\(\int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx\) [127]

Optimal result

Integrand size = 15, antiderivative size = 279 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx=-\frac {(b c-a d)^{10}}{21 b^{11} (a+b x)^{21}}-\frac {d (b c-a d)^9}{2 b^{11} (a+b x)^{20}}-\frac {45 d^2 (b c-a d)^8}{19 b^{11} (a+b x)^{19}}-\frac {20 d^3 (b c-a d)^7}{3 b^{11} (a+b x)^{18}}-\frac {210 d^4 (b c-a d)^6}{17 b^{11} (a+b x)^{17}}-\frac {63 d^5 (b c-a d)^5}{4 b^{11} (a+b x)^{16}}-\frac {14 d^6 (b c-a d)^4}{b^{11} (a+b x)^{15}}-\frac {60 d^7 (b c-a d)^3}{7 b^{11} (a+b x)^{14}}-\frac {45 d^8 (b c-a d)^2}{13 b^{11} (a+b x)^{13}}-\frac {5 d^9 (b c-a d)}{6 b^{11} (a+b x)^{12}}-\frac {d^{10}}{11 b^{11} (a+b x)^{11}} \] Output:

-1/21*(-a*d+b*c)^10/b^11/(b*x+a)^21-1/2*d*(-a*d+b*c)^9/b^11/(b*x+a)^20-45/ 
19*d^2*(-a*d+b*c)^8/b^11/(b*x+a)^19-20/3*d^3*(-a*d+b*c)^7/b^11/(b*x+a)^18- 
210/17*d^4*(-a*d+b*c)^6/b^11/(b*x+a)^17-63/4*d^5*(-a*d+b*c)^5/b^11/(b*x+a) 
^16-14*d^6*(-a*d+b*c)^4/b^11/(b*x+a)^15-60/7*d^7*(-a*d+b*c)^3/b^11/(b*x+a) 
^14-45/13*d^8*(-a*d+b*c)^2/b^11/(b*x+a)^13-5/6*d^9*(-a*d+b*c)/b^11/(b*x+a) 
^12-1/11*d^10/b^11/(b*x+a)^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(692\) vs. \(2(279)=558\).

Time = 0.17 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.48 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx=-\frac {a^{10} d^{10}+a^9 b d^9 (11 c+21 d x)+3 a^8 b^2 d^8 \left (22 c^2+77 c d x+70 d^2 x^2\right )+2 a^7 b^3 d^7 \left (143 c^3+693 c^2 d x+1155 c d^2 x^2+665 d^3 x^3\right )+7 a^6 b^4 d^6 \left (143 c^4+858 c^3 d x+1980 c^2 d^2 x^2+2090 c d^3 x^3+855 d^4 x^4\right )+21 a^5 b^5 d^5 \left (143 c^5+1001 c^4 d x+2860 c^3 d^2 x^2+4180 c^2 d^3 x^3+3135 c d^4 x^4+969 d^5 x^5\right )+7 a^4 b^6 d^4 \left (1144 c^6+9009 c^5 d x+30030 c^4 d^2 x^2+54340 c^3 d^3 x^3+56430 c^2 d^4 x^4+31977 c d^5 x^5+7752 d^6 x^6\right )+2 a^3 b^7 d^3 \left (9724 c^7+84084 c^6 d x+315315 c^5 d^2 x^2+665665 c^4 d^3 x^3+855855 c^3 d^4 x^4+671517 c^2 d^5 x^5+298452 c d^6 x^6+58140 d^7 x^7\right )+3 a^2 b^8 d^2 \left (14586 c^8+136136 c^7 d x+560560 c^6 d^2 x^2+1331330 c^5 d^3 x^3+1996995 c^4 d^4 x^4+1939938 c^3 d^5 x^5+1193808 c^2 d^6 x^6+426360 c d^7 x^7+67830 d^8 x^8\right )+a b^9 d \left (92378 c^9+918918 c^8 d x+4084080 c^7 d^2 x^2+10650640 c^6 d^3 x^3+17972955 c^5 d^4 x^4+20369349 c^4 d^5 x^5+15519504 c^3 d^6 x^6+7674480 c^2 d^7 x^7+2238390 c d^8 x^8+293930 d^9 x^9\right )+b^{10} \left (184756 c^{10}+1939938 c^9 d x+9189180 c^8 d^2 x^2+25865840 c^7 d^3 x^3+47927880 c^6 d^4 x^4+61108047 c^5 d^5 x^5+54318264 c^4 d^6 x^6+33256080 c^3 d^7 x^7+13430340 c^2 d^8 x^8+3233230 c d^9 x^9+352716 d^{10} x^{10}\right )}{3879876 b^{11} (a+b x)^{21}} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^22,x]
 

Output:

-1/3879876*(a^10*d^10 + a^9*b*d^9*(11*c + 21*d*x) + 3*a^8*b^2*d^8*(22*c^2 
+ 77*c*d*x + 70*d^2*x^2) + 2*a^7*b^3*d^7*(143*c^3 + 693*c^2*d*x + 1155*c*d 
^2*x^2 + 665*d^3*x^3) + 7*a^6*b^4*d^6*(143*c^4 + 858*c^3*d*x + 1980*c^2*d^ 
2*x^2 + 2090*c*d^3*x^3 + 855*d^4*x^4) + 21*a^5*b^5*d^5*(143*c^5 + 1001*c^4 
*d*x + 2860*c^3*d^2*x^2 + 4180*c^2*d^3*x^3 + 3135*c*d^4*x^4 + 969*d^5*x^5) 
 + 7*a^4*b^6*d^4*(1144*c^6 + 9009*c^5*d*x + 30030*c^4*d^2*x^2 + 54340*c^3* 
d^3*x^3 + 56430*c^2*d^4*x^4 + 31977*c*d^5*x^5 + 7752*d^6*x^6) + 2*a^3*b^7* 
d^3*(9724*c^7 + 84084*c^6*d*x + 315315*c^5*d^2*x^2 + 665665*c^4*d^3*x^3 + 
855855*c^3*d^4*x^4 + 671517*c^2*d^5*x^5 + 298452*c*d^6*x^6 + 58140*d^7*x^7 
) + 3*a^2*b^8*d^2*(14586*c^8 + 136136*c^7*d*x + 560560*c^6*d^2*x^2 + 13313 
30*c^5*d^3*x^3 + 1996995*c^4*d^4*x^4 + 1939938*c^3*d^5*x^5 + 1193808*c^2*d 
^6*x^6 + 426360*c*d^7*x^7 + 67830*d^8*x^8) + a*b^9*d*(92378*c^9 + 918918*c 
^8*d*x + 4084080*c^7*d^2*x^2 + 10650640*c^6*d^3*x^3 + 17972955*c^5*d^4*x^4 
 + 20369349*c^4*d^5*x^5 + 15519504*c^3*d^6*x^6 + 7674480*c^2*d^7*x^7 + 223 
8390*c*d^8*x^8 + 293930*d^9*x^9) + b^10*(184756*c^10 + 1939938*c^9*d*x + 9 
189180*c^8*d^2*x^2 + 25865840*c^7*d^3*x^3 + 47927880*c^6*d^4*x^4 + 6110804 
7*c^5*d^5*x^5 + 54318264*c^4*d^6*x^6 + 33256080*c^3*d^7*x^7 + 13430340*c^2 
*d^8*x^8 + 3233230*c*d^9*x^9 + 352716*d^10*x^10))/(b^11*(a + b*x)^21)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 279, 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 = {53, 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.

\(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx\)

\(\Big \downarrow \) 53

\(\displaystyle \int \left (\frac {10 d^9 (b c-a d)}{b^{10} (a+b x)^{13}}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)^{14}}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^{15}}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^{16}}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^{17}}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^{18}}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^{19}}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^{20}}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^{21}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^{22}}+\frac {d^{10}}{b^{10} (a+b x)^{12}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5 d^9 (b c-a d)}{6 b^{11} (a+b x)^{12}}-\frac {45 d^8 (b c-a d)^2}{13 b^{11} (a+b x)^{13}}-\frac {60 d^7 (b c-a d)^3}{7 b^{11} (a+b x)^{14}}-\frac {14 d^6 (b c-a d)^4}{b^{11} (a+b x)^{15}}-\frac {63 d^5 (b c-a d)^5}{4 b^{11} (a+b x)^{16}}-\frac {210 d^4 (b c-a d)^6}{17 b^{11} (a+b x)^{17}}-\frac {20 d^3 (b c-a d)^7}{3 b^{11} (a+b x)^{18}}-\frac {45 d^2 (b c-a d)^8}{19 b^{11} (a+b x)^{19}}-\frac {d (b c-a d)^9}{2 b^{11} (a+b x)^{20}}-\frac {(b c-a d)^{10}}{21 b^{11} (a+b x)^{21}}-\frac {d^{10}}{11 b^{11} (a+b x)^{11}}\)

Input:

Int[(c + d*x)^10/(a + b*x)^22,x]
 

Output:

-1/21*(b*c - a*d)^10/(b^11*(a + b*x)^21) - (d*(b*c - a*d)^9)/(2*b^11*(a + 
b*x)^20) - (45*d^2*(b*c - a*d)^8)/(19*b^11*(a + b*x)^19) - (20*d^3*(b*c - 
a*d)^7)/(3*b^11*(a + b*x)^18) - (210*d^4*(b*c - a*d)^6)/(17*b^11*(a + b*x) 
^17) - (63*d^5*(b*c - a*d)^5)/(4*b^11*(a + b*x)^16) - (14*d^6*(b*c - a*d)^ 
4)/(b^11*(a + b*x)^15) - (60*d^7*(b*c - a*d)^3)/(7*b^11*(a + b*x)^14) - (4 
5*d^8*(b*c - a*d)^2)/(13*b^11*(a + b*x)^13) - (5*d^9*(b*c - a*d))/(6*b^11* 
(a + b*x)^12) - d^10/(11*b^11*(a + b*x)^11)
 

Defintions of rubi rules used

rule 53

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] && 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])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

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

Time = 0.29 (sec) , antiderivative size = 831, normalized size of antiderivative = 2.98

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+11 a^{9} b c \,d^{9}+66 a^{8} b^{2} c^{2} d^{8}+286 a^{7} b^{3} c^{3} d^{7}+1001 a^{6} b^{4} c^{4} d^{6}+3003 a^{5} b^{5} c^{5} d^{5}+8008 a^{4} b^{6} c^{6} d^{4}+19448 a^{3} b^{7} c^{7} d^{3}+43758 a^{2} b^{8} c^{8} d^{2}+92378 a \,b^{9} c^{9} d +184756 b^{10} c^{10}}{3879876 b^{11}}-\frac {d \left (a^{9} d^{9}+11 a^{8} b c \,d^{8}+66 a^{7} b^{2} c^{2} d^{7}+286 a^{6} b^{3} c^{3} d^{6}+1001 a^{5} b^{4} c^{4} d^{5}+3003 a^{4} b^{5} c^{5} d^{4}+8008 a^{3} b^{6} c^{6} d^{3}+19448 a^{2} b^{7} c^{7} d^{2}+43758 a \,b^{8} c^{8} d +92378 c^{9} b^{9}\right ) x}{184756 b^{10}}-\frac {5 d^{2} \left (a^{8} d^{8}+11 a^{7} b c \,d^{7}+66 a^{6} b^{2} c^{2} d^{6}+286 a^{5} b^{3} c^{3} d^{5}+1001 a^{4} b^{4} c^{4} d^{4}+3003 a^{3} b^{5} c^{5} d^{3}+8008 a^{2} b^{6} c^{6} d^{2}+19448 a \,b^{7} c^{7} d +43758 c^{8} b^{8}\right ) x^{2}}{92378 b^{9}}-\frac {5 d^{3} \left (a^{7} d^{7}+11 a^{6} b c \,d^{6}+66 a^{5} b^{2} c^{2} d^{5}+286 a^{4} b^{3} c^{3} d^{4}+1001 a^{3} b^{4} c^{4} d^{3}+3003 a^{2} b^{5} c^{5} d^{2}+8008 a \,b^{6} c^{6} d +19448 b^{7} c^{7}\right ) x^{3}}{14586 b^{8}}-\frac {15 d^{4} \left (a^{6} d^{6}+11 a^{5} b c \,d^{5}+66 a^{4} b^{2} c^{2} d^{4}+286 a^{3} b^{3} c^{3} d^{3}+1001 a^{2} b^{4} c^{4} d^{2}+3003 a \,b^{5} c^{5} d +8008 c^{6} b^{6}\right ) x^{4}}{9724 b^{7}}-\frac {3 d^{5} \left (a^{5} d^{5}+11 a^{4} b c \,d^{4}+66 a^{3} b^{2} c^{2} d^{3}+286 a^{2} b^{3} c^{3} d^{2}+1001 a \,b^{4} c^{4} d +3003 c^{5} b^{5}\right ) x^{5}}{572 b^{6}}-\frac {2 d^{6} \left (d^{4} a^{4}+11 a^{3} b c \,d^{3}+66 a^{2} b^{2} c^{2} d^{2}+286 a \,b^{3} c^{3} d +1001 c^{4} b^{4}\right ) x^{6}}{143 b^{5}}-\frac {30 d^{7} \left (a^{3} d^{3}+11 a^{2} b c \,d^{2}+66 a \,b^{2} c^{2} d +286 b^{3} c^{3}\right ) x^{7}}{1001 b^{4}}-\frac {15 d^{8} \left (a^{2} d^{2}+11 a b c d +66 b^{2} c^{2}\right ) x^{8}}{286 b^{3}}-\frac {5 d^{9} \left (a d +11 b c \right ) x^{9}}{66 b^{2}}-\frac {d^{10} x^{10}}{11 b}}{\left (b x +a \right )^{21}}\)	\(831\)
default	\(-\frac {210 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 +c^{6} b^{6}\right )}{17 b^{11} \left (b x +a \right )^{17}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{13 b^{11} \left (b x +a \right )^{13}}+\frac {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 -c^{9} b^{9}\right )}{2 b^{11} \left (b x +a \right )^{20}}+\frac {5 d^{9} \left (a d -b c \right )}{6 b^{11} \left (b x +a \right )^{12}}-\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 +c^{8} b^{8}\right )}{19 b^{11} \left (b x +a \right )^{19}}-\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}}{21 b^{11} \left (b x +a \right )^{21}}+\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 )}{7 b^{11} \left (b x +a \right )^{14}}-\frac {14 d^{6} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{b^{11} \left (b x +a \right )^{15}}+\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 -c^{5} b^{5}\right )}{4 b^{11} \left (b x +a \right )^{16}}+\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 )}{3 b^{11} \left (b x +a \right )^{18}}-\frac {d^{10}}{11 b^{11} \left (b x +a \right )^{11}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{10} d^{10}-11 a^{9} b^{11} c \,d^{9}-66 a^{8} b^{12} c^{2} d^{8}-286 a^{7} b^{13} c^{3} d^{7}-1001 a^{6} b^{14} c^{4} d^{6}-3003 a^{5} b^{15} c^{5} d^{5}-8008 a^{4} b^{16} c^{6} d^{4}-19448 a^{3} c^{7} d^{3} b^{17}-43758 a^{2} b^{18} c^{8} d^{2}-92378 a \,b^{19} c^{9} d -184756 b^{20} c^{10}}{3879876 b^{21}}+\frac {\left (-a^{9} b^{10} d^{10}-11 a^{8} b^{11} c \,d^{9}-66 a^{7} b^{12} c^{2} d^{8}-286 a^{6} b^{13} c^{3} d^{7}-1001 a^{5} b^{14} c^{4} d^{6}-3003 a^{4} b^{15} c^{5} d^{5}-8008 a^{3} b^{16} c^{6} d^{4}-19448 a^{2} c^{7} d^{3} b^{17}-43758 a \,b^{18} c^{8} d^{2}-92378 b^{19} c^{9} d \right ) x}{184756 b^{20}}+\frac {5 \left (-a^{8} b^{10} d^{10}-11 a^{7} b^{11} c \,d^{9}-66 a^{6} b^{12} c^{2} d^{8}-286 a^{5} b^{13} c^{3} d^{7}-1001 a^{4} b^{14} c^{4} d^{6}-3003 a^{3} b^{15} c^{5} d^{5}-8008 a^{2} b^{16} c^{6} d^{4}-19448 a \,c^{7} d^{3} b^{17}-43758 b^{18} c^{8} d^{2}\right ) x^{2}}{92378 b^{19}}+\frac {5 \left (-a^{7} b^{10} d^{10}-11 a^{6} b^{11} c \,d^{9}-66 a^{5} b^{12} c^{2} d^{8}-286 a^{4} b^{13} c^{3} d^{7}-1001 a^{3} b^{14} c^{4} d^{6}-3003 a^{2} b^{15} c^{5} d^{5}-8008 a \,b^{16} c^{6} d^{4}-19448 b^{17} c^{7} d^{3}\right ) x^{3}}{14586 b^{18}}+\frac {15 \left (-a^{6} b^{10} d^{10}-11 a^{5} b^{11} c \,d^{9}-66 a^{4} b^{12} c^{2} d^{8}-286 a^{3} b^{13} c^{3} d^{7}-1001 a^{2} b^{14} c^{4} d^{6}-3003 a \,b^{15} c^{5} d^{5}-8008 b^{16} c^{6} d^{4}\right ) x^{4}}{9724 b^{17}}+\frac {3 \left (-a^{5} b^{10} d^{10}-11 a^{4} b^{11} c \,d^{9}-66 a^{3} b^{12} c^{2} d^{8}-286 a^{2} b^{13} c^{3} d^{7}-1001 a \,b^{14} c^{4} d^{6}-3003 b^{15} c^{5} d^{5}\right ) x^{5}}{572 b^{16}}+\frac {2 \left (-a^{4} b^{10} d^{10}-11 a^{3} b^{11} c \,d^{9}-66 a^{2} b^{12} c^{2} d^{8}-286 a \,b^{13} c^{3} d^{7}-1001 b^{14} c^{4} d^{6}\right ) x^{6}}{143 b^{15}}+\frac {30 \left (-a^{3} b^{10} d^{10}-11 a^{2} b^{11} c \,d^{9}-66 a \,b^{12} c^{2} d^{8}-286 b^{13} c^{3} d^{7}\right ) x^{7}}{1001 b^{14}}+\frac {15 \left (-a^{2} b^{10} d^{10}-11 a \,b^{11} c \,d^{9}-66 b^{12} c^{2} d^{8}\right ) x^{8}}{286 b^{13}}+\frac {5 \left (-a \,b^{10} d^{10}-11 b^{11} c \,d^{9}\right ) x^{9}}{66 b^{12}}-\frac {d^{10} x^{10}}{11 b}}{\left (b x +a \right )^{21}}\)	\(909\)
gosper	\(-\frac {352716 x^{10} d^{10} b^{10}+293930 x^{9} a \,b^{9} d^{10}+3233230 x^{9} b^{10} c \,d^{9}+203490 x^{8} a^{2} b^{8} d^{10}+2238390 x^{8} a \,b^{9} c \,d^{9}[phone] x^{8} b^{10} c^{2} d^{8}+116280 x^{7} a^{3} b^{7} d^{10}+1279080 x^{7} a^{2} b^{8} c \,d^{9}+7674480 x^{7} a \,b^{9} c^{2} d^{8}[phone] x^{7} b^{10} c^{3} d^{7}+54264 x^{6} a^{4} b^{6} d^{10}+596904 x^{6} a^{3} b^{7} c \,d^{9}+3581424 x^{6} a^{2} b^{8} c^{2} d^{8}[phone] x^{6} a \,b^{9} c^{3} d^{7}[phone] x^{6} b^{10} c^{4} d^{6}+20349 x^{5} a^{5} b^{5} d^{10}+223839 x^{5} a^{4} b^{6} c \,d^{9}+1343034 x^{5} a^{3} b^{7} c^{2} d^{8}+5819814 x^{5} a^{2} b^{8} c^{3} d^{7}[phone] x^{5} a \,b^{9} c^{4} d^{6}[phone] x^{5} b^{10} c^{5} d^{5}+5985 x^{4} a^{6} b^{4} d^{10}+65835 x^{4} a^{5} b^{5} c \,d^{9}+395010 x^{4} a^{4} b^{6} c^{2} d^{8}+1711710 x^{4} a^{3} b^{7} c^{3} d^{7}+5990985 x^{4} a^{2} b^{8} c^{4} d^{6}[phone] x^{4} a \,b^{9} c^{5} d^{5}[phone] x^{4} b^{10} c^{6} d^{4}+1330 x^{3} a^{7} b^{3} d^{10}+14630 x^{3} a^{6} b^{4} c \,d^{9}+87780 x^{3} a^{5} b^{5} c^{2} d^{8}+380380 x^{3} a^{4} b^{6} c^{3} d^{7}+1331330 x^{3} a^{3} b^{7} c^{4} d^{6}+3993990 x^{3} a^{2} b^{8} c^{5} d^{5}[phone] x^{3} a \,b^{9} c^{6} d^{4}[phone] x^{3} b^{10} c^{7} d^{3}+210 x^{2} a^{8} b^{2} d^{10}+2310 x^{2} a^{7} b^{3} c \,d^{9}+13860 x^{2} a^{6} b^{4} c^{2} d^{8}+60060 x^{2} a^{5} b^{5} c^{3} d^{7}+210210 x^{2} a^{4} b^{6} c^{4} d^{6}+630630 x^{2} a^{3} b^{7} c^{5} d^{5}+1681680 x^{2} a^{2} b^{8} c^{6} d^{4}+4084080 x^{2} a \,b^{9} c^{7} d^{3}+9189180 x^{2} b^{10} c^{8} d^{2}+21 x \,a^{9} b \,d^{10}+231 x \,a^{8} b^{2} c \,d^{9}+1386 x \,a^{7} b^{3} c^{2} d^{8}+6006 x \,a^{6} b^{4} c^{3} d^{7}+21021 x \,a^{5} b^{5} c^{4} d^{6}+63063 x \,a^{4} b^{6} c^{5} d^{5}+168168 x \,a^{3} b^{7} c^{6} d^{4}+408408 x \,a^{2} b^{8} c^{7} d^{3}+918918 x a \,b^{9} c^{8} d^{2}+1939938 x \,b^{10} c^{9} d +a^{10} d^{10}+11 a^{9} b c \,d^{9}+66 a^{8} b^{2} c^{2} d^{8}+286 a^{7} b^{3} c^{3} d^{7}+1001 a^{6} b^{4} c^{4} d^{6}+3003 a^{5} b^{5} c^{5} d^{5}+8008 a^{4} b^{6} c^{6} d^{4}+19448 a^{3} b^{7} c^{7} d^{3}+43758 a^{2} b^{8} c^{8} d^{2}+92378 a \,b^{9} c^{9} d +184756 b^{10} c^{10}}{3879876 b^{11} \left (b x +a \right )^{21}}\)	\(962\)
orering	\(-\frac {352716 x^{10} d^{10} b^{10}+293930 x^{9} a \,b^{9} d^{10}+3233230 x^{9} b^{10} c \,d^{9}+203490 x^{8} a^{2} b^{8} d^{10}+2238390 x^{8} a \,b^{9} c \,d^{9}[phone] x^{8} b^{10} c^{2} d^{8}+116280 x^{7} a^{3} b^{7} d^{10}+1279080 x^{7} a^{2} b^{8} c \,d^{9}+7674480 x^{7} a \,b^{9} c^{2} d^{8}[phone] x^{7} b^{10} c^{3} d^{7}+54264 x^{6} a^{4} b^{6} d^{10}+596904 x^{6} a^{3} b^{7} c \,d^{9}+3581424 x^{6} a^{2} b^{8} c^{2} d^{8}[phone] x^{6} a \,b^{9} c^{3} d^{7}[phone] x^{6} b^{10} c^{4} d^{6}+20349 x^{5} a^{5} b^{5} d^{10}+223839 x^{5} a^{4} b^{6} c \,d^{9}+1343034 x^{5} a^{3} b^{7} c^{2} d^{8}+5819814 x^{5} a^{2} b^{8} c^{3} d^{7}[phone] x^{5} a \,b^{9} c^{4} d^{6}[phone] x^{5} b^{10} c^{5} d^{5}+5985 x^{4} a^{6} b^{4} d^{10}+65835 x^{4} a^{5} b^{5} c \,d^{9}+395010 x^{4} a^{4} b^{6} c^{2} d^{8}+1711710 x^{4} a^{3} b^{7} c^{3} d^{7}+5990985 x^{4} a^{2} b^{8} c^{4} d^{6}[phone] x^{4} a \,b^{9} c^{5} d^{5}[phone] x^{4} b^{10} c^{6} d^{4}+1330 x^{3} a^{7} b^{3} d^{10}+14630 x^{3} a^{6} b^{4} c \,d^{9}+87780 x^{3} a^{5} b^{5} c^{2} d^{8}+380380 x^{3} a^{4} b^{6} c^{3} d^{7}+1331330 x^{3} a^{3} b^{7} c^{4} d^{6}+3993990 x^{3} a^{2} b^{8} c^{5} d^{5}[phone] x^{3} a \,b^{9} c^{6} d^{4}[phone] x^{3} b^{10} c^{7} d^{3}+210 x^{2} a^{8} b^{2} d^{10}+2310 x^{2} a^{7} b^{3} c \,d^{9}+13860 x^{2} a^{6} b^{4} c^{2} d^{8}+60060 x^{2} a^{5} b^{5} c^{3} d^{7}+210210 x^{2} a^{4} b^{6} c^{4} d^{6}+630630 x^{2} a^{3} b^{7} c^{5} d^{5}+1681680 x^{2} a^{2} b^{8} c^{6} d^{4}+4084080 x^{2} a \,b^{9} c^{7} d^{3}+9189180 x^{2} b^{10} c^{8} d^{2}+21 x \,a^{9} b \,d^{10}+231 x \,a^{8} b^{2} c \,d^{9}+1386 x \,a^{7} b^{3} c^{2} d^{8}+6006 x \,a^{6} b^{4} c^{3} d^{7}+21021 x \,a^{5} b^{5} c^{4} d^{6}+63063 x \,a^{4} b^{6} c^{5} d^{5}+168168 x \,a^{3} b^{7} c^{6} d^{4}+408408 x \,a^{2} b^{8} c^{7} d^{3}+918918 x a \,b^{9} c^{8} d^{2}+1939938 x \,b^{10} c^{9} d +a^{10} d^{10}+11 a^{9} b c \,d^{9}+66 a^{8} b^{2} c^{2} d^{8}+286 a^{7} b^{3} c^{3} d^{7}+1001 a^{6} b^{4} c^{4} d^{6}+3003 a^{5} b^{5} c^{5} d^{5}+8008 a^{4} b^{6} c^{6} d^{4}+19448 a^{3} b^{7} c^{7} d^{3}+43758 a^{2} b^{8} c^{8} d^{2}+92378 a \,b^{9} c^{9} d +184756 b^{10} c^{10}}{3879876 b^{11} \left (b x +a \right )^{21}}\)	\(962\)
parallelrisch	\(\frac {-352716 d^{10} x^{10} b^{20}-293930 a \,b^{19} d^{10} x^{9}-3233230 b^{20} c \,d^{9} x^{9}-203490 a^{2} b^{18} d^{10} x^{8}-2238390 a \,b^{19} c \,d^{9} x^{8}-13430340 b^{20} c^{2} d^{8} x^{8}-116280 a^{3} b^{17} d^{10} x^{7}-1279080 a^{2} b^{18} c \,d^{9} x^{7}-7674480 a \,b^{19} c^{2} d^{8} x^{7}-33256080 b^{20} c^{3} d^{7} x^{7}-54264 a^{4} b^{16} d^{10} x^{6}-596904 a^{3} b^{17} c \,d^{9} x^{6}-3581424 a^{2} b^{18} c^{2} d^{8} x^{6}-15519504 a \,b^{19} c^{3} d^{7} x^{6}-54318264 b^{20} c^{4} d^{6} x^{6}-20349 a^{5} b^{15} d^{10} x^{5}-223839 a^{4} b^{16} c \,d^{9} x^{5}-1343034 a^{3} b^{17} c^{2} d^{8} x^{5}-5819814 a^{2} b^{18} c^{3} d^{7} x^{5}-20369349 a \,b^{19} c^{4} d^{6} x^{5}-61108047 b^{20} c^{5} d^{5} x^{5}-5985 a^{6} b^{14} d^{10} x^{4}-65835 a^{5} b^{15} c \,d^{9} x^{4}-395010 a^{4} b^{16} c^{2} d^{8} x^{4}-1711710 a^{3} b^{17} c^{3} d^{7} x^{4}-5990985 a^{2} b^{18} c^{4} d^{6} x^{4}-17972955 a \,b^{19} c^{5} d^{5} x^{4}-47927880 b^{20} c^{6} d^{4} x^{4}-1330 a^{7} b^{13} d^{10} x^{3}-14630 a^{6} b^{14} c \,d^{9} x^{3}-87780 a^{5} b^{15} c^{2} d^{8} x^{3}-380380 a^{4} b^{16} c^{3} d^{7} x^{3}-1331330 a^{3} b^{17} c^{4} d^{6} x^{3}-3993990 a^{2} b^{18} c^{5} d^{5} x^{3}-10650640 a \,b^{19} c^{6} d^{4} x^{3}-25865840 b^{20} c^{7} d^{3} x^{3}-210 a^{8} b^{12} d^{10} x^{2}-2310 a^{7} b^{13} c \,d^{9} x^{2}-13860 a^{6} b^{14} c^{2} d^{8} x^{2}-60060 a^{5} b^{15} c^{3} d^{7} x^{2}-210210 a^{4} b^{16} c^{4} d^{6} x^{2}-630630 a^{3} b^{17} c^{5} d^{5} x^{2}-1681680 a^{2} b^{18} c^{6} d^{4} x^{2}-4084080 a \,b^{19} c^{7} d^{3} x^{2}-9189180 b^{20} c^{8} d^{2} x^{2}-21 a^{9} b^{11} d^{10} x -231 a^{8} b^{12} c \,d^{9} x -1386 a^{7} b^{13} c^{2} d^{8} x -6006 a^{6} b^{14} c^{3} d^{7} x -21021 a^{5} b^{15} c^{4} d^{6} x -63063 a^{4} b^{16} c^{5} d^{5} x -168168 a^{3} b^{17} c^{6} d^{4} x -408408 a^{2} b^{18} c^{7} d^{3} x -918918 a \,b^{19} c^{8} d^{2} x -1939938 b^{20} c^{9} d x -a^{10} b^{10} d^{10}-11 a^{9} b^{11} c \,d^{9}-66 a^{8} b^{12} c^{2} d^{8}-286 a^{7} b^{13} c^{3} d^{7}-1001 a^{6} b^{14} c^{4} d^{6}-3003 a^{5} b^{15} c^{5} d^{5}-8008 a^{4} b^{16} c^{6} d^{4}-19448 a^{3} c^{7} d^{3} b^{17}-43758 a^{2} b^{18} c^{8} d^{2}-92378 a \,b^{19} c^{9} d -184756 b^{20} c^{10}}{3879876 b^{21} \left (b x +a \right )^{21}}\)	\(970\)

Input:

int((d*x+c)^10/(b*x+a)^22,x,method=_RETURNVERBOSE)
 

Output:

(-1/3879876/b^11*(a^10*d^10+11*a^9*b*c*d^9+66*a^8*b^2*c^2*d^8+286*a^7*b^3* 
c^3*d^7+1001*a^6*b^4*c^4*d^6+3003*a^5*b^5*c^5*d^5+8008*a^4*b^6*c^6*d^4+194 
48*a^3*b^7*c^7*d^3+43758*a^2*b^8*c^8*d^2+92378*a*b^9*c^9*d+184756*b^10*c^1 
0)-1/184756/b^10*d*(a^9*d^9+11*a^8*b*c*d^8+66*a^7*b^2*c^2*d^7+286*a^6*b^3* 
c^3*d^6+1001*a^5*b^4*c^4*d^5+3003*a^4*b^5*c^5*d^4+8008*a^3*b^6*c^6*d^3+194 
48*a^2*b^7*c^7*d^2+43758*a*b^8*c^8*d+92378*b^9*c^9)*x-5/92378/b^9*d^2*(a^8 
*d^8+11*a^7*b*c*d^7+66*a^6*b^2*c^2*d^6+286*a^5*b^3*c^3*d^5+1001*a^4*b^4*c^ 
4*d^4+3003*a^3*b^5*c^5*d^3+8008*a^2*b^6*c^6*d^2+19448*a*b^7*c^7*d+43758*b^ 
8*c^8)*x^2-5/14586/b^8*d^3*(a^7*d^7+11*a^6*b*c*d^6+66*a^5*b^2*c^2*d^5+286* 
a^4*b^3*c^3*d^4+1001*a^3*b^4*c^4*d^3+3003*a^2*b^5*c^5*d^2+8008*a*b^6*c^6*d 
+19448*b^7*c^7)*x^3-15/9724/b^7*d^4*(a^6*d^6+11*a^5*b*c*d^5+66*a^4*b^2*c^2 
*d^4+286*a^3*b^3*c^3*d^3+1001*a^2*b^4*c^4*d^2+3003*a*b^5*c^5*d+8008*b^6*c^ 
6)*x^4-3/572/b^6*d^5*(a^5*d^5+11*a^4*b*c*d^4+66*a^3*b^2*c^2*d^3+286*a^2*b^ 
3*c^3*d^2+1001*a*b^4*c^4*d+3003*b^5*c^5)*x^5-2/143/b^5*d^6*(a^4*d^4+11*a^3 
*b*c*d^3+66*a^2*b^2*c^2*d^2+286*a*b^3*c^3*d+1001*b^4*c^4)*x^6-30/1001/b^4* 
d^7*(a^3*d^3+11*a^2*b*c*d^2+66*a*b^2*c^2*d+286*b^3*c^3)*x^7-15/286/b^3*d^8 
*(a^2*d^2+11*a*b*c*d+66*b^2*c^2)*x^8-5/66/b^2*d^9*(a*d+11*b*c)*x^9-1/11/b* 
d^10*x^10)/(b*x+a)^21
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (259) = 518\).

Time = 0.10 (sec) , antiderivative size = 1085, normalized size of antiderivative = 3.89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^22,x, algorithm="fricas")
 

Output:

-1/3879876*(352716*b^10*d^10*x^10 + 184756*b^10*c^10 + 92378*a*b^9*c^9*d + 
 43758*a^2*b^8*c^8*d^2 + 19448*a^3*b^7*c^7*d^3 + 8008*a^4*b^6*c^6*d^4 + 30 
03*a^5*b^5*c^5*d^5 + 1001*a^6*b^4*c^4*d^6 + 286*a^7*b^3*c^3*d^7 + 66*a^8*b 
^2*c^2*d^8 + 11*a^9*b*c*d^9 + a^10*d^10 + 293930*(11*b^10*c*d^9 + a*b^9*d^ 
10)*x^9 + 203490*(66*b^10*c^2*d^8 + 11*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 1 
16280*(286*b^10*c^3*d^7 + 66*a*b^9*c^2*d^8 + 11*a^2*b^8*c*d^9 + a^3*b^7*d^ 
10)*x^7 + 54264*(1001*b^10*c^4*d^6 + 286*a*b^9*c^3*d^7 + 66*a^2*b^8*c^2*d^ 
8 + 11*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 20349*(3003*b^10*c^5*d^5 + 1001 
*a*b^9*c^4*d^6 + 286*a^2*b^8*c^3*d^7 + 66*a^3*b^7*c^2*d^8 + 11*a^4*b^6*c*d 
^9 + a^5*b^5*d^10)*x^5 + 5985*(8008*b^10*c^6*d^4 + 3003*a*b^9*c^5*d^5 + 10 
01*a^2*b^8*c^4*d^6 + 286*a^3*b^7*c^3*d^7 + 66*a^4*b^6*c^2*d^8 + 11*a^5*b^5 
*c*d^9 + a^6*b^4*d^10)*x^4 + 1330*(19448*b^10*c^7*d^3 + 8008*a*b^9*c^6*d^4 
 + 3003*a^2*b^8*c^5*d^5 + 1001*a^3*b^7*c^4*d^6 + 286*a^4*b^6*c^3*d^7 + 66* 
a^5*b^5*c^2*d^8 + 11*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 210*(43758*b^10*c 
^8*d^2 + 19448*a*b^9*c^7*d^3 + 8008*a^2*b^8*c^6*d^4 + 3003*a^3*b^7*c^5*d^5 
 + 1001*a^4*b^6*c^4*d^6 + 286*a^5*b^5*c^3*d^7 + 66*a^6*b^4*c^2*d^8 + 11*a^ 
7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 21*(92378*b^10*c^9*d + 43758*a*b^9*c^8*d 
^2 + 19448*a^2*b^8*c^7*d^3 + 8008*a^3*b^7*c^6*d^4 + 3003*a^4*b^6*c^5*d^5 + 
 1001*a^5*b^5*c^4*d^6 + 286*a^6*b^4*c^3*d^7 + 66*a^7*b^3*c^2*d^8 + 11*a^8* 
b^2*c*d^9 + a^9*b*d^10)*x)/(b^32*x^21 + 21*a*b^31*x^20 + 210*a^2*b^30*x...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**10/(b*x+a)**22,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1085 vs. \(2 (259) = 518\).

Time = 0.13 (sec) , antiderivative size = 1085, normalized size of antiderivative = 3.89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^22,x, algorithm="maxima")
 

Output:

-1/3879876*(352716*b^10*d^10*x^10 + 184756*b^10*c^10 + 92378*a*b^9*c^9*d + 
 43758*a^2*b^8*c^8*d^2 + 19448*a^3*b^7*c^7*d^3 + 8008*a^4*b^6*c^6*d^4 + 30 
03*a^5*b^5*c^5*d^5 + 1001*a^6*b^4*c^4*d^6 + 286*a^7*b^3*c^3*d^7 + 66*a^8*b 
^2*c^2*d^8 + 11*a^9*b*c*d^9 + a^10*d^10 + 293930*(11*b^10*c*d^9 + a*b^9*d^ 
10)*x^9 + 203490*(66*b^10*c^2*d^8 + 11*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 1 
16280*(286*b^10*c^3*d^7 + 66*a*b^9*c^2*d^8 + 11*a^2*b^8*c*d^9 + a^3*b^7*d^ 
10)*x^7 + 54264*(1001*b^10*c^4*d^6 + 286*a*b^9*c^3*d^7 + 66*a^2*b^8*c^2*d^ 
8 + 11*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 20349*(3003*b^10*c^5*d^5 + 1001 
*a*b^9*c^4*d^6 + 286*a^2*b^8*c^3*d^7 + 66*a^3*b^7*c^2*d^8 + 11*a^4*b^6*c*d 
^9 + a^5*b^5*d^10)*x^5 + 5985*(8008*b^10*c^6*d^4 + 3003*a*b^9*c^5*d^5 + 10 
01*a^2*b^8*c^4*d^6 + 286*a^3*b^7*c^3*d^7 + 66*a^4*b^6*c^2*d^8 + 11*a^5*b^5 
*c*d^9 + a^6*b^4*d^10)*x^4 + 1330*(19448*b^10*c^7*d^3 + 8008*a*b^9*c^6*d^4 
 + 3003*a^2*b^8*c^5*d^5 + 1001*a^3*b^7*c^4*d^6 + 286*a^4*b^6*c^3*d^7 + 66* 
a^5*b^5*c^2*d^8 + 11*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 210*(43758*b^10*c 
^8*d^2 + 19448*a*b^9*c^7*d^3 + 8008*a^2*b^8*c^6*d^4 + 3003*a^3*b^7*c^5*d^5 
 + 1001*a^4*b^6*c^4*d^6 + 286*a^5*b^5*c^3*d^7 + 66*a^6*b^4*c^2*d^8 + 11*a^ 
7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 21*(92378*b^10*c^9*d + 43758*a*b^9*c^8*d 
^2 + 19448*a^2*b^8*c^7*d^3 + 8008*a^3*b^7*c^6*d^4 + 3003*a^4*b^6*c^5*d^5 + 
 1001*a^5*b^5*c^4*d^6 + 286*a^6*b^4*c^3*d^7 + 66*a^7*b^3*c^2*d^8 + 11*a^8* 
b^2*c*d^9 + a^9*b*d^10)*x)/(b^32*x^21 + 21*a*b^31*x^20 + 210*a^2*b^30*x...
 

Giac [B] (verification not implemented)

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

Time = 0.13 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.44 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^22,x, algorithm="giac")
 

Output:

-1/3879876*(352716*b^10*d^10*x^10 + 3233230*b^10*c*d^9*x^9 + 293930*a*b^9* 
d^10*x^9 + 13430340*b^10*c^2*d^8*x^8 + 2238390*a*b^9*c*d^9*x^8 + 203490*a^ 
2*b^8*d^10*x^8 + 33256080*b^10*c^3*d^7*x^7 + 7674480*a*b^9*c^2*d^8*x^7 + 1 
279080*a^2*b^8*c*d^9*x^7 + 116280*a^3*b^7*d^10*x^7 + 54318264*b^10*c^4*d^6 
*x^6 + 15519504*a*b^9*c^3*d^7*x^6 + 3581424*a^2*b^8*c^2*d^8*x^6 + 596904*a 
^3*b^7*c*d^9*x^6 + 54264*a^4*b^6*d^10*x^6 + 61108047*b^10*c^5*d^5*x^5 + 20 
369349*a*b^9*c^4*d^6*x^5 + 5819814*a^2*b^8*c^3*d^7*x^5 + 1343034*a^3*b^7*c 
^2*d^8*x^5 + 223839*a^4*b^6*c*d^9*x^5 + 20349*a^5*b^5*d^10*x^5 + 47927880* 
b^10*c^6*d^4*x^4 + 17972955*a*b^9*c^5*d^5*x^4 + 5990985*a^2*b^8*c^4*d^6*x^ 
4 + 1711710*a^3*b^7*c^3*d^7*x^4 + 395010*a^4*b^6*c^2*d^8*x^4 + 65835*a^5*b 
^5*c*d^9*x^4 + 5985*a^6*b^4*d^10*x^4 + 25865840*b^10*c^7*d^3*x^3 + 1065064 
0*a*b^9*c^6*d^4*x^3 + 3993990*a^2*b^8*c^5*d^5*x^3 + 1331330*a^3*b^7*c^4*d^ 
6*x^3 + 380380*a^4*b^6*c^3*d^7*x^3 + 87780*a^5*b^5*c^2*d^8*x^3 + 14630*a^6 
*b^4*c*d^9*x^3 + 1330*a^7*b^3*d^10*x^3 + 9189180*b^10*c^8*d^2*x^2 + 408408 
0*a*b^9*c^7*d^3*x^2 + 1681680*a^2*b^8*c^6*d^4*x^2 + 630630*a^3*b^7*c^5*d^5 
*x^2 + 210210*a^4*b^6*c^4*d^6*x^2 + 60060*a^5*b^5*c^3*d^7*x^2 + 13860*a^6* 
b^4*c^2*d^8*x^2 + 2310*a^7*b^3*c*d^9*x^2 + 210*a^8*b^2*d^10*x^2 + 1939938* 
b^10*c^9*d*x + 918918*a*b^9*c^8*d^2*x + 408408*a^2*b^8*c^7*d^3*x + 168168* 
a^3*b^7*c^6*d^4*x + 63063*a^4*b^6*c^5*d^5*x + 21021*a^5*b^5*c^4*d^6*x + 60 
06*a^6*b^4*c^3*d^7*x + 1386*a^7*b^3*c^2*d^8*x + 231*a^8*b^2*c*d^9*x + 2...
 

Mupad [B] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 1186, normalized size of antiderivative = 4.25 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx =\text {Too large to display} \] Input:

int((c + d*x)^10/(a + b*x)^22,x)
 

Output:

-(a^10*d^10 + 184756*b^10*c^10 + 352716*b^10*d^10*x^10 + 293930*a*b^9*d^10 
*x^9 + 3233230*b^10*c*d^9*x^9 + 43758*a^2*b^8*c^8*d^2 + 19448*a^3*b^7*c^7* 
d^3 + 8008*a^4*b^6*c^6*d^4 + 3003*a^5*b^5*c^5*d^5 + 1001*a^6*b^4*c^4*d^6 + 
 286*a^7*b^3*c^3*d^7 + 66*a^8*b^2*c^2*d^8 + 210*a^8*b^2*d^10*x^2 + 1330*a^ 
7*b^3*d^10*x^3 + 5985*a^6*b^4*d^10*x^4 + 20349*a^5*b^5*d^10*x^5 + 54264*a^ 
4*b^6*d^10*x^6 + 116280*a^3*b^7*d^10*x^7 + 203490*a^2*b^8*d^10*x^8 + 91891 
80*b^10*c^8*d^2*x^2 + 25865840*b^10*c^7*d^3*x^3 + 47927880*b^10*c^6*d^4*x^ 
4 + 61108047*b^10*c^5*d^5*x^5 + 54318264*b^10*c^4*d^6*x^6 + 33256080*b^10* 
c^3*d^7*x^7 + 13430340*b^10*c^2*d^8*x^8 + 92378*a*b^9*c^9*d + 11*a^9*b*c*d 
^9 + 21*a^9*b*d^10*x + 1939938*b^10*c^9*d*x + 1681680*a^2*b^8*c^6*d^4*x^2 
+ 630630*a^3*b^7*c^5*d^5*x^2 + 210210*a^4*b^6*c^4*d^6*x^2 + 60060*a^5*b^5* 
c^3*d^7*x^2 + 13860*a^6*b^4*c^2*d^8*x^2 + 3993990*a^2*b^8*c^5*d^5*x^3 + 13 
31330*a^3*b^7*c^4*d^6*x^3 + 380380*a^4*b^6*c^3*d^7*x^3 + 87780*a^5*b^5*c^2 
*d^8*x^3 + 5990985*a^2*b^8*c^4*d^6*x^4 + 1711710*a^3*b^7*c^3*d^7*x^4 + 395 
010*a^4*b^6*c^2*d^8*x^4 + 5819814*a^2*b^8*c^3*d^7*x^5 + 1343034*a^3*b^7*c^ 
2*d^8*x^5 + 3581424*a^2*b^8*c^2*d^8*x^6 + 918918*a*b^9*c^8*d^2*x + 231*a^8 
*b^2*c*d^9*x + 2238390*a*b^9*c*d^9*x^8 + 408408*a^2*b^8*c^7*d^3*x + 168168 
*a^3*b^7*c^6*d^4*x + 63063*a^4*b^6*c^5*d^5*x + 21021*a^5*b^5*c^4*d^6*x + 6 
006*a^6*b^4*c^3*d^7*x + 1386*a^7*b^3*c^2*d^8*x + 4084080*a*b^9*c^7*d^3*x^2 
 + 2310*a^7*b^3*c*d^9*x^2 + 10650640*a*b^9*c^6*d^4*x^3 + 14630*a^6*b^4*...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1182, normalized size of antiderivative = 4.24 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^22,x)
 

Output:

( - a**10*d**10 - 11*a**9*b*c*d**9 - 21*a**9*b*d**10*x - 66*a**8*b**2*c**2 
*d**8 - 231*a**8*b**2*c*d**9*x - 210*a**8*b**2*d**10*x**2 - 286*a**7*b**3* 
c**3*d**7 - 1386*a**7*b**3*c**2*d**8*x - 2310*a**7*b**3*c*d**9*x**2 - 1330 
*a**7*b**3*d**10*x**3 - 1001*a**6*b**4*c**4*d**6 - 6006*a**6*b**4*c**3*d** 
7*x - 13860*a**6*b**4*c**2*d**8*x**2 - 14630*a**6*b**4*c*d**9*x**3 - 5985* 
a**6*b**4*d**10*x**4 - 3003*a**5*b**5*c**5*d**5 - 21021*a**5*b**5*c**4*d** 
6*x - 60060*a**5*b**5*c**3*d**7*x**2 - 87780*a**5*b**5*c**2*d**8*x**3 - 65 
835*a**5*b**5*c*d**9*x**4 - 20349*a**5*b**5*d**10*x**5 - 8008*a**4*b**6*c* 
*6*d**4 - 63063*a**4*b**6*c**5*d**5*x - 210210*a**4*b**6*c**4*d**6*x**2 - 
380380*a**4*b**6*c**3*d**7*x**3 - 395010*a**4*b**6*c**2*d**8*x**4 - 223839 
*a**4*b**6*c*d**9*x**5 - 54264*a**4*b**6*d**10*x**6 - 19448*a**3*b**7*c**7 
*d**3 - 168168*a**3*b**7*c**6*d**4*x - 630630*a**3*b**7*c**5*d**5*x**2 - 1 
331330*a**3*b**7*c**4*d**6*x**3 - 1711710*a**3*b**7*c**3*d**7*x**4 - 13430 
34*a**3*b**7*c**2*d**8*x**5 - 596904*a**3*b**7*c*d**9*x**6 - 116280*a**3*b 
**7*d**10*x**7 - 43758*a**2*b**8*c**8*d**2 - 408408*a**2*b**8*c**7*d**3*x 
- 1681680*a**2*b**8*c**6*d**4*x**2 - 3993990*a**2*b**8*c**5*d**5*x**3 - 59 
90985*a**2*b**8*c**4*d**6*x**4 - 5819814*a**2*b**8*c**3*d**7*x**5 - 358142 
4*a**2*b**8*c**2*d**8*x**6 - 1279080*a**2*b**8*c*d**9*x**7 - 203490*a**2*b 
**8*d**10*x**8 - 92378*a*b**9*c**9*d - 918918*a*b**9*c**8*d**2*x - 4084080 
*a*b**9*c**7*d**3*x**2 - 10650640*a*b**9*c**6*d**4*x**3 - 17972955*a*b*...