\(\int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx\) [111]

Optimal result

Integrand size = 15, antiderivative size = 260 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\frac {210 d^6 (b c-a d)^4 x}{b^{10}}-\frac {(b c-a d)^{10}}{5 b^{11} (a+b x)^5}-\frac {5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac {15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}-\frac {60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac {210 d^4 (b c-a d)^6}{b^{11} (a+b x)}+\frac {60 d^7 (b c-a d)^3 (a+b x)^2}{b^{11}}+\frac {15 d^8 (b c-a d)^2 (a+b x)^3}{b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^4}{2 b^{11}}+\frac {d^{10} (a+b x)^5}{5 b^{11}}+\frac {252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}} \] Output:

210*d^6*(-a*d+b*c)^4*x/b^10-1/5*(-a*d+b*c)^10/b^11/(b*x+a)^5-5/2*d*(-a*d+b 
*c)^9/b^11/(b*x+a)^4-15*d^2*(-a*d+b*c)^8/b^11/(b*x+a)^3-60*d^3*(-a*d+b*c)^ 
7/b^11/(b*x+a)^2-210*d^4*(-a*d+b*c)^6/b^11/(b*x+a)+60*d^7*(-a*d+b*c)^3*(b* 
x+a)^2/b^11+15*d^8*(-a*d+b*c)^2*(b*x+a)^3/b^11+5/2*d^9*(-a*d+b*c)*(b*x+a)^ 
4/b^11+1/5*d^10*(b*x+a)^5/b^11+252*d^5*(-a*d+b*c)^5*ln(b*x+a)/b^11
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.17 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\frac {10 b d^6 \left (210 b^4 c^4-720 a b^3 c^3 d+945 a^2 b^2 c^2 d^2-560 a^3 b c d^3+126 a^4 d^4\right ) x+10 b^2 d^7 \left (60 b^3 c^3-135 a b^2 c^2 d+105 a^2 b c d^2-28 a^3 d^3\right ) x^2+10 b^3 d^8 \left (15 b^2 c^2-20 a b c d+7 a^2 d^2\right ) x^3+5 b^4 d^9 (5 b c-3 a d) x^4+2 b^5 d^{10} x^5-\frac {2 (b c-a d)^{10}}{(a+b x)^5}+\frac {25 d (-b c+a d)^9}{(a+b x)^4}-\frac {150 d^2 (b c-a d)^8}{(a+b x)^3}+\frac {600 d^3 (-b c+a d)^7}{(a+b x)^2}-\frac {2100 d^4 (b c-a d)^6}{a+b x}+2520 d^5 (b c-a d)^5 \log (a+b x)}{10 b^{11}} \] Input:

Integrate[(c + d*x)^10/(a + b*x)^6,x]
 

Output:

(10*b*d^6*(210*b^4*c^4 - 720*a*b^3*c^3*d + 945*a^2*b^2*c^2*d^2 - 560*a^3*b 
*c*d^3 + 126*a^4*d^4)*x + 10*b^2*d^7*(60*b^3*c^3 - 135*a*b^2*c^2*d + 105*a 
^2*b*c*d^2 - 28*a^3*d^3)*x^2 + 10*b^3*d^8*(15*b^2*c^2 - 20*a*b*c*d + 7*a^2 
*d^2)*x^3 + 5*b^4*d^9*(5*b*c - 3*a*d)*x^4 + 2*b^5*d^10*x^5 - (2*(b*c - a*d 
)^10)/(a + b*x)^5 + (25*d*(-(b*c) + a*d)^9)/(a + b*x)^4 - (150*d^2*(b*c - 
a*d)^8)/(a + b*x)^3 + (600*d^3*(-(b*c) + a*d)^7)/(a + b*x)^2 - (2100*d^4*( 
b*c - a*d)^6)/(a + b*x) + 2520*d^5*(b*c - a*d)^5*Log[a + b*x])/(10*b^11)
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 260, 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.

\(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx\)

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {5 d^9 (a+b x)^4 (b c-a d)}{2 b^{11}}+\frac {15 d^8 (a+b x)^3 (b c-a d)^2}{b^{11}}+\frac {60 d^7 (a+b x)^2 (b c-a d)^3}{b^{11}}+\frac {252 d^5 (b c-a d)^5 \log (a+b x)}{b^{11}}-\frac {210 d^4 (b c-a d)^6}{b^{11} (a+b x)}-\frac {60 d^3 (b c-a d)^7}{b^{11} (a+b x)^2}-\frac {15 d^2 (b c-a d)^8}{b^{11} (a+b x)^3}-\frac {5 d (b c-a d)^9}{2 b^{11} (a+b x)^4}-\frac {(b c-a d)^{10}}{5 b^{11} (a+b x)^5}+\frac {d^{10} (a+b x)^5}{5 b^{11}}+\frac {210 d^6 x (b c-a d)^4}{b^{10}}\)

Input:

Int[(c + d*x)^10/(a + b*x)^6,x]
 

Output:

(210*d^6*(b*c - a*d)^4*x)/b^10 - (b*c - a*d)^10/(5*b^11*(a + b*x)^5) - (5* 
d*(b*c - a*d)^9)/(2*b^11*(a + b*x)^4) - (15*d^2*(b*c - a*d)^8)/(b^11*(a + 
b*x)^3) - (60*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^2) - (210*d^4*(b*c - a*d) 
^6)/(b^11*(a + b*x)) + (60*d^7*(b*c - a*d)^3*(a + b*x)^2)/b^11 + (15*d^8*( 
b*c - a*d)^2*(a + b*x)^3)/b^11 + (5*d^9*(b*c - a*d)*(a + b*x)^4)/(2*b^11) 
+ (d^10*(a + b*x)^5)/(5*b^11) + (252*d^5*(b*c - a*d)^5*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. \(846\) vs. \(2(252)=504\).

Time = 0.14 (sec) , antiderivative size = 847, normalized size of antiderivative = 3.26

method	result	size
norman	\(\frac {-\frac {5754 a^{10} d^{10}-28770 a^{9} b c \,d^{9}+57540 a^{8} b^{2} c^{2} d^{8}-57540 a^{7} b^{3} c^{3} d^{7}+28770 a^{6} b^{4} c^{4} d^{6}-5754 a^{5} b^{5} c^{5} d^{5}+420 a^{4} b^{6} c^{6} d^{4}+60 a^{3} b^{7} c^{7} d^{3}+15 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +2 b^{10} c^{10}}{10 b^{11}}+\frac {d^{10} x^{10}}{5 b}-\frac {5 \left (252 a^{6} d^{10}-1260 a^{5} b c \,d^{9}+2520 a^{4} b^{2} c^{2} d^{8}-2520 a^{3} b^{3} c^{3} d^{7}+1260 a^{2} b^{4} c^{4} d^{6}-252 a \,b^{5} c^{5} d^{5}+42 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}-\frac {10 \left (378 a^{7} d^{10}-1890 a^{6} b c \,d^{9}+3780 a^{5} b^{2} c^{2} d^{8}-3780 a^{4} b^{3} c^{3} d^{7}+1890 a^{3} b^{4} c^{4} d^{6}-378 a^{2} b^{5} c^{5} d^{5}+42 a \,b^{6} c^{6} d^{4}+6 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}-\frac {5 \left (924 a^{8} d^{10}-4620 a^{7} b c \,d^{9}+9240 a^{6} b^{2} c^{2} d^{8}-9240 a^{5} b^{3} c^{3} d^{7}+4620 a^{4} b^{4} c^{4} d^{6}-924 a^{3} b^{5} c^{5} d^{5}+84 a^{2} b^{6} c^{6} d^{4}+12 a \,b^{7} c^{7} d^{3}+3 b^{8} c^{8} d^{2}\right ) x^{2}}{b^{9}}-\frac {5 \left (1050 a^{9} d^{10}-5250 a^{8} b c \,d^{9}+10500 a^{7} b^{2} c^{2} d^{8}-10500 a^{6} b^{3} c^{3} d^{7}+5250 a^{5} b^{4} c^{4} d^{6}-1050 a^{4} b^{5} c^{5} d^{5}+84 a^{3} b^{6} c^{6} d^{4}+12 a^{2} b^{7} c^{7} d^{3}+3 a \,b^{8} c^{8} d^{2}+b^{9} c^{9} d \right ) x}{2 b^{10}}+\frac {42 d^{6} \left (d^{4} a^{4}-5 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}-10 a \,b^{3} c^{3} d +5 c^{4} b^{4}\right ) x^{6}}{b^{5}}-\frac {6 d^{7} \left (a^{3} d^{3}-5 a^{2} b c \,d^{2}+10 a \,b^{2} c^{2} d -10 b^{3} c^{3}\right ) x^{7}}{b^{4}}+\frac {3 d^{8} \left (a^{2} d^{2}-5 a b c d +10 b^{2} c^{2}\right ) x^{8}}{2 b^{3}}-\frac {d^{9} \left (a d -5 b c \right ) x^{9}}{2 b^{2}}}{\left (b x +a \right )^{5}}-\frac {252 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right ) \ln \left (b x +a \right )}{b^{11}}\)	\(847\)
default	\(\frac {d^{6} \left (\frac {1}{5} d^{4} x^{5} b^{4}-\frac {3}{2} a \,b^{3} d^{4} x^{4}+\frac {5}{2} b^{4} c \,d^{3} x^{4}+7 a^{2} b^{2} d^{4} x^{3}-20 a \,b^{3} c \,d^{3} x^{3}+15 b^{4} c^{2} d^{2} x^{3}-28 a^{3} b \,d^{4} x^{2}+105 a^{2} b^{2} c \,d^{3} x^{2}-135 a \,b^{3} c^{2} d^{2} x^{2}+60 b^{4} c^{3} d \,x^{2}+126 a^{4} d^{4} x -560 a^{3} b c \,d^{3} x +945 a^{2} b^{2} c^{2} d^{2} x -720 a \,b^{3} c^{3} d x +210 c^{4} b^{4} x \right )}{b^{10}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{5 b^{11} \left (b x +a \right )^{5}}+\frac {5 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -c^{9} b^{9}\right )}{2 b^{11} \left (b x +a \right )^{4}}+\frac {60 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{2}}-\frac {210 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{b^{11} \left (b x +a \right )}-\frac {252 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right ) \ln \left (b x +a \right )}{b^{11}}-\frac {15 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +c^{8} b^{8}\right )}{b^{11} \left (b x +a \right )^{3}}\)	\(870\)
risch	\(\frac {d^{10} x^{5}}{5 b^{6}}-\frac {3 d^{10} a \,x^{4}}{2 b^{7}}+\frac {5 d^{9} c \,x^{4}}{2 b^{6}}+\frac {7 d^{10} a^{2} x^{3}}{b^{8}}-\frac {20 d^{9} a c \,x^{3}}{b^{7}}+\frac {15 d^{8} c^{2} x^{3}}{b^{6}}-\frac {28 d^{10} a^{3} x^{2}}{b^{9}}+\frac {105 d^{9} a^{2} c \,x^{2}}{b^{8}}-\frac {135 d^{8} a \,c^{2} x^{2}}{b^{7}}+\frac {60 d^{7} c^{3} x^{2}}{b^{6}}+\frac {126 d^{10} a^{4} x}{b^{10}}-\frac {560 d^{9} a^{3} c x}{b^{9}}+\frac {945 d^{8} a^{2} c^{2} x}{b^{8}}-\frac {720 d^{7} a \,c^{3} x}{b^{7}}+\frac {210 d^{6} c^{4} x}{b^{6}}+\frac {\left (-210 a^{6} b^{3} d^{10}+1260 a^{5} c \,d^{9} b^{4}-3150 a^{4} c^{2} d^{8} b^{5}+4200 a^{3} c^{3} d^{7} b^{6}-3150 a^{2} b^{7} c^{4} d^{6}+1260 a \,b^{8} c^{5} d^{5}-210 c^{6} d^{4} b^{9}\right ) x^{4}-60 b^{2} d^{3} \left (13 a^{7} d^{7}-77 a^{6} b c \,d^{6}+189 a^{5} b^{2} c^{2} d^{5}-245 a^{4} b^{3} c^{3} d^{4}+175 a^{3} b^{4} c^{4} d^{3}-63 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +b^{7} c^{7}\right ) x^{3}-15 b \,d^{2} \left (73 a^{8} d^{8}-428 a^{7} b c \,d^{7}+1036 a^{6} b^{2} c^{2} d^{6}-1316 a^{5} b^{3} c^{3} d^{5}+910 a^{4} b^{4} c^{4} d^{4}-308 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}+4 a \,b^{7} c^{7} d +c^{8} b^{8}\right ) x^{2}-\frac {5 d \left (275 a^{9} d^{9}-1599 a^{8} b c \,d^{8}+3828 a^{7} b^{2} c^{2} d^{7}-4788 a^{6} b^{3} c^{3} d^{6}+3234 a^{5} b^{4} c^{4} d^{5}-1050 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}+12 a^{2} b^{7} c^{7} d^{2}+3 a \,b^{8} c^{8} d +c^{9} b^{9}\right ) x}{2}-\frac {1627 a^{10} d^{10}-9395 a^{9} b c \,d^{9}+22290 a^{8} b^{2} c^{2} d^{8}-27540 a^{7} b^{3} c^{3} d^{7}+18270 a^{6} b^{4} c^{4} d^{6}-5754 a^{5} b^{5} c^{5} d^{5}+420 a^{4} b^{6} c^{6} d^{4}+60 a^{3} b^{7} c^{7} d^{3}+15 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +2 b^{10} c^{10}}{10 b}}{b^{10} \left (b x +a \right )^{5}}-\frac {252 d^{10} \ln \left (b x +a \right ) a^{5}}{b^{11}}+\frac {1260 d^{9} \ln \left (b x +a \right ) a^{4} c}{b^{10}}-\frac {2520 d^{8} \ln \left (b x +a \right ) a^{3} c^{2}}{b^{9}}+\frac {2520 d^{7} \ln \left (b x +a \right ) a^{2} c^{3}}{b^{8}}-\frac {1260 d^{6} \ln \left (b x +a \right ) a \,c^{4}}{b^{7}}+\frac {252 d^{5} \ln \left (b x +a \right ) c^{5}}{b^{6}}\)	\(898\)
parallelrisch	\(\text {Expression too large to display}\)	\(1625\)

Input:

int((d*x+c)^10/(b*x+a)^6,x,method=_RETURNVERBOSE)
 

Output:

(-1/10*(5754*a^10*d^10-28770*a^9*b*c*d^9+57540*a^8*b^2*c^2*d^8-57540*a^7*b 
^3*c^3*d^7+28770*a^6*b^4*c^4*d^6-5754*a^5*b^5*c^5*d^5+420*a^4*b^6*c^6*d^4+ 
60*a^3*b^7*c^7*d^3+15*a^2*b^8*c^8*d^2+5*a*b^9*c^9*d+2*b^10*c^10)/b^11+1/5/ 
b*d^10*x^10-5*(252*a^6*d^10-1260*a^5*b*c*d^9+2520*a^4*b^2*c^2*d^8-2520*a^3 
*b^3*c^3*d^7+1260*a^2*b^4*c^4*d^6-252*a*b^5*c^5*d^5+42*b^6*c^6*d^4)/b^7*x^ 
4-10*(378*a^7*d^10-1890*a^6*b*c*d^9+3780*a^5*b^2*c^2*d^8-3780*a^4*b^3*c^3* 
d^7+1890*a^3*b^4*c^4*d^6-378*a^2*b^5*c^5*d^5+42*a*b^6*c^6*d^4+6*b^7*c^7*d^ 
3)/b^8*x^3-5*(924*a^8*d^10-4620*a^7*b*c*d^9+9240*a^6*b^2*c^2*d^8-9240*a^5* 
b^3*c^3*d^7+4620*a^4*b^4*c^4*d^6-924*a^3*b^5*c^5*d^5+84*a^2*b^6*c^6*d^4+12 
*a*b^7*c^7*d^3+3*b^8*c^8*d^2)/b^9*x^2-5/2*(1050*a^9*d^10-5250*a^8*b*c*d^9+ 
10500*a^7*b^2*c^2*d^8-10500*a^6*b^3*c^3*d^7+5250*a^5*b^4*c^4*d^6-1050*a^4* 
b^5*c^5*d^5+84*a^3*b^6*c^6*d^4+12*a^2*b^7*c^7*d^3+3*a*b^8*c^8*d^2+b^9*c^9* 
d)/b^10*x+42*d^6*(a^4*d^4-5*a^3*b*c*d^3+10*a^2*b^2*c^2*d^2-10*a*b^3*c^3*d+ 
5*b^4*c^4)/b^5*x^6-6*d^7*(a^3*d^3-5*a^2*b*c*d^2+10*a*b^2*c^2*d-10*b^3*c^3) 
/b^4*x^7+3/2*d^8*(a^2*d^2-5*a*b*c*d+10*b^2*c^2)/b^3*x^8-1/2*d^9*(a*d-5*b*c 
)/b^2*x^9)/(b*x+a)^5-252/b^11*d^5*(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)*ln(b*x+a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1395 vs. \(2 (252) = 504\).

Time = 0.08 (sec) , antiderivative size = 1395, normalized size of antiderivative = 5.37 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^6,x, algorithm="fricas")
 

Output:

1/10*(2*b^10*d^10*x^10 - 2*b^10*c^10 - 5*a*b^9*c^9*d - 15*a^2*b^8*c^8*d^2 
- 60*a^3*b^7*c^7*d^3 - 420*a^4*b^6*c^6*d^4 + 5754*a^5*b^5*c^5*d^5 - 18270* 
a^6*b^4*c^4*d^6 + 27540*a^7*b^3*c^3*d^7 - 22290*a^8*b^2*c^2*d^8 + 9395*a^9 
*b*c*d^9 - 1627*a^10*d^10 + 5*(5*b^10*c*d^9 - a*b^9*d^10)*x^9 + 15*(10*b^1 
0*c^2*d^8 - 5*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 60*(10*b^10*c^3*d^7 - 10*a 
*b^9*c^2*d^8 + 5*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 420*(5*b^10*c^4*d^6 - 
 10*a*b^9*c^3*d^7 + 10*a^2*b^8*c^2*d^8 - 5*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x 
^6 + (10500*a*b^9*c^4*d^6 - 30000*a^2*b^8*c^3*d^7 + 35250*a^3*b^7*c^2*d^8 
- 19375*a^4*b^6*c*d^9 + 4127*a^5*b^5*d^10)*x^5 - 5*(420*b^10*c^6*d^4 - 252 
0*a*b^9*c^5*d^5 + 2100*a^2*b^8*c^4*d^6 + 4800*a^3*b^7*c^3*d^7 - 10050*a^4* 
b^6*c^2*d^8 + 6775*a^5*b^5*c*d^9 - 1607*a^6*b^4*d^10)*x^4 - 10*(60*b^10*c^ 
7*d^3 + 420*a*b^9*c^6*d^4 - 3780*a^2*b^8*c^5*d^5 + 8400*a^3*b^7*c^4*d^6 - 
7800*a^4*b^6*c^3*d^7 + 2550*a^5*b^5*c^2*d^8 + 475*a^6*b^4*c*d^9 - 347*a^7* 
b^3*d^10)*x^3 - 10*(15*b^10*c^8*d^2 + 60*a*b^9*c^7*d^3 + 420*a^2*b^8*c^6*d 
^4 - 4620*a^3*b^7*c^5*d^5 + 12600*a^4*b^6*c^4*d^6 - 16200*a^5*b^5*c^3*d^7 
+ 10950*a^6*b^4*c^2*d^8 - 3725*a^7*b^3*c*d^9 + 493*a^8*b^2*d^10)*x^2 - 5*( 
5*b^10*c^9*d + 15*a*b^9*c^8*d^2 + 60*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 
 - 5250*a^4*b^6*c^5*d^5 + 15750*a^5*b^5*c^4*d^6 - 22500*a^6*b^4*c^3*d^7 + 
17250*a^7*b^3*c^2*d^8 - 6875*a^8*b^2*c*d^9 + 1123*a^9*b*d^10)*x + 2520*(a^ 
5*b^5*c^5*d^5 - 5*a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 - 10*a^8*b^2*c^2...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**10/(b*x+a)**6,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (252) = 504\).

Time = 0.08 (sec) , antiderivative size = 912, normalized size of antiderivative = 3.51 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^6,x, algorithm="maxima")
 

Output:

-1/10*(2*b^10*c^10 + 5*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d 
^3 + 420*a^4*b^6*c^6*d^4 - 5754*a^5*b^5*c^5*d^5 + 18270*a^6*b^4*c^4*d^6 - 
27540*a^7*b^3*c^3*d^7 + 22290*a^8*b^2*c^2*d^8 - 9395*a^9*b*c*d^9 + 1627*a^ 
10*d^10 + 2100*(b^10*c^6*d^4 - 6*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 + 15*a^4*b^6*c^2*d^8 - 6*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 
+ 600*(b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 - 63*a^2*b^8*c^5*d^5 + 175*a^3*b^7*c 
^4*d^6 - 245*a^4*b^6*c^3*d^7 + 189*a^5*b^5*c^2*d^8 - 77*a^6*b^4*c*d^9 + 13 
*a^7*b^3*d^10)*x^3 + 150*(b^10*c^8*d^2 + 4*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6* 
d^4 - 308*a^3*b^7*c^5*d^5 + 910*a^4*b^6*c^4*d^6 - 1316*a^5*b^5*c^3*d^7 + 1 
036*a^6*b^4*c^2*d^8 - 428*a^7*b^3*c*d^9 + 73*a^8*b^2*d^10)*x^2 + 25*(b^10* 
c^9*d + 3*a*b^9*c^8*d^2 + 12*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 - 1050*a 
^4*b^6*c^5*d^5 + 3234*a^5*b^5*c^4*d^6 - 4788*a^6*b^4*c^3*d^7 + 3828*a^7*b^ 
3*c^2*d^8 - 1599*a^8*b^2*c*d^9 + 275*a^9*b*d^10)*x)/(b^16*x^5 + 5*a*b^15*x 
^4 + 10*a^2*b^14*x^3 + 10*a^3*b^13*x^2 + 5*a^4*b^12*x + a^5*b^11) + 1/10*( 
2*b^4*d^10*x^5 + 5*(5*b^4*c*d^9 - 3*a*b^3*d^10)*x^4 + 10*(15*b^4*c^2*d^8 - 
 20*a*b^3*c*d^9 + 7*a^2*b^2*d^10)*x^3 + 10*(60*b^4*c^3*d^7 - 135*a*b^3*c^2 
*d^8 + 105*a^2*b^2*c*d^9 - 28*a^3*b*d^10)*x^2 + 10*(210*b^4*c^4*d^6 - 720* 
a*b^3*c^3*d^7 + 945*a^2*b^2*c^2*d^8 - 560*a^3*b*c*d^9 + 126*a^4*d^10)*x)/b 
^10 + 252*(b^5*c^5*d^5 - 5*a*b^4*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10*a^3*b^2 
*c^2*d^8 + 5*a^4*b*c*d^9 - a^5*d^10)*log(b*x + a)/b^11
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 883 vs. \(2 (252) = 504\).

Time = 0.15 (sec) , antiderivative size = 883, normalized size of antiderivative = 3.40 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^10/(b*x+a)^6,x, algorithm="giac")
 

Output:

252*(b^5*c^5*d^5 - 5*a*b^4*c^4*d^6 + 10*a^2*b^3*c^3*d^7 - 10*a^3*b^2*c^2*d 
^8 + 5*a^4*b*c*d^9 - a^5*d^10)*log(abs(b*x + a))/b^11 - 1/10*(2*b^10*c^10 
+ 5*a*b^9*c^9*d + 15*a^2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 420*a^4*b^6*c^ 
6*d^4 - 5754*a^5*b^5*c^5*d^5 + 18270*a^6*b^4*c^4*d^6 - 27540*a^7*b^3*c^3*d 
^7 + 22290*a^8*b^2*c^2*d^8 - 9395*a^9*b*c*d^9 + 1627*a^10*d^10 + 2100*(b^1 
0*c^6*d^4 - 6*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 + 15 
*a^4*b^6*c^2*d^8 - 6*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 600*(b^10*c^7*d^3 
 + 7*a*b^9*c^6*d^4 - 63*a^2*b^8*c^5*d^5 + 175*a^3*b^7*c^4*d^6 - 245*a^4*b^ 
6*c^3*d^7 + 189*a^5*b^5*c^2*d^8 - 77*a^6*b^4*c*d^9 + 13*a^7*b^3*d^10)*x^3 
+ 150*(b^10*c^8*d^2 + 4*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 - 308*a^3*b^7*c 
^5*d^5 + 910*a^4*b^6*c^4*d^6 - 1316*a^5*b^5*c^3*d^7 + 1036*a^6*b^4*c^2*d^8 
 - 428*a^7*b^3*c*d^9 + 73*a^8*b^2*d^10)*x^2 + 25*(b^10*c^9*d + 3*a*b^9*c^8 
*d^2 + 12*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 - 1050*a^4*b^6*c^5*d^5 + 32 
34*a^5*b^5*c^4*d^6 - 4788*a^6*b^4*c^3*d^7 + 3828*a^7*b^3*c^2*d^8 - 1599*a^ 
8*b^2*c*d^9 + 275*a^9*b*d^10)*x)/((b*x + a)^5*b^11) + 1/10*(2*b^24*d^10*x^ 
5 + 25*b^24*c*d^9*x^4 - 15*a*b^23*d^10*x^4 + 150*b^24*c^2*d^8*x^3 - 200*a* 
b^23*c*d^9*x^3 + 70*a^2*b^22*d^10*x^3 + 600*b^24*c^3*d^7*x^2 - 1350*a*b^23 
*c^2*d^8*x^2 + 1050*a^2*b^22*c*d^9*x^2 - 280*a^3*b^21*d^10*x^2 + 2100*b^24 
*c^4*d^6*x - 7200*a*b^23*c^3*d^7*x + 9450*a^2*b^22*c^2*d^8*x - 5600*a^3*b^ 
21*c*d^9*x + 1260*a^4*b^20*d^10*x)/b^30
 

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 1141, normalized size of antiderivative = 4.39 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx =\text {Too large to display} \] Input:

int((c + d*x)^10/(a + b*x)^6,x)
 

Output:

x^3*((2*a*((6*a*d^10)/b^7 - (10*c*d^9)/b^6))/b - (5*a^2*d^10)/b^8 + (15*c^ 
2*d^8)/b^6) - x^2*((3*a*((6*a*((6*a*d^10)/b^7 - (10*c*d^9)/b^6))/b - (15*a 
^2*d^10)/b^8 + (45*c^2*d^8)/b^6))/b + (10*a^3*d^10)/b^9 - (60*c^3*d^7)/b^6 
 - (15*a^2*((6*a*d^10)/b^7 - (10*c*d^9)/b^6))/(2*b^2)) - x^4*((3*a*d^10)/( 
2*b^7) - (5*c*d^9)/(2*b^6)) - (x^4*(210*a^6*b^3*d^10 + 210*b^9*c^6*d^4 - 1 
260*a*b^8*c^5*d^5 - 1260*a^5*b^4*c*d^9 + 3150*a^2*b^7*c^4*d^6 - 4200*a^3*b 
^6*c^3*d^7 + 3150*a^4*b^5*c^2*d^8) + (1627*a^10*d^10 + 2*b^10*c^10 + 15*a^ 
2*b^8*c^8*d^2 + 60*a^3*b^7*c^7*d^3 + 420*a^4*b^6*c^6*d^4 - 5754*a^5*b^5*c^ 
5*d^5 + 18270*a^6*b^4*c^4*d^6 - 27540*a^7*b^3*c^3*d^7 + 22290*a^8*b^2*c^2* 
d^8 + 5*a*b^9*c^9*d - 9395*a^9*b*c*d^9)/(10*b) + x*((1375*a^9*d^10)/2 + (5 
*b^9*c^9*d)/2 + (15*a*b^8*c^8*d^2)/2 + 30*a^2*b^7*c^7*d^3 + 210*a^3*b^6*c^ 
6*d^4 - 2625*a^4*b^5*c^5*d^5 + 8085*a^5*b^4*c^4*d^6 - 11970*a^6*b^3*c^3*d^ 
7 + 9570*a^7*b^2*c^2*d^8 - (7995*a^8*b*c*d^9)/2) + x^3*(780*a^7*b^2*d^10 + 
 60*b^9*c^7*d^3 + 420*a*b^8*c^6*d^4 - 4620*a^6*b^3*c*d^9 - 3780*a^2*b^7*c^ 
5*d^5 + 10500*a^3*b^6*c^4*d^6 - 14700*a^4*b^5*c^3*d^7 + 11340*a^5*b^4*c^2* 
d^8) + x^2*(1095*a^8*b*d^10 + 15*b^9*c^8*d^2 + 60*a*b^8*c^7*d^3 - 6420*a^7 
*b^2*c*d^9 + 420*a^2*b^7*c^6*d^4 - 4620*a^3*b^6*c^5*d^5 + 13650*a^4*b^5*c^ 
4*d^6 - 19740*a^5*b^4*c^3*d^7 + 15540*a^6*b^3*c^2*d^8))/(a^5*b^10 + b^15*x 
^5 + 5*a^4*b^11*x + 5*a*b^14*x^4 + 10*a^3*b^12*x^2 + 10*a^2*b^13*x^3) + x* 
((6*a*((6*a*((6*a*((6*a*d^10)/b^7 - (10*c*d^9)/b^6))/b - (15*a^2*d^10)/...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1638, normalized size of antiderivative = 6.30 \[ \int \frac {(c+d x)^{10}}{(a+b x)^6} \, dx =\text {Too large to display} \] Input:

int((d*x+c)^10/(b*x+a)^6,x)
                                                                                    
                                                                                    
 

Output:

( - 2520*log(a + b*x)*a**11*d**10 + 12600*log(a + b*x)*a**10*b*c*d**9 - 12 
600*log(a + b*x)*a**10*b*d**10*x - 25200*log(a + b*x)*a**9*b**2*c**2*d**8 
+ 63000*log(a + b*x)*a**9*b**2*c*d**9*x - 25200*log(a + b*x)*a**9*b**2*d** 
10*x**2 + 25200*log(a + b*x)*a**8*b**3*c**3*d**7 - 126000*log(a + b*x)*a** 
8*b**3*c**2*d**8*x + 126000*log(a + b*x)*a**8*b**3*c*d**9*x**2 - 25200*log 
(a + b*x)*a**8*b**3*d**10*x**3 - 12600*log(a + b*x)*a**7*b**4*c**4*d**6 + 
126000*log(a + b*x)*a**7*b**4*c**3*d**7*x - 252000*log(a + b*x)*a**7*b**4* 
c**2*d**8*x**2 + 126000*log(a + b*x)*a**7*b**4*c*d**9*x**3 - 12600*log(a + 
 b*x)*a**7*b**4*d**10*x**4 + 2520*log(a + b*x)*a**6*b**5*c**5*d**5 - 63000 
*log(a + b*x)*a**6*b**5*c**4*d**6*x + 252000*log(a + b*x)*a**6*b**5*c**3*d 
**7*x**2 - 252000*log(a + b*x)*a**6*b**5*c**2*d**8*x**3 + 63000*log(a + b* 
x)*a**6*b**5*c*d**9*x**4 - 2520*log(a + b*x)*a**6*b**5*d**10*x**5 + 12600* 
log(a + b*x)*a**5*b**6*c**5*d**5*x - 126000*log(a + b*x)*a**5*b**6*c**4*d* 
*6*x**2 + 252000*log(a + b*x)*a**5*b**6*c**3*d**7*x**3 - 126000*log(a + b* 
x)*a**5*b**6*c**2*d**8*x**4 + 12600*log(a + b*x)*a**5*b**6*c*d**9*x**5 + 2 
5200*log(a + b*x)*a**4*b**7*c**5*d**5*x**2 - 126000*log(a + b*x)*a**4*b**7 
*c**4*d**6*x**3 + 126000*log(a + b*x)*a**4*b**7*c**3*d**7*x**4 - 25200*log 
(a + b*x)*a**4*b**7*c**2*d**8*x**5 + 25200*log(a + b*x)*a**3*b**8*c**5*d** 
5*x**3 - 63000*log(a + b*x)*a**3*b**8*c**4*d**6*x**4 + 25200*log(a + b*x)* 
a**3*b**8*c**3*d**7*x**5 + 12600*log(a + b*x)*a**2*b**9*c**5*d**5*x**4 ...