\(\int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx\) [1332]

Optimal result

Integrand size = 15, antiderivative size = 279 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx=-\frac {(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac {10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac {5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac {120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac {105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac {84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac {15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac {120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac {15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac {10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac {d^{10}}{10 b^{11} (a+b x)^{10}} \]

[Out]

-1/20*(-a*d+b*c)^10/b^11/(b*x+a)^20-10/19*d*(-a*d+b*c)^9/b^11/(b*x+a)^19-5/2*d^2*(-a*d+b*c)^8/b^11/(b*x+a)^18-
120/17*d^3*(-a*d+b*c)^7/b^11/(b*x+a)^17-105/8*d^4*(-a*d+b*c)^6/b^11/(b*x+a)^16-84/5*d^5*(-a*d+b*c)^5/b^11/(b*x
+a)^15-15*d^6*(-a*d+b*c)^4/b^11/(b*x+a)^14-120/13*d^7*(-a*d+b*c)^3/b^11/(b*x+a)^13-15/4*d^8*(-a*d+b*c)^2/b^11/
(b*x+a)^12-10/11*d^9*(-a*d+b*c)/b^11/(b*x+a)^11-1/10*d^10/b^11/(b*x+a)^10

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} \[ \int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx=-\frac {10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac {15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac {120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac {15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac {84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac {105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac {120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac {5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac {10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac {(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac {d^{10}}{10 b^{11} (a+b x)^{10}} \]

[In]

Int[(c + d*x)^10/(a + b*x)^21,x]

[Out]

-1/20*(b*c - a*d)^10/(b^11*(a + b*x)^20) - (10*d*(b*c - a*d)^9)/(19*b^11*(a + b*x)^19) - (5*d^2*(b*c - a*d)^8)
/(2*b^11*(a + b*x)^18) - (120*d^3*(b*c - a*d)^7)/(17*b^11*(a + b*x)^17) - (105*d^4*(b*c - a*d)^6)/(8*b^11*(a +
b*x)^16) - (84*d^5*(b*c - a*d)^5)/(5*b^11*(a + b*x)^15) - (15*d^6*(b*c - a*d)^4)/(b^11*(a + b*x)^14) - (120*d
^7*(b*c - a*d)^3)/(13*b^11*(a + b*x)^13) - (15*d^8*(b*c - a*d)^2)/(4*b^11*(a + b*x)^12) - (10*d^9*(b*c - a*d))
/(11*b^11*(a + b*x)^11) - d^10/(10*b^11*(a + b*x)^10)

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)^{21}}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^{20}}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^{19}}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^{18}}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^{17}}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^{16}}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^{15}}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^{14}}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)^{13}}+\frac {10 d^9 (b c-a d)}{b^{10} (a+b x)^{12}}+\frac {d^{10}}{b^{10} (a+b x)^{11}}\right ) \, dx \\ & = -\frac {(b c-a d)^{10}}{20 b^{11} (a+b x)^{20}}-\frac {10 d (b c-a d)^9}{19 b^{11} (a+b x)^{19}}-\frac {5 d^2 (b c-a d)^8}{2 b^{11} (a+b x)^{18}}-\frac {120 d^3 (b c-a d)^7}{17 b^{11} (a+b x)^{17}}-\frac {105 d^4 (b c-a d)^6}{8 b^{11} (a+b x)^{16}}-\frac {84 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^{15}}-\frac {15 d^6 (b c-a d)^4}{b^{11} (a+b x)^{14}}-\frac {120 d^7 (b c-a d)^3}{13 b^{11} (a+b x)^{13}}-\frac {15 d^8 (b c-a d)^2}{4 b^{11} (a+b x)^{12}}-\frac {10 d^9 (b c-a d)}{11 b^{11} (a+b x)^{11}}-\frac {d^{10}}{10 b^{11} (a+b x)^{10}} \\ \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 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx=-\frac {a^{10} d^{10}+10 a^9 b d^9 (c+2 d x)+5 a^8 b^2 d^8 \left (11 c^2+40 c d x+38 d^2 x^2\right )+20 a^7 b^3 d^7 \left (11 c^3+55 c^2 d x+95 c d^2 x^2+57 d^3 x^3\right )+5 a^6 b^4 d^6 \left (143 c^4+880 c^3 d x+2090 c^2 d^2 x^2+2280 c d^3 x^3+969 d^4 x^4\right )+2 a^5 b^5 d^5 \left (1001 c^5+7150 c^4 d x+20900 c^3 d^2 x^2+31350 c^2 d^3 x^3+24225 c d^4 x^4+7752 d^5 x^5\right )+5 a^4 b^6 d^4 \left (1001 c^6+8008 c^5 d x+27170 c^4 d^2 x^2+50160 c^3 d^3 x^3+53295 c^2 d^4 x^4+31008 c d^5 x^5+7752 d^6 x^6\right )+20 a^3 b^7 d^3 \left (572 c^7+5005 c^6 d x+19019 c^5 d^2 x^2+40755 c^4 d^3 x^3+53295 c^3 d^4 x^4+42636 c^2 d^5 x^5+19380 c d^6 x^6+3876 d^7 x^7\right )+5 a^2 b^8 d^2 \left (4862 c^8+45760 c^7 d x+190190 c^6 d^2 x^2+456456 c^5 d^3 x^3+692835 c^4 d^4 x^4+682176 c^3 d^5 x^5+426360 c^2 d^6 x^6+155040 c d^7 x^7+25194 d^8 x^8\right )+10 a b^9 d \left (4862 c^9+48620 c^8 d x+217360 c^7 d^2 x^2+570570 c^6 d^3 x^3+969969 c^5 d^4 x^4+1108536 c^4 d^5 x^5+852720 c^3 d^6 x^6+426360 c^2 d^7 x^7+125970 c d^8 x^8+16796 d^9 x^9\right )+b^{10} \left (92378 c^{10}+972400 c^9 d x+4618900 c^8 d^2 x^2+13041600 c^7 d^3 x^3+24249225 c^6 d^4 x^4+31039008 c^5 d^5 x^5+27713400 c^4 d^6 x^6+17054400 c^3 d^7 x^7+6928350 c^2 d^8 x^8+1679600 c d^9 x^9+184756 d^{10} x^{10}\right )}{1847560 b^{11} (a+b x)^{20}} \]

[In]

Integrate[(c + d*x)^10/(a + b*x)^21,x]

[Out]

-1/1847560*(a^10*d^10 + 10*a^9*b*d^9*(c + 2*d*x) + 5*a^8*b^2*d^8*(11*c^2 + 40*c*d*x + 38*d^2*x^2) + 20*a^7*b^3
*d^7*(11*c^3 + 55*c^2*d*x + 95*c*d^2*x^2 + 57*d^3*x^3) + 5*a^6*b^4*d^6*(143*c^4 + 880*c^3*d*x + 2090*c^2*d^2*x
^2 + 2280*c*d^3*x^3 + 969*d^4*x^4) + 2*a^5*b^5*d^5*(1001*c^5 + 7150*c^4*d*x + 20900*c^3*d^2*x^2 + 31350*c^2*d^
3*x^3 + 24225*c*d^4*x^4 + 7752*d^5*x^5) + 5*a^4*b^6*d^4*(1001*c^6 + 8008*c^5*d*x + 27170*c^4*d^2*x^2 + 50160*c
^3*d^3*x^3 + 53295*c^2*d^4*x^4 + 31008*c*d^5*x^5 + 7752*d^6*x^6) + 20*a^3*b^7*d^3*(572*c^7 + 5005*c^6*d*x + 19
019*c^5*d^2*x^2 + 40755*c^4*d^3*x^3 + 53295*c^3*d^4*x^4 + 42636*c^2*d^5*x^5 + 19380*c*d^6*x^6 + 3876*d^7*x^7)
+ 5*a^2*b^8*d^2*(4862*c^8 + 45760*c^7*d*x + 190190*c^6*d^2*x^2 + 456456*c^5*d^3*x^3 + 692835*c^4*d^4*x^4 + 682
176*c^3*d^5*x^5 + 426360*c^2*d^6*x^6 + 155040*c*d^7*x^7 + 25194*d^8*x^8) + 10*a*b^9*d*(4862*c^9 + 48620*c^8*d*
x + 217360*c^7*d^2*x^2 + 570570*c^6*d^3*x^3 + 969969*c^5*d^4*x^4 + 1108536*c^4*d^5*x^5 + 852720*c^3*d^6*x^6 +
426360*c^2*d^7*x^7 + 125970*c*d^8*x^8 + 16796*d^9*x^9) + b^10*(92378*c^10 + 972400*c^9*d*x + 4618900*c^8*d^2*x
^2 + 13041600*c^7*d^3*x^3 + 24249225*c^6*d^4*x^4 + 31039008*c^5*d^5*x^5 + 27713400*c^4*d^6*x^6 + 17054400*c^3*
d^7*x^7 + 6928350*c^2*d^8*x^8 + 1679600*c*d^9*x^9 + 184756*d^10*x^10))/(b^11*(a + b*x)^20)

Maple [B] (verified)

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

Time = 0.27 (sec) , antiderivative size = 831, normalized size of antiderivative = 2.98

method	result	size
risch	\(\frac {-\frac {a^{10} d^{10}+10 a^{9} b c \,d^{9}+55 a^{8} b^{2} c^{2} d^{8}+220 a^{7} b^{3} c^{3} d^{7}+715 a^{6} b^{4} c^{4} d^{6}+2002 a^{5} b^{5} c^{5} d^{5}+5005 a^{4} b^{6} c^{6} d^{4}+11440 a^{3} b^{7} c^{7} d^{3}+24310 a^{2} b^{8} c^{8} d^{2}+48620 a \,b^{9} c^{9} d +92378 b^{10} c^{10}}{1847560 b^{11}}-\frac {d \left (a^{9} d^{9}+10 a^{8} b c \,d^{8}+55 a^{7} b^{2} c^{2} d^{7}+220 a^{6} b^{3} c^{3} d^{6}+715 a^{5} b^{4} c^{4} d^{5}+2002 a^{4} b^{5} c^{5} d^{4}+5005 a^{3} b^{6} c^{6} d^{3}+11440 a^{2} b^{7} c^{7} d^{2}+24310 a \,b^{8} c^{8} d +48620 b^{9} c^{9}\right ) x}{92378 b^{10}}-\frac {d^{2} \left (a^{8} d^{8}+10 a^{7} b c \,d^{7}+55 a^{6} b^{2} c^{2} d^{6}+220 a^{5} b^{3} c^{3} d^{5}+715 a^{4} b^{4} c^{4} d^{4}+2002 a^{3} b^{5} c^{5} d^{3}+5005 a^{2} b^{6} c^{6} d^{2}+11440 a \,b^{7} c^{7} d +24310 b^{8} c^{8}\right ) x^{2}}{9724 b^{9}}-\frac {3 d^{3} \left (a^{7} d^{7}+10 a^{6} b c \,d^{6}+55 a^{5} b^{2} c^{2} d^{5}+220 a^{4} b^{3} c^{3} d^{4}+715 a^{3} b^{4} c^{4} d^{3}+2002 a^{2} b^{5} c^{5} d^{2}+5005 a \,b^{6} c^{6} d +11440 b^{7} c^{7}\right ) x^{3}}{4862 b^{8}}-\frac {3 d^{4} \left (a^{6} d^{6}+10 a^{5} b c \,d^{5}+55 a^{4} b^{2} c^{2} d^{4}+220 a^{3} b^{3} c^{3} d^{3}+715 a^{2} b^{4} c^{4} d^{2}+2002 a \,b^{5} c^{5} d +5005 b^{6} c^{6}\right ) x^{4}}{1144 b^{7}}-\frac {6 d^{5} \left (a^{5} d^{5}+10 a^{4} b c \,d^{4}+55 a^{3} b^{2} c^{2} d^{3}+220 a^{2} b^{3} c^{3} d^{2}+715 a \,b^{4} c^{4} d +2002 b^{5} c^{5}\right ) x^{5}}{715 b^{6}}-\frac {3 d^{6} \left (a^{4} d^{4}+10 a^{3} b c \,d^{3}+55 a^{2} b^{2} c^{2} d^{2}+220 a \,b^{3} c^{3} d +715 b^{4} c^{4}\right ) x^{6}}{143 b^{5}}-\frac {6 d^{7} \left (a^{3} d^{3}+10 a^{2} b c \,d^{2}+55 a \,b^{2} c^{2} d +220 b^{3} c^{3}\right ) x^{7}}{143 b^{4}}-\frac {3 d^{8} \left (a^{2} d^{2}+10 a b c d +55 b^{2} c^{2}\right ) x^{8}}{44 b^{3}}-\frac {d^{9} \left (a d +10 b c \right ) x^{9}}{11 b^{2}}-\frac {d^{10} x^{10}}{10 b}}{\left (b x +a \right )^{20}}\)	\(831\)
default	\(\frac {10 d^{9} \left (a d -b c \right )}{11 b^{11} \left (b x +a \right )^{11}}+\frac {120 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 )}{13 b^{11} \left (b x +a \right )^{13}}+\frac {10 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 )}{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}}{20 b^{11} \left (b x +a \right )^{20}}-\frac {15 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 )^{14}}-\frac {15 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 b^{11} \left (b x +a \right )^{12}}-\frac {d^{10}}{10 b^{11} \left (b x +a \right )^{10}}-\frac {105 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 )}{8 b^{11} \left (b x +a \right )^{16}}-\frac {5 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 )}{2 b^{11} \left (b x +a \right )^{18}}+\frac {84 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 )}{5 b^{11} \left (b x +a \right )^{15}}+\frac {120 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 )}{17 b^{11} \left (b x +a \right )^{17}}\)	\(867\)
norman	\(\frac {\frac {-a^{10} b^{9} d^{10}-10 a^{9} b^{10} c \,d^{9}-55 a^{8} b^{11} c^{2} d^{8}-220 a^{7} b^{12} c^{3} d^{7}-715 a^{6} b^{13} c^{4} d^{6}-2002 a^{5} b^{14} c^{5} d^{5}-5005 a^{4} b^{15} c^{6} d^{4}-11440 a^{3} b^{16} c^{7} d^{3}-24310 a^{2} c^{8} d^{2} b^{17}-48620 a \,b^{18} c^{9} d -92378 b^{19} c^{10}}{1847560 b^{20}}+\frac {\left (-a^{9} b^{9} d^{10}-10 a^{8} b^{10} c \,d^{9}-55 a^{7} b^{11} c^{2} d^{8}-220 a^{6} b^{12} c^{3} d^{7}-715 a^{5} b^{13} c^{4} d^{6}-2002 a^{4} b^{14} c^{5} d^{5}-5005 a^{3} b^{15} c^{6} d^{4}-11440 a^{2} b^{16} c^{7} d^{3}-24310 a \,c^{8} d^{2} b^{17}-48620 b^{18} c^{9} d \right ) x}{92378 b^{19}}+\frac {\left (-a^{8} b^{9} d^{10}-10 a^{7} b^{10} c \,d^{9}-55 a^{6} b^{11} c^{2} d^{8}-220 a^{5} b^{12} c^{3} d^{7}-715 a^{4} b^{13} c^{4} d^{6}-2002 a^{3} b^{14} c^{5} d^{5}-5005 a^{2} b^{15} c^{6} d^{4}-11440 a \,b^{16} c^{7} d^{3}-24310 b^{17} c^{8} d^{2}\right ) x^{2}}{9724 b^{18}}+\frac {3 \left (-a^{7} b^{9} d^{10}-10 a^{6} b^{10} c \,d^{9}-55 a^{5} b^{11} c^{2} d^{8}-220 a^{4} b^{12} c^{3} d^{7}-715 a^{3} b^{13} c^{4} d^{6}-2002 a^{2} b^{14} c^{5} d^{5}-5005 a \,b^{15} c^{6} d^{4}-11440 b^{16} c^{7} d^{3}\right ) x^{3}}{4862 b^{17}}+\frac {3 \left (-a^{6} b^{9} d^{10}-10 a^{5} b^{10} c \,d^{9}-55 a^{4} b^{11} c^{2} d^{8}-220 a^{3} b^{12} c^{3} d^{7}-715 a^{2} b^{13} c^{4} d^{6}-2002 a \,b^{14} c^{5} d^{5}-5005 b^{15} c^{6} d^{4}\right ) x^{4}}{1144 b^{16}}+\frac {6 \left (-a^{5} b^{9} d^{10}-10 a^{4} b^{10} c \,d^{9}-55 a^{3} b^{11} c^{2} d^{8}-220 a^{2} b^{12} c^{3} d^{7}-715 a \,b^{13} c^{4} d^{6}-2002 b^{14} c^{5} d^{5}\right ) x^{5}}{715 b^{15}}+\frac {3 \left (-a^{4} b^{9} d^{10}-10 a^{3} b^{10} c \,d^{9}-55 a^{2} b^{11} c^{2} d^{8}-220 a \,b^{12} c^{3} d^{7}-715 b^{13} c^{4} d^{6}\right ) x^{6}}{143 b^{14}}+\frac {6 \left (-a^{3} b^{9} d^{10}-10 a^{2} b^{10} c \,d^{9}-55 a \,b^{11} c^{2} d^{8}-220 b^{12} c^{3} d^{7}\right ) x^{7}}{143 b^{13}}+\frac {3 \left (-a^{2} b^{9} d^{10}-10 a \,b^{10} c \,d^{9}-55 b^{11} c^{2} d^{8}\right ) x^{8}}{44 b^{12}}+\frac {\left (-a \,b^{9} d^{10}-10 b^{10} c \,d^{9}\right ) x^{9}}{11 b^{11}}-\frac {d^{10} x^{10}}{10 b}}{\left (b x +a \right )^{20}}\)	\(909\)
gosper	\(-\frac {184756 x^{10} d^{10} b^{10}+167960 x^{9} a \,b^{9} d^{10}+1679600 x^{9} b^{10} c \,d^{9}+125970 x^{8} a^{2} b^{8} d^{10}+1259700 x^{8} a \,b^{9} c \,d^{9}+6928350 x^{8} b^{10} c^{2} d^{8}+77520 x^{7} a^{3} b^{7} d^{10}+775200 x^{7} a^{2} b^{8} c \,d^{9}+4263600 x^{7} a \,b^{9} c^{2} d^{8}[phone] x^{7} b^{10} c^{3} d^{7}+38760 x^{6} a^{4} b^{6} d^{10}+387600 x^{6} a^{3} b^{7} c \,d^{9}+2131800 x^{6} a^{2} b^{8} c^{2} d^{8}+8527200 x^{6} a \,b^{9} c^{3} d^{7}[phone] x^{6} b^{10} c^{4} d^{6}+15504 x^{5} a^{5} b^{5} d^{10}+155040 x^{5} a^{4} b^{6} c \,d^{9}+852720 x^{5} a^{3} b^{7} c^{2} d^{8}+3410880 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}+4845 x^{4} a^{6} b^{4} d^{10}+48450 x^{4} a^{5} b^{5} c \,d^{9}+266475 x^{4} a^{4} b^{6} c^{2} d^{8}+1065900 x^{4} a^{3} b^{7} c^{3} d^{7}+3464175 x^{4} a^{2} b^{8} c^{4} d^{6}+9699690 x^{4} a \,b^{9} c^{5} d^{5}[phone] x^{4} b^{10} c^{6} d^{4}+1140 x^{3} a^{7} b^{3} d^{10}+11400 x^{3} a^{6} b^{4} c \,d^{9}+62700 x^{3} a^{5} b^{5} c^{2} d^{8}+250800 x^{3} a^{4} b^{6} c^{3} d^{7}+815100 x^{3} a^{3} b^{7} c^{4} d^{6}+2282280 x^{3} a^{2} b^{8} c^{5} d^{5}+5705700 x^{3} a \,b^{9} c^{6} d^{4}[phone] x^{3} b^{10} c^{7} d^{3}+190 x^{2} a^{8} b^{2} d^{10}+1900 x^{2} a^{7} b^{3} c \,d^{9}+10450 x^{2} a^{6} b^{4} c^{2} d^{8}+41800 x^{2} a^{5} b^{5} c^{3} d^{7}+135850 x^{2} a^{4} b^{6} c^{4} d^{6}+380380 x^{2} a^{3} b^{7} c^{5} d^{5}+950950 x^{2} a^{2} b^{8} c^{6} d^{4}+2173600 x^{2} a \,b^{9} c^{7} d^{3}+4618900 x^{2} b^{10} c^{8} d^{2}+20 x \,a^{9} b \,d^{10}+200 x \,a^{8} b^{2} c \,d^{9}+1100 x \,a^{7} b^{3} c^{2} d^{8}+4400 x \,a^{6} b^{4} c^{3} d^{7}+14300 x \,a^{5} b^{5} c^{4} d^{6}+40040 x \,a^{4} b^{6} c^{5} d^{5}+100100 x \,a^{3} b^{7} c^{6} d^{4}+228800 x \,a^{2} b^{8} c^{7} d^{3}+486200 x a \,b^{9} c^{8} d^{2}+972400 x \,b^{10} c^{9} d +a^{10} d^{10}+10 a^{9} b c \,d^{9}+55 a^{8} b^{2} c^{2} d^{8}+220 a^{7} b^{3} c^{3} d^{7}+715 a^{6} b^{4} c^{4} d^{6}+2002 a^{5} b^{5} c^{5} d^{5}+5005 a^{4} b^{6} c^{6} d^{4}+11440 a^{3} b^{7} c^{7} d^{3}+24310 a^{2} b^{8} c^{8} d^{2}+48620 a \,b^{9} c^{9} d +92378 b^{10} c^{10}}{1847560 b^{11} \left (b x +a \right )^{20}}\)	\(962\)
parallelrisch	\(\frac {-184756 d^{10} x^{10} b^{19}-167960 a \,b^{18} d^{10} x^{9}-1679600 b^{19} c \,d^{9} x^{9}-125970 a^{2} b^{17} d^{10} x^{8}-1259700 a \,b^{18} c \,d^{9} x^{8}-6928350 b^{19} c^{2} d^{8} x^{8}-77520 a^{3} b^{16} d^{10} x^{7}-775200 a^{2} b^{17} c \,d^{9} x^{7}-4263600 a \,b^{18} c^{2} d^{8} x^{7}-17054400 b^{19} c^{3} d^{7} x^{7}-38760 a^{4} b^{15} d^{10} x^{6}-387600 a^{3} b^{16} c \,d^{9} x^{6}-2131800 a^{2} b^{17} c^{2} d^{8} x^{6}-8527200 a \,b^{18} c^{3} d^{7} x^{6}-27713400 b^{19} c^{4} d^{6} x^{6}-15504 a^{5} b^{14} d^{10} x^{5}-155040 a^{4} b^{15} c \,d^{9} x^{5}-852720 a^{3} b^{16} c^{2} d^{8} x^{5}-3410880 a^{2} b^{17} c^{3} d^{7} x^{5}-11085360 a \,b^{18} c^{4} d^{6} x^{5}-31039008 b^{19} c^{5} d^{5} x^{5}-4845 a^{6} b^{13} d^{10} x^{4}-48450 a^{5} b^{14} c \,d^{9} x^{4}-266475 a^{4} b^{15} c^{2} d^{8} x^{4}-1065900 a^{3} b^{16} c^{3} d^{7} x^{4}-3464175 a^{2} b^{17} c^{4} d^{6} x^{4}-9699690 a \,b^{18} c^{5} d^{5} x^{4}-24249225 b^{19} c^{6} d^{4} x^{4}-1140 a^{7} b^{12} d^{10} x^{3}-11400 a^{6} b^{13} c \,d^{9} x^{3}-62700 a^{5} b^{14} c^{2} d^{8} x^{3}-250800 a^{4} b^{15} c^{3} d^{7} x^{3}-815100 a^{3} b^{16} c^{4} d^{6} x^{3}-2282280 a^{2} b^{17} c^{5} d^{5} x^{3}-5705700 a \,b^{18} c^{6} d^{4} x^{3}-13041600 b^{19} c^{7} d^{3} x^{3}-190 a^{8} b^{11} d^{10} x^{2}-1900 a^{7} b^{12} c \,d^{9} x^{2}-10450 a^{6} b^{13} c^{2} d^{8} x^{2}-41800 a^{5} b^{14} c^{3} d^{7} x^{2}-135850 a^{4} b^{15} c^{4} d^{6} x^{2}-380380 a^{3} b^{16} c^{5} d^{5} x^{2}-950950 a^{2} b^{17} c^{6} d^{4} x^{2}-2173600 a \,b^{18} c^{7} d^{3} x^{2}-4618900 b^{19} c^{8} d^{2} x^{2}-20 a^{9} b^{10} d^{10} x -200 a^{8} b^{11} c \,d^{9} x -1100 a^{7} b^{12} c^{2} d^{8} x -4400 a^{6} b^{13} c^{3} d^{7} x -14300 a^{5} b^{14} c^{4} d^{6} x -40040 a^{4} b^{15} c^{5} d^{5} x -100100 a^{3} b^{16} c^{6} d^{4} x -228800 a^{2} b^{17} c^{7} d^{3} x -486200 a \,b^{18} c^{8} d^{2} x -972400 b^{19} c^{9} d x -a^{10} b^{9} d^{10}-10 a^{9} b^{10} c \,d^{9}-55 a^{8} b^{11} c^{2} d^{8}-220 a^{7} b^{12} c^{3} d^{7}-715 a^{6} b^{13} c^{4} d^{6}-2002 a^{5} b^{14} c^{5} d^{5}-5005 a^{4} b^{15} c^{6} d^{4}-11440 a^{3} b^{16} c^{7} d^{3}-24310 a^{2} c^{8} d^{2} b^{17}-48620 a \,b^{18} c^{9} d -92378 b^{19} c^{10}}{1847560 b^{20} \left (b x +a \right )^{20}}\)	\(970\)

[In]

int((d*x+c)^10/(b*x+a)^21,x,method=_RETURNVERBOSE)

[Out]

(-1/1847560/b^11*(a^10*d^10+10*a^9*b*c*d^9+55*a^8*b^2*c^2*d^8+220*a^7*b^3*c^3*d^7+715*a^6*b^4*c^4*d^6+2002*a^5
*b^5*c^5*d^5+5005*a^4*b^6*c^6*d^4+11440*a^3*b^7*c^7*d^3+24310*a^2*b^8*c^8*d^2+48620*a*b^9*c^9*d+92378*b^10*c^1
0)-1/92378/b^10*d*(a^9*d^9+10*a^8*b*c*d^8+55*a^7*b^2*c^2*d^7+220*a^6*b^3*c^3*d^6+715*a^5*b^4*c^4*d^5+2002*a^4*
b^5*c^5*d^4+5005*a^3*b^6*c^6*d^3+11440*a^2*b^7*c^7*d^2+24310*a*b^8*c^8*d+48620*b^9*c^9)*x-1/9724/b^9*d^2*(a^8*
d^8+10*a^7*b*c*d^7+55*a^6*b^2*c^2*d^6+220*a^5*b^3*c^3*d^5+715*a^4*b^4*c^4*d^4+2002*a^3*b^5*c^5*d^3+5005*a^2*b^
6*c^6*d^2+11440*a*b^7*c^7*d+24310*b^8*c^8)*x^2-3/4862/b^8*d^3*(a^7*d^7+10*a^6*b*c*d^6+55*a^5*b^2*c^2*d^5+220*a
^4*b^3*c^3*d^4+715*a^3*b^4*c^4*d^3+2002*a^2*b^5*c^5*d^2+5005*a*b^6*c^6*d+11440*b^7*c^7)*x^3-3/1144/b^7*d^4*(a^
6*d^6+10*a^5*b*c*d^5+55*a^4*b^2*c^2*d^4+220*a^3*b^3*c^3*d^3+715*a^2*b^4*c^4*d^2+2002*a*b^5*c^5*d+5005*b^6*c^6)
*x^4-6/715/b^6*d^5*(a^5*d^5+10*a^4*b*c*d^4+55*a^3*b^2*c^2*d^3+220*a^2*b^3*c^3*d^2+715*a*b^4*c^4*d+2002*b^5*c^5
)*x^5-3/143/b^5*d^6*(a^4*d^4+10*a^3*b*c*d^3+55*a^2*b^2*c^2*d^2+220*a*b^3*c^3*d+715*b^4*c^4)*x^6-6/143/b^4*d^7*
(a^3*d^3+10*a^2*b*c*d^2+55*a*b^2*c^2*d+220*b^3*c^3)*x^7-3/44/b^3*d^8*(a^2*d^2+10*a*b*c*d+55*b^2*c^2)*x^8-1/11/
b^2*d^9*(a*d+10*b*c)*x^9-1/10/b*d^10*x^10)/(b*x+a)^20

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (259) = 518\).

Time = 0.23 (sec) , antiderivative size = 1074, normalized size of antiderivative = 3.85 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx=-\frac {184756 \, b^{10} d^{10} x^{10} + 92378 \, b^{10} c^{10} + 48620 \, a b^{9} c^{9} d + 24310 \, a^{2} b^{8} c^{8} d^{2} + 11440 \, a^{3} b^{7} c^{7} d^{3} + 5005 \, a^{4} b^{6} c^{6} d^{4} + 2002 \, a^{5} b^{5} c^{5} d^{5} + 715 \, a^{6} b^{4} c^{4} d^{6} + 220 \, a^{7} b^{3} c^{3} d^{7} + 55 \, a^{8} b^{2} c^{2} d^{8} + 10 \, a^{9} b c d^{9} + a^{10} d^{10} + 167960 \, {\left (10 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 125970 \, {\left (55 \, b^{10} c^{2} d^{8} + 10 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 77520 \, {\left (220 \, b^{10} c^{3} d^{7} + 55 \, a b^{9} c^{2} d^{8} + 10 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 38760 \, {\left (715 \, b^{10} c^{4} d^{6} + 220 \, a b^{9} c^{3} d^{7} + 55 \, a^{2} b^{8} c^{2} d^{8} + 10 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 15504 \, {\left (2002 \, b^{10} c^{5} d^{5} + 715 \, a b^{9} c^{4} d^{6} + 220 \, a^{2} b^{8} c^{3} d^{7} + 55 \, a^{3} b^{7} c^{2} d^{8} + 10 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 4845 \, {\left (5005 \, b^{10} c^{6} d^{4} + 2002 \, a b^{9} c^{5} d^{5} + 715 \, a^{2} b^{8} c^{4} d^{6} + 220 \, a^{3} b^{7} c^{3} d^{7} + 55 \, a^{4} b^{6} c^{2} d^{8} + 10 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 1140 \, {\left (11440 \, b^{10} c^{7} d^{3} + 5005 \, a b^{9} c^{6} d^{4} + 2002 \, a^{2} b^{8} c^{5} d^{5} + 715 \, a^{3} b^{7} c^{4} d^{6} + 220 \, a^{4} b^{6} c^{3} d^{7} + 55 \, a^{5} b^{5} c^{2} d^{8} + 10 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 190 \, {\left (24310 \, b^{10} c^{8} d^{2} + 11440 \, a b^{9} c^{7} d^{3} + 5005 \, a^{2} b^{8} c^{6} d^{4} + 2002 \, a^{3} b^{7} c^{5} d^{5} + 715 \, a^{4} b^{6} c^{4} d^{6} + 220 \, a^{5} b^{5} c^{3} d^{7} + 55 \, a^{6} b^{4} c^{2} d^{8} + 10 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 20 \, {\left (48620 \, b^{10} c^{9} d + 24310 \, a b^{9} c^{8} d^{2} + 11440 \, a^{2} b^{8} c^{7} d^{3} + 5005 \, a^{3} b^{7} c^{6} d^{4} + 2002 \, a^{4} b^{6} c^{5} d^{5} + 715 \, a^{5} b^{5} c^{4} d^{6} + 220 \, a^{6} b^{4} c^{3} d^{7} + 55 \, a^{7} b^{3} c^{2} d^{8} + 10 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{1847560 \, {\left (b^{31} x^{20} + 20 \, a b^{30} x^{19} + 190 \, a^{2} b^{29} x^{18} + 1140 \, a^{3} b^{28} x^{17} + 4845 \, a^{4} b^{27} x^{16} + 15504 \, a^{5} b^{26} x^{15} + 38760 \, a^{6} b^{25} x^{14} + 77520 \, a^{7} b^{24} x^{13} + 125970 \, a^{8} b^{23} x^{12} + 167960 \, a^{9} b^{22} x^{11} + 184756 \, a^{10} b^{21} x^{10} + 167960 \, a^{11} b^{20} x^{9} + 125970 \, a^{12} b^{19} x^{8} + 77520 \, a^{13} b^{18} x^{7} + 38760 \, a^{14} b^{17} x^{6} + 15504 \, a^{15} b^{16} x^{5} + 4845 \, a^{16} b^{15} x^{4} + 1140 \, a^{17} b^{14} x^{3} + 190 \, a^{18} b^{13} x^{2} + 20 \, a^{19} b^{12} x + a^{20} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^21,x, algorithm="fricas")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*a^2*b^8*c^8*d^2 + 11440*a^3*b^
7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*c^3*d^7 + 55*a^8*b
^2*c^2*d^8 + 10*a^9*b*c*d^9 + a^10*d^10 + 167960*(10*b^10*c*d^9 + a*b^9*d^10)*x^9 + 125970*(55*b^10*c^2*d^8 +
10*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 77520*(220*b^10*c^3*d^7 + 55*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 + a^3*b^7*d
^10)*x^7 + 38760*(715*b^10*c^4*d^6 + 220*a*b^9*c^3*d^7 + 55*a^2*b^8*c^2*d^8 + 10*a^3*b^7*c*d^9 + a^4*b^6*d^10)
*x^6 + 15504*(2002*b^10*c^5*d^5 + 715*a*b^9*c^4*d^6 + 220*a^2*b^8*c^3*d^7 + 55*a^3*b^7*c^2*d^8 + 10*a^4*b^6*c*
d^9 + a^5*b^5*d^10)*x^5 + 4845*(5005*b^10*c^6*d^4 + 2002*a*b^9*c^5*d^5 + 715*a^2*b^8*c^4*d^6 + 220*a^3*b^7*c^3
*d^7 + 55*a^4*b^6*c^2*d^8 + 10*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1140*(11440*b^10*c^7*d^3 + 5005*a*b^9*c^6*d
^4 + 2002*a^2*b^8*c^5*d^5 + 715*a^3*b^7*c^4*d^6 + 220*a^4*b^6*c^3*d^7 + 55*a^5*b^5*c^2*d^8 + 10*a^6*b^4*c*d^9
+ a^7*b^3*d^10)*x^3 + 190*(24310*b^10*c^8*d^2 + 11440*a*b^9*c^7*d^3 + 5005*a^2*b^8*c^6*d^4 + 2002*a^3*b^7*c^5*
d^5 + 715*a^4*b^6*c^4*d^6 + 220*a^5*b^5*c^3*d^7 + 55*a^6*b^4*c^2*d^8 + 10*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 +
20*(48620*b^10*c^9*d + 24310*a*b^9*c^8*d^2 + 11440*a^2*b^8*c^7*d^3 + 5005*a^3*b^7*c^6*d^4 + 2002*a^4*b^6*c^5*d
^5 + 715*a^5*b^5*c^4*d^6 + 220*a^6*b^4*c^3*d^7 + 55*a^7*b^3*c^2*d^8 + 10*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^31*
x^20 + 20*a*b^30*x^19 + 190*a^2*b^29*x^18 + 1140*a^3*b^28*x^17 + 4845*a^4*b^27*x^16 + 15504*a^5*b^26*x^15 + 38
760*a^6*b^25*x^14 + 77520*a^7*b^24*x^13 + 125970*a^8*b^23*x^12 + 167960*a^9*b^22*x^11 + 184756*a^10*b^21*x^10
+ 167960*a^11*b^20*x^9 + 125970*a^12*b^19*x^8 + 77520*a^13*b^18*x^7 + 38760*a^14*b^17*x^6 + 15504*a^15*b^16*x^
5 + 4845*a^16*b^15*x^4 + 1140*a^17*b^14*x^3 + 190*a^18*b^13*x^2 + 20*a^19*b^12*x + a^20*b^11)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**10/(b*x+a)**21,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (259) = 518\).

Time = 0.28 (sec) , antiderivative size = 1074, normalized size of antiderivative = 3.85 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx=-\frac {184756 \, b^{10} d^{10} x^{10} + 92378 \, b^{10} c^{10} + 48620 \, a b^{9} c^{9} d + 24310 \, a^{2} b^{8} c^{8} d^{2} + 11440 \, a^{3} b^{7} c^{7} d^{3} + 5005 \, a^{4} b^{6} c^{6} d^{4} + 2002 \, a^{5} b^{5} c^{5} d^{5} + 715 \, a^{6} b^{4} c^{4} d^{6} + 220 \, a^{7} b^{3} c^{3} d^{7} + 55 \, a^{8} b^{2} c^{2} d^{8} + 10 \, a^{9} b c d^{9} + a^{10} d^{10} + 167960 \, {\left (10 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 125970 \, {\left (55 \, b^{10} c^{2} d^{8} + 10 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 77520 \, {\left (220 \, b^{10} c^{3} d^{7} + 55 \, a b^{9} c^{2} d^{8} + 10 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 38760 \, {\left (715 \, b^{10} c^{4} d^{6} + 220 \, a b^{9} c^{3} d^{7} + 55 \, a^{2} b^{8} c^{2} d^{8} + 10 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 15504 \, {\left (2002 \, b^{10} c^{5} d^{5} + 715 \, a b^{9} c^{4} d^{6} + 220 \, a^{2} b^{8} c^{3} d^{7} + 55 \, a^{3} b^{7} c^{2} d^{8} + 10 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 4845 \, {\left (5005 \, b^{10} c^{6} d^{4} + 2002 \, a b^{9} c^{5} d^{5} + 715 \, a^{2} b^{8} c^{4} d^{6} + 220 \, a^{3} b^{7} c^{3} d^{7} + 55 \, a^{4} b^{6} c^{2} d^{8} + 10 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 1140 \, {\left (11440 \, b^{10} c^{7} d^{3} + 5005 \, a b^{9} c^{6} d^{4} + 2002 \, a^{2} b^{8} c^{5} d^{5} + 715 \, a^{3} b^{7} c^{4} d^{6} + 220 \, a^{4} b^{6} c^{3} d^{7} + 55 \, a^{5} b^{5} c^{2} d^{8} + 10 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 190 \, {\left (24310 \, b^{10} c^{8} d^{2} + 11440 \, a b^{9} c^{7} d^{3} + 5005 \, a^{2} b^{8} c^{6} d^{4} + 2002 \, a^{3} b^{7} c^{5} d^{5} + 715 \, a^{4} b^{6} c^{4} d^{6} + 220 \, a^{5} b^{5} c^{3} d^{7} + 55 \, a^{6} b^{4} c^{2} d^{8} + 10 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 20 \, {\left (48620 \, b^{10} c^{9} d + 24310 \, a b^{9} c^{8} d^{2} + 11440 \, a^{2} b^{8} c^{7} d^{3} + 5005 \, a^{3} b^{7} c^{6} d^{4} + 2002 \, a^{4} b^{6} c^{5} d^{5} + 715 \, a^{5} b^{5} c^{4} d^{6} + 220 \, a^{6} b^{4} c^{3} d^{7} + 55 \, a^{7} b^{3} c^{2} d^{8} + 10 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{1847560 \, {\left (b^{31} x^{20} + 20 \, a b^{30} x^{19} + 190 \, a^{2} b^{29} x^{18} + 1140 \, a^{3} b^{28} x^{17} + 4845 \, a^{4} b^{27} x^{16} + 15504 \, a^{5} b^{26} x^{15} + 38760 \, a^{6} b^{25} x^{14} + 77520 \, a^{7} b^{24} x^{13} + 125970 \, a^{8} b^{23} x^{12} + 167960 \, a^{9} b^{22} x^{11} + 184756 \, a^{10} b^{21} x^{10} + 167960 \, a^{11} b^{20} x^{9} + 125970 \, a^{12} b^{19} x^{8} + 77520 \, a^{13} b^{18} x^{7} + 38760 \, a^{14} b^{17} x^{6} + 15504 \, a^{15} b^{16} x^{5} + 4845 \, a^{16} b^{15} x^{4} + 1140 \, a^{17} b^{14} x^{3} + 190 \, a^{18} b^{13} x^{2} + 20 \, a^{19} b^{12} x + a^{20} b^{11}\right )}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^21,x, algorithm="maxima")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*a^2*b^8*c^8*d^2 + 11440*a^3*b^
7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*c^3*d^7 + 55*a^8*b
^2*c^2*d^8 + 10*a^9*b*c*d^9 + a^10*d^10 + 167960*(10*b^10*c*d^9 + a*b^9*d^10)*x^9 + 125970*(55*b^10*c^2*d^8 +
10*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 77520*(220*b^10*c^3*d^7 + 55*a*b^9*c^2*d^8 + 10*a^2*b^8*c*d^9 + a^3*b^7*d
^10)*x^7 + 38760*(715*b^10*c^4*d^6 + 220*a*b^9*c^3*d^7 + 55*a^2*b^8*c^2*d^8 + 10*a^3*b^7*c*d^9 + a^4*b^6*d^10)
*x^6 + 15504*(2002*b^10*c^5*d^5 + 715*a*b^9*c^4*d^6 + 220*a^2*b^8*c^3*d^7 + 55*a^3*b^7*c^2*d^8 + 10*a^4*b^6*c*
d^9 + a^5*b^5*d^10)*x^5 + 4845*(5005*b^10*c^6*d^4 + 2002*a*b^9*c^5*d^5 + 715*a^2*b^8*c^4*d^6 + 220*a^3*b^7*c^3
*d^7 + 55*a^4*b^6*c^2*d^8 + 10*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 1140*(11440*b^10*c^7*d^3 + 5005*a*b^9*c^6*d
^4 + 2002*a^2*b^8*c^5*d^5 + 715*a^3*b^7*c^4*d^6 + 220*a^4*b^6*c^3*d^7 + 55*a^5*b^5*c^2*d^8 + 10*a^6*b^4*c*d^9
+ a^7*b^3*d^10)*x^3 + 190*(24310*b^10*c^8*d^2 + 11440*a*b^9*c^7*d^3 + 5005*a^2*b^8*c^6*d^4 + 2002*a^3*b^7*c^5*
d^5 + 715*a^4*b^6*c^4*d^6 + 220*a^5*b^5*c^3*d^7 + 55*a^6*b^4*c^2*d^8 + 10*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 +
20*(48620*b^10*c^9*d + 24310*a*b^9*c^8*d^2 + 11440*a^2*b^8*c^7*d^3 + 5005*a^3*b^7*c^6*d^4 + 2002*a^4*b^6*c^5*d
^5 + 715*a^5*b^5*c^4*d^6 + 220*a^6*b^4*c^3*d^7 + 55*a^7*b^3*c^2*d^8 + 10*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^31*
x^20 + 20*a*b^30*x^19 + 190*a^2*b^29*x^18 + 1140*a^3*b^28*x^17 + 4845*a^4*b^27*x^16 + 15504*a^5*b^26*x^15 + 38
760*a^6*b^25*x^14 + 77520*a^7*b^24*x^13 + 125970*a^8*b^23*x^12 + 167960*a^9*b^22*x^11 + 184756*a^10*b^21*x^10
+ 167960*a^11*b^20*x^9 + 125970*a^12*b^19*x^8 + 77520*a^13*b^18*x^7 + 38760*a^14*b^17*x^6 + 15504*a^15*b^16*x^
5 + 4845*a^16*b^15*x^4 + 1140*a^17*b^14*x^3 + 190*a^18*b^13*x^2 + 20*a^19*b^12*x + a^20*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 = 0.32 (sec) , antiderivative size = 961, normalized size of antiderivative = 3.44 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx=-\frac {184756 \, b^{10} d^{10} x^{10} + 1679600 \, b^{10} c d^{9} x^{9} + 167960 \, a b^{9} d^{10} x^{9} + 6928350 \, b^{10} c^{2} d^{8} x^{8} + 1259700 \, a b^{9} c d^{9} x^{8} + 125970 \, a^{2} b^{8} d^{10} x^{8} + 17054400 \, b^{10} c^{3} d^{7} x^{7} + 4263600 \, a b^{9} c^{2} d^{8} x^{7} + 775200 \, a^{2} b^{8} c d^{9} x^{7} + 77520 \, a^{3} b^{7} d^{10} x^{7} + 27713400 \, b^{10} c^{4} d^{6} x^{6} + 8527200 \, a b^{9} c^{3} d^{7} x^{6} + 2131800 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 387600 \, a^{3} b^{7} c d^{9} x^{6} + 38760 \, a^{4} b^{6} d^{10} x^{6} + 31039008 \, b^{10} c^{5} d^{5} x^{5} + 11085360 \, a b^{9} c^{4} d^{6} x^{5} + 3410880 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 852720 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 155040 \, a^{4} b^{6} c d^{9} x^{5} + 15504 \, a^{5} b^{5} d^{10} x^{5} + 24249225 \, b^{10} c^{6} d^{4} x^{4} + 9699690 \, a b^{9} c^{5} d^{5} x^{4} + 3464175 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 1065900 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 266475 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 48450 \, a^{5} b^{5} c d^{9} x^{4} + 4845 \, a^{6} b^{4} d^{10} x^{4} + 13041600 \, b^{10} c^{7} d^{3} x^{3} + 5705700 \, a b^{9} c^{6} d^{4} x^{3} + 2282280 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 815100 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 250800 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 62700 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 11400 \, a^{6} b^{4} c d^{9} x^{3} + 1140 \, a^{7} b^{3} d^{10} x^{3} + 4618900 \, b^{10} c^{8} d^{2} x^{2} + 2173600 \, a b^{9} c^{7} d^{3} x^{2} + 950950 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 380380 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 135850 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 41800 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 10450 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 1900 \, a^{7} b^{3} c d^{9} x^{2} + 190 \, a^{8} b^{2} d^{10} x^{2} + 972400 \, b^{10} c^{9} d x + 486200 \, a b^{9} c^{8} d^{2} x + 228800 \, a^{2} b^{8} c^{7} d^{3} x + 100100 \, a^{3} b^{7} c^{6} d^{4} x + 40040 \, a^{4} b^{6} c^{5} d^{5} x + 14300 \, a^{5} b^{5} c^{4} d^{6} x + 4400 \, a^{6} b^{4} c^{3} d^{7} x + 1100 \, a^{7} b^{3} c^{2} d^{8} x + 200 \, a^{8} b^{2} c d^{9} x + 20 \, a^{9} b d^{10} x + 92378 \, b^{10} c^{10} + 48620 \, a b^{9} c^{9} d + 24310 \, a^{2} b^{8} c^{8} d^{2} + 11440 \, a^{3} b^{7} c^{7} d^{3} + 5005 \, a^{4} b^{6} c^{6} d^{4} + 2002 \, a^{5} b^{5} c^{5} d^{5} + 715 \, a^{6} b^{4} c^{4} d^{6} + 220 \, a^{7} b^{3} c^{3} d^{7} + 55 \, a^{8} b^{2} c^{2} d^{8} + 10 \, a^{9} b c d^{9} + a^{10} d^{10}}{1847560 \, {\left (b x + a\right )}^{20} b^{11}} \]

[In]

integrate((d*x+c)^10/(b*x+a)^21,x, algorithm="giac")

[Out]

-1/1847560*(184756*b^10*d^10*x^10 + 1679600*b^10*c*d^9*x^9 + 167960*a*b^9*d^10*x^9 + 6928350*b^10*c^2*d^8*x^8
+ 1259700*a*b^9*c*d^9*x^8 + 125970*a^2*b^8*d^10*x^8 + 17054400*b^10*c^3*d^7*x^7 + 4263600*a*b^9*c^2*d^8*x^7 +
775200*a^2*b^8*c*d^9*x^7 + 77520*a^3*b^7*d^10*x^7 + 27713400*b^10*c^4*d^6*x^6 + 8527200*a*b^9*c^3*d^7*x^6 + 21
31800*a^2*b^8*c^2*d^8*x^6 + 387600*a^3*b^7*c*d^9*x^6 + 38760*a^4*b^6*d^10*x^6 + 31039008*b^10*c^5*d^5*x^5 + 11
085360*a*b^9*c^4*d^6*x^5 + 3410880*a^2*b^8*c^3*d^7*x^5 + 852720*a^3*b^7*c^2*d^8*x^5 + 155040*a^4*b^6*c*d^9*x^5
+ 15504*a^5*b^5*d^10*x^5 + 24249225*b^10*c^6*d^4*x^4 + 9699690*a*b^9*c^5*d^5*x^4 + 3464175*a^2*b^8*c^4*d^6*x^
4 + 1065900*a^3*b^7*c^3*d^7*x^4 + 266475*a^4*b^6*c^2*d^8*x^4 + 48450*a^5*b^5*c*d^9*x^4 + 4845*a^6*b^4*d^10*x^4
+ 13041600*b^10*c^7*d^3*x^3 + 5705700*a*b^9*c^6*d^4*x^3 + 2282280*a^2*b^8*c^5*d^5*x^3 + 815100*a^3*b^7*c^4*d^
6*x^3 + 250800*a^4*b^6*c^3*d^7*x^3 + 62700*a^5*b^5*c^2*d^8*x^3 + 11400*a^6*b^4*c*d^9*x^3 + 1140*a^7*b^3*d^10*x
^3 + 4618900*b^10*c^8*d^2*x^2 + 2173600*a*b^9*c^7*d^3*x^2 + 950950*a^2*b^8*c^6*d^4*x^2 + 380380*a^3*b^7*c^5*d^
5*x^2 + 135850*a^4*b^6*c^4*d^6*x^2 + 41800*a^5*b^5*c^3*d^7*x^2 + 10450*a^6*b^4*c^2*d^8*x^2 + 1900*a^7*b^3*c*d^
9*x^2 + 190*a^8*b^2*d^10*x^2 + 972400*b^10*c^9*d*x + 486200*a*b^9*c^8*d^2*x + 228800*a^2*b^8*c^7*d^3*x + 10010
0*a^3*b^7*c^6*d^4*x + 40040*a^4*b^6*c^5*d^5*x + 14300*a^5*b^5*c^4*d^6*x + 4400*a^6*b^4*c^3*d^7*x + 1100*a^7*b^
3*c^2*d^8*x + 200*a^8*b^2*c*d^9*x + 20*a^9*b*d^10*x + 92378*b^10*c^10 + 48620*a*b^9*c^9*d + 24310*a^2*b^8*c^8*
d^2 + 11440*a^3*b^7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 + 220*a^7*b^3*
c^3*d^7 + 55*a^8*b^2*c^2*d^8 + 10*a^9*b*c*d^9 + a^10*d^10)/((b*x + a)^20*b^11)

Mupad [B] (verification not implemented)

Time = 0.90 (sec) , antiderivative size = 1175, normalized size of antiderivative = 4.21 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{21}} \, dx=-\frac {a^{10}\,d^{10}+10\,a^9\,b\,c\,d^9+20\,a^9\,b\,d^{10}\,x+55\,a^8\,b^2\,c^2\,d^8+200\,a^8\,b^2\,c\,d^9\,x+190\,a^8\,b^2\,d^{10}\,x^2+220\,a^7\,b^3\,c^3\,d^7+1100\,a^7\,b^3\,c^2\,d^8\,x+1900\,a^7\,b^3\,c\,d^9\,x^2+1140\,a^7\,b^3\,d^{10}\,x^3+715\,a^6\,b^4\,c^4\,d^6+4400\,a^6\,b^4\,c^3\,d^7\,x+10450\,a^6\,b^4\,c^2\,d^8\,x^2+11400\,a^6\,b^4\,c\,d^9\,x^3+4845\,a^6\,b^4\,d^{10}\,x^4+2002\,a^5\,b^5\,c^5\,d^5+14300\,a^5\,b^5\,c^4\,d^6\,x+41800\,a^5\,b^5\,c^3\,d^7\,x^2+62700\,a^5\,b^5\,c^2\,d^8\,x^3+48450\,a^5\,b^5\,c\,d^9\,x^4+15504\,a^5\,b^5\,d^{10}\,x^5+5005\,a^4\,b^6\,c^6\,d^4+40040\,a^4\,b^6\,c^5\,d^5\,x+135850\,a^4\,b^6\,c^4\,d^6\,x^2+250800\,a^4\,b^6\,c^3\,d^7\,x^3+266475\,a^4\,b^6\,c^2\,d^8\,x^4+155040\,a^4\,b^6\,c\,d^9\,x^5+38760\,a^4\,b^6\,d^{10}\,x^6+11440\,a^3\,b^7\,c^7\,d^3+100100\,a^3\,b^7\,c^6\,d^4\,x+380380\,a^3\,b^7\,c^5\,d^5\,x^2+815100\,a^3\,b^7\,c^4\,d^6\,x^3+1065900\,a^3\,b^7\,c^3\,d^7\,x^4+852720\,a^3\,b^7\,c^2\,d^8\,x^5+387600\,a^3\,b^7\,c\,d^9\,x^6+77520\,a^3\,b^7\,d^{10}\,x^7+24310\,a^2\,b^8\,c^8\,d^2+228800\,a^2\,b^8\,c^7\,d^3\,x+950950\,a^2\,b^8\,c^6\,d^4\,x^2+2282280\,a^2\,b^8\,c^5\,d^5\,x^3+3464175\,a^2\,b^8\,c^4\,d^6\,x^4+3410880\,a^2\,b^8\,c^3\,d^7\,x^5+2131800\,a^2\,b^8\,c^2\,d^8\,x^6+775200\,a^2\,b^8\,c\,d^9\,x^7+125970\,a^2\,b^8\,d^{10}\,x^8+48620\,a\,b^9\,c^9\,d+486200\,a\,b^9\,c^8\,d^2\,x+2173600\,a\,b^9\,c^7\,d^3\,x^2+5705700\,a\,b^9\,c^6\,d^4\,x^3+9699690\,a\,b^9\,c^5\,d^5\,x^4+11085360\,a\,b^9\,c^4\,d^6\,x^5+8527200\,a\,b^9\,c^3\,d^7\,x^6+4263600\,a\,b^9\,c^2\,d^8\,x^7+1259700\,a\,b^9\,c\,d^9\,x^8+167960\,a\,b^9\,d^{10}\,x^9+92378\,b^{10}\,c^{10}+972400\,b^{10}\,c^9\,d\,x+4618900\,b^{10}\,c^8\,d^2\,x^2+13041600\,b^{10}\,c^7\,d^3\,x^3+24249225\,b^{10}\,c^6\,d^4\,x^4+31039008\,b^{10}\,c^5\,d^5\,x^5+27713400\,b^{10}\,c^4\,d^6\,x^6+17054400\,b^{10}\,c^3\,d^7\,x^7+6928350\,b^{10}\,c^2\,d^8\,x^8+1679600\,b^{10}\,c\,d^9\,x^9+184756\,b^{10}\,d^{10}\,x^{10}}{1847560\,a^{20}\,b^{11}[phone]\,a^{19}\,b^{12}\,x+351036400\,a^{18}\,b^{13}\,x^2+2106218400\,a^{17}\,b^{14}\,x^3+8951428200\,a^{16}\,b^{15}\,x^4+28644570240\,a^{15}\,b^{16}\,x^5+71611425600\,a^{14}\,b^{17}\,x^6+143222851200\,a^{13}\,b^{18}\,x^7+232737133200\,a^{12}\,b^{19}\,x^8+310316177600\,a^{11}\,b^{20}\,x^9+341347795360\,a^{10}\,b^{21}\,x^{10}[phone]\,a^9\,b^{22}\,x^{11}[phone]\,a^8\,b^{23}\,x^{12}[phone]\,a^7\,b^{24}\,x^{13}[phone]\,a^6\,b^{25}\,x^{14}[phone]\,a^5\,b^{26}\,x^{15}[phone]\,a^4\,b^{27}\,x^{16}[phone]\,a^3\,b^{28}\,x^{17}[phone]\,a^2\,b^{29}\,x^{18}[phone]\,a\,b^{30}\,x^{19}+1847560\,b^{31}\,x^{20}} \]

[In]

int((c + d*x)^10/(a + b*x)^21,x)

[Out]

-(a^10*d^10 + 92378*b^10*c^10 + 184756*b^10*d^10*x^10 + 167960*a*b^9*d^10*x^9 + 1679600*b^10*c*d^9*x^9 + 24310
*a^2*b^8*c^8*d^2 + 11440*a^3*b^7*c^7*d^3 + 5005*a^4*b^6*c^6*d^4 + 2002*a^5*b^5*c^5*d^5 + 715*a^6*b^4*c^4*d^6 +
220*a^7*b^3*c^3*d^7 + 55*a^8*b^2*c^2*d^8 + 190*a^8*b^2*d^10*x^2 + 1140*a^7*b^3*d^10*x^3 + 4845*a^6*b^4*d^10*x
^4 + 15504*a^5*b^5*d^10*x^5 + 38760*a^4*b^6*d^10*x^6 + 77520*a^3*b^7*d^10*x^7 + 125970*a^2*b^8*d^10*x^8 + 4618
900*b^10*c^8*d^2*x^2 + 13041600*b^10*c^7*d^3*x^3 + 24249225*b^10*c^6*d^4*x^4 + 31039008*b^10*c^5*d^5*x^5 + 277
13400*b^10*c^4*d^6*x^6 + 17054400*b^10*c^3*d^7*x^7 + 6928350*b^10*c^2*d^8*x^8 + 48620*a*b^9*c^9*d + 10*a^9*b*c
*d^9 + 20*a^9*b*d^10*x + 972400*b^10*c^9*d*x + 950950*a^2*b^8*c^6*d^4*x^2 + 380380*a^3*b^7*c^5*d^5*x^2 + 13585
0*a^4*b^6*c^4*d^6*x^2 + 41800*a^5*b^5*c^3*d^7*x^2 + 10450*a^6*b^4*c^2*d^8*x^2 + 2282280*a^2*b^8*c^5*d^5*x^3 +
815100*a^3*b^7*c^4*d^6*x^3 + 250800*a^4*b^6*c^3*d^7*x^3 + 62700*a^5*b^5*c^2*d^8*x^3 + 3464175*a^2*b^8*c^4*d^6*
x^4 + 1065900*a^3*b^7*c^3*d^7*x^4 + 266475*a^4*b^6*c^2*d^8*x^4 + 3410880*a^2*b^8*c^3*d^7*x^5 + 852720*a^3*b^7*
c^2*d^8*x^5 + 2131800*a^2*b^8*c^2*d^8*x^6 + 486200*a*b^9*c^8*d^2*x + 200*a^8*b^2*c*d^9*x + 1259700*a*b^9*c*d^9
*x^8 + 228800*a^2*b^8*c^7*d^3*x + 100100*a^3*b^7*c^6*d^4*x + 40040*a^4*b^6*c^5*d^5*x + 14300*a^5*b^5*c^4*d^6*x
+ 4400*a^6*b^4*c^3*d^7*x + 1100*a^7*b^3*c^2*d^8*x + 2173600*a*b^9*c^7*d^3*x^2 + 1900*a^7*b^3*c*d^9*x^2 + 5705
700*a*b^9*c^6*d^4*x^3 + 11400*a^6*b^4*c*d^9*x^3 + 9699690*a*b^9*c^5*d^5*x^4 + 48450*a^5*b^5*c*d^9*x^4 + 110853
60*a*b^9*c^4*d^6*x^5 + 155040*a^4*b^6*c*d^9*x^5 + 8527200*a*b^9*c^3*d^7*x^6 + 387600*a^3*b^7*c*d^9*x^6 + 42636
00*a*b^9*c^2*d^8*x^7 + 775200*a^2*b^8*c*d^9*x^7)/(1847560*a^20*b^11 + 1847560*b^31*x^20 + 36951200*a^19*b^12*x
+ 36951200*a*b^30*x^19 + 351036400*a^18*b^13*x^2 + 2106218400*a^17*b^14*x^3 + 8951428200*a^16*b^15*x^4 + 2864
4570240*a^15*b^16*x^5 + 71611425600*a^14*b^17*x^6 + 143222851200*a^13*b^18*x^7 + 232737133200*a^12*b^19*x^8 +
310316177600*a^11*b^20*x^9 + 341347795360*a^10*b^21*x^10 + 310316177600*a^9*b^22*x^11 + 232737133200*a^8*b^23*
x^12 + 143222851200*a^7*b^24*x^13 + 71611425600*a^6*b^25*x^14 + 28644570240*a^5*b^26*x^15 + 8951428200*a^4*b^2
7*x^16 + 2106218400*a^3*b^28*x^17 + 351036400*a^2*b^29*x^18)