\(\int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx\) [116]

Optimal result

Integrand size = 15, antiderivative size = 271 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=-\frac {(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}-\frac {10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac {45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac {120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac {35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac {252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac {105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac {40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac {45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac {10 d^9 (b c-a d)}{b^{11} (a+b x)}+\frac {d^{10} \log (a+b x)}{b^{11}} \] Output:

-1/10*(-a*d+b*c)^10/b^11/(b*x+a)^10-10/9*d*(-a*d+b*c)^9/b^11/(b*x+a)^9-45/ 
8*d^2*(-a*d+b*c)^8/b^11/(b*x+a)^8-120/7*d^3*(-a*d+b*c)^7/b^11/(b*x+a)^7-35 
*d^4*(-a*d+b*c)^6/b^11/(b*x+a)^6-252/5*d^5*(-a*d+b*c)^5/b^11/(b*x+a)^5-105 
/2*d^6*(-a*d+b*c)^4/b^11/(b*x+a)^4-40*d^7*(-a*d+b*c)^3/b^11/(b*x+a)^3-45/2 
*d^8*(-a*d+b*c)^2/b^11/(b*x+a)^2-10*d^9*(-a*d+b*c)/b^11/(b*x+a)+d^10*ln(b* 
x+a)/b^11
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(591\) vs. \(2(271)=542\).

Time = 0.21 (sec) , antiderivative size = 591, normalized size of antiderivative = 2.18 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=-\frac {(b c-a d) \left (7381 a^9 d^9+a^8 b d^8 (4861 c+71290 d x)+a^7 b^2 d^7 \left (3601 c^2+46090 c d x+308205 d^2 x^2\right )+a^6 b^3 d^6 \left (2761 c^3+33490 c^2 d x+194805 c d^2 x^2+784080 d^3 x^3\right )+a^5 b^4 d^5 \left (2131 c^4+25090 c^3 d x+138105 c^2 d^2 x^2+481680 c d^3 x^3+1296540 d^4 x^4\right )+a^4 b^5 d^4 \left (1627 c^5+18790 c^4 d x+100305 c^3 d^2 x^2+330480 c^2 d^3 x^3+767340 c d^4 x^4+1450008 d^5 x^5\right )+a^3 b^6 d^3 \left (1207 c^6+13750 c^5 d x+71955 c^4 d^2 x^2+229680 c^3 d^3 x^3+502740 c^2 d^4 x^4+814968 c d^5 x^5+1102500 d^6 x^6\right )+a^2 b^7 d^2 \left (847 c^7+9550 c^6 d x+49275 c^5 d^2 x^2+154080 c^4 d^3 x^3+326340 c^3 d^4 x^4+497448 c^2 d^5 x^5+573300 c d^6 x^6+554400 d^7 x^7\right )+a b^8 d \left (532 c^8+5950 c^7 d x+30375 c^6 d^2 x^2+93600 c^5 d^3 x^3+194040 c^4 d^4 x^4+285768 c^3 d^5 x^5+308700 c^2 d^6 x^6+252000 c d^7 x^7+170100 d^8 x^8\right )+b^9 \left (252 c^9+2800 c^8 d x+14175 c^7 d^2 x^2+43200 c^6 d^3 x^3+88200 c^5 d^4 x^4+127008 c^4 d^5 x^5+132300 c^3 d^6 x^6+100800 c^2 d^7 x^7+56700 c d^8 x^8+25200 d^9 x^9\right )\right )}{2520 b^{11} (a+b x)^{10}}+\frac {d^{10} \log (a+b x)}{b^{11}} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^11,x]
 

Output:

-1/2520*((b*c - a*d)*(7381*a^9*d^9 + a^8*b*d^8*(4861*c + 71290*d*x) + a^7* 
b^2*d^7*(3601*c^2 + 46090*c*d*x + 308205*d^2*x^2) + a^6*b^3*d^6*(2761*c^3 
+ 33490*c^2*d*x + 194805*c*d^2*x^2 + 784080*d^3*x^3) + a^5*b^4*d^5*(2131*c 
^4 + 25090*c^3*d*x + 138105*c^2*d^2*x^2 + 481680*c*d^3*x^3 + 1296540*d^4*x 
^4) + a^4*b^5*d^4*(1627*c^5 + 18790*c^4*d*x + 100305*c^3*d^2*x^2 + 330480* 
c^2*d^3*x^3 + 767340*c*d^4*x^4 + 1450008*d^5*x^5) + a^3*b^6*d^3*(1207*c^6 
+ 13750*c^5*d*x + 71955*c^4*d^2*x^2 + 229680*c^3*d^3*x^3 + 502740*c^2*d^4* 
x^4 + 814968*c*d^5*x^5 + 1102500*d^6*x^6) + a^2*b^7*d^2*(847*c^7 + 9550*c^ 
6*d*x + 49275*c^5*d^2*x^2 + 154080*c^4*d^3*x^3 + 326340*c^3*d^4*x^4 + 4974 
48*c^2*d^5*x^5 + 573300*c*d^6*x^6 + 554400*d^7*x^7) + a*b^8*d*(532*c^8 + 5 
950*c^7*d*x + 30375*c^6*d^2*x^2 + 93600*c^5*d^3*x^3 + 194040*c^4*d^4*x^4 + 
 285768*c^3*d^5*x^5 + 308700*c^2*d^6*x^6 + 252000*c*d^7*x^7 + 170100*d^8*x 
^8) + b^9*(252*c^9 + 2800*c^8*d*x + 14175*c^7*d^2*x^2 + 43200*c^6*d^3*x^3 
+ 88200*c^5*d^4*x^4 + 127008*c^4*d^5*x^5 + 132300*c^3*d^6*x^6 + 100800*c^2 
*d^7*x^7 + 56700*c*d^8*x^8 + 25200*d^9*x^9)))/(b^11*(a + b*x)^10) + (d^10* 
Log[a + b*x])/b^11
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {49, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx\)

\(\Big \downarrow \) 49

\(\displaystyle \int \left (\frac {10 d^9 (b c-a d)}{b^{10} (a+b x)^2}+\frac {45 d^8 (b c-a d)^2}{b^{10} (a+b x)^3}+\frac {120 d^7 (b c-a d)^3}{b^{10} (a+b x)^4}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)^5}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^6}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^7}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^8}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^9}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^{10}}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^{11}}+\frac {d^{10}}{b^{10} (a+b x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {10 d^9 (b c-a d)}{b^{11} (a+b x)}-\frac {45 d^8 (b c-a d)^2}{2 b^{11} (a+b x)^2}-\frac {40 d^7 (b c-a d)^3}{b^{11} (a+b x)^3}-\frac {105 d^6 (b c-a d)^4}{2 b^{11} (a+b x)^4}-\frac {252 d^5 (b c-a d)^5}{5 b^{11} (a+b x)^5}-\frac {35 d^4 (b c-a d)^6}{b^{11} (a+b x)^6}-\frac {120 d^3 (b c-a d)^7}{7 b^{11} (a+b x)^7}-\frac {45 d^2 (b c-a d)^8}{8 b^{11} (a+b x)^8}-\frac {10 d (b c-a d)^9}{9 b^{11} (a+b x)^9}-\frac {(b c-a d)^{10}}{10 b^{11} (a+b x)^{10}}+\frac {d^{10} \log (a+b x)}{b^{11}}\)

Input:

Int[(c + d*x)^10/(a + b*x)^11,x]
 

Output:

-1/10*(b*c - a*d)^10/(b^11*(a + b*x)^10) - (10*d*(b*c - a*d)^9)/(9*b^11*(a 
 + b*x)^9) - (45*d^2*(b*c - a*d)^8)/(8*b^11*(a + b*x)^8) - (120*d^3*(b*c - 
 a*d)^7)/(7*b^11*(a + b*x)^7) - (35*d^4*(b*c - a*d)^6)/(b^11*(a + b*x)^6) 
- (252*d^5*(b*c - a*d)^5)/(5*b^11*(a + b*x)^5) - (105*d^6*(b*c - a*d)^4)/( 
2*b^11*(a + b*x)^4) - (40*d^7*(b*c - a*d)^3)/(b^11*(a + b*x)^3) - (45*d^8* 
(b*c - a*d)^2)/(2*b^11*(a + b*x)^2) - (10*d^9*(b*c - a*d))/(b^11*(a + b*x) 
) + (d^10*Log[a + b*x])/b^11
 

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

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. \(842\) vs. \(2(257)=514\).

Time = 0.16 (sec) , antiderivative size = 843, normalized size of antiderivative = 3.11

method	result	size
risch	\(\frac {\frac {10 d^{9} \left (a d -b c \right ) x^{9}}{b^{2}}+\frac {45 d^{8} \left (3 a^{2} d^{2}-2 a b c d -b^{2} c^{2}\right ) x^{8}}{2 b^{3}}+\frac {20 d^{7} \left (11 a^{3} d^{3}-6 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d -2 b^{3} c^{3}\right ) x^{7}}{b^{4}}+\frac {35 d^{6} \left (25 d^{4} a^{4}-12 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -3 c^{4} b^{4}\right ) x^{6}}{2 b^{5}}+\frac {21 d^{5} \left (137 a^{5} d^{5}-60 a^{4} b c \,d^{4}-30 a^{3} b^{2} c^{2} d^{3}-20 a^{2} b^{3} c^{3} d^{2}-15 a \,b^{4} c^{4} d -12 c^{5} b^{5}\right ) x^{5}}{5 b^{6}}+\frac {7 d^{4} \left (147 a^{6} d^{6}-60 a^{5} b c \,d^{5}-30 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}-15 a^{2} b^{4} c^{4} d^{2}-12 a \,b^{5} c^{5} d -10 c^{6} b^{6}\right ) x^{4}}{2 b^{7}}+\frac {2 d^{3} \left (1089 a^{7} d^{7}-420 a^{6} b c \,d^{6}-210 a^{5} b^{2} c^{2} d^{5}-140 a^{4} b^{3} c^{3} d^{4}-105 a^{3} b^{4} c^{4} d^{3}-84 a^{2} b^{5} c^{5} d^{2}-70 a \,b^{6} c^{6} d -60 b^{7} c^{7}\right ) x^{3}}{7 b^{8}}+\frac {3 d^{2} \left (2283 a^{8} d^{8}-840 a^{7} b c \,d^{7}-420 a^{6} b^{2} c^{2} d^{6}-280 a^{5} b^{3} c^{3} d^{5}-210 a^{4} b^{4} c^{4} d^{4}-168 a^{3} b^{5} c^{5} d^{3}-140 a^{2} b^{6} c^{6} d^{2}-120 a \,b^{7} c^{7} d -105 c^{8} b^{8}\right ) x^{2}}{56 b^{9}}+\frac {d \left (7129 a^{9} d^{9}-2520 a^{8} b c \,d^{8}-1260 a^{7} b^{2} c^{2} d^{7}-840 a^{6} b^{3} c^{3} d^{6}-630 a^{5} b^{4} c^{4} d^{5}-504 a^{4} b^{5} c^{5} d^{4}-420 a^{3} b^{6} c^{6} d^{3}-360 a^{2} b^{7} c^{7} d^{2}-315 a \,b^{8} c^{8} d -280 c^{9} b^{9}\right ) x}{252 b^{10}}+\frac {7381 a^{10} d^{10}-2520 a^{9} b c \,d^{9}-1260 a^{8} b^{2} c^{2} d^{8}-840 a^{7} b^{3} c^{3} d^{7}-630 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-420 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}-315 a^{2} b^{8} c^{8} d^{2}-280 a \,b^{9} c^{9} d -252 b^{10} c^{10}}{2520 b^{11}}}{\left (b x +a \right )^{10}}+\frac {d^{10} \ln \left (b x +a \right )}{b^{11}}\)	\(843\)
norman	\(\frac {\frac {7381 a^{10} d^{10}-2520 a^{9} b c \,d^{9}-1260 a^{8} b^{2} c^{2} d^{8}-840 a^{7} b^{3} c^{3} d^{7}-630 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-420 a^{4} b^{6} c^{6} d^{4}-360 a^{3} b^{7} c^{7} d^{3}-315 a^{2} b^{8} c^{8} d^{2}-280 a \,b^{9} c^{9} d -252 b^{10} c^{10}}{2520 b^{11}}+\frac {10 \left (a \,d^{10}-b c \,d^{9}\right ) x^{9}}{b^{2}}+\frac {45 \left (3 a^{2} d^{10}-2 a b c \,d^{9}-b^{2} c^{2} d^{8}\right ) x^{8}}{2 b^{3}}+\frac {20 \left (11 a^{3} d^{10}-6 a^{2} b c \,d^{9}-3 a \,b^{2} c^{2} d^{8}-2 b^{3} c^{3} d^{7}\right ) x^{7}}{b^{4}}+\frac {35 \left (25 a^{4} d^{10}-12 a^{3} b c \,d^{9}-6 a^{2} b^{2} c^{2} d^{8}-4 a \,b^{3} c^{3} d^{7}-3 b^{4} c^{4} d^{6}\right ) x^{6}}{2 b^{5}}+\frac {21 \left (137 a^{5} d^{10}-60 a^{4} b c \,d^{9}-30 a^{3} b^{2} c^{2} d^{8}-20 a^{2} b^{3} c^{3} d^{7}-15 a \,b^{4} c^{4} d^{6}-12 b^{5} c^{5} d^{5}\right ) x^{5}}{5 b^{6}}+\frac {7 \left (147 a^{6} d^{10}-60 a^{5} b c \,d^{9}-30 a^{4} b^{2} c^{2} d^{8}-20 a^{3} b^{3} c^{3} d^{7}-15 a^{2} b^{4} c^{4} d^{6}-12 a \,b^{5} c^{5} d^{5}-10 b^{6} c^{6} d^{4}\right ) x^{4}}{2 b^{7}}+\frac {2 \left (1089 a^{7} d^{10}-420 a^{6} b c \,d^{9}-210 a^{5} b^{2} c^{2} d^{8}-140 a^{4} b^{3} c^{3} d^{7}-105 a^{3} b^{4} c^{4} d^{6}-84 a^{2} b^{5} c^{5} d^{5}-70 a \,b^{6} c^{6} d^{4}-60 b^{7} c^{7} d^{3}\right ) x^{3}}{7 b^{8}}+\frac {3 \left (2283 a^{8} d^{10}-840 a^{7} b c \,d^{9}-420 a^{6} b^{2} c^{2} d^{8}-280 a^{5} b^{3} c^{3} d^{7}-210 a^{4} b^{4} c^{4} d^{6}-168 a^{3} b^{5} c^{5} d^{5}-140 a^{2} b^{6} c^{6} d^{4}-120 a \,b^{7} c^{7} d^{3}-105 b^{8} c^{8} d^{2}\right ) x^{2}}{56 b^{9}}+\frac {\left (7129 a^{9} d^{10}-2520 a^{8} b c \,d^{9}-1260 a^{7} b^{2} c^{2} d^{8}-840 a^{6} b^{3} c^{3} d^{7}-630 a^{5} b^{4} c^{4} d^{6}-504 a^{4} b^{5} c^{5} d^{5}-420 a^{3} b^{6} c^{6} d^{4}-360 a^{2} b^{7} c^{7} d^{3}-315 a \,b^{8} c^{8} d^{2}-280 b^{9} c^{9} d \right ) x}{252 b^{10}}}{\left (b x +a \right )^{10}}+\frac {d^{10} \ln \left (b x +a \right )}{b^{11}}\)	\(861\)
default	\(\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 -c^{9} b^{9}\right )}{9 b^{11} \left (b x +a \right )^{9}}+\frac {252 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 )}{5 b^{11} \left (b x +a \right )^{5}}-\frac {105 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 )}{2 b^{11} \left (b x +a \right )^{4}}+\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 )}{7 b^{11} \left (b x +a \right )^{7}}-\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 )}{8 b^{11} \left (b x +a \right )^{8}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 b^{11} \left (b x +a \right )^{2}}+\frac {10 d^{9} \left (a d -b c \right )}{b^{11} \left (b x +a \right )}+\frac {d^{10} \ln \left (b x +a \right )}{b^{11}}-\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}}{10 b^{11} \left (b x +a \right )^{10}}+\frac {40 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{11} \left (b x +a \right )^{3}}-\frac {35 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 )}{b^{11} \left (b x +a \right )^{6}}\)	\(865\)
parallelrisch	\(\text {Expression too large to display}\)	\(1157\)

Input:

int((d*x+c)^10/(b*x+a)^11,x,method=_RETURNVERBOSE)
 

Output:

(10*d^9*(a*d-b*c)/b^2*x^9+45/2*d^8*(3*a^2*d^2-2*a*b*c*d-b^2*c^2)/b^3*x^8+2 
0*d^7*(11*a^3*d^3-6*a^2*b*c*d^2-3*a*b^2*c^2*d-2*b^3*c^3)/b^4*x^7+35/2*d^6* 
(25*a^4*d^4-12*a^3*b*c*d^3-6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d-3*b^4*c^4)/b^5* 
x^6+21/5*d^5*(137*a^5*d^5-60*a^4*b*c*d^4-30*a^3*b^2*c^2*d^3-20*a^2*b^3*c^3 
*d^2-15*a*b^4*c^4*d-12*b^5*c^5)/b^6*x^5+7/2*d^4*(147*a^6*d^6-60*a^5*b*c*d^ 
5-30*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^2-12*a*b^5*c^5*d- 
10*b^6*c^6)/b^7*x^4+2/7*d^3*(1089*a^7*d^7-420*a^6*b*c*d^6-210*a^5*b^2*c^2* 
d^5-140*a^4*b^3*c^3*d^4-105*a^3*b^4*c^4*d^3-84*a^2*b^5*c^5*d^2-70*a*b^6*c^ 
6*d-60*b^7*c^7)/b^8*x^3+3/56*d^2*(2283*a^8*d^8-840*a^7*b*c*d^7-420*a^6*b^2 
*c^2*d^6-280*a^5*b^3*c^3*d^5-210*a^4*b^4*c^4*d^4-168*a^3*b^5*c^5*d^3-140*a 
^2*b^6*c^6*d^2-120*a*b^7*c^7*d-105*b^8*c^8)/b^9*x^2+1/252*d*(7129*a^9*d^9- 
2520*a^8*b*c*d^8-1260*a^7*b^2*c^2*d^7-840*a^6*b^3*c^3*d^6-630*a^5*b^4*c^4* 
d^5-504*a^4*b^5*c^5*d^4-420*a^3*b^6*c^6*d^3-360*a^2*b^7*c^7*d^2-315*a*b^8* 
c^8*d-280*b^9*c^9)/b^10*x+1/2520*(7381*a^10*d^10-2520*a^9*b*c*d^9-1260*a^8 
*b^2*c^2*d^8-840*a^7*b^3*c^3*d^7-630*a^6*b^4*c^4*d^6-504*a^5*b^5*c^5*d^5-4 
20*a^4*b^6*c^6*d^4-360*a^3*b^7*c^7*d^3-315*a^2*b^8*c^8*d^2-280*a*b^9*c^9*d 
-252*b^10*c^10)/b^11)/(b*x+a)^10+d^10*ln(b*x+a)/b^11
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1107 vs. \(2 (257) = 514\).

Time = 0.10 (sec) , antiderivative size = 1107, normalized size of antiderivative = 4.08 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^11,x, algorithm="fricas")
 

Output:

-1/2520*(252*b^10*c^10 + 280*a*b^9*c^9*d + 315*a^2*b^8*c^8*d^2 + 360*a^3*b 
^7*c^7*d^3 + 420*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d 
^6 + 840*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 2520*a^9*b*c*d^9 - 7381* 
a^10*d^10 + 25200*(b^10*c*d^9 - a*b^9*d^10)*x^9 + 56700*(b^10*c^2*d^8 + 2* 
a*b^9*c*d^9 - 3*a^2*b^8*d^10)*x^8 + 50400*(2*b^10*c^3*d^7 + 3*a*b^9*c^2*d^ 
8 + 6*a^2*b^8*c*d^9 - 11*a^3*b^7*d^10)*x^7 + 44100*(3*b^10*c^4*d^6 + 4*a*b 
^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 12*a^3*b^7*c*d^9 - 25*a^4*b^6*d^10)*x^6 + 
 10584*(12*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 + 30*a^3*b 
^7*c^2*d^8 + 60*a^4*b^6*c*d^9 - 137*a^5*b^5*d^10)*x^5 + 8820*(10*b^10*c^6* 
d^4 + 12*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 + 30*a^4* 
b^6*c^2*d^8 + 60*a^5*b^5*c*d^9 - 147*a^6*b^4*d^10)*x^4 + 720*(60*b^10*c^7* 
d^3 + 70*a*b^9*c^6*d^4 + 84*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6 + 140*a^ 
4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 + 420*a^6*b^4*c*d^9 - 1089*a^7*b^3*d^1 
0)*x^3 + 135*(105*b^10*c^8*d^2 + 120*a*b^9*c^7*d^3 + 140*a^2*b^8*c^6*d^4 + 
 168*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7 + 420*a^6 
*b^4*c^2*d^8 + 840*a^7*b^3*c*d^9 - 2283*a^8*b^2*d^10)*x^2 + 10*(280*b^10*c 
^9*d + 315*a*b^9*c^8*d^2 + 360*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 + 504 
*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 + 1260*a^7*b^ 
3*c^2*d^8 + 2520*a^8*b^2*c*d^9 - 7129*a^9*b*d^10)*x - 2520*(b^10*d^10*x^10 
 + 10*a*b^9*d^10*x^9 + 45*a^2*b^8*d^10*x^8 + 120*a^3*b^7*d^10*x^7 + 210...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**10/(b*x+a)**11,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (257) = 514\).

Time = 0.12 (sec) , antiderivative size = 975, normalized size of antiderivative = 3.60 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^11,x, algorithm="maxima")
 

Output:

-1/2520*(252*b^10*c^10 + 280*a*b^9*c^9*d + 315*a^2*b^8*c^8*d^2 + 360*a^3*b 
^7*c^7*d^3 + 420*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d 
^6 + 840*a^7*b^3*c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 2520*a^9*b*c*d^9 - 7381* 
a^10*d^10 + 25200*(b^10*c*d^9 - a*b^9*d^10)*x^9 + 56700*(b^10*c^2*d^8 + 2* 
a*b^9*c*d^9 - 3*a^2*b^8*d^10)*x^8 + 50400*(2*b^10*c^3*d^7 + 3*a*b^9*c^2*d^ 
8 + 6*a^2*b^8*c*d^9 - 11*a^3*b^7*d^10)*x^7 + 44100*(3*b^10*c^4*d^6 + 4*a*b 
^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 12*a^3*b^7*c*d^9 - 25*a^4*b^6*d^10)*x^6 + 
 10584*(12*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6 + 20*a^2*b^8*c^3*d^7 + 30*a^3*b 
^7*c^2*d^8 + 60*a^4*b^6*c*d^9 - 137*a^5*b^5*d^10)*x^5 + 8820*(10*b^10*c^6* 
d^4 + 12*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 20*a^3*b^7*c^3*d^7 + 30*a^4* 
b^6*c^2*d^8 + 60*a^5*b^5*c*d^9 - 147*a^6*b^4*d^10)*x^4 + 720*(60*b^10*c^7* 
d^3 + 70*a*b^9*c^6*d^4 + 84*a^2*b^8*c^5*d^5 + 105*a^3*b^7*c^4*d^6 + 140*a^ 
4*b^6*c^3*d^7 + 210*a^5*b^5*c^2*d^8 + 420*a^6*b^4*c*d^9 - 1089*a^7*b^3*d^1 
0)*x^3 + 135*(105*b^10*c^8*d^2 + 120*a*b^9*c^7*d^3 + 140*a^2*b^8*c^6*d^4 + 
 168*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 + 280*a^5*b^5*c^3*d^7 + 420*a^6 
*b^4*c^2*d^8 + 840*a^7*b^3*c*d^9 - 2283*a^8*b^2*d^10)*x^2 + 10*(280*b^10*c 
^9*d + 315*a*b^9*c^8*d^2 + 360*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 + 504 
*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 + 840*a^6*b^4*c^3*d^7 + 1260*a^7*b^ 
3*c^2*d^8 + 2520*a^8*b^2*c*d^9 - 7129*a^9*b*d^10)*x)/(b^21*x^10 + 10*a*b^2 
0*x^9 + 45*a^2*b^19*x^8 + 120*a^3*b^18*x^7 + 210*a^4*b^17*x^6 + 252*a^5...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (257) = 514\).

Time = 0.13 (sec) , antiderivative size = 874, normalized size of antiderivative = 3.23 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^11,x, algorithm="giac")
 

Output:

d^10*log(abs(b*x + a))/b^11 - 1/2520*(25200*(b^9*c*d^9 - a*b^8*d^10)*x^9 + 
 56700*(b^9*c^2*d^8 + 2*a*b^8*c*d^9 - 3*a^2*b^7*d^10)*x^8 + 50400*(2*b^9*c 
^3*d^7 + 3*a*b^8*c^2*d^8 + 6*a^2*b^7*c*d^9 - 11*a^3*b^6*d^10)*x^7 + 44100* 
(3*b^9*c^4*d^6 + 4*a*b^8*c^3*d^7 + 6*a^2*b^7*c^2*d^8 + 12*a^3*b^6*c*d^9 - 
25*a^4*b^5*d^10)*x^6 + 10584*(12*b^9*c^5*d^5 + 15*a*b^8*c^4*d^6 + 20*a^2*b 
^7*c^3*d^7 + 30*a^3*b^6*c^2*d^8 + 60*a^4*b^5*c*d^9 - 137*a^5*b^4*d^10)*x^5 
 + 8820*(10*b^9*c^6*d^4 + 12*a*b^8*c^5*d^5 + 15*a^2*b^7*c^4*d^6 + 20*a^3*b 
^6*c^3*d^7 + 30*a^4*b^5*c^2*d^8 + 60*a^5*b^4*c*d^9 - 147*a^6*b^3*d^10)*x^4 
 + 720*(60*b^9*c^7*d^3 + 70*a*b^8*c^6*d^4 + 84*a^2*b^7*c^5*d^5 + 105*a^3*b 
^6*c^4*d^6 + 140*a^4*b^5*c^3*d^7 + 210*a^5*b^4*c^2*d^8 + 420*a^6*b^3*c*d^9 
 - 1089*a^7*b^2*d^10)*x^3 + 135*(105*b^9*c^8*d^2 + 120*a*b^8*c^7*d^3 + 140 
*a^2*b^7*c^6*d^4 + 168*a^3*b^6*c^5*d^5 + 210*a^4*b^5*c^4*d^6 + 280*a^5*b^4 
*c^3*d^7 + 420*a^6*b^3*c^2*d^8 + 840*a^7*b^2*c*d^9 - 2283*a^8*b*d^10)*x^2 
+ 10*(280*b^9*c^9*d + 315*a*b^8*c^8*d^2 + 360*a^2*b^7*c^7*d^3 + 420*a^3*b^ 
6*c^6*d^4 + 504*a^4*b^5*c^5*d^5 + 630*a^5*b^4*c^4*d^6 + 840*a^6*b^3*c^3*d^ 
7 + 1260*a^7*b^2*c^2*d^8 + 2520*a^8*b*c*d^9 - 7129*a^9*d^10)*x + (252*b^10 
*c^10 + 280*a*b^9*c^9*d + 315*a^2*b^8*c^8*d^2 + 360*a^3*b^7*c^7*d^3 + 420* 
a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 + 840*a^7*b^3* 
c^3*d^7 + 1260*a^8*b^2*c^2*d^8 + 2520*a^9*b*c*d^9 - 7381*a^10*d^10)/b)/((b 
*x + a)^10*b^10)
 

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 866, normalized size of antiderivative = 3.20 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx=\frac {d^{10}\,\ln \left (a+b\,x\right )}{b^{11}}-\frac {x^4\,\left (-\frac {1029\,a^6\,b^4\,d^{10}}{2}+210\,a^5\,b^5\,c\,d^9+105\,a^4\,b^6\,c^2\,d^8+70\,a^3\,b^7\,c^3\,d^7+\frac {105\,a^2\,b^8\,c^4\,d^6}{2}+42\,a\,b^9\,c^5\,d^5+35\,b^{10}\,c^6\,d^4\right )-x^9\,\left (10\,a\,b^9\,d^{10}-10\,b^{10}\,c\,d^9\right )+x\,\left (-\frac {7129\,a^9\,b\,d^{10}}{252}+10\,a^8\,b^2\,c\,d^9+5\,a^7\,b^3\,c^2\,d^8+\frac {10\,a^6\,b^4\,c^3\,d^7}{3}+\frac {5\,a^5\,b^5\,c^4\,d^6}{2}+2\,a^4\,b^6\,c^5\,d^5+\frac {5\,a^3\,b^7\,c^6\,d^4}{3}+\frac {10\,a^2\,b^8\,c^7\,d^3}{7}+\frac {5\,a\,b^9\,c^8\,d^2}{4}+\frac {10\,b^{10}\,c^9\,d}{9}\right )+x^6\,\left (-\frac {875\,a^4\,b^6\,d^{10}}{2}+210\,a^3\,b^7\,c\,d^9+105\,a^2\,b^8\,c^2\,d^8+70\,a\,b^9\,c^3\,d^7+\frac {105\,b^{10}\,c^4\,d^6}{2}\right )+x^8\,\left (-\frac {135\,a^2\,b^8\,d^{10}}{2}+45\,a\,b^9\,c\,d^9+\frac {45\,b^{10}\,c^2\,d^8}{2}\right )+x^3\,\left (-\frac {2178\,a^7\,b^3\,d^{10}}{7}+120\,a^6\,b^4\,c\,d^9+60\,a^5\,b^5\,c^2\,d^8+40\,a^4\,b^6\,c^3\,d^7+30\,a^3\,b^7\,c^4\,d^6+24\,a^2\,b^8\,c^5\,d^5+20\,a\,b^9\,c^6\,d^4+\frac {120\,b^{10}\,c^7\,d^3}{7}\right )+x^5\,\left (-\frac {2877\,a^5\,b^5\,d^{10}}{5}+252\,a^4\,b^6\,c\,d^9+126\,a^3\,b^7\,c^2\,d^8+84\,a^2\,b^8\,c^3\,d^7+63\,a\,b^9\,c^4\,d^6+\frac {252\,b^{10}\,c^5\,d^5}{5}\right )-\frac {7381\,a^{10}\,d^{10}}{2520}+\frac {b^{10}\,c^{10}}{10}+x^7\,\left (-220\,a^3\,b^7\,d^{10}+120\,a^2\,b^8\,c\,d^9+60\,a\,b^9\,c^2\,d^8+40\,b^{10}\,c^3\,d^7\right )+x^2\,\left (-\frac {6849\,a^8\,b^2\,d^{10}}{56}+45\,a^7\,b^3\,c\,d^9+\frac {45\,a^6\,b^4\,c^2\,d^8}{2}+15\,a^5\,b^5\,c^3\,d^7+\frac {45\,a^4\,b^6\,c^4\,d^6}{4}+9\,a^3\,b^7\,c^5\,d^5+\frac {15\,a^2\,b^8\,c^6\,d^4}{2}+\frac {45\,a\,b^9\,c^7\,d^3}{7}+\frac {45\,b^{10}\,c^8\,d^2}{8}\right )+\frac {a^2\,b^8\,c^8\,d^2}{8}+\frac {a^3\,b^7\,c^7\,d^3}{7}+\frac {a^4\,b^6\,c^6\,d^4}{6}+\frac {a^5\,b^5\,c^5\,d^5}{5}+\frac {a^6\,b^4\,c^4\,d^6}{4}+\frac {a^7\,b^3\,c^3\,d^7}{3}+\frac {a^8\,b^2\,c^2\,d^8}{2}+\frac {a\,b^9\,c^9\,d}{9}+a^9\,b\,c\,d^9}{b^{11}\,{\left (a+b\,x\right )}^{10}} \] Input:

int((c + d*x)^10/(a + b*x)^11,x)
 

Output:

(d^10*log(a + b*x))/b^11 - (x^4*(35*b^10*c^6*d^4 - (1029*a^6*b^4*d^10)/2 + 
 42*a*b^9*c^5*d^5 + 210*a^5*b^5*c*d^9 + (105*a^2*b^8*c^4*d^6)/2 + 70*a^3*b 
^7*c^3*d^7 + 105*a^4*b^6*c^2*d^8) - x^9*(10*a*b^9*d^10 - 10*b^10*c*d^9) + 
x*((10*b^10*c^9*d)/9 - (7129*a^9*b*d^10)/252 + (5*a*b^9*c^8*d^2)/4 + 10*a^ 
8*b^2*c*d^9 + (10*a^2*b^8*c^7*d^3)/7 + (5*a^3*b^7*c^6*d^4)/3 + 2*a^4*b^6*c 
^5*d^5 + (5*a^5*b^5*c^4*d^6)/2 + (10*a^6*b^4*c^3*d^7)/3 + 5*a^7*b^3*c^2*d^ 
8) + x^6*((105*b^10*c^4*d^6)/2 - (875*a^4*b^6*d^10)/2 + 70*a*b^9*c^3*d^7 + 
 210*a^3*b^7*c*d^9 + 105*a^2*b^8*c^2*d^8) + x^8*((45*b^10*c^2*d^8)/2 - (13 
5*a^2*b^8*d^10)/2 + 45*a*b^9*c*d^9) + x^3*((120*b^10*c^7*d^3)/7 - (2178*a^ 
7*b^3*d^10)/7 + 20*a*b^9*c^6*d^4 + 120*a^6*b^4*c*d^9 + 24*a^2*b^8*c^5*d^5 
+ 30*a^3*b^7*c^4*d^6 + 40*a^4*b^6*c^3*d^7 + 60*a^5*b^5*c^2*d^8) + x^5*((25 
2*b^10*c^5*d^5)/5 - (2877*a^5*b^5*d^10)/5 + 63*a*b^9*c^4*d^6 + 252*a^4*b^6 
*c*d^9 + 84*a^2*b^8*c^3*d^7 + 126*a^3*b^7*c^2*d^8) - (7381*a^10*d^10)/2520 
 + (b^10*c^10)/10 + x^7*(40*b^10*c^3*d^7 - 220*a^3*b^7*d^10 + 60*a*b^9*c^2 
*d^8 + 120*a^2*b^8*c*d^9) + x^2*((45*b^10*c^8*d^2)/8 - (6849*a^8*b^2*d^10) 
/56 + (45*a*b^9*c^7*d^3)/7 + 45*a^7*b^3*c*d^9 + (15*a^2*b^8*c^6*d^4)/2 + 9 
*a^3*b^7*c^5*d^5 + (45*a^4*b^6*c^4*d^6)/4 + 15*a^5*b^5*c^3*d^7 + (45*a^6*b 
^4*c^2*d^8)/2) + (a^2*b^8*c^8*d^2)/8 + (a^3*b^7*c^7*d^3)/7 + (a^4*b^6*c^6* 
d^4)/6 + (a^5*b^5*c^5*d^5)/5 + (a^6*b^4*c^4*d^6)/4 + (a^7*b^3*c^3*d^7)/3 + 
 (a^8*b^2*c^2*d^8)/2 + (a*b^9*c^9*d)/9 + a^9*b*c*d^9)/(b^11*(a + b*x)^1...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1160, normalized size of antiderivative = 4.28 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{11}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^11,x)
                                                                                    
                                                                                    
 

Output:

(2520*log(a + b*x)*a**11*d**10 + 25200*log(a + b*x)*a**10*b*d**10*x + 1134 
00*log(a + b*x)*a**9*b**2*d**10*x**2 + 302400*log(a + b*x)*a**8*b**3*d**10 
*x**3 + 529200*log(a + b*x)*a**7*b**4*d**10*x**4 + 635040*log(a + b*x)*a** 
6*b**5*d**10*x**5 + 529200*log(a + b*x)*a**5*b**6*d**10*x**6 + 302400*log( 
a + b*x)*a**4*b**7*d**10*x**7 + 113400*log(a + b*x)*a**3*b**8*d**10*x**8 + 
 25200*log(a + b*x)*a**2*b**9*d**10*x**9 + 2520*log(a + b*x)*a*b**10*d**10 
*x**10 + 4861*a**11*d**10 + 46090*a**10*b*d**10*x - 1260*a**9*b**2*c**2*d* 
*8 + 194805*a**9*b**2*d**10*x**2 - 840*a**8*b**3*c**3*d**7 - 12600*a**8*b* 
*3*c**2*d**8*x + 481680*a**8*b**3*d**10*x**3 - 630*a**7*b**4*c**4*d**6 - 8 
400*a**7*b**4*c**3*d**7*x - 56700*a**7*b**4*c**2*d**8*x**2 + 767340*a**7*b 
**4*d**10*x**4 - 504*a**6*b**5*c**5*d**5 - 6300*a**6*b**5*c**4*d**6*x - 37 
800*a**6*b**5*c**3*d**7*x**2 - 151200*a**6*b**5*c**2*d**8*x**3 + 814968*a* 
*6*b**5*d**10*x**5 - 420*a**5*b**6*c**6*d**4 - 5040*a**5*b**6*c**5*d**5*x 
- 28350*a**5*b**6*c**4*d**6*x**2 - 100800*a**5*b**6*c**3*d**7*x**3 - 26460 
0*a**5*b**6*c**2*d**8*x**4 + 573300*a**5*b**6*d**10*x**6 - 360*a**4*b**7*c 
**7*d**3 - 4200*a**4*b**7*c**6*d**4*x - 22680*a**4*b**7*c**5*d**5*x**2 - 7 
5600*a**4*b**7*c**4*d**6*x**3 - 176400*a**4*b**7*c**3*d**7*x**4 - 317520*a 
**4*b**7*c**2*d**8*x**5 + 252000*a**4*b**7*d**10*x**7 - 315*a**3*b**8*c**8 
*d**2 - 3600*a**3*b**8*c**7*d**3*x - 18900*a**3*b**8*c**6*d**4*x**2 - 6048 
0*a**3*b**8*c**5*d**5*x**3 - 132300*a**3*b**8*c**4*d**6*x**4 - 211680*a...