + 9*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 116
28*(1287*b^10*c^5*d^5 + 495*a*b^9*c^4*d^6 + 165*a^2*b^8*c^3*d^7 + 45*a^3*b^7*c^2*d^8 + 9*a^4*b^6*c*d^9 + a^5*b
^5*d^10)*x^5 + 3876*(3003*b^10*c^6*d^4 + 1287*a*b^9*c^5*d^5 + 495*a^2*b^8*c^4*d^6 + 165*a^3*b^7*c^3*d^7 + 45*a
^4*b^6*c^2*d^8 + 9*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 969*(6435*b^10*c^7*d^3 + 3003*a*b^9*c^6*d^4 + 1287*a^2*
b^8*c^5*d^5 + 495*a^3*b^7*c^4*d^6 + 165*a^4*b^6*c^3*d^7 + 45*a^5*b^5*c^2*d^8 + 9*a^6*b^4*c*d^9 + a^7*b^3*d^10)
*x^3 + 171*(12870*b^10*c^8*d^2 + 6435*a*b^9*c^7*d^3 + 3003*a^2*b^8*c^6*d^4 + 1287*a^3*b^7*c^5*d^5 + 495*a^4*b^
6*c^4*d^6 + 165*a^5*b^5*c^3*d^7 + 45*a^6*b^4*c^2*d^8 + 9*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 19*(24310*b^10*c^
9*d + 12870*a*b^9*c^8*d^2 + 6435*a^2*b^8*c^7*d^3 + 3003*a^3*b^7*c^6*d^4 + 1287*a^4*b^6*c^5*d^5 + 495*a^5*b^5*c
^4*d^6 + 165*a^6*b^4*c^3*d^7 + 45*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^30*x^19 + 19*a*b^29*x^
18 + 171*a^2*b^28*x^17 + 969*a^3*b^27*x^16 + 3876*a^4*b^26*x^15 + 11628*a^5*b^25*x^14 + 27132*a^6*b^24*x^13 +
50388*a^7*b^23*x^12 + 75582*a^8*b^22*x^11 + 92378*a^9*b^21*x^10 + 92378*a^10*b^20*x^9 + 75582*a^11*b^19*x^8 +
50388*a^12*b^18*x^7 + 27132*a^13*b^17*x^6 + 11628*a^14*b^16*x^5 + 3876*a^15*b^15*x^4 + 969*a^16*b^14*x^3 + 171
*a^17*b^13*x^2 + 19*a^18*b^12*x + a^19*b^11)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**10/(b*x+a)**20,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1063 vs. \(2 (259) = 518\).

Time = 0.27 (sec) , antiderivative size = 1063, normalized size of antiderivative = 3.89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx=-\frac {92378 \, b^{10} d^{10} x^{10} + 43758 \, b^{10} c^{10} + 24310 \, a b^{9} c^{9} d + 12870 \, a^{2} b^{8} c^{8} d^{2} + 6435 \, a^{3} b^{7} c^{7} d^{3} + 3003 \, a^{4} b^{6} c^{6} d^{4} + 1287 \, a^{5} b^{5} c^{5} d^{5} + 495 \, a^{6} b^{4} c^{4} d^{6} + 165 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} + 9 \, a^{9} b c d^{9} + a^{10} d^{10} + 92378 \, {\left (9 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 75582 \, {\left (45 \, b^{10} c^{2} d^{8} + 9 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 50388 \, {\left (165 \, b^{10} c^{3} d^{7} + 45 \, a b^{9} c^{2} d^{8} + 9 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 27132 \, {\left (495 \, b^{10} c^{4} d^{6} + 165 \, a b^{9} c^{3} d^{7} + 45 \, a^{2} b^{8} c^{2} d^{8} + 9 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 11628 \, {\left (1287 \, b^{10} c^{5} d^{5} + 495 \, a b^{9} c^{4} d^{6} + 165 \, a^{2} b^{8} c^{3} d^{7} + 45 \, a^{3} b^{7} c^{2} d^{8} + 9 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 3876 \, {\left (3003 \, b^{10} c^{6} d^{4} + 1287 \, a b^{9} c^{5} d^{5} + 495 \, a^{2} b^{8} c^{4} d^{6} + 165 \, a^{3} b^{7} c^{3} d^{7} + 45 \, a^{4} b^{6} c^{2} d^{8} + 9 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 969 \, {\left (6435 \, b^{10} c^{7} d^{3} + 3003 \, a b^{9} c^{6} d^{4} + 1287 \, a^{2} b^{8} c^{5} d^{5} + 495 \, a^{3} b^{7} c^{4} d^{6} + 165 \, a^{4} b^{6} c^{3} d^{7} + 45 \, a^{5} b^{5} c^{2} d^{8} + 9 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 171 \, {\left (12870 \, b^{10} c^{8} d^{2} + 6435 \, a b^{9} c^{7} d^{3} + 3003 \, a^{2} b^{8} c^{6} d^{4} + 1287 \, a^{3} b^{7} c^{5} d^{5} + 495 \, a^{4} b^{6} c^{4} d^{6} + 165 \, a^{5} b^{5} c^{3} d^{7} + 45 \, a^{6} b^{4} c^{2} d^{8} + 9 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 19 \, {\left (24310 \, b^{10} c^{9} d + 12870 \, a b^{9} c^{8} d^{2} + 6435 \, a^{2} b^{8} c^{7} d^{3} + 3003 \, a^{3} b^{7} c^{6} d^{4} + 1287 \, a^{4} b^{6} c^{5} d^{5} + 495 \, a^{5} b^{5} c^{4} d^{6} + 165 \, a^{6} b^{4} c^{3} d^{7} + 45 \, a^{7} b^{3} c^{2} d^{8} + 9 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{831402 \, {\left (b^{30} x^{19} + 19 \, a b^{29} x^{18} + 171 \, a^{2} b^{28} x^{17} + 969 \, a^{3} b^{27} x^{16} + 3876 \, a^{4} b^{26} x^{15} + 11628 \, a^{5} b^{25} x^{14} + 27132 \, a^{6} b^{24} x^{13} + 50388 \, a^{7} b^{23} x^{12} + 75582 \, a^{8} b^{22} x^{11} + 92378 \, a^{9} b^{21} x^{10} + 92378 \, a^{10} b^{20} x^{9} + 75582 \, a^{11} b^{19} x^{8} + 50388 \, a^{12} b^{18} x^{7} + 27132 \, a^{13} b^{17} x^{6} + 11628 \, a^{14} b^{16} x^{5} + 3876 \, a^{15} b^{15} x^{4} + 969 \, a^{16} b^{14} x^{3} + 171 \, a^{17} b^{13} x^{2} + 19 \, a^{18} b^{12} x + a^{19} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^20,x, algorithm="maxima")

[Out]

-1/831402*(92378*b^10*d^10*x^10 + 43758*b^10*c^10 + 24310*a*b^9*c^9*d + 12870*a^2*b^8*c^8*d^2 + 6435*a^3*b^7*c
^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165*a^7*b^3*c^3*d^7 + 45*a^8*b^2*
c^2*d^8 + 9*a^9*b*c*d^9 + a^10*d^10 + 92378*(9*b^10*c*d^9 + a*b^9*d^10)*x^9 + 75582*(45*b^10*c^2*d^8 + 9*a*b^9
*c*d^9 + a^2*b^8*d^10)*x^8 + 50388*(165*b^10*c^3*d^7 + 45*a*b^9*c^2*d^8 + 9*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7
+ 27132*(495*b^10*c^4*d^6 + 165*a*b^9*c^3*d^7 + 45*a^2*b^8*c^2*d^8 + 9*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 116
28*(1287*b^10*c^5*d^5 + 495*a*b^9*c^4*d^6 + 165*a^2*b^8*c^3*d^7 + 45*a^3*b^7*c^2*d^8 + 9*a^4*b^6*c*d^9 + a^5*b
^5*d^10)*x^5 + 3876*(3003*b^10*c^6*d^4 + 1287*a*b^9*c^5*d^5 + 495*a^2*b^8*c^4*d^6 + 165*a^3*b^7*c^3*d^7 + 45*a
^4*b^6*c^2*d^8 + 9*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 969*(6435*b^10*c^7*d^3 + 3003*a*b^9*c^6*d^4 + 1287*a^2*
b^8*c^5*d^5 + 495*a^3*b^7*c^4*d^6 + 165*a^4*b^6*c^3*d^7 + 45*a^5*b^5*c^2*d^8 + 9*a^6*b^4*c*d^9 + a^7*b^3*d^10)
*x^3 + 171*(12870*b^10*c^8*d^2 + 6435*a*b^9*c^7*d^3 + 3003*a^2*b^8*c^6*d^4 + 1287*a^3*b^7*c^5*d^5 + 495*a^4*b^
6*c^4*d^6 + 165*a^5*b^5*c^3*d^7 + 45*a^6*b^4*c^2*d^8 + 9*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 19*(24310*b^10*c^
9*d + 12870*a*b^9*c^8*d^2 + 6435*a^2*b^8*c^7*d^3 + 3003*a^3*b^7*c^6*d^4 + 1287*a^4*b^6*c^5*d^5 + 495*a^5*b^5*c
^4*d^6 + 165*a^6*b^4*c^3*d^7 + 45*a^7*b^3*c^2*d^8 + 9*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^30*x^19 + 19*a*b^29*x^
18 + 171*a^2*b^28*x^17 + 969*a^3*b^27*x^16 + 3876*a^4*b^26*x^15 + 11628*a^5*b^25*x^14 + 27132*a^6*b^24*x^13 +
50388*a^7*b^23*x^12 + 75582*a^8*b^22*x^11 + 92378*a^9*b^21*x^10 + 92378*a^10*b^20*x^9 + 75582*a^11*b^19*x^8 +
50388*a^12*b^18*x^7 + 27132*a^13*b^17*x^6 + 11628*a^14*b^16*x^5 + 3876*a^15*b^15*x^4 + 969*a^16*b^14*x^3 + 171
*a^17*b^13*x^2 + 19*a^18*b^12*x + a^19*b^11)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (259) = 518\).

Time = 0.30 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.52 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx=-\frac {92378 \, b^{10} d^{10} x^{10} + 831402 \, b^{10} c d^{9} x^{9} + 92378 \, a b^{9} d^{10} x^{9} + 3401190 \, b^{10} c^{2} d^{8} x^{8} + 680238 \, a b^{9} c d^{9} x^{8} + 75582 \, a^{2} b^{8} d^{10} x^{8} + 8314020 \, b^{10} c^{3} d^{7} x^{7} + 2267460 \, a b^{9} c^{2} d^{8} x^{7} + 453492 \, a^{2} b^{8} c d^{9} x^{7} + 50388 \, a^{3} b^{7} d^{10} x^{7} + 13430340 \, b^{10} c^{4} d^{6} x^{6} + 4476780 \, a b^{9} c^{3} d^{7} x^{6} + 1220940 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 244188 \, a^{3} b^{7} c d^{9} x^{6} + 27132 \, a^{4} b^{6} d^{10} x^{6} + 14965236 \, b^{10} c^{5} d^{5} x^{5} + 5755860 \, a b^{9} c^{4} d^{6} x^{5} + 1918620 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 523260 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 104652 \, a^{4} b^{6} c d^{9} x^{5} + 11628 \, a^{5} b^{5} d^{10} x^{5} + 11639628 \, b^{10} c^{6} d^{4} x^{4} + 4988412 \, a b^{9} c^{5} d^{5} x^{4} + 1918620 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 639540 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 174420 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 34884 \, a^{5} b^{5} c d^{9} x^{4} + 3876 \, a^{6} b^{4} d^{10} x^{4} + 6235515 \, b^{10} c^{7} d^{3} x^{3} + 2909907 \, a b^{9} c^{6} d^{4} x^{3} + 1247103 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 479655 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 159885 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 43605 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 8721 \, a^{6} b^{4} c d^{9} x^{3} + 969 \, a^{7} b^{3} d^{10} x^{3} + 2200770 \, b^{10} c^{8} d^{2} x^{2} + 1100385 \, a b^{9} c^{7} d^{3} x^{2} + 513513 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 220077 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 84645 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 28215 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 7695 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 1539 \, a^{7} b^{3} c d^{9} x^{2} + 171 \, a^{8} b^{2} d^{10} x^{2} + 461890 \, b^{10} c^{9} d x + 244530 \, a b^{9} c^{8} d^{2} x + 122265 \, a^{2} b^{8} c^{7} d^{3} x + 57057 \, a^{3} b^{7} c^{6} d^{4} x + 24453 \, a^{4} b^{6} c^{5} d^{5} x + 9405 \, a^{5} b^{5} c^{4} d^{6} x + 3135 \, a^{6} b^{4} c^{3} d^{7} x + 855 \, a^{7} b^{3} c^{2} d^{8} x + 171 \, a^{8} b^{2} c d^{9} x + 19 \, a^{9} b d^{10} x + 43758 \, b^{10} c^{10} + 24310 \, a b^{9} c^{9} d + 12870 \, a^{2} b^{8} c^{8} d^{2} + 6435 \, a^{3} b^{7} c^{7} d^{3} + 3003 \, a^{4} b^{6} c^{6} d^{4} + 1287 \, a^{5} b^{5} c^{5} d^{5} + 495 \, a^{6} b^{4} c^{4} d^{6} + 165 \, a^{7} b^{3} c^{3} d^{7} + 45 \, a^{8} b^{2} c^{2} d^{8} + 9 \, a^{9} b c d^{9} + a^{10} d^{10}}{831402 \, {\left (b x + a\right )}^{19} b^{11}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^20,x, algorithm="giac")

[Out]

-1/831402*(92378*b^10*d^10*x^10 + 831402*b^10*c*d^9*x^9 + 92378*a*b^9*d^10*x^9 + 3401190*b^10*c^2*d^8*x^8 + 68
0238*a*b^9*c*d^9*x^8 + 75582*a^2*b^8*d^10*x^8 + 8314020*b^10*c^3*d^7*x^7 + 2267460*a*b^9*c^2*d^8*x^7 + 453492*
a^2*b^8*c*d^9*x^7 + 50388*a^3*b^7*d^10*x^7 + 13430340*b^10*c^4*d^6*x^6 + 4476780*a*b^9*c^3*d^7*x^6 + 1220940*a
^2*b^8*c^2*d^8*x^6 + 244188*a^3*b^7*c*d^9*x^6 + 27132*a^4*b^6*d^10*x^6 + 14965236*b^10*c^5*d^5*x^5 + 5755860*a
*b^9*c^4*d^6*x^5 + 1918620*a^2*b^8*c^3*d^7*x^5 + 523260*a^3*b^7*c^2*d^8*x^5 + 104652*a^4*b^6*c*d^9*x^5 + 11628
*a^5*b^5*d^10*x^5 + 11639628*b^10*c^6*d^4*x^4 + 4988412*a*b^9*c^5*d^5*x^4 + 1918620*a^2*b^8*c^4*d^6*x^4 + 6395
40*a^3*b^7*c^3*d^7*x^4 + 174420*a^4*b^6*c^2*d^8*x^4 + 34884*a^5*b^5*c*d^9*x^4 + 3876*a^6*b^4*d^10*x^4 + 623551
5*b^10*c^7*d^3*x^3 + 2909907*a*b^9*c^6*d^4*x^3 + 1247103*a^2*b^8*c^5*d^5*x^3 + 479655*a^3*b^7*c^4*d^6*x^3 + 15
9885*a^4*b^6*c^3*d^7*x^3 + 43605*a^5*b^5*c^2*d^8*x^3 + 8721*a^6*b^4*c*d^9*x^3 + 969*a^7*b^3*d^10*x^3 + 2200770
*b^10*c^8*d^2*x^2 + 1100385*a*b^9*c^7*d^3*x^2 + 513513*a^2*b^8*c^6*d^4*x^2 + 220077*a^3*b^7*c^5*d^5*x^2 + 8464
5*a^4*b^6*c^4*d^6*x^2 + 28215*a^5*b^5*c^3*d^7*x^2 + 7695*a^6*b^4*c^2*d^8*x^2 + 1539*a^7*b^3*c*d^9*x^2 + 171*a^
8*b^2*d^10*x^2 + 461890*b^10*c^9*d*x + 244530*a*b^9*c^8*d^2*x + 122265*a^2*b^8*c^7*d^3*x + 57057*a^3*b^7*c^6*d
^4*x + 24453*a^4*b^6*c^5*d^5*x + 9405*a^5*b^5*c^4*d^6*x + 3135*a^6*b^4*c^3*d^7*x + 855*a^7*b^3*c^2*d^8*x + 171
*a^8*b^2*c*d^9*x + 19*a^9*b*d^10*x + 43758*b^10*c^10 + 24310*a*b^9*c^9*d + 12870*a^2*b^8*c^8*d^2 + 6435*a^3*b^
7*c^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165*a^7*b^3*c^3*d^7 + 45*a^8*b
^2*c^2*d^8 + 9*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^19*b^11)

Mupad [B] (verification not implemented)

Time = 25.16 (sec) , antiderivative size = 1164, normalized size of antiderivative = 4.26 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{20}} \, dx=-\frac {a^{10}\,d^{10}+9\,a^9\,b\,c\,d^9+19\,a^9\,b\,d^{10}\,x+45\,a^8\,b^2\,c^2\,d^8+171\,a^8\,b^2\,c\,d^9\,x+171\,a^8\,b^2\,d^{10}\,x^2+165\,a^7\,b^3\,c^3\,d^7+855\,a^7\,b^3\,c^2\,d^8\,x+1539\,a^7\,b^3\,c\,d^9\,x^2+969\,a^7\,b^3\,d^{10}\,x^3+495\,a^6\,b^4\,c^4\,d^6+3135\,a^6\,b^4\,c^3\,d^7\,x+7695\,a^6\,b^4\,c^2\,d^8\,x^2+8721\,a^6\,b^4\,c\,d^9\,x^3+3876\,a^6\,b^4\,d^{10}\,x^4+1287\,a^5\,b^5\,c^5\,d^5+9405\,a^5\,b^5\,c^4\,d^6\,x+28215\,a^5\,b^5\,c^3\,d^7\,x^2+43605\,a^5\,b^5\,c^2\,d^8\,x^3+34884\,a^5\,b^5\,c\,d^9\,x^4+11628\,a^5\,b^5\,d^{10}\,x^5+3003\,a^4\,b^6\,c^6\,d^4+24453\,a^4\,b^6\,c^5\,d^5\,x+84645\,a^4\,b^6\,c^4\,d^6\,x^2+159885\,a^4\,b^6\,c^3\,d^7\,x^3+174420\,a^4\,b^6\,c^2\,d^8\,x^4+104652\,a^4\,b^6\,c\,d^9\,x^5+27132\,a^4\,b^6\,d^{10}\,x^6+6435\,a^3\,b^7\,c^7\,d^3+57057\,a^3\,b^7\,c^6\,d^4\,x+220077\,a^3\,b^7\,c^5\,d^5\,x^2+479655\,a^3\,b^7\,c^4\,d^6\,x^3+639540\,a^3\,b^7\,c^3\,d^7\,x^4+523260\,a^3\,b^7\,c^2\,d^8\,x^5+244188\,a^3\,b^7\,c\,d^9\,x^6+50388\,a^3\,b^7\,d^{10}\,x^7+12870\,a^2\,b^8\,c^8\,d^2+122265\,a^2\,b^8\,c^7\,d^3\,x+513513\,a^2\,b^8\,c^6\,d^4\,x^2+1247103\,a^2\,b^8\,c^5\,d^5\,x^3+1918620\,a^2\,b^8\,c^4\,d^6\,x^4+1918620\,a^2\,b^8\,c^3\,d^7\,x^5+1220940\,a^2\,b^8\,c^2\,d^8\,x^6+453492\,a^2\,b^8\,c\,d^9\,x^7+75582\,a^2\,b^8\,d^{10}\,x^8+24310\,a\,b^9\,c^9\,d+244530\,a\,b^9\,c^8\,d^2\,x+1100385\,a\,b^9\,c^7\,d^3\,x^2+2909907\,a\,b^9\,c^6\,d^4\,x^3+4988412\,a\,b^9\,c^5\,d^5\,x^4+5755860\,a\,b^9\,c^4\,d^6\,x^5+4476780\,a\,b^9\,c^3\,d^7\,x^6+2267460\,a\,b^9\,c^2\,d^8\,x^7+680238\,a\,b^9\,c\,d^9\,x^8+92378\,a\,b^9\,d^{10}\,x^9+43758\,b^{10}\,c^{10}+461890\,b^{10}\,c^9\,d\,x+2200770\,b^{10}\,c^8\,d^2\,x^2+6235515\,b^{10}\,c^7\,d^3\,x^3+11639628\,b^{10}\,c^6\,d^4\,x^4+14965236\,b^{10}\,c^5\,d^5\,x^5+13430340\,b^{10}\,c^4\,d^6\,x^6+8314020\,b^{10}\,c^3\,d^7\,x^7+3401190\,b^{10}\,c^2\,d^8\,x^8+831402\,b^{10}\,c\,d^9\,x^9+92378\,b^{10}\,d^{10}\,x^{10}}{831402\,a^{19}\,b^{11}[phone]\,a^{18}\,b^{12}\,x+142169742\,a^{17}\,b^{13}\,x^2+805628538\,a^{16}\,b^{14}\,x^3+3222514152\,a^{15}\,b^{15}\,x^4+9667542456\,a^{14}\,b^{16}\,x^5+22557599064\,a^{13}\,b^{17}\,x^6+41892683976\,a^{12}\,b^{18}\,x^7+62839025964\,a^{11}\,b^{19}\,x^8+76803253956\,a^{10}\,b^{20}\,x^9+76803253956\,a^9\,b^{21}\,x^{10}[phone]\,a^8\,b^{22}\,x^{11}[phone]\,a^7\,b^{23}\,x^{12}[phone]\,a^6\,b^{24}\,x^{13}[phone]\,a^5\,b^{25}\,x^{14}[phone]\,a^4\,b^{26}\,x^{15}[phone]\,a^3\,b^{27}\,x^{16}[phone]\,a^2\,b^{28}\,x^{17}[phone]\,a\,b^{29}\,x^{18}+831402\,b^{30}\,x^{19}} \]

[In]

int((c + d*x)^10/(a + b*x)^20,x)

[Out]

-(a^10*d^10 + 43758*b^10*c^10 + 92378*b^10*d^10*x^10 + 92378*a*b^9*d^10*x^9 + 831402*b^10*c*d^9*x^9 + 12870*a^
2*b^8*c^8*d^2 + 6435*a^3*b^7*c^7*d^3 + 3003*a^4*b^6*c^6*d^4 + 1287*a^5*b^5*c^5*d^5 + 495*a^6*b^4*c^4*d^6 + 165
*a^7*b^3*c^3*d^7 + 45*a^8*b^2*c^2*d^8 + 171*a^8*b^2*d^10*x^2 + 969*a^7*b^3*d^10*x^3 + 3876*a^6*b^4*d^10*x^4 +
11628*a^5*b^5*d^10*x^5 + 27132*a^4*b^6*d^10*x^6 + 50388*a^3*b^7*d^10*x^7 + 75582*a^2*b^8*d^10*x^8 + 2200770*b^
10*c^8*d^2*x^2 + 6235515*b^10*c^7*d^3*x^3 + 11639628*b^10*c^6*d^4*x^4 + 14965236*b^10*c^5*d^5*x^5 + 13430340*b
^10*c^4*d^6*x^6 + 8314020*b^10*c^3*d^7*x^7 + 3401190*b^10*c^2*d^8*x^8 + 24310*a*b^9*c^9*d + 9*a^9*b*c*d^9 + 19
*a^9*b*d^10*x + 461890*b^10*c^9*d*x + 513513*a^2*b^8*c^6*d^4*x^2 + 220077*a^3*b^7*c^5*d^5*x^2 + 84645*a^4*b^6*
c^4*d^6*x^2 + 28215*a^5*b^5*c^3*d^7*x^2 + 7695*a^6*b^4*c^2*d^8*x^2 + 1247103*a^2*b^8*c^5*d^5*x^3 + 479655*a^3*
b^7*c^4*d^6*x^3 + 159885*a^4*b^6*c^3*d^7*x^3 + 43605*a^5*b^5*c^2*d^8*x^3 + 1918620*a^2*b^8*c^4*d^6*x^4 + 63954
0*a^3*b^7*c^3*d^7*x^4 + 174420*a^4*b^6*c^2*d^8*x^4 + 1918620*a^2*b^8*c^3*d^7*x^5 + 523260*a^3*b^7*c^2*d^8*x^5
+ 1220940*a^2*b^8*c^2*d^8*x^6 + 244530*a*b^9*c^8*d^2*x + 171*a^8*b^2*c*d^9*x + 680238*a*b^9*c*d^9*x^8 + 122265
*a^2*b^8*c^7*d^3*x + 57057*a^3*b^7*c^6*d^4*x + 24453*a^4*b^6*c^5*d^5*x + 9405*a^5*b^5*c^4*d^6*x + 3135*a^6*b^4
*c^3*d^7*x + 855*a^7*b^3*c^2*d^8*x + 1100385*a*b^9*c^7*d^3*x^2 + 1539*a^7*b^3*c*d^9*x^2 + 2909907*a*b^9*c^6*d^
4*x^3 + 8721*a^6*b^4*c*d^9*x^3 + 4988412*a*b^9*c^5*d^5*x^4 + 34884*a^5*b^5*c*d^9*x^4 + 5755860*a*b^9*c^4*d^6*x
^5 + 104652*a^4*b^6*c*d^9*x^5 + 4476780*a*b^9*c^3*d^7*x^6 + 244188*a^3*b^7*c*d^9*x^6 + 2267460*a*b^9*c^2*d^8*x
^7 + 453492*a^2*b^8*c*d^9*x^7)/(831402*a^19*b^11 + 831402*b^30*x^19 + 15796638*a^18*b^12*x + 15796638*a*b^29*x
^18 + 142169742*a^17*b^13*x^2 + 805628538*a^16*b^14*x^3 + 3222514152*a^15*b^15*x^4 + 9667542456*a^14*b^16*x^5
+ 22557599064*a^13*b^17*x^6 + 41892683976*a^12*b^18*x^7 + 62839025964*a^11*b^19*x^8 + 76803253956*a^10*b^20*x^
9 + 76803253956*a^9*b^21*x^10 + 62839025964*a^8*b^22*x^11 + 41892683976*a^7*b^23*x^12 + 22557599064*a^6*b^24*x
^13 + 9667542456*a^5*b^25*x^14 + 3222514152*a^4*b^26*x^15 + 805628538*a^3*b^27*x^16 + 142169742*a^2*b^28*x^17)