Integrand size = 15, antiderivative size = 28 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {(c+d x)^{11}}{11 (b c-a d) (a+b x)^{11}} \] Output:
-1/11*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^11
Leaf count is larger than twice the leaf count of optimal. \(665\) vs. \(2(28)=56\).
Time = 0.16 (sec) , antiderivative size = 665, normalized size of antiderivative = 23.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {a^{10} d^{10}+a^9 b d^9 (c+11 d x)+a^8 b^2 d^8 \left (c^2+11 c d x+55 d^2 x^2\right )+a^7 b^3 d^7 \left (c^3+11 c^2 d x+55 c d^2 x^2+165 d^3 x^3\right )+a^6 b^4 d^6 \left (c^4+11 c^3 d x+55 c^2 d^2 x^2+165 c d^3 x^3+330 d^4 x^4\right )+a^5 b^5 d^5 \left (c^5+11 c^4 d x+55 c^3 d^2 x^2+165 c^2 d^3 x^3+330 c d^4 x^4+462 d^5 x^5\right )+a^4 b^6 d^4 \left (c^6+11 c^5 d x+55 c^4 d^2 x^2+165 c^3 d^3 x^3+330 c^2 d^4 x^4+462 c d^5 x^5+462 d^6 x^6\right )+a^3 b^7 d^3 \left (c^7+11 c^6 d x+55 c^5 d^2 x^2+165 c^4 d^3 x^3+330 c^3 d^4 x^4+462 c^2 d^5 x^5+462 c d^6 x^6+330 d^7 x^7\right )+a^2 b^8 d^2 \left (c^8+11 c^7 d x+55 c^6 d^2 x^2+165 c^5 d^3 x^3+330 c^4 d^4 x^4+462 c^3 d^5 x^5+462 c^2 d^6 x^6+330 c d^7 x^7+165 d^8 x^8\right )+a b^9 d \left (c^9+11 c^8 d x+55 c^7 d^2 x^2+165 c^6 d^3 x^3+330 c^5 d^4 x^4+462 c^4 d^5 x^5+462 c^3 d^6 x^6+330 c^2 d^7 x^7+165 c d^8 x^8+55 d^9 x^9\right )+b^{10} \left (c^{10}+11 c^9 d x+55 c^8 d^2 x^2+165 c^7 d^3 x^3+330 c^6 d^4 x^4+462 c^5 d^5 x^5+462 c^4 d^6 x^6+330 c^3 d^7 x^7+165 c^2 d^8 x^8+55 c d^9 x^9+11 d^{10} x^{10}\right )}{11 b^{11} (a+b x)^{11}} \] Input:
Integrate[(c + d*x)^10/(a + b*x)^12,x]
Output:
-1/11*(a^10*d^10 + a^9*b*d^9*(c + 11*d*x) + a^8*b^2*d^8*(c^2 + 11*c*d*x + 55*d^2*x^2) + a^7*b^3*d^7*(c^3 + 11*c^2*d*x + 55*c*d^2*x^2 + 165*d^3*x^3) + a^6*b^4*d^6*(c^4 + 11*c^3*d*x + 55*c^2*d^2*x^2 + 165*c*d^3*x^3 + 330*d^4 *x^4) + a^5*b^5*d^5*(c^5 + 11*c^4*d*x + 55*c^3*d^2*x^2 + 165*c^2*d^3*x^3 + 330*c*d^4*x^4 + 462*d^5*x^5) + a^4*b^6*d^4*(c^6 + 11*c^5*d*x + 55*c^4*d^2 *x^2 + 165*c^3*d^3*x^3 + 330*c^2*d^4*x^4 + 462*c*d^5*x^5 + 462*d^6*x^6) + a^3*b^7*d^3*(c^7 + 11*c^6*d*x + 55*c^5*d^2*x^2 + 165*c^4*d^3*x^3 + 330*c^3 *d^4*x^4 + 462*c^2*d^5*x^5 + 462*c*d^6*x^6 + 330*d^7*x^7) + a^2*b^8*d^2*(c ^8 + 11*c^7*d*x + 55*c^6*d^2*x^2 + 165*c^5*d^3*x^3 + 330*c^4*d^4*x^4 + 462 *c^3*d^5*x^5 + 462*c^2*d^6*x^6 + 330*c*d^7*x^7 + 165*d^8*x^8) + a*b^9*d*(c ^9 + 11*c^8*d*x + 55*c^7*d^2*x^2 + 165*c^6*d^3*x^3 + 330*c^5*d^4*x^4 + 462 *c^4*d^5*x^5 + 462*c^3*d^6*x^6 + 330*c^2*d^7*x^7 + 165*c*d^8*x^8 + 55*d^9* x^9) + b^10*(c^10 + 11*c^9*d*x + 55*c^8*d^2*x^2 + 165*c^7*d^3*x^3 + 330*c^ 6*d^4*x^4 + 462*c^5*d^5*x^5 + 462*c^4*d^6*x^6 + 330*c^3*d^7*x^7 + 165*c^2* d^8*x^8 + 55*c*d^9*x^9 + 11*d^10*x^10))/(b^11*(a + b*x)^11)
Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {48}
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.
| \(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {(c+d x)^{11}}{11 (a+b x)^{11} (b c-a d)}\) |
Input:
Int[(c + d*x)^10/(a + b*x)^12,x]
Output:
-1/11*(c + d*x)^11/((b*c - a*d)*(a + b*x)^11)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(775\) vs. \(2(26)=52\).
Time = 0.16 (sec) , antiderivative size = 776, normalized size of antiderivative = 27.71
| method | result | size |
| risch | \(\frac {-\frac {d^{10} x^{10}}{b}-\frac {5 d^{9} \left (a d +b c \right ) x^{9}}{b^{2}}-\frac {15 d^{8} \left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{8}}{b^{3}}-\frac {30 d^{7} \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{7}}{b^{4}}-\frac {42 d^{6} \left (d^{4} a^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +c^{4} b^{4}\right ) x^{6}}{b^{5}}-\frac {42 d^{5} \left (a^{5} d^{5}+a^{4} b c \,d^{4}+a^{3} b^{2} c^{2} d^{3}+a^{2} b^{3} c^{3} d^{2}+a \,b^{4} c^{4} d +c^{5} b^{5}\right ) x^{5}}{b^{6}}-\frac {30 d^{4} \left (a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +c^{6} b^{6}\right ) x^{4}}{b^{7}}-\frac {15 d^{3} \left (a^{7} d^{7}+a^{6} b c \,d^{6}+a^{5} b^{2} c^{2} d^{5}+a^{4} b^{3} c^{3} d^{4}+a^{3} b^{4} c^{4} d^{3}+a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +b^{7} c^{7}\right ) x^{3}}{b^{8}}-\frac {5 d^{2} \left (a^{8} d^{8}+a^{7} b c \,d^{7}+a^{6} b^{2} c^{2} d^{6}+a^{5} b^{3} c^{3} d^{5}+a^{4} b^{4} c^{4} d^{4}+a^{3} b^{5} c^{5} d^{3}+a^{2} b^{6} c^{6} d^{2}+a \,b^{7} c^{7} d +c^{8} b^{8}\right ) x^{2}}{b^{9}}-\frac {d \left (a^{9} d^{9}+a^{8} b c \,d^{8}+a^{7} b^{2} c^{2} d^{7}+a^{6} b^{3} c^{3} d^{6}+a^{5} b^{4} c^{4} d^{5}+a^{4} b^{5} c^{5} d^{4}+a^{3} b^{6} c^{6} d^{3}+a^{2} b^{7} c^{7} d^{2}+a \,b^{8} c^{8} d +c^{9} b^{9}\right ) x}{b^{10}}-\frac {a^{10} d^{10}+a^{9} b c \,d^{9}+a^{8} b^{2} c^{2} d^{8}+a^{7} b^{3} c^{3} d^{7}+a^{6} b^{4} c^{4} d^{6}+a^{5} b^{5} c^{5} d^{5}+a^{4} b^{6} c^{6} d^{4}+a^{3} b^{7} c^{7} d^{3}+a^{2} b^{8} c^{8} d^{2}+a \,b^{9} c^{9} d +b^{10} c^{10}}{11 b^{11}}}{\left (b x +a \right )^{11}}\) | \(776\) |
| norman | \(\frac {-\frac {d^{10} x^{10}}{b}+\frac {5 \left (-a \,d^{10}-b c \,d^{9}\right ) x^{9}}{b^{2}}+\frac {15 \left (-a^{2} d^{10}-a b c \,d^{9}-b^{2} c^{2} d^{8}\right ) x^{8}}{b^{3}}+\frac {30 \left (-a^{3} d^{10}-a^{2} b c \,d^{9}-a \,b^{2} c^{2} d^{8}-b^{3} c^{3} d^{7}\right ) x^{7}}{b^{4}}+\frac {42 \left (-a^{4} d^{10}-a^{3} b c \,d^{9}-a^{2} b^{2} c^{2} d^{8}-a \,b^{3} c^{3} d^{7}-b^{4} c^{4} d^{6}\right ) x^{6}}{b^{5}}+\frac {42 \left (-a^{5} d^{10}-a^{4} b c \,d^{9}-a^{3} b^{2} c^{2} d^{8}-a^{2} b^{3} c^{3} d^{7}-a \,b^{4} c^{4} d^{6}-b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}+\frac {30 \left (-a^{6} d^{10}-a^{5} b c \,d^{9}-a^{4} b^{2} c^{2} d^{8}-a^{3} b^{3} c^{3} d^{7}-a^{2} b^{4} c^{4} d^{6}-a \,b^{5} c^{5} d^{5}-b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}+\frac {15 \left (-a^{7} d^{10}-a^{6} b c \,d^{9}-a^{5} b^{2} c^{2} d^{8}-a^{4} b^{3} c^{3} d^{7}-a^{3} b^{4} c^{4} d^{6}-a^{2} b^{5} c^{5} d^{5}-a \,b^{6} c^{6} d^{4}-b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}+\frac {5 \left (-a^{8} d^{10}-a^{7} b c \,d^{9}-a^{6} b^{2} c^{2} d^{8}-a^{5} b^{3} c^{3} d^{7}-a^{4} b^{4} c^{4} d^{6}-a^{3} b^{5} c^{5} d^{5}-a^{2} b^{6} c^{6} d^{4}-a \,b^{7} c^{7} d^{3}-b^{8} c^{8} d^{2}\right ) x^{2}}{b^{9}}+\frac {\left (-a^{9} d^{10}-a^{8} b c \,d^{9}-a^{7} b^{2} c^{2} d^{8}-a^{6} b^{3} c^{3} d^{7}-a^{5} b^{4} c^{4} d^{6}-a^{4} b^{5} c^{5} d^{5}-a^{3} b^{6} c^{6} d^{4}-a^{2} b^{7} c^{7} d^{3}-a \,b^{8} c^{8} d^{2}-b^{9} c^{9} d \right ) x}{b^{10}}+\frac {-a^{10} d^{10}-a^{9} b c \,d^{9}-a^{8} b^{2} c^{2} d^{8}-a^{7} b^{3} c^{3} d^{7}-a^{6} b^{4} c^{4} d^{6}-a^{5} b^{5} c^{5} d^{5}-a^{4} b^{6} c^{6} d^{4}-a^{3} b^{7} c^{7} d^{3}-a^{2} b^{8} c^{8} d^{2}-a \,b^{9} c^{9} d -b^{10} c^{10}}{11 b^{11}}}{\left (b x +a \right )^{11}}\) | \(858\) |
| default | \(-\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 +c^{8} b^{8}\right )}{b^{11} \left (b x +a \right )^{9}}-\frac {42 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 )^{5}}+\frac {30 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 )^{4}}-\frac {30 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 )^{7}}+\frac {15 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 )^{8}}+\frac {5 d^{9} \left (a d -b c \right )}{b^{11} \left (b x +a \right )^{2}}-\frac {d^{10}}{b^{11} \left (b x +a \right )}+\frac {d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -c^{9} b^{9}\right )}{b^{11} \left (b x +a \right )^{10}}-\frac {15 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{11} \left (b x +a \right )^{3}}+\frac {42 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 )^{6}}-\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}}{11 b^{11} \left (b x +a \right )^{11}}\) | \(866\) |
| gosper | \(-\frac {11 x^{10} d^{10} b^{10}+55 x^{9} a \,b^{9} d^{10}+55 x^{9} b^{10} c \,d^{9}+165 x^{8} a^{2} b^{8} d^{10}+165 x^{8} a \,b^{9} c \,d^{9}+165 x^{8} b^{10} c^{2} d^{8}+330 x^{7} a^{3} b^{7} d^{10}+330 x^{7} a^{2} b^{8} c \,d^{9}+330 x^{7} a \,b^{9} c^{2} d^{8}+330 x^{7} b^{10} c^{3} d^{7}+462 x^{6} a^{4} b^{6} d^{10}+462 x^{6} a^{3} b^{7} c \,d^{9}+462 x^{6} a^{2} b^{8} c^{2} d^{8}+462 x^{6} a \,b^{9} c^{3} d^{7}+462 x^{6} b^{10} c^{4} d^{6}+462 x^{5} a^{5} b^{5} d^{10}+462 x^{5} a^{4} b^{6} c \,d^{9}+462 x^{5} a^{3} b^{7} c^{2} d^{8}+462 x^{5} a^{2} b^{8} c^{3} d^{7}+462 x^{5} a \,b^{9} c^{4} d^{6}+462 x^{5} b^{10} c^{5} d^{5}+330 x^{4} a^{6} b^{4} d^{10}+330 x^{4} a^{5} b^{5} c \,d^{9}+330 x^{4} a^{4} b^{6} c^{2} d^{8}+330 x^{4} a^{3} b^{7} c^{3} d^{7}+330 x^{4} a^{2} b^{8} c^{4} d^{6}+330 x^{4} a \,b^{9} c^{5} d^{5}+330 x^{4} b^{10} c^{6} d^{4}+165 x^{3} a^{7} b^{3} d^{10}+165 x^{3} a^{6} b^{4} c \,d^{9}+165 x^{3} a^{5} b^{5} c^{2} d^{8}+165 x^{3} a^{4} b^{6} c^{3} d^{7}+165 x^{3} a^{3} b^{7} c^{4} d^{6}+165 x^{3} a^{2} b^{8} c^{5} d^{5}+165 x^{3} a \,b^{9} c^{6} d^{4}+165 x^{3} b^{10} c^{7} d^{3}+55 x^{2} a^{8} b^{2} d^{10}+55 x^{2} a^{7} b^{3} c \,d^{9}+55 x^{2} a^{6} b^{4} c^{2} d^{8}+55 x^{2} a^{5} b^{5} c^{3} d^{7}+55 x^{2} a^{4} b^{6} c^{4} d^{6}+55 x^{2} a^{3} b^{7} c^{5} d^{5}+55 x^{2} a^{2} b^{8} c^{6} d^{4}+55 x^{2} a \,b^{9} c^{7} d^{3}+55 x^{2} b^{10} c^{8} d^{2}+11 x \,a^{9} b \,d^{10}+11 x \,a^{8} b^{2} c \,d^{9}+11 x \,a^{7} b^{3} c^{2} d^{8}+11 x \,a^{6} b^{4} c^{3} d^{7}+11 x \,a^{5} b^{5} c^{4} d^{6}+11 x \,a^{4} b^{6} c^{5} d^{5}+11 x \,a^{3} b^{7} c^{6} d^{4}+11 x \,a^{2} b^{8} c^{7} d^{3}+11 x a \,b^{9} c^{8} d^{2}+11 x \,b^{10} c^{9} d +a^{10} d^{10}+a^{9} b c \,d^{9}+a^{8} b^{2} c^{2} d^{8}+a^{7} b^{3} c^{3} d^{7}+a^{6} b^{4} c^{4} d^{6}+a^{5} b^{5} c^{5} d^{5}+a^{4} b^{6} c^{6} d^{4}+a^{3} b^{7} c^{7} d^{3}+a^{2} b^{8} c^{8} d^{2}+a \,b^{9} c^{9} d +b^{10} c^{10}}{11 \left (b x +a \right )^{11} b^{11}}\) | \(952\) |
| orering | \(-\frac {11 x^{10} d^{10} b^{10}+55 x^{9} a \,b^{9} d^{10}+55 x^{9} b^{10} c \,d^{9}+165 x^{8} a^{2} b^{8} d^{10}+165 x^{8} a \,b^{9} c \,d^{9}+165 x^{8} b^{10} c^{2} d^{8}+330 x^{7} a^{3} b^{7} d^{10}+330 x^{7} a^{2} b^{8} c \,d^{9}+330 x^{7} a \,b^{9} c^{2} d^{8}+330 x^{7} b^{10} c^{3} d^{7}+462 x^{6} a^{4} b^{6} d^{10}+462 x^{6} a^{3} b^{7} c \,d^{9}+462 x^{6} a^{2} b^{8} c^{2} d^{8}+462 x^{6} a \,b^{9} c^{3} d^{7}+462 x^{6} b^{10} c^{4} d^{6}+462 x^{5} a^{5} b^{5} d^{10}+462 x^{5} a^{4} b^{6} c \,d^{9}+462 x^{5} a^{3} b^{7} c^{2} d^{8}+462 x^{5} a^{2} b^{8} c^{3} d^{7}+462 x^{5} a \,b^{9} c^{4} d^{6}+462 x^{5} b^{10} c^{5} d^{5}+330 x^{4} a^{6} b^{4} d^{10}+330 x^{4} a^{5} b^{5} c \,d^{9}+330 x^{4} a^{4} b^{6} c^{2} d^{8}+330 x^{4} a^{3} b^{7} c^{3} d^{7}+330 x^{4} a^{2} b^{8} c^{4} d^{6}+330 x^{4} a \,b^{9} c^{5} d^{5}+330 x^{4} b^{10} c^{6} d^{4}+165 x^{3} a^{7} b^{3} d^{10}+165 x^{3} a^{6} b^{4} c \,d^{9}+165 x^{3} a^{5} b^{5} c^{2} d^{8}+165 x^{3} a^{4} b^{6} c^{3} d^{7}+165 x^{3} a^{3} b^{7} c^{4} d^{6}+165 x^{3} a^{2} b^{8} c^{5} d^{5}+165 x^{3} a \,b^{9} c^{6} d^{4}+165 x^{3} b^{10} c^{7} d^{3}+55 x^{2} a^{8} b^{2} d^{10}+55 x^{2} a^{7} b^{3} c \,d^{9}+55 x^{2} a^{6} b^{4} c^{2} d^{8}+55 x^{2} a^{5} b^{5} c^{3} d^{7}+55 x^{2} a^{4} b^{6} c^{4} d^{6}+55 x^{2} a^{3} b^{7} c^{5} d^{5}+55 x^{2} a^{2} b^{8} c^{6} d^{4}+55 x^{2} a \,b^{9} c^{7} d^{3}+55 x^{2} b^{10} c^{8} d^{2}+11 x \,a^{9} b \,d^{10}+11 x \,a^{8} b^{2} c \,d^{9}+11 x \,a^{7} b^{3} c^{2} d^{8}+11 x \,a^{6} b^{4} c^{3} d^{7}+11 x \,a^{5} b^{5} c^{4} d^{6}+11 x \,a^{4} b^{6} c^{5} d^{5}+11 x \,a^{3} b^{7} c^{6} d^{4}+11 x \,a^{2} b^{8} c^{7} d^{3}+11 x a \,b^{9} c^{8} d^{2}+11 x \,b^{10} c^{9} d +a^{10} d^{10}+a^{9} b c \,d^{9}+a^{8} b^{2} c^{2} d^{8}+a^{7} b^{3} c^{3} d^{7}+a^{6} b^{4} c^{4} d^{6}+a^{5} b^{5} c^{5} d^{5}+a^{4} b^{6} c^{6} d^{4}+a^{3} b^{7} c^{7} d^{3}+a^{2} b^{8} c^{8} d^{2}+a \,b^{9} c^{9} d +b^{10} c^{10}}{11 \left (b x +a \right )^{11} b^{11}}\) | \(952\) |
| parallelrisch | \(\frac {-11 x^{10} d^{10} b^{10}-55 x^{9} a \,b^{9} d^{10}-55 x^{9} b^{10} c \,d^{9}-165 x^{8} a^{2} b^{8} d^{10}-165 x^{8} a \,b^{9} c \,d^{9}-165 x^{8} b^{10} c^{2} d^{8}-330 x^{7} a^{3} b^{7} d^{10}-330 x^{7} a^{2} b^{8} c \,d^{9}-330 x^{7} a \,b^{9} c^{2} d^{8}-330 x^{7} b^{10} c^{3} d^{7}-462 x^{6} a^{4} b^{6} d^{10}-462 x^{6} a^{3} b^{7} c \,d^{9}-462 x^{6} a^{2} b^{8} c^{2} d^{8}-462 x^{6} a \,b^{9} c^{3} d^{7}-462 x^{6} b^{10} c^{4} d^{6}-462 x^{5} a^{5} b^{5} d^{10}-462 x^{5} a^{4} b^{6} c \,d^{9}-462 x^{5} a^{3} b^{7} c^{2} d^{8}-462 x^{5} a^{2} b^{8} c^{3} d^{7}-462 x^{5} a \,b^{9} c^{4} d^{6}-462 x^{5} b^{10} c^{5} d^{5}-330 x^{4} a^{6} b^{4} d^{10}-330 x^{4} a^{5} b^{5} c \,d^{9}-330 x^{4} a^{4} b^{6} c^{2} d^{8}-330 x^{4} a^{3} b^{7} c^{3} d^{7}-330 x^{4} a^{2} b^{8} c^{4} d^{6}-330 x^{4} a \,b^{9} c^{5} d^{5}-330 x^{4} b^{10} c^{6} d^{4}-165 x^{3} a^{7} b^{3} d^{10}-165 x^{3} a^{6} b^{4} c \,d^{9}-165 x^{3} a^{5} b^{5} c^{2} d^{8}-165 x^{3} a^{4} b^{6} c^{3} d^{7}-165 x^{3} a^{3} b^{7} c^{4} d^{6}-165 x^{3} a^{2} b^{8} c^{5} d^{5}-165 x^{3} a \,b^{9} c^{6} d^{4}-165 x^{3} b^{10} c^{7} d^{3}-55 x^{2} a^{8} b^{2} d^{10}-55 x^{2} a^{7} b^{3} c \,d^{9}-55 x^{2} a^{6} b^{4} c^{2} d^{8}-55 x^{2} a^{5} b^{5} c^{3} d^{7}-55 x^{2} a^{4} b^{6} c^{4} d^{6}-55 x^{2} a^{3} b^{7} c^{5} d^{5}-55 x^{2} a^{2} b^{8} c^{6} d^{4}-55 x^{2} a \,b^{9} c^{7} d^{3}-55 x^{2} b^{10} c^{8} d^{2}-11 x \,a^{9} b \,d^{10}-11 x \,a^{8} b^{2} c \,d^{9}-11 x \,a^{7} b^{3} c^{2} d^{8}-11 x \,a^{6} b^{4} c^{3} d^{7}-11 x \,a^{5} b^{5} c^{4} d^{6}-11 x \,a^{4} b^{6} c^{5} d^{5}-11 x \,a^{3} b^{7} c^{6} d^{4}-11 x \,a^{2} b^{8} c^{7} d^{3}-11 x a \,b^{9} c^{8} d^{2}-11 x \,b^{10} c^{9} d -a^{10} d^{10}-a^{9} b c \,d^{9}-a^{8} b^{2} c^{2} d^{8}-a^{7} b^{3} c^{3} d^{7}-a^{6} b^{4} c^{4} d^{6}-a^{5} b^{5} c^{5} d^{5}-a^{4} b^{6} c^{6} d^{4}-a^{3} b^{7} c^{7} d^{3}-a^{2} b^{8} c^{8} d^{2}-a \,b^{9} c^{9} d -b^{10} c^{10}}{11 b^{11} \left (b x +a \right )^{11}}\) | \(963\) |
Input:
int((d*x+c)^10/(b*x+a)^12,x,method=_RETURNVERBOSE)
Output:
(-1/b*d^10*x^10-5*d^9*(a*d+b*c)/b^2*x^9-15*d^8*(a^2*d^2+a*b*c*d+b^2*c^2)/b ^3*x^8-30*d^7*(a^3*d^3+a^2*b*c*d^2+a*b^2*c^2*d+b^3*c^3)/b^4*x^7-42*d^6*(a^ 4*d^4+a^3*b*c*d^3+a^2*b^2*c^2*d^2+a*b^3*c^3*d+b^4*c^4)/b^5*x^6-42*d^5*(a^5 *d^5+a^4*b*c*d^4+a^3*b^2*c^2*d^3+a^2*b^3*c^3*d^2+a*b^4*c^4*d+b^5*c^5)/b^6* x^5-30*d^4*(a^6*d^6+a^5*b*c*d^5+a^4*b^2*c^2*d^4+a^3*b^3*c^3*d^3+a^2*b^4*c^ 4*d^2+a*b^5*c^5*d+b^6*c^6)/b^7*x^4-15*d^3*(a^7*d^7+a^6*b*c*d^6+a^5*b^2*c^2 *d^5+a^4*b^3*c^3*d^4+a^3*b^4*c^4*d^3+a^2*b^5*c^5*d^2+a*b^6*c^6*d+b^7*c^7)/ b^8*x^3-5*d^2*(a^8*d^8+a^7*b*c*d^7+a^6*b^2*c^2*d^6+a^5*b^3*c^3*d^5+a^4*b^4 *c^4*d^4+a^3*b^5*c^5*d^3+a^2*b^6*c^6*d^2+a*b^7*c^7*d+b^8*c^8)/b^9*x^2-d*(a ^9*d^9+a^8*b*c*d^8+a^7*b^2*c^2*d^7+a^6*b^3*c^3*d^6+a^5*b^4*c^4*d^5+a^4*b^5 *c^5*d^4+a^3*b^6*c^6*d^3+a^2*b^7*c^7*d^2+a*b^8*c^8*d+b^9*c^9)/b^10*x-1/11* (a^10*d^10+a^9*b*c*d^9+a^8*b^2*c^2*d^8+a^7*b^3*c^3*d^7+a^6*b^4*c^4*d^6+a^5 *b^5*c^5*d^5+a^4*b^6*c^6*d^4+a^3*b^7*c^7*d^3+a^2*b^8*c^8*d^2+a*b^9*c^9*d+b ^10*c^10)/b^11)/(b*x+a)^11
Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 920, normalized size of antiderivative = 32.86 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^10/(b*x+a)^12,x, algorithm="fricas")
Output:
-1/11*(11*b^10*d^10*x^10 + b^10*c^10 + a*b^9*c^9*d + a^2*b^8*c^8*d^2 + a^3 *b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b ^3*c^3*d^7 + a^8*b^2*c^2*d^8 + a^9*b*c*d^9 + a^10*d^10 + 55*(b^10*c*d^9 + a*b^9*d^10)*x^9 + 165*(b^10*c^2*d^8 + a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 33 0*(b^10*c^3*d^7 + a*b^9*c^2*d^8 + a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 462* (b^10*c^4*d^6 + a*b^9*c^3*d^7 + a^2*b^8*c^2*d^8 + a^3*b^7*c*d^9 + a^4*b^6* d^10)*x^6 + 462*(b^10*c^5*d^5 + a*b^9*c^4*d^6 + a^2*b^8*c^3*d^7 + a^3*b^7* c^2*d^8 + a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 330*(b^10*c^6*d^4 + a*b^9*c^ 5*d^5 + a^2*b^8*c^4*d^6 + a^3*b^7*c^3*d^7 + a^4*b^6*c^2*d^8 + a^5*b^5*c*d^ 9 + a^6*b^4*d^10)*x^4 + 165*(b^10*c^7*d^3 + a*b^9*c^6*d^4 + a^2*b^8*c^5*d^ 5 + a^3*b^7*c^4*d^6 + a^4*b^6*c^3*d^7 + a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 55*(b^10*c^8*d^2 + a*b^9*c^7*d^3 + a^2*b^8*c^6*d^4 + a ^3*b^7*c^5*d^5 + a^4*b^6*c^4*d^6 + a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + a^7 *b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 11*(b^10*c^9*d + a*b^9*c^8*d^2 + a^2*b^8* c^7*d^3 + a^3*b^7*c^6*d^4 + a^4*b^6*c^5*d^5 + a^5*b^5*c^4*d^6 + a^6*b^4*c^ 3*d^7 + a^7*b^3*c^2*d^8 + a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^22*x^11 + 11*a *b^21*x^10 + 55*a^2*b^20*x^9 + 165*a^3*b^19*x^8 + 330*a^4*b^18*x^7 + 462*a ^5*b^17*x^6 + 462*a^6*b^16*x^5 + 330*a^7*b^15*x^4 + 165*a^8*b^14*x^3 + 55* a^9*b^13*x^2 + 11*a^10*b^12*x + a^11*b^11)
Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**10/(b*x+a)**12,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (26) = 52\).
Time = 0.07 (sec) , antiderivative size = 920, normalized size of antiderivative = 32.86 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^10/(b*x+a)^12,x, algorithm="maxima")
Output:
-1/11*(11*b^10*d^10*x^10 + b^10*c^10 + a*b^9*c^9*d + a^2*b^8*c^8*d^2 + a^3 *b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5*b^5*c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b ^3*c^3*d^7 + a^8*b^2*c^2*d^8 + a^9*b*c*d^9 + a^10*d^10 + 55*(b^10*c*d^9 + a*b^9*d^10)*x^9 + 165*(b^10*c^2*d^8 + a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 33 0*(b^10*c^3*d^7 + a*b^9*c^2*d^8 + a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 462* (b^10*c^4*d^6 + a*b^9*c^3*d^7 + a^2*b^8*c^2*d^8 + a^3*b^7*c*d^9 + a^4*b^6* d^10)*x^6 + 462*(b^10*c^5*d^5 + a*b^9*c^4*d^6 + a^2*b^8*c^3*d^7 + a^3*b^7* c^2*d^8 + a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 330*(b^10*c^6*d^4 + a*b^9*c^ 5*d^5 + a^2*b^8*c^4*d^6 + a^3*b^7*c^3*d^7 + a^4*b^6*c^2*d^8 + a^5*b^5*c*d^ 9 + a^6*b^4*d^10)*x^4 + 165*(b^10*c^7*d^3 + a*b^9*c^6*d^4 + a^2*b^8*c^5*d^ 5 + a^3*b^7*c^4*d^6 + a^4*b^6*c^3*d^7 + a^5*b^5*c^2*d^8 + a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 55*(b^10*c^8*d^2 + a*b^9*c^7*d^3 + a^2*b^8*c^6*d^4 + a ^3*b^7*c^5*d^5 + a^4*b^6*c^4*d^6 + a^5*b^5*c^3*d^7 + a^6*b^4*c^2*d^8 + a^7 *b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 11*(b^10*c^9*d + a*b^9*c^8*d^2 + a^2*b^8* c^7*d^3 + a^3*b^7*c^6*d^4 + a^4*b^6*c^5*d^5 + a^5*b^5*c^4*d^6 + a^6*b^4*c^ 3*d^7 + a^7*b^3*c^2*d^8 + a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^22*x^11 + 11*a *b^21*x^10 + 55*a^2*b^20*x^9 + 165*a^3*b^19*x^8 + 330*a^4*b^18*x^7 + 462*a ^5*b^17*x^6 + 462*a^6*b^16*x^5 + 330*a^7*b^15*x^4 + 165*a^8*b^14*x^3 + 55* a^9*b^13*x^2 + 11*a^10*b^12*x + a^11*b^11)
Leaf count of result is larger than twice the leaf count of optimal. 951 vs. \(2 (26) = 52\).
Time = 0.13 (sec) , antiderivative size = 951, normalized size of antiderivative = 33.96 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^10/(b*x+a)^12,x, algorithm="giac")
Output:
-1/11*(11*b^10*d^10*x^10 + 55*b^10*c*d^9*x^9 + 55*a*b^9*d^10*x^9 + 165*b^1 0*c^2*d^8*x^8 + 165*a*b^9*c*d^9*x^8 + 165*a^2*b^8*d^10*x^8 + 330*b^10*c^3* d^7*x^7 + 330*a*b^9*c^2*d^8*x^7 + 330*a^2*b^8*c*d^9*x^7 + 330*a^3*b^7*d^10 *x^7 + 462*b^10*c^4*d^6*x^6 + 462*a*b^9*c^3*d^7*x^6 + 462*a^2*b^8*c^2*d^8* x^6 + 462*a^3*b^7*c*d^9*x^6 + 462*a^4*b^6*d^10*x^6 + 462*b^10*c^5*d^5*x^5 + 462*a*b^9*c^4*d^6*x^5 + 462*a^2*b^8*c^3*d^7*x^5 + 462*a^3*b^7*c^2*d^8*x^ 5 + 462*a^4*b^6*c*d^9*x^5 + 462*a^5*b^5*d^10*x^5 + 330*b^10*c^6*d^4*x^4 + 330*a*b^9*c^5*d^5*x^4 + 330*a^2*b^8*c^4*d^6*x^4 + 330*a^3*b^7*c^3*d^7*x^4 + 330*a^4*b^6*c^2*d^8*x^4 + 330*a^5*b^5*c*d^9*x^4 + 330*a^6*b^4*d^10*x^4 + 165*b^10*c^7*d^3*x^3 + 165*a*b^9*c^6*d^4*x^3 + 165*a^2*b^8*c^5*d^5*x^3 + 165*a^3*b^7*c^4*d^6*x^3 + 165*a^4*b^6*c^3*d^7*x^3 + 165*a^5*b^5*c^2*d^8*x^ 3 + 165*a^6*b^4*c*d^9*x^3 + 165*a^7*b^3*d^10*x^3 + 55*b^10*c^8*d^2*x^2 + 5 5*a*b^9*c^7*d^3*x^2 + 55*a^2*b^8*c^6*d^4*x^2 + 55*a^3*b^7*c^5*d^5*x^2 + 55 *a^4*b^6*c^4*d^6*x^2 + 55*a^5*b^5*c^3*d^7*x^2 + 55*a^6*b^4*c^2*d^8*x^2 + 5 5*a^7*b^3*c*d^9*x^2 + 55*a^8*b^2*d^10*x^2 + 11*b^10*c^9*d*x + 11*a*b^9*c^8 *d^2*x + 11*a^2*b^8*c^7*d^3*x + 11*a^3*b^7*c^6*d^4*x + 11*a^4*b^6*c^5*d^5* x + 11*a^5*b^5*c^4*d^6*x + 11*a^6*b^4*c^3*d^7*x + 11*a^7*b^3*c^2*d^8*x + 1 1*a^8*b^2*c*d^9*x + 11*a^9*b*d^10*x + b^10*c^10 + a*b^9*c^9*d + a^2*b^8*c^ 8*d^2 + a^3*b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5*b^5*c^5*d^5 + a^6*b^4*c^4* d^6 + a^7*b^3*c^3*d^7 + a^8*b^2*c^2*d^8 + a^9*b*c*d^9 + a^10*d^10)/((b*...
Time = 0.28 (sec) , antiderivative size = 1066, normalized size of antiderivative = 38.07 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=-\frac {a^{10}\,d^{10}+a^9\,b\,c\,d^9+11\,a^9\,b\,d^{10}\,x+a^8\,b^2\,c^2\,d^8+11\,a^8\,b^2\,c\,d^9\,x+55\,a^8\,b^2\,d^{10}\,x^2+a^7\,b^3\,c^3\,d^7+11\,a^7\,b^3\,c^2\,d^8\,x+55\,a^7\,b^3\,c\,d^9\,x^2+165\,a^7\,b^3\,d^{10}\,x^3+a^6\,b^4\,c^4\,d^6+11\,a^6\,b^4\,c^3\,d^7\,x+55\,a^6\,b^4\,c^2\,d^8\,x^2+165\,a^6\,b^4\,c\,d^9\,x^3+330\,a^6\,b^4\,d^{10}\,x^4+a^5\,b^5\,c^5\,d^5+11\,a^5\,b^5\,c^4\,d^6\,x+55\,a^5\,b^5\,c^3\,d^7\,x^2+165\,a^5\,b^5\,c^2\,d^8\,x^3+330\,a^5\,b^5\,c\,d^9\,x^4+462\,a^5\,b^5\,d^{10}\,x^5+a^4\,b^6\,c^6\,d^4+11\,a^4\,b^6\,c^5\,d^5\,x+55\,a^4\,b^6\,c^4\,d^6\,x^2+165\,a^4\,b^6\,c^3\,d^7\,x^3+330\,a^4\,b^6\,c^2\,d^8\,x^4+462\,a^4\,b^6\,c\,d^9\,x^5+462\,a^4\,b^6\,d^{10}\,x^6+a^3\,b^7\,c^7\,d^3+11\,a^3\,b^7\,c^6\,d^4\,x+55\,a^3\,b^7\,c^5\,d^5\,x^2+165\,a^3\,b^7\,c^4\,d^6\,x^3+330\,a^3\,b^7\,c^3\,d^7\,x^4+462\,a^3\,b^7\,c^2\,d^8\,x^5+462\,a^3\,b^7\,c\,d^9\,x^6+330\,a^3\,b^7\,d^{10}\,x^7+a^2\,b^8\,c^8\,d^2+11\,a^2\,b^8\,c^7\,d^3\,x+55\,a^2\,b^8\,c^6\,d^4\,x^2+165\,a^2\,b^8\,c^5\,d^5\,x^3+330\,a^2\,b^8\,c^4\,d^6\,x^4+462\,a^2\,b^8\,c^3\,d^7\,x^5+462\,a^2\,b^8\,c^2\,d^8\,x^6+330\,a^2\,b^8\,c\,d^9\,x^7+165\,a^2\,b^8\,d^{10}\,x^8+a\,b^9\,c^9\,d+11\,a\,b^9\,c^8\,d^2\,x+55\,a\,b^9\,c^7\,d^3\,x^2+165\,a\,b^9\,c^6\,d^4\,x^3+330\,a\,b^9\,c^5\,d^5\,x^4+462\,a\,b^9\,c^4\,d^6\,x^5+462\,a\,b^9\,c^3\,d^7\,x^6+330\,a\,b^9\,c^2\,d^8\,x^7+165\,a\,b^9\,c\,d^9\,x^8+55\,a\,b^9\,d^{10}\,x^9+b^{10}\,c^{10}+11\,b^{10}\,c^9\,d\,x+55\,b^{10}\,c^8\,d^2\,x^2+165\,b^{10}\,c^7\,d^3\,x^3+330\,b^{10}\,c^6\,d^4\,x^4+462\,b^{10}\,c^5\,d^5\,x^5+462\,b^{10}\,c^4\,d^6\,x^6+330\,b^{10}\,c^3\,d^7\,x^7+165\,b^{10}\,c^2\,d^8\,x^8+55\,b^{10}\,c\,d^9\,x^9+11\,b^{10}\,d^{10}\,x^{10}}{11\,a^{11}\,b^{11}+121\,a^{10}\,b^{12}\,x+605\,a^9\,b^{13}\,x^2+1815\,a^8\,b^{14}\,x^3+3630\,a^7\,b^{15}\,x^4+5082\,a^6\,b^{16}\,x^5+5082\,a^5\,b^{17}\,x^6+3630\,a^4\,b^{18}\,x^7+1815\,a^3\,b^{19}\,x^8+605\,a^2\,b^{20}\,x^9+121\,a\,b^{21}\,x^{10}+11\,b^{22}\,x^{11}} \] Input:
int((c + d*x)^10/(a + b*x)^12,x)
Output:
-(a^10*d^10 + b^10*c^10 + 11*b^10*d^10*x^10 + 55*a*b^9*d^10*x^9 + 55*b^10* c*d^9*x^9 + a^2*b^8*c^8*d^2 + a^3*b^7*c^7*d^3 + a^4*b^6*c^6*d^4 + a^5*b^5* c^5*d^5 + a^6*b^4*c^4*d^6 + a^7*b^3*c^3*d^7 + a^8*b^2*c^2*d^8 + 55*a^8*b^2 *d^10*x^2 + 165*a^7*b^3*d^10*x^3 + 330*a^6*b^4*d^10*x^4 + 462*a^5*b^5*d^10 *x^5 + 462*a^4*b^6*d^10*x^6 + 330*a^3*b^7*d^10*x^7 + 165*a^2*b^8*d^10*x^8 + 55*b^10*c^8*d^2*x^2 + 165*b^10*c^7*d^3*x^3 + 330*b^10*c^6*d^4*x^4 + 462* b^10*c^5*d^5*x^5 + 462*b^10*c^4*d^6*x^6 + 330*b^10*c^3*d^7*x^7 + 165*b^10* c^2*d^8*x^8 + a*b^9*c^9*d + a^9*b*c*d^9 + 11*a^9*b*d^10*x + 11*b^10*c^9*d* x + 55*a^2*b^8*c^6*d^4*x^2 + 55*a^3*b^7*c^5*d^5*x^2 + 55*a^4*b^6*c^4*d^6*x ^2 + 55*a^5*b^5*c^3*d^7*x^2 + 55*a^6*b^4*c^2*d^8*x^2 + 165*a^2*b^8*c^5*d^5 *x^3 + 165*a^3*b^7*c^4*d^6*x^3 + 165*a^4*b^6*c^3*d^7*x^3 + 165*a^5*b^5*c^2 *d^8*x^3 + 330*a^2*b^8*c^4*d^6*x^4 + 330*a^3*b^7*c^3*d^7*x^4 + 330*a^4*b^6 *c^2*d^8*x^4 + 462*a^2*b^8*c^3*d^7*x^5 + 462*a^3*b^7*c^2*d^8*x^5 + 462*a^2 *b^8*c^2*d^8*x^6 + 11*a*b^9*c^8*d^2*x + 11*a^8*b^2*c*d^9*x + 165*a*b^9*c*d ^9*x^8 + 11*a^2*b^8*c^7*d^3*x + 11*a^3*b^7*c^6*d^4*x + 11*a^4*b^6*c^5*d^5* x + 11*a^5*b^5*c^4*d^6*x + 11*a^6*b^4*c^3*d^7*x + 11*a^7*b^3*c^2*d^8*x + 5 5*a*b^9*c^7*d^3*x^2 + 55*a^7*b^3*c*d^9*x^2 + 165*a*b^9*c^6*d^4*x^3 + 165*a ^6*b^4*c*d^9*x^3 + 330*a*b^9*c^5*d^5*x^4 + 330*a^5*b^5*c*d^9*x^4 + 462*a*b ^9*c^4*d^6*x^5 + 462*a^4*b^6*c*d^9*x^5 + 462*a*b^9*c^3*d^7*x^6 + 462*a^3*b ^7*c*d^9*x^6 + 330*a*b^9*c^2*d^8*x^7 + 330*a^2*b^8*c*d^9*x^7)/(11*a^11*...
Time = 0.17 (sec) , antiderivative size = 969, normalized size of antiderivative = 34.61 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{12}} \, dx=\frac {b^{10} d^{10} x^{11}-55 a \,b^{9} c \,d^{9} x^{9}-165 a^{2} b^{8} c \,d^{9} x^{8}-165 a \,b^{9} c^{2} d^{8} x^{8}-330 a^{3} b^{7} c \,d^{9} x^{7}-330 a^{2} b^{8} c^{2} d^{8} x^{7}-330 a \,b^{9} c^{3} d^{7} x^{7}-462 a^{4} b^{6} c \,d^{9} x^{6}-462 a^{3} b^{7} c^{2} d^{8} x^{6}-462 a^{2} b^{8} c^{3} d^{7} x^{6}-462 a \,b^{9} c^{4} d^{6} x^{6}-462 a^{5} b^{5} c \,d^{9} x^{5}-462 a^{4} b^{6} c^{2} d^{8} x^{5}-462 a^{3} b^{7} c^{3} d^{7} x^{5}-462 a^{2} b^{8} c^{4} d^{6} x^{5}-462 a \,b^{9} c^{5} d^{5} x^{5}-330 a^{6} b^{4} c \,d^{9} x^{4}-330 a^{5} b^{5} c^{2} d^{8} x^{4}-330 a^{4} b^{6} c^{3} d^{7} x^{4}-330 a^{3} b^{7} c^{4} d^{6} x^{4}-330 a^{2} b^{8} c^{5} d^{5} x^{4}-330 a \,b^{9} c^{6} d^{4} x^{4}-165 a^{7} b^{3} c \,d^{9} x^{3}-165 a^{6} b^{4} c^{2} d^{8} x^{3}-165 a^{5} b^{5} c^{3} d^{7} x^{3}-165 a^{4} b^{6} c^{4} d^{6} x^{3}-165 a^{3} b^{7} c^{5} d^{5} x^{3}-165 a^{2} b^{8} c^{6} d^{4} x^{3}-165 a \,b^{9} c^{7} d^{3} x^{3}-55 a^{8} b^{2} c \,d^{9} x^{2}-55 a^{7} b^{3} c^{2} d^{8} x^{2}-55 a^{6} b^{4} c^{3} d^{7} x^{2}-55 a^{5} b^{5} c^{4} d^{6} x^{2}-55 a^{4} b^{6} c^{5} d^{5} x^{2}-55 a^{3} b^{7} c^{6} d^{4} x^{2}-55 a^{2} b^{8} c^{7} d^{3} x^{2}-55 a \,b^{9} c^{8} d^{2} x^{2}-11 a^{9} b c \,d^{9} x -11 a^{8} b^{2} c^{2} d^{8} x -11 a^{7} b^{3} c^{3} d^{7} x -11 a^{6} b^{4} c^{4} d^{6} x -11 a^{5} b^{5} c^{5} d^{5} x -11 a^{4} b^{6} c^{6} d^{4} x -11 a^{3} b^{7} c^{7} d^{3} x -11 a^{2} b^{8} c^{8} d^{2} x -11 a \,b^{9} c^{9} d x -a^{10} c \,d^{9}-a^{9} b \,c^{2} d^{8}-a^{8} b^{2} c^{3} d^{7}-a^{7} b^{3} c^{4} d^{6}-a^{6} b^{4} c^{5} d^{5}-a^{5} b^{5} c^{6} d^{4}-a^{4} b^{6} c^{7} d^{3}-a^{3} b^{7} c^{8} d^{2}-a^{2} b^{8} c^{9} d -a \,b^{9} c^{10}}{11 a \,b^{10} \left (b^{11} x^{11}+11 a \,b^{10} x^{10}+55 a^{2} b^{9} x^{9}+165 a^{3} b^{8} x^{8}+330 a^{4} b^{7} x^{7}+462 a^{5} b^{6} x^{6}+462 a^{6} b^{5} x^{5}+330 a^{7} b^{4} x^{4}+165 a^{8} b^{3} x^{3}+55 a^{9} b^{2} x^{2}+11 a^{10} b x +a^{11}\right )} \] Input:
int((d*x+c)^10/(b*x+a)^12,x)
Output:
( - a**10*c*d**9 - a**9*b*c**2*d**8 - 11*a**9*b*c*d**9*x - a**8*b**2*c**3* d**7 - 11*a**8*b**2*c**2*d**8*x - 55*a**8*b**2*c*d**9*x**2 - a**7*b**3*c** 4*d**6 - 11*a**7*b**3*c**3*d**7*x - 55*a**7*b**3*c**2*d**8*x**2 - 165*a**7 *b**3*c*d**9*x**3 - a**6*b**4*c**5*d**5 - 11*a**6*b**4*c**4*d**6*x - 55*a* *6*b**4*c**3*d**7*x**2 - 165*a**6*b**4*c**2*d**8*x**3 - 330*a**6*b**4*c*d* *9*x**4 - a**5*b**5*c**6*d**4 - 11*a**5*b**5*c**5*d**5*x - 55*a**5*b**5*c* *4*d**6*x**2 - 165*a**5*b**5*c**3*d**7*x**3 - 330*a**5*b**5*c**2*d**8*x**4 - 462*a**5*b**5*c*d**9*x**5 - a**4*b**6*c**7*d**3 - 11*a**4*b**6*c**6*d** 4*x - 55*a**4*b**6*c**5*d**5*x**2 - 165*a**4*b**6*c**4*d**6*x**3 - 330*a** 4*b**6*c**3*d**7*x**4 - 462*a**4*b**6*c**2*d**8*x**5 - 462*a**4*b**6*c*d** 9*x**6 - a**3*b**7*c**8*d**2 - 11*a**3*b**7*c**7*d**3*x - 55*a**3*b**7*c** 6*d**4*x**2 - 165*a**3*b**7*c**5*d**5*x**3 - 330*a**3*b**7*c**4*d**6*x**4 - 462*a**3*b**7*c**3*d**7*x**5 - 462*a**3*b**7*c**2*d**8*x**6 - 330*a**3*b **7*c*d**9*x**7 - a**2*b**8*c**9*d - 11*a**2*b**8*c**8*d**2*x - 55*a**2*b* *8*c**7*d**3*x**2 - 165*a**2*b**8*c**6*d**4*x**3 - 330*a**2*b**8*c**5*d**5 *x**4 - 462*a**2*b**8*c**4*d**6*x**5 - 462*a**2*b**8*c**3*d**7*x**6 - 330* a**2*b**8*c**2*d**8*x**7 - 165*a**2*b**8*c*d**9*x**8 - a*b**9*c**10 - 11*a *b**9*c**9*d*x - 55*a*b**9*c**8*d**2*x**2 - 165*a*b**9*c**7*d**3*x**3 - 33 0*a*b**9*c**6*d**4*x**4 - 462*a*b**9*c**5*d**5*x**5 - 462*a*b**9*c**4*d**6 *x**6 - 330*a*b**9*c**3*d**7*x**7 - 165*a*b**9*c**2*d**8*x**8 - 55*a*b*...