\(\int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx\) [1333]

Optimal result

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

[Out]

-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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 279, 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} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx=-\frac {5 d^9 (b c-a d)}{6 b^{11} (a+b x)^{12}}-\frac {45 d^8 (b c-a d)^2}{13 b^{11} (a+b x)^{13}}-\frac {60 d^7 (b c-a d)^3}{7 b^{11} (a+b x)^{14}}-\frac {14 d^6 (b c-a d)^4}{b^{11} (a+b x)^{15}}-\frac {63 d^5 (b c-a d)^5}{4 b^{11} (a+b x)^{16}}-\frac {210 d^4 (b c-a d)^6}{17 b^{11} (a+b x)^{17}}-\frac {20 d^3 (b c-a d)^7}{3 b^{11} (a+b x)^{18}}-\frac {45 d^2 (b c-a d)^8}{19 b^{11} (a+b x)^{19}}-\frac {d (b c-a d)^9}{2 b^{11} (a+b x)^{20}}-\frac {(b c-a d)^{10}}{21 b^{11} (a+b x)^{21}}-\frac {d^{10}}{11 b^{11} (a+b x)^{11}} \]

[In]

Int[(c + d*x)^10/(a + b*x)^22,x]

[Out]

-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)/(1
9*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) - (45*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)

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

Mathematica [B] (verified)

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

Time = 0.18 (sec) , antiderivative size = 692, normalized size of antiderivative = 2.48 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx=-\frac {a^{10} d^{10}+a^9 b d^9 (11 c+21 d x)+3 a^8 b^2 d^8 \left (22 c^2+77 c d x+70 d^2 x^2\right )+2 a^7 b^3 d^7 \left (143 c^3+693 c^2 d x+1155 c d^2 x^2+665 d^3 x^3\right )+7 a^6 b^4 d^6 \left (143 c^4+858 c^3 d x+1980 c^2 d^2 x^2+2090 c d^3 x^3+855 d^4 x^4\right )+21 a^5 b^5 d^5 \left (143 c^5+1001 c^4 d x+2860 c^3 d^2 x^2+4180 c^2 d^3 x^3+3135 c d^4 x^4+969 d^5 x^5\right )+7 a^4 b^6 d^4 \left (1144 c^6+9009 c^5 d x+30030 c^4 d^2 x^2+54340 c^3 d^3 x^3+56430 c^2 d^4 x^4+31977 c d^5 x^5+7752 d^6 x^6\right )+2 a^3 b^7 d^3 \left (9724 c^7+84084 c^6 d x+315315 c^5 d^2 x^2+665665 c^4 d^3 x^3+855855 c^3 d^4 x^4+671517 c^2 d^5 x^5+298452 c d^6 x^6+58140 d^7 x^7\right )+3 a^2 b^8 d^2 \left (14586 c^8+136136 c^7 d x+560560 c^6 d^2 x^2+1331330 c^5 d^3 x^3+1996995 c^4 d^4 x^4+1939938 c^3 d^5 x^5+1193808 c^2 d^6 x^6+426360 c d^7 x^7+67830 d^8 x^8\right )+a b^9 d \left (92378 c^9+918918 c^8 d x+4084080 c^7 d^2 x^2+10650640 c^6 d^3 x^3+17972955 c^5 d^4 x^4+20369349 c^4 d^5 x^5+15519504 c^3 d^6 x^6+7674480 c^2 d^7 x^7+2238390 c d^8 x^8+293930 d^9 x^9\right )+b^{10} \left (184756 c^{10}+1939938 c^9 d x+9189180 c^8 d^2 x^2+25865840 c^7 d^3 x^3+47927880 c^6 d^4 x^4+61108047 c^5 d^5 x^5+54318264 c^4 d^6 x^6+33256080 c^3 d^7 x^7+13430340 c^2 d^8 x^8+3233230 c d^9 x^9+352716 d^{10} x^{10}\right )}{3879876 b^{11} (a+b x)^{21}} \]

[In]

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

[Out]

-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 + 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) + 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 + 15
519504*c^3*d^6*x^6 + 7674480*c^2*d^7*x^7 + 2238390*c*d^8*x^8 + 293930*d^9*x^9) + b^10*(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))/(b^11*(a + b
*x)^21)

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 = 2.98

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+11 a^{9} b c \,d^{9}+66 a^{8} b^{2} c^{2} d^{8}+286 a^{7} b^{3} c^{3} d^{7}+1001 a^{6} b^{4} c^{4} d^{6}+3003 a^{5} b^{5} c^{5} d^{5}+8008 a^{4} b^{6} c^{6} d^{4}+19448 a^{3} b^{7} c^{7} d^{3}+43758 a^{2} b^{8} c^{8} d^{2}+92378 a \,b^{9} c^{9} d +184756 b^{10} c^{10}}{3879876 b^{11}}-\frac {d \left (a^{9} d^{9}+11 a^{8} b c \,d^{8}+66 a^{7} b^{2} c^{2} d^{7}+286 a^{6} b^{3} c^{3} d^{6}+1001 a^{5} b^{4} c^{4} d^{5}+3003 a^{4} b^{5} c^{5} d^{4}+8008 a^{3} b^{6} c^{6} d^{3}+19448 a^{2} b^{7} c^{7} d^{2}+43758 a \,b^{8} c^{8} d +92378 b^{9} c^{9}\right ) x}{184756 b^{10}}-\frac {5 d^{2} \left (a^{8} d^{8}+11 a^{7} b c \,d^{7}+66 a^{6} b^{2} c^{2} d^{6}+286 a^{5} b^{3} c^{3} d^{5}+1001 a^{4} b^{4} c^{4} d^{4}+3003 a^{3} b^{5} c^{5} d^{3}+8008 a^{2} b^{6} c^{6} d^{2}+19448 a \,b^{7} c^{7} d +43758 b^{8} c^{8}\right ) x^{2}}{92378 b^{9}}-\frac {5 d^{3} \left (a^{7} d^{7}+11 a^{6} b c \,d^{6}+66 a^{5} b^{2} c^{2} d^{5}+286 a^{4} b^{3} c^{3} d^{4}+1001 a^{3} b^{4} c^{4} d^{3}+3003 a^{2} b^{5} c^{5} d^{2}+8008 a \,b^{6} c^{6} d +19448 b^{7} c^{7}\right ) x^{3}}{14586 b^{8}}-\frac {15 d^{4} \left (a^{6} d^{6}+11 a^{5} b c \,d^{5}+66 a^{4} b^{2} c^{2} d^{4}+286 a^{3} b^{3} c^{3} d^{3}+1001 a^{2} b^{4} c^{4} d^{2}+3003 a \,b^{5} c^{5} d +8008 b^{6} c^{6}\right ) x^{4}}{9724 b^{7}}-\frac {3 d^{5} \left (a^{5} d^{5}+11 a^{4} b c \,d^{4}+66 a^{3} b^{2} c^{2} d^{3}+286 a^{2} b^{3} c^{3} d^{2}+1001 a \,b^{4} c^{4} d +3003 b^{5} c^{5}\right ) x^{5}}{572 b^{6}}-\frac {2 d^{6} \left (a^{4} d^{4}+11 a^{3} b c \,d^{3}+66 a^{2} b^{2} c^{2} d^{2}+286 a \,b^{3} c^{3} d +1001 b^{4} c^{4}\right ) x^{6}}{143 b^{5}}-\frac {30 d^{7} \left (a^{3} d^{3}+11 a^{2} b c \,d^{2}+66 a \,b^{2} c^{2} d +286 b^{3} c^{3}\right ) x^{7}}{1001 b^{4}}-\frac {15 d^{8} \left (a^{2} d^{2}+11 a b c d +66 b^{2} c^{2}\right ) x^{8}}{286 b^{3}}-\frac {5 d^{9} \left (a d +11 b c \right ) x^{9}}{66 b^{2}}-\frac {d^{10} x^{10}}{11 b}}{\left (b x +a \right )^{21}}\)	\(831\)
default	\(-\frac {d^{10}}{11 b^{11} \left (b x +a \right )^{11}}-\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 {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 )}{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 {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 )}{2 b^{11} \left (b x +a \right )^{20}}+\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 {5 d^{9} \left (a d -b c \right )}{6 b^{11} \left (b x +a \right )^{12}}+\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 )}{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 {14 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 )^{15}}-\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 +b^{6} c^{6}\right )}{17 b^{11} \left (b x +a \right )^{17}}\)	\(867\)
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}}\)	\(962\)
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}}\)	\(970\)

[In]

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

[Out]

(-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+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)-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+19448*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+800
8*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+3
003*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 = 0.23 (sec) , antiderivative size = 1085, normalized size of antiderivative = 3.89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx=-\frac {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} + 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} + 11 \, a^{9} b c d^{9} + a^{10} d^{10} + 293930 \, {\left (11 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 203490 \, {\left (66 \, b^{10} c^{2} d^{8} + 11 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 116280 \, {\left (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}\right )} x^{7} + 54264 \, {\left (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}\right )} x^{6} + 20349 \, {\left (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}\right )} x^{5} + 5985 \, {\left (8008 \, b^{10} c^{6} d^{4} + 3003 \, a b^{9} c^{5} d^{5} + 1001 \, 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}\right )} x^{4} + 1330 \, {\left (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}\right )} x^{3} + 210 \, {\left (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}\right )} x^{2} + 21 \, {\left (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}\right )} x}{3879876 \, {\left (b^{32} x^{21} + 21 \, a b^{31} x^{20} + 210 \, a^{2} b^{30} x^{19} + 1330 \, a^{3} b^{29} x^{18} + 5985 \, a^{4} b^{28} x^{17} + 20349 \, a^{5} b^{27} x^{16} + 54264 \, a^{6} b^{26} x^{15} + 116280 \, a^{7} b^{25} x^{14} + 203490 \, a^{8} b^{24} x^{13} + 293930 \, a^{9} b^{23} x^{12} + 352716 \, a^{10} b^{22} x^{11} + 352716 \, a^{11} b^{21} x^{10} + 293930 \, a^{12} b^{20} x^{9} + 203490 \, a^{13} b^{19} x^{8} + 116280 \, a^{14} b^{18} x^{7} + 54264 \, a^{15} b^{17} x^{6} + 20349 \, a^{16} b^{16} x^{5} + 5985 \, a^{17} b^{15} x^{4} + 1330 \, a^{18} b^{14} x^{3} + 210 \, a^{19} b^{13} x^{2} + 21 \, a^{20} b^{12} x + a^{21} b^{11}\right )}} \]

[In]

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

[Out]

-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 + 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 + 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 + 116280*(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 + 1001*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^19 + 1330*a^3*b^29*x^18 + 5985*a^4*b^28*x^17 + 20349*a^5*b^27*
x^16 + 54264*a^6*b^26*x^15 + 116280*a^7*b^25*x^14 + 203490*a^8*b^24*x^13 + 293930*a^9*b^23*x^12 + 352716*a^10*
b^22*x^11 + 352716*a^11*b^21*x^10 + 293930*a^12*b^20*x^9 + 203490*a^13*b^19*x^8 + 116280*a^14*b^18*x^7 + 54264
*a^15*b^17*x^6 + 20349*a^16*b^16*x^5 + 5985*a^17*b^15*x^4 + 1330*a^18*b^14*x^3 + 210*a^19*b^13*x^2 + 21*a^20*b
^12*x + a^21*b^11)

Sympy [F(-1)]

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

[In]

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

[Out]

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 = 0.30 (sec) , antiderivative size = 1085, normalized size of antiderivative = 3.89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx=-\frac {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} + 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} + 11 \, a^{9} b c d^{9} + a^{10} d^{10} + 293930 \, {\left (11 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 203490 \, {\left (66 \, b^{10} c^{2} d^{8} + 11 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 116280 \, {\left (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}\right )} x^{7} + 54264 \, {\left (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}\right )} x^{6} + 20349 \, {\left (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}\right )} x^{5} + 5985 \, {\left (8008 \, b^{10} c^{6} d^{4} + 3003 \, a b^{9} c^{5} d^{5} + 1001 \, 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}\right )} x^{4} + 1330 \, {\left (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}\right )} x^{3} + 210 \, {\left (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}\right )} x^{2} + 21 \, {\left (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}\right )} x}{3879876 \, {\left (b^{32} x^{21} + 21 \, a b^{31} x^{20} + 210 \, a^{2} b^{30} x^{19} + 1330 \, a^{3} b^{29} x^{18} + 5985 \, a^{4} b^{28} x^{17} + 20349 \, a^{5} b^{27} x^{16} + 54264 \, a^{6} b^{26} x^{15} + 116280 \, a^{7} b^{25} x^{14} + 203490 \, a^{8} b^{24} x^{13} + 293930 \, a^{9} b^{23} x^{12} + 352716 \, a^{10} b^{22} x^{11} + 352716 \, a^{11} b^{21} x^{10} + 293930 \, a^{12} b^{20} x^{9} + 203490 \, a^{13} b^{19} x^{8} + 116280 \, a^{14} b^{18} x^{7} + 54264 \, a^{15} b^{17} x^{6} + 20349 \, a^{16} b^{16} x^{5} + 5985 \, a^{17} b^{15} x^{4} + 1330 \, a^{18} b^{14} x^{3} + 210 \, a^{19} b^{13} x^{2} + 21 \, a^{20} b^{12} x + a^{21} b^{11}\right )}} \]

[In]

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

[Out]

-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 + 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 + 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 + 116280*(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 + 1001*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^19 + 1330*a^3*b^29*x^18 + 5985*a^4*b^28*x^17 + 20349*a^5*b^27*
x^16 + 54264*a^6*b^26*x^15 + 116280*a^7*b^25*x^14 + 203490*a^8*b^24*x^13 + 293930*a^9*b^23*x^12 + 352716*a^10*
b^22*x^11 + 352716*a^11*b^21*x^10 + 293930*a^12*b^20*x^9 + 203490*a^13*b^19*x^8 + 116280*a^14*b^18*x^7 + 54264
*a^15*b^17*x^6 + 20349*a^16*b^16*x^5 + 5985*a^17*b^15*x^4 + 1330*a^18*b^14*x^3 + 210*a^19*b^13*x^2 + 21*a^20*b
^12*x + a^21*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 = 3.44 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx=-\frac {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} + 1279080 \, 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} + 20369349 \, 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} + 10650640 \, 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} + 4084080 \, 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 + 6006 \, 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 + 21 \, a^{9} b d^{10} x + 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} + 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} + 11 \, a^{9} b c d^{9} + a^{10} d^{10}}{3879876 \, {\left (b x + a\right )}^{21} b^{11}} \]

[In]

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

[Out]

-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 +
1279080*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
+ 20369349*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 + 10650640*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 + 4084080*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 + 6006*a^6*b^4*c^3*d^7*x + 1
386*a^7*b^3*c^2*d^8*x + 231*a^8*b^2*c*d^9*x + 21*a^9*b*d^10*x + 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 + 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 + 11*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^21*b^11)

Mupad [B] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 1186, normalized size of antiderivative = 4.25 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{22}} \, dx=-\frac {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+65835\,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+1331330\,a^3\,b^7\,c^4\,d^6\,x^3+1711710\,a^3\,b^7\,c^3\,d^7\,x^4+1343034\,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+5990985\,a^2\,b^8\,c^4\,d^6\,x^4+5819814\,a^2\,b^8\,c^3\,d^7\,x^5+3581424\,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^9\,c^5\,d^5\,x^4+20369349\,a\,b^9\,c^4\,d^6\,x^5+15519504\,a\,b^9\,c^3\,d^7\,x^6+7674480\,a\,b^9\,c^2\,d^8\,x^7+2238390\,a\,b^9\,c\,d^9\,x^8+293930\,a\,b^9\,d^{10}\,x^9+184756\,b^{10}\,c^{10}+1939938\,b^{10}\,c^9\,d\,x+9189180\,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+3233230\,b^{10}\,c\,d^9\,x^9+352716\,b^{10}\,d^{10}\,x^{10}}{3879876\,a^{21}\,b^{11}[phone]\,a^{20}\,b^{12}\,x+814773960\,a^{19}\,b^{13}\,x^2+5160235080\,a^{18}\,b^{14}\,x^3+23221057860\,a^{17}\,b^{15}\,x^4+78951596724\,a^{16}\,b^{16}\,x^5+210537591264\,a^{15}\,b^{17}\,x^6+451151981280\,a^{14}\,b^{18}\,x^7+789515967240\,a^{13}\,b^{19}\,x^8+1140411952680\,a^{12}\,b^{20}\,x^9+1368494343216\,a^{11}\,b^{21}\,x^{10}[phone]\,a^{10}\,b^{22}\,x^{11}[phone]\,a^9\,b^{23}\,x^{12}[phone]\,a^8\,b^{24}\,x^{13}[phone]\,a^7\,b^{25}\,x^{14}[phone]\,a^6\,b^{26}\,x^{15}[phone]\,a^5\,b^{27}\,x^{16}[phone]\,a^4\,b^{28}\,x^{17}[phone]\,a^3\,b^{29}\,x^{18}[phone]\,a^2\,b^{30}\,x^{19}[phone]\,a\,b^{31}\,x^{20}+3879876\,b^{32}\,x^{21}} \]

[In]

int((c + d*x)^10/(a + b*x)^22,x)

[Out]

-(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 + 4375
8*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 + 9
189180*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 + 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 + 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 + 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 + 6006*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*c*d^9*x^3 + 17972955*a*b^9*c^5*d^5*x^4 + 65835*a^5*b^5*c*d^9*x^
4 + 20369349*a*b^9*c^4*d^6*x^5 + 223839*a^4*b^6*c*d^9*x^5 + 15519504*a*b^9*c^3*d^7*x^6 + 596904*a^3*b^7*c*d^9*
x^6 + 7674480*a*b^9*c^2*d^8*x^7 + 1279080*a^2*b^8*c*d^9*x^7)/(3879876*a^21*b^11 + 3879876*b^32*x^21 + 81477396
*a^20*b^12*x + 81477396*a*b^31*x^20 + 814773960*a^19*b^13*x^2 + 5160235080*a^18*b^14*x^3 + 23221057860*a^17*b^
15*x^4 + 78951596724*a^16*b^16*x^5 + 210537591264*a^15*b^17*x^6 + 451151981280*a^14*b^18*x^7 + 789515967240*a^
13*b^19*x^8 + 1140411952680*a^12*b^20*x^9 + 1368494343216*a^11*b^21*x^10 + 1368494343216*a^10*b^22*x^11 + 1140
411952680*a^9*b^23*x^12 + 789515967240*a^8*b^24*x^13 + 451151981280*a^7*b^25*x^14 + 210537591264*a^6*b^26*x^15
+ 78951596724*a^5*b^27*x^16 + 23221057860*a^4*b^28*x^17 + 5160235080*a^3*b^29*x^18 + 814773960*a^2*b^30*x^19)