\(\int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx\) [113]

Optimal result

Integrand size = 15, antiderivative size = 258 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\frac {45 d^8 (b c-a d)^2 x}{b^{10}}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}+\frac {5 d^9 (b c-a d) (a+b x)^2}{b^{11}}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}} \] Output:

45*d^8*(-a*d+b*c)^2*x/b^10-1/7*(-a*d+b*c)^10/b^11/(b*x+a)^7-5/3*d*(-a*d+b* 
c)^9/b^11/(b*x+a)^6-9*d^2*(-a*d+b*c)^8/b^11/(b*x+a)^5-30*d^3*(-a*d+b*c)^7/ 
b^11/(b*x+a)^4-70*d^4*(-a*d+b*c)^6/b^11/(b*x+a)^3-126*d^5*(-a*d+b*c)^5/b^1 
1/(b*x+a)^2-210*d^6*(-a*d+b*c)^4/b^11/(b*x+a)+5*d^9*(-a*d+b*c)*(b*x+a)^2/b 
^11+1/3*d^10*(b*x+a)^3/b^11+120*d^7*(-a*d+b*c)^3*ln(b*x+a)/b^11
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\frac {21 b d^8 \left (45 b^2 c^2-80 a b c d+36 a^2 d^2\right ) x+21 b^2 d^9 (5 b c-4 a d) x^2+7 b^3 d^{10} x^3-\frac {3 (b c-a d)^{10}}{(a+b x)^7}+\frac {35 d (-b c+a d)^9}{(a+b x)^6}-\frac {189 d^2 (b c-a d)^8}{(a+b x)^5}+\frac {630 d^3 (-b c+a d)^7}{(a+b x)^4}-\frac {1470 d^4 (b c-a d)^6}{(a+b x)^3}+\frac {2646 d^5 (-b c+a d)^5}{(a+b x)^2}-\frac {4410 d^6 (b c-a d)^4}{a+b x}+2520 d^7 (b c-a d)^3 \log (a+b x)}{21 b^{11}} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^8,x]
 

Output:

(21*b*d^8*(45*b^2*c^2 - 80*a*b*c*d + 36*a^2*d^2)*x + 21*b^2*d^9*(5*b*c - 4 
*a*d)*x^2 + 7*b^3*d^10*x^3 - (3*(b*c - a*d)^10)/(a + b*x)^7 + (35*d*(-(b*c 
) + a*d)^9)/(a + b*x)^6 - (189*d^2*(b*c - a*d)^8)/(a + b*x)^5 + (630*d^3*( 
-(b*c) + a*d)^7)/(a + b*x)^4 - (1470*d^4*(b*c - a*d)^6)/(a + b*x)^3 + (264 
6*d^5*(-(b*c) + a*d)^5)/(a + b*x)^2 - (4410*d^6*(b*c - a*d)^4)/(a + b*x) + 
 2520*d^7*(b*c - a*d)^3*Log[a + b*x])/(21*b^11)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 258, 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.

\(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx\)

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 d^9 (a+b x)^2 (b c-a d)}{b^{11}}+\frac {120 d^7 (b c-a d)^3 \log (a+b x)}{b^{11}}-\frac {210 d^6 (b c-a d)^4}{b^{11} (a+b x)}-\frac {126 d^5 (b c-a d)^5}{b^{11} (a+b x)^2}-\frac {70 d^4 (b c-a d)^6}{b^{11} (a+b x)^3}-\frac {30 d^3 (b c-a d)^7}{b^{11} (a+b x)^4}-\frac {9 d^2 (b c-a d)^8}{b^{11} (a+b x)^5}-\frac {5 d (b c-a d)^9}{3 b^{11} (a+b x)^6}-\frac {(b c-a d)^{10}}{7 b^{11} (a+b x)^7}+\frac {d^{10} (a+b x)^3}{3 b^{11}}+\frac {45 d^8 x (b c-a d)^2}{b^{10}}\)

Input:

Int[(c + d*x)^10/(a + b*x)^8,x]
 

Output:

(45*d^8*(b*c - a*d)^2*x)/b^10 - (b*c - a*d)^10/(7*b^11*(a + b*x)^7) - (5*d 
*(b*c - a*d)^9)/(3*b^11*(a + b*x)^6) - (9*d^2*(b*c - a*d)^8)/(b^11*(a + b* 
x)^5) - (30*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^4) - (70*d^4*(b*c - a*d)^6) 
/(b^11*(a + b*x)^3) - (126*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)^2) - (210*d^ 
6*(b*c - a*d)^4)/(b^11*(a + b*x)) + (5*d^9*(b*c - a*d)*(a + b*x)^2)/b^11 + 
 (d^10*(a + b*x)^3)/(3*b^11) + (120*d^7*(b*c - a*d)^3*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. \(853\) vs. \(2(252)=504\).

Time = 0.15 (sec) , antiderivative size = 854, normalized size of antiderivative = 3.31

method	result	size
norman	\(\frac {-\frac {6534 a^{10} d^{10}-19602 a^{9} b c \,d^{9}+19602 a^{8} b^{2} c^{2} d^{8}-6534 a^{7} b^{3} c^{3} d^{7}+630 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+42 a^{4} b^{6} c^{6} d^{4}+18 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +3 b^{10} c^{10}}{21 b^{11}}+\frac {d^{10} x^{10}}{3 b}-\frac {7 \left (120 a^{4} d^{10}-360 a^{3} b c \,d^{9}+360 a^{2} b^{2} c^{2} d^{8}-120 a \,b^{3} c^{3} d^{7}+30 b^{4} c^{4} d^{6}\right ) x^{6}}{b^{5}}-\frac {21 \left (180 a^{5} d^{10}-540 a^{4} b c \,d^{9}+540 a^{3} b^{2} c^{2} d^{8}-180 a^{2} b^{3} c^{3} d^{7}+30 a \,b^{4} c^{4} d^{6}+6 b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}-\frac {35 \left (220 a^{6} d^{10}-660 a^{5} b c \,d^{9}+660 a^{4} b^{2} c^{2} d^{8}-220 a^{3} b^{3} c^{3} d^{7}+30 a^{2} b^{4} c^{4} d^{6}+6 a \,b^{5} c^{5} d^{5}+2 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}-\frac {5 \left (1750 a^{7} d^{10}-5250 a^{6} b c \,d^{9}+5250 a^{5} b^{2} c^{2} d^{8}-1750 a^{4} b^{3} c^{3} d^{7}+210 a^{3} b^{4} c^{4} d^{6}+42 a^{2} b^{5} c^{5} d^{5}+14 a \,b^{6} c^{6} d^{4}+6 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}-\frac {3 \left (1918 a^{8} d^{10}-5754 a^{7} b c \,d^{9}+5754 a^{6} b^{2} c^{2} d^{8}-1918 a^{5} b^{3} c^{3} d^{7}+210 a^{4} b^{4} c^{4} d^{6}+42 a^{3} b^{5} c^{5} d^{5}+14 a^{2} b^{6} c^{6} d^{4}+6 a \,b^{7} c^{7} d^{3}+3 b^{8} c^{8} d^{2}\right ) x^{2}}{b^{9}}-\frac {\left (6174 a^{9} d^{10}-18522 a^{8} b c \,d^{9}+18522 a^{7} b^{2} c^{2} d^{8}-6174 a^{6} b^{3} c^{3} d^{7}+630 a^{5} b^{4} c^{4} d^{6}+126 a^{4} b^{5} c^{5} d^{5}+42 a^{3} b^{6} c^{6} d^{4}+18 a^{2} b^{7} c^{7} d^{3}+9 a \,b^{8} c^{8} d^{2}+5 b^{9} c^{9} d \right ) x}{3 b^{10}}+\frac {15 d^{8} \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{8}}{b^{3}}-\frac {5 d^{9} \left (a d -3 b c \right ) x^{9}}{3 b^{2}}}{\left (b x +a \right )^{7}}-\frac {120 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(854\)
default	\(\frac {d^{8} \left (\frac {1}{3} d^{2} x^{3} b^{2}-4 x^{2} a b \,d^{2}+5 x^{2} b^{2} c d +36 a^{2} d^{2} x -80 a b c d x +45 b^{2} c^{2} x \right )}{b^{10}}-\frac {9 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +c^{8} b^{8}\right )}{b^{11} \left (b x +a \right )^{5}}+\frac {30 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{4}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{7 b^{11} \left (b x +a \right )^{7}}+\frac {126 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{b^{11} \left (b x +a \right )^{2}}-\frac {210 d^{6} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{b^{11} \left (b x +a \right )}-\frac {120 d^{7} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{11}}-\frac {70 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{b^{11} \left (b x +a \right )^{3}}+\frac {5 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -c^{9} b^{9}\right )}{3 b^{11} \left (b x +a \right )^{6}}\)	\(856\)
risch	\(\frac {d^{10} x^{3}}{3 b^{8}}-\frac {4 d^{10} x^{2} a}{b^{9}}+\frac {5 d^{9} x^{2} c}{b^{8}}+\frac {36 d^{10} a^{2} x}{b^{10}}-\frac {80 d^{9} a c x}{b^{9}}+\frac {45 d^{8} c^{2} x}{b^{8}}+\frac {\left (-210 a^{4} b^{5} d^{10}+840 a^{3} b^{6} c \,d^{9}-1260 a^{2} b^{7} c^{2} d^{8}+840 a \,b^{8} c^{3} d^{7}-210 b^{9} c^{4} d^{6}\right ) x^{6}-126 b^{4} d^{5} \left (9 a^{5} d^{5}-35 a^{4} b c \,d^{4}+50 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +c^{5} b^{5}\right ) x^{5}-70 b^{3} d^{4} \left (37 a^{6} d^{6}-141 a^{5} b c \,d^{5}+195 a^{4} b^{2} c^{2} d^{4}-110 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +c^{6} b^{6}\right ) x^{4}-10 b^{2} d^{3} \left (319 a^{7} d^{7}-1197 a^{6} b c \,d^{6}+1617 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +3 b^{7} c^{7}\right ) x^{3}-3 b \,d^{2} \left (743 a^{8} d^{8}-2754 a^{7} b c \,d^{7}+3654 a^{6} b^{2} c^{2} d^{6}-1918 a^{5} b^{3} c^{3} d^{5}+210 a^{4} b^{4} c^{4} d^{4}+42 a^{3} b^{5} c^{5} d^{3}+14 a^{2} b^{6} c^{6} d^{2}+6 a \,b^{7} c^{7} d +3 c^{8} b^{8}\right ) x^{2}-\frac {d \left (2509 a^{9} d^{9}-9207 a^{8} b c \,d^{8}+12042 a^{7} b^{2} c^{2} d^{7}-6174 a^{6} b^{3} c^{3} d^{6}+630 a^{5} b^{4} c^{4} d^{5}+126 a^{4} b^{5} c^{5} d^{4}+42 a^{3} b^{6} c^{6} d^{3}+18 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d +5 c^{9} b^{9}\right ) x}{3}-\frac {2761 a^{10} d^{10}-10047 a^{9} b c \,d^{9}+12987 a^{8} b^{2} c^{2} d^{8}-6534 a^{7} b^{3} c^{3} d^{7}+630 a^{6} b^{4} c^{4} d^{6}+126 a^{5} b^{5} c^{5} d^{5}+42 a^{4} b^{6} c^{6} d^{4}+18 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +3 b^{10} c^{10}}{21 b}}{b^{10} \left (b x +a \right )^{7}}-\frac {120 d^{10} \ln \left (b x +a \right ) a^{3}}{b^{11}}+\frac {360 d^{9} \ln \left (b x +a \right ) a^{2} c}{b^{10}}-\frac {360 d^{8} \ln \left (b x +a \right ) a \,c^{2}}{b^{9}}+\frac {120 d^{7} \ln \left (b x +a \right ) c^{3}}{b^{8}}\)	\(868\)
parallelrisch	\(\text {Expression too large to display}\)	\(1567\)

Input:

int((d*x+c)^10/(b*x+a)^8,x,method=_RETURNVERBOSE)
 

Output:

(-1/21*(6534*a^10*d^10-19602*a^9*b*c*d^9+19602*a^8*b^2*c^2*d^8-6534*a^7*b^ 
3*c^3*d^7+630*a^6*b^4*c^4*d^6+126*a^5*b^5*c^5*d^5+42*a^4*b^6*c^6*d^4+18*a^ 
3*b^7*c^7*d^3+9*a^2*b^8*c^8*d^2+5*a*b^9*c^9*d+3*b^10*c^10)/b^11+1/3/b*d^10 
*x^10-7*(120*a^4*d^10-360*a^3*b*c*d^9+360*a^2*b^2*c^2*d^8-120*a*b^3*c^3*d^ 
7+30*b^4*c^4*d^6)/b^5*x^6-21*(180*a^5*d^10-540*a^4*b*c*d^9+540*a^3*b^2*c^2 
*d^8-180*a^2*b^3*c^3*d^7+30*a*b^4*c^4*d^6+6*b^5*c^5*d^5)/b^6*x^5-35*(220*a 
^6*d^10-660*a^5*b*c*d^9+660*a^4*b^2*c^2*d^8-220*a^3*b^3*c^3*d^7+30*a^2*b^4 
*c^4*d^6+6*a*b^5*c^5*d^5+2*b^6*c^6*d^4)/b^7*x^4-5*(1750*a^7*d^10-5250*a^6* 
b*c*d^9+5250*a^5*b^2*c^2*d^8-1750*a^4*b^3*c^3*d^7+210*a^3*b^4*c^4*d^6+42*a 
^2*b^5*c^5*d^5+14*a*b^6*c^6*d^4+6*b^7*c^7*d^3)/b^8*x^3-3*(1918*a^8*d^10-57 
54*a^7*b*c*d^9+5754*a^6*b^2*c^2*d^8-1918*a^5*b^3*c^3*d^7+210*a^4*b^4*c^4*d 
^6+42*a^3*b^5*c^5*d^5+14*a^2*b^6*c^6*d^4+6*a*b^7*c^7*d^3+3*b^8*c^8*d^2)/b^ 
9*x^2-1/3*(6174*a^9*d^10-18522*a^8*b*c*d^9+18522*a^7*b^2*c^2*d^8-6174*a^6* 
b^3*c^3*d^7+630*a^5*b^4*c^4*d^6+126*a^4*b^5*c^5*d^5+42*a^3*b^6*c^6*d^4+18* 
a^2*b^7*c^7*d^3+9*a*b^8*c^8*d^2+5*b^9*c^9*d)/b^10*x+15*d^8*(a^2*d^2-3*a*b* 
c*d+3*b^2*c^2)/b^3*x^8-5/3*d^9*(a*d-3*b*c)/b^2*x^9)/(b*x+a)^7-120/b^11*d^7 
*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*ln(b*x+a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1362 vs. \(2 (252) = 504\).

Time = 0.15 (sec) , antiderivative size = 1362, normalized size of antiderivative = 5.28 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^8,x, algorithm="fricas")
 

Output:

1/21*(7*b^10*d^10*x^10 - 3*b^10*c^10 - 5*a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 - 
 18*a^3*b^7*c^7*d^3 - 42*a^4*b^6*c^6*d^4 - 126*a^5*b^5*c^5*d^5 - 630*a^6*b 
^4*c^4*d^6 + 6534*a^7*b^3*c^3*d^7 - 12987*a^8*b^2*c^2*d^8 + 10047*a^9*b*c* 
d^9 - 2761*a^10*d^10 + 35*(3*b^10*c*d^9 - a*b^9*d^10)*x^9 + 315*(3*b^10*c^ 
2*d^8 - 3*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 49*(135*a*b^9*c^2*d^8 - 195*a^ 
2*b^8*c*d^9 + 77*a^3*b^7*d^10)*x^7 - 49*(90*b^10*c^4*d^6 - 360*a*b^9*c^3*d 
^7 + 135*a^2*b^8*c^2*d^8 + 285*a^3*b^7*c*d^9 - 179*a^4*b^6*d^10)*x^6 - 147 
*(18*b^10*c^5*d^5 + 90*a*b^9*c^4*d^6 - 540*a^2*b^8*c^3*d^7 + 675*a^3*b^7*c 
^2*d^8 - 255*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 - 245*(6*b^10*c^6*d^4 + 18* 
a*b^9*c^5*d^5 + 90*a^2*b^8*c^4*d^6 - 660*a^3*b^7*c^3*d^7 + 1035*a^4*b^6*c^ 
2*d^8 - 615*a^5*b^5*c*d^9 + 121*a^6*b^4*d^10)*x^4 - 35*(18*b^10*c^7*d^3 + 
42*a*b^9*c^6*d^4 + 126*a^2*b^8*c^5*d^5 + 630*a^3*b^7*c^4*d^6 - 5250*a^4*b^ 
6*c^3*d^7 + 9135*a^5*b^5*c^2*d^8 - 6195*a^6*b^4*c*d^9 + 1477*a^7*b^3*d^10) 
*x^3 - 21*(9*b^10*c^8*d^2 + 18*a*b^9*c^7*d^3 + 42*a^2*b^8*c^6*d^4 + 126*a^ 
3*b^7*c^5*d^5 + 630*a^4*b^6*c^4*d^6 - 5754*a^5*b^5*c^3*d^7 + 10647*a^6*b^4 
*c^2*d^8 - 7707*a^7*b^3*c*d^9 + 1981*a^8*b^2*d^10)*x^2 - 7*(5*b^10*c^9*d + 
 9*a*b^9*c^8*d^2 + 18*a^2*b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 + 126*a^4*b^6*c 
^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d^7 + 11907*a^7*b^3*c^2*d^ 
8 - 8967*a^8*b^2*c*d^9 + 2401*a^9*b*d^10)*x + 2520*(a^7*b^3*c^3*d^7 - 3*a^ 
8*b^2*c^2*d^8 + 3*a^9*b*c*d^9 - a^10*d^10 + (b^10*c^3*d^7 - 3*a*b^9*c^2...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**10/(b*x+a)**8,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 934 vs. \(2 (252) = 504\).

Time = 0.09 (sec) , antiderivative size = 934, normalized size of antiderivative = 3.62 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^8,x, algorithm="maxima")
 

Output:

-1/21*(3*b^10*c^10 + 5*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^ 
3 + 42*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 - 6534* 
a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 - 10047*a^9*b*c*d^9 + 2761*a^10*d^ 
10 + 4410*(b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 - 4*a^3*b^7* 
c*d^9 + a^4*b^6*d^10)*x^6 + 2646*(b^10*c^5*d^5 + 5*a*b^9*c^4*d^6 - 30*a^2* 
b^8*c^3*d^7 + 50*a^3*b^7*c^2*d^8 - 35*a^4*b^6*c*d^9 + 9*a^5*b^5*d^10)*x^5 
+ 1470*(b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 110*a^3*b^7* 
c^3*d^7 + 195*a^4*b^6*c^2*d^8 - 141*a^5*b^5*c*d^9 + 37*a^6*b^4*d^10)*x^4 + 
 210*(3*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 105*a^3*b^7* 
c^4*d^6 - 875*a^4*b^6*c^3*d^7 + 1617*a^5*b^5*c^2*d^8 - 1197*a^6*b^4*c*d^9 
+ 319*a^7*b^3*d^10)*x^3 + 63*(3*b^10*c^8*d^2 + 6*a*b^9*c^7*d^3 + 14*a^2*b^ 
8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 1918*a^5*b^5*c^3*d^ 
7 + 3654*a^6*b^4*c^2*d^8 - 2754*a^7*b^3*c*d^9 + 743*a^8*b^2*d^10)*x^2 + 7* 
(5*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 18*a^2*b^8*c^7*d^3 + 42*a^3*b^7*c^6*d^4 
+ 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^3*d^7 + 12042 
*a^7*b^3*c^2*d^8 - 9207*a^8*b^2*c*d^9 + 2509*a^9*b*d^10)*x)/(b^18*x^7 + 7* 
a*b^17*x^6 + 21*a^2*b^16*x^5 + 35*a^3*b^15*x^4 + 35*a^4*b^14*x^3 + 21*a^5* 
b^13*x^2 + 7*a^6*b^12*x + a^7*b^11) + 1/3*(b^2*d^10*x^3 + 3*(5*b^2*c*d^9 - 
 4*a*b*d^10)*x^2 + 3*(45*b^2*c^2*d^8 - 80*a*b*c*d^9 + 36*a^2*d^10)*x)/b^10 
 + 120*(b^3*c^3*d^7 - 3*a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10)*log(b...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 872 vs. \(2 (252) = 504\).

Time = 0.12 (sec) , antiderivative size = 872, normalized size of antiderivative = 3.38 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^8,x, algorithm="giac")
 

Output:

120*(b^3*c^3*d^7 - 3*a*b^2*c^2*d^8 + 3*a^2*b*c*d^9 - a^3*d^10)*log(abs(b*x 
 + a))/b^11 - 1/21*(3*b^10*c^10 + 5*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 18*a 
^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 126*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^ 
4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^8*b^2*c^2*d^8 - 10047*a^9*b*c*d^9 + 
 2761*a^10*d^10 + 4410*(b^10*c^4*d^6 - 4*a*b^9*c^3*d^7 + 6*a^2*b^8*c^2*d^8 
 - 4*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 2646*(b^10*c^5*d^5 + 5*a*b^9*c^4* 
d^6 - 30*a^2*b^8*c^3*d^7 + 50*a^3*b^7*c^2*d^8 - 35*a^4*b^6*c*d^9 + 9*a^5*b 
^5*d^10)*x^5 + 1470*(b^10*c^6*d^4 + 3*a*b^9*c^5*d^5 + 15*a^2*b^8*c^4*d^6 - 
 110*a^3*b^7*c^3*d^7 + 195*a^4*b^6*c^2*d^8 - 141*a^5*b^5*c*d^9 + 37*a^6*b^ 
4*d^10)*x^4 + 210*(3*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 21*a^2*b^8*c^5*d^5 + 
 105*a^3*b^7*c^4*d^6 - 875*a^4*b^6*c^3*d^7 + 1617*a^5*b^5*c^2*d^8 - 1197*a 
^6*b^4*c*d^9 + 319*a^7*b^3*d^10)*x^3 + 63*(3*b^10*c^8*d^2 + 6*a*b^9*c^7*d^ 
3 + 14*a^2*b^8*c^6*d^4 + 42*a^3*b^7*c^5*d^5 + 210*a^4*b^6*c^4*d^6 - 1918*a 
^5*b^5*c^3*d^7 + 3654*a^6*b^4*c^2*d^8 - 2754*a^7*b^3*c*d^9 + 743*a^8*b^2*d 
^10)*x^2 + 7*(5*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 18*a^2*b^8*c^7*d^3 + 42*a^3 
*b^7*c^6*d^4 + 126*a^4*b^6*c^5*d^5 + 630*a^5*b^5*c^4*d^6 - 6174*a^6*b^4*c^ 
3*d^7 + 12042*a^7*b^3*c^2*d^8 - 9207*a^8*b^2*c*d^9 + 2509*a^9*b*d^10)*x)/( 
(b*x + a)^7*b^11) + 1/3*(b^16*d^10*x^3 + 15*b^16*c*d^9*x^2 - 12*a*b^15*d^1 
0*x^2 + 135*b^16*c^2*d^8*x - 240*a*b^15*c*d^9*x + 108*a^2*b^14*d^10*x)/b^2 
4
 

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 950, normalized size of antiderivative = 3.68 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx=x\,\left (\frac {8\,a\,\left (\frac {8\,a\,d^{10}}{b^9}-\frac {10\,c\,d^9}{b^8}\right )}{b}-\frac {28\,a^2\,d^{10}}{b^{10}}+\frac {45\,c^2\,d^8}{b^8}\right )-\frac {x^4\,\left (2590\,a^6\,b^3\,d^{10}-9870\,a^5\,b^4\,c\,d^9+13650\,a^4\,b^5\,c^2\,d^8-7700\,a^3\,b^6\,c^3\,d^7+1050\,a^2\,b^7\,c^4\,d^6+210\,a\,b^8\,c^5\,d^5+70\,b^9\,c^6\,d^4\right )+x^6\,\left (210\,a^4\,b^5\,d^{10}-840\,a^3\,b^6\,c\,d^9+1260\,a^2\,b^7\,c^2\,d^8-840\,a\,b^8\,c^3\,d^7+210\,b^9\,c^4\,d^6\right )+\frac {2761\,a^{10}\,d^{10}-10047\,a^9\,b\,c\,d^9+12987\,a^8\,b^2\,c^2\,d^8-6534\,a^7\,b^3\,c^3\,d^7+630\,a^6\,b^4\,c^4\,d^6+126\,a^5\,b^5\,c^5\,d^5+42\,a^4\,b^6\,c^6\,d^4+18\,a^3\,b^7\,c^7\,d^3+9\,a^2\,b^8\,c^8\,d^2+5\,a\,b^9\,c^9\,d+3\,b^{10}\,c^{10}}{21\,b}+x\,\left (\frac {2509\,a^9\,d^{10}}{3}-3069\,a^8\,b\,c\,d^9+4014\,a^7\,b^2\,c^2\,d^8-2058\,a^6\,b^3\,c^3\,d^7+210\,a^5\,b^4\,c^4\,d^6+42\,a^4\,b^5\,c^5\,d^5+14\,a^3\,b^6\,c^6\,d^4+6\,a^2\,b^7\,c^7\,d^3+3\,a\,b^8\,c^8\,d^2+\frac {5\,b^9\,c^9\,d}{3}\right )+x^3\,\left (3190\,a^7\,b^2\,d^{10}-11970\,a^6\,b^3\,c\,d^9+16170\,a^5\,b^4\,c^2\,d^8-8750\,a^4\,b^5\,c^3\,d^7+1050\,a^3\,b^6\,c^4\,d^6+210\,a^2\,b^7\,c^5\,d^5+70\,a\,b^8\,c^6\,d^4+30\,b^9\,c^7\,d^3\right )+x^2\,\left (2229\,a^8\,b\,d^{10}-8262\,a^7\,b^2\,c\,d^9+10962\,a^6\,b^3\,c^2\,d^8-5754\,a^5\,b^4\,c^3\,d^7+630\,a^4\,b^5\,c^4\,d^6+126\,a^3\,b^6\,c^5\,d^5+42\,a^2\,b^7\,c^6\,d^4+18\,a\,b^8\,c^7\,d^3+9\,b^9\,c^8\,d^2\right )+x^5\,\left (1134\,a^5\,b^4\,d^{10}-4410\,a^4\,b^5\,c\,d^9+6300\,a^3\,b^6\,c^2\,d^8-3780\,a^2\,b^7\,c^3\,d^7+630\,a\,b^8\,c^4\,d^6+126\,b^9\,c^5\,d^5\right )}{a^7\,b^{10}+7\,a^6\,b^{11}\,x+21\,a^5\,b^{12}\,x^2+35\,a^4\,b^{13}\,x^3+35\,a^3\,b^{14}\,x^4+21\,a^2\,b^{15}\,x^5+7\,a\,b^{16}\,x^6+b^{17}\,x^7}-x^2\,\left (\frac {4\,a\,d^{10}}{b^9}-\frac {5\,c\,d^9}{b^8}\right )-\frac {\ln \left (a+b\,x\right )\,\left (120\,a^3\,d^{10}-360\,a^2\,b\,c\,d^9+360\,a\,b^2\,c^2\,d^8-120\,b^3\,c^3\,d^7\right )}{b^{11}}+\frac {d^{10}\,x^3}{3\,b^8} \] Input:

int((c + d*x)^10/(a + b*x)^8,x)
 

Output:

x*((8*a*((8*a*d^10)/b^9 - (10*c*d^9)/b^8))/b - (28*a^2*d^10)/b^10 + (45*c^ 
2*d^8)/b^8) - (x^4*(2590*a^6*b^3*d^10 + 70*b^9*c^6*d^4 + 210*a*b^8*c^5*d^5 
 - 9870*a^5*b^4*c*d^9 + 1050*a^2*b^7*c^4*d^6 - 7700*a^3*b^6*c^3*d^7 + 1365 
0*a^4*b^5*c^2*d^8) + x^6*(210*a^4*b^5*d^10 + 210*b^9*c^4*d^6 - 840*a*b^8*c 
^3*d^7 - 840*a^3*b^6*c*d^9 + 1260*a^2*b^7*c^2*d^8) + (2761*a^10*d^10 + 3*b 
^10*c^10 + 9*a^2*b^8*c^8*d^2 + 18*a^3*b^7*c^7*d^3 + 42*a^4*b^6*c^6*d^4 + 1 
26*a^5*b^5*c^5*d^5 + 630*a^6*b^4*c^4*d^6 - 6534*a^7*b^3*c^3*d^7 + 12987*a^ 
8*b^2*c^2*d^8 + 5*a*b^9*c^9*d - 10047*a^9*b*c*d^9)/(21*b) + x*((2509*a^9*d 
^10)/3 + (5*b^9*c^9*d)/3 + 3*a*b^8*c^8*d^2 + 6*a^2*b^7*c^7*d^3 + 14*a^3*b^ 
6*c^6*d^4 + 42*a^4*b^5*c^5*d^5 + 210*a^5*b^4*c^4*d^6 - 2058*a^6*b^3*c^3*d^ 
7 + 4014*a^7*b^2*c^2*d^8 - 3069*a^8*b*c*d^9) + x^3*(3190*a^7*b^2*d^10 + 30 
*b^9*c^7*d^3 + 70*a*b^8*c^6*d^4 - 11970*a^6*b^3*c*d^9 + 210*a^2*b^7*c^5*d^ 
5 + 1050*a^3*b^6*c^4*d^6 - 8750*a^4*b^5*c^3*d^7 + 16170*a^5*b^4*c^2*d^8) + 
 x^2*(2229*a^8*b*d^10 + 9*b^9*c^8*d^2 + 18*a*b^8*c^7*d^3 - 8262*a^7*b^2*c* 
d^9 + 42*a^2*b^7*c^6*d^4 + 126*a^3*b^6*c^5*d^5 + 630*a^4*b^5*c^4*d^6 - 575 
4*a^5*b^4*c^3*d^7 + 10962*a^6*b^3*c^2*d^8) + x^5*(1134*a^5*b^4*d^10 + 126* 
b^9*c^5*d^5 + 630*a*b^8*c^4*d^6 - 4410*a^4*b^5*c*d^9 - 3780*a^2*b^7*c^3*d^ 
7 + 6300*a^3*b^6*c^2*d^8))/(a^7*b^10 + b^17*x^7 + 7*a^6*b^11*x + 7*a*b^16* 
x^6 + 21*a^5*b^12*x^2 + 35*a^4*b^13*x^3 + 35*a^3*b^14*x^4 + 21*a^2*b^15*x^ 
5) - x^2*((4*a*d^10)/b^9 - (5*c*d^9)/b^8) - (log(a + b*x)*(120*a^3*d^10...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1568, normalized size of antiderivative = 6.08 \[ \int \frac {(c+d x)^{10}}{(a+b x)^8} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^8,x)
                                                                                    
                                                                                    
 

Output:

( - 2520*log(a + b*x)*a**11*d**10 + 7560*log(a + b*x)*a**10*b*c*d**9 - 176 
40*log(a + b*x)*a**10*b*d**10*x - 7560*log(a + b*x)*a**9*b**2*c**2*d**8 + 
52920*log(a + b*x)*a**9*b**2*c*d**9*x - 52920*log(a + b*x)*a**9*b**2*d**10 
*x**2 + 2520*log(a + b*x)*a**8*b**3*c**3*d**7 - 52920*log(a + b*x)*a**8*b* 
*3*c**2*d**8*x + 158760*log(a + b*x)*a**8*b**3*c*d**9*x**2 - 88200*log(a + 
 b*x)*a**8*b**3*d**10*x**3 + 17640*log(a + b*x)*a**7*b**4*c**3*d**7*x - 15 
8760*log(a + b*x)*a**7*b**4*c**2*d**8*x**2 + 264600*log(a + b*x)*a**7*b**4 
*c*d**9*x**3 - 88200*log(a + b*x)*a**7*b**4*d**10*x**4 + 52920*log(a + b*x 
)*a**6*b**5*c**3*d**7*x**2 - 264600*log(a + b*x)*a**6*b**5*c**2*d**8*x**3 
+ 264600*log(a + b*x)*a**6*b**5*c*d**9*x**4 - 52920*log(a + b*x)*a**6*b**5 
*d**10*x**5 + 88200*log(a + b*x)*a**5*b**6*c**3*d**7*x**3 - 264600*log(a + 
 b*x)*a**5*b**6*c**2*d**8*x**4 + 158760*log(a + b*x)*a**5*b**6*c*d**9*x**5 
 - 17640*log(a + b*x)*a**5*b**6*d**10*x**6 + 88200*log(a + b*x)*a**4*b**7* 
c**3*d**7*x**4 - 158760*log(a + b*x)*a**4*b**7*c**2*d**8*x**5 + 52920*log( 
a + b*x)*a**4*b**7*c*d**9*x**6 - 2520*log(a + b*x)*a**4*b**7*d**10*x**7 + 
52920*log(a + b*x)*a**3*b**8*c**3*d**7*x**5 - 52920*log(a + b*x)*a**3*b**8 
*c**2*d**8*x**6 + 7560*log(a + b*x)*a**3*b**8*c*d**9*x**7 + 17640*log(a + 
b*x)*a**2*b**9*c**3*d**7*x**6 - 7560*log(a + b*x)*a**2*b**9*c**2*d**8*x**7 
 + 2520*log(a + b*x)*a*b**10*c**3*d**7*x**7 - 4014*a**11*d**10 + 12042*a** 
10*b*c*d**9 - 25578*a**10*b*d**10*x - 12042*a**9*b**2*c**2*d**8 + 76734...