Integrand size = 15, antiderivative size = 89 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=-\frac {(c+d x)^{11}}{13 (b c-a d) (a+b x)^{13}}+\frac {d (c+d x)^{11}}{78 (b c-a d)^2 (a+b x)^{12}}-\frac {d^2 (c+d x)^{11}}{858 (b c-a d)^3 (a+b x)^{11}} \]
-1/13*(d*x+c)^11/(-a*d+b*c)/(b*x+a)^13+1/78*d*(d*x+c)^11/(-a*d+b*c)^2/(b*x +a)^12-1/858*d^2*(d*x+c)^11/(-a*d+b*c)^3/(b*x+a)^11
Leaf count is larger than twice the leaf count of optimal. \(690\) vs. \(2(89)=178\).
Time = 0.18 (sec) , antiderivative size = 690, normalized size of antiderivative = 7.75 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=-\frac {a^{10} d^{10}+a^9 b d^9 (3 c+13 d x)+3 a^8 b^2 d^8 \left (2 c^2+13 c d x+26 d^2 x^2\right )+2 a^7 b^3 d^7 \left (5 c^3+39 c^2 d x+117 c d^2 x^2+143 d^3 x^3\right )+a^6 b^4 d^6 \left (15 c^4+130 c^3 d x+468 c^2 d^2 x^2+858 c d^3 x^3+715 d^4 x^4\right )+3 a^5 b^5 d^5 \left (7 c^5+65 c^4 d x+260 c^3 d^2 x^2+572 c^2 d^3 x^3+715 c d^4 x^4+429 d^5 x^5\right )+a^4 b^6 d^4 \left (28 c^6+273 c^5 d x+1170 c^4 d^2 x^2+2860 c^3 d^3 x^3+4290 c^2 d^4 x^4+3861 c d^5 x^5+1716 d^6 x^6\right )+2 a^3 b^7 d^3 \left (18 c^7+182 c^6 d x+819 c^5 d^2 x^2+2145 c^4 d^3 x^3+3575 c^3 d^4 x^4+3861 c^2 d^5 x^5+2574 c d^6 x^6+858 d^7 x^7\right )+3 a^2 b^8 d^2 \left (15 c^8+156 c^7 d x+728 c^6 d^2 x^2+2002 c^5 d^3 x^3+3575 c^4 d^4 x^4+4290 c^3 d^5 x^5+3432 c^2 d^6 x^6+1716 c d^7 x^7+429 d^8 x^8\right )+a b^9 d \left (55 c^9+585 c^8 d x+2808 c^7 d^2 x^2+8008 c^6 d^3 x^3+15015 c^5 d^4 x^4+19305 c^4 d^5 x^5+17160 c^3 d^6 x^6+10296 c^2 d^7 x^7+3861 c d^8 x^8+715 d^9 x^9\right )+b^{10} \left (66 c^{10}+715 c^9 d x+3510 c^8 d^2 x^2+10296 c^7 d^3 x^3+20020 c^6 d^4 x^4+27027 c^5 d^5 x^5+25740 c^4 d^6 x^6+17160 c^3 d^7 x^7+7722 c^2 d^8 x^8+2145 c d^9 x^9+286 d^{10} x^{10}\right )}{858 b^{11} (a+b x)^{13}} \]
-1/858*(a^10*d^10 + a^9*b*d^9*(3*c + 13*d*x) + 3*a^8*b^2*d^8*(2*c^2 + 13*c *d*x + 26*d^2*x^2) + 2*a^7*b^3*d^7*(5*c^3 + 39*c^2*d*x + 117*c*d^2*x^2 + 1 43*d^3*x^3) + a^6*b^4*d^6*(15*c^4 + 130*c^3*d*x + 468*c^2*d^2*x^2 + 858*c* d^3*x^3 + 715*d^4*x^4) + 3*a^5*b^5*d^5*(7*c^5 + 65*c^4*d*x + 260*c^3*d^2*x ^2 + 572*c^2*d^3*x^3 + 715*c*d^4*x^4 + 429*d^5*x^5) + a^4*b^6*d^4*(28*c^6 + 273*c^5*d*x + 1170*c^4*d^2*x^2 + 2860*c^3*d^3*x^3 + 4290*c^2*d^4*x^4 + 3 861*c*d^5*x^5 + 1716*d^6*x^6) + 2*a^3*b^7*d^3*(18*c^7 + 182*c^6*d*x + 819* c^5*d^2*x^2 + 2145*c^4*d^3*x^3 + 3575*c^3*d^4*x^4 + 3861*c^2*d^5*x^5 + 257 4*c*d^6*x^6 + 858*d^7*x^7) + 3*a^2*b^8*d^2*(15*c^8 + 156*c^7*d*x + 728*c^6 *d^2*x^2 + 2002*c^5*d^3*x^3 + 3575*c^4*d^4*x^4 + 4290*c^3*d^5*x^5 + 3432*c ^2*d^6*x^6 + 1716*c*d^7*x^7 + 429*d^8*x^8) + a*b^9*d*(55*c^9 + 585*c^8*d*x + 2808*c^7*d^2*x^2 + 8008*c^6*d^3*x^3 + 15015*c^5*d^4*x^4 + 19305*c^4*d^5 *x^5 + 17160*c^3*d^6*x^6 + 10296*c^2*d^7*x^7 + 3861*c*d^8*x^8 + 715*d^9*x^ 9) + b^10*(66*c^10 + 715*c^9*d*x + 3510*c^8*d^2*x^2 + 10296*c^7*d^3*x^3 + 20020*c^6*d^4*x^4 + 27027*c^5*d^5*x^5 + 25740*c^4*d^6*x^6 + 17160*c^3*d^7* x^7 + 7722*c^2*d^8*x^8 + 2145*c*d^9*x^9 + 286*d^10*x^10))/(b^11*(a + b*x)^ 13)
Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {55, 55, 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.
| \(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {2 d \int \frac {(c+d x)^{10}}{(a+b x)^{13}}dx}{13 (b c-a d)}-\frac {(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {2 d \left (-\frac {d \int \frac {(c+d x)^{10}}{(a+b x)^{12}}dx}{12 (b c-a d)}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)}\right )}{13 (b c-a d)}-\frac {(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {(c+d x)^{11}}{13 (a+b x)^{13} (b c-a d)}-\frac {2 d \left (\frac {d (c+d x)^{11}}{132 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)}\right )}{13 (b c-a d)}\) |
-1/13*(c + d*x)^11/((b*c - a*d)*(a + b*x)^13) - (2*d*(-1/12*(c + d*x)^11/( (b*c - a*d)*(a + b*x)^12) + (d*(c + d*x)^11)/(132*(b*c - a*d)^2*(a + b*x)^ 11)))/(13*(b*c - a*d))
3.14.25.3.1 Defintions of rubi rules used
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]
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] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Leaf count of result is larger than twice the leaf count of optimal. \(830\) vs. \(2(83)=166\).
Time = 0.23 (sec) , antiderivative size = 831, normalized size of antiderivative = 9.34
| method | result | size |
| risch | \(\frac {-\frac {d^{10} x^{10}}{3 b}-\frac {5 d^{9} \left (a d +3 b c \right ) x^{9}}{6 b^{2}}-\frac {3 d^{8} \left (a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}\right ) x^{8}}{2 b^{3}}-\frac {2 d^{7} \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +10 b^{3} c^{3}\right ) x^{7}}{b^{4}}-\frac {2 d^{6} \left (a^{4} d^{4}+3 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+10 a \,b^{3} c^{3} d +15 b^{4} c^{4}\right ) x^{6}}{b^{5}}-\frac {3 d^{5} \left (a^{5} d^{5}+3 a^{4} b c \,d^{4}+6 a^{3} b^{2} c^{2} d^{3}+10 a^{2} b^{3} c^{3} d^{2}+15 a \,b^{4} c^{4} d +21 b^{5} c^{5}\right ) x^{5}}{2 b^{6}}-\frac {5 d^{4} \left (a^{6} d^{6}+3 a^{5} b c \,d^{5}+6 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}+21 a \,b^{5} c^{5} d +28 b^{6} c^{6}\right ) x^{4}}{6 b^{7}}-\frac {d^{3} \left (a^{7} d^{7}+3 a^{6} b c \,d^{6}+6 a^{5} b^{2} c^{2} d^{5}+10 a^{4} b^{3} c^{3} d^{4}+15 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+28 a \,b^{6} c^{6} d +36 b^{7} c^{7}\right ) x^{3}}{3 b^{8}}-\frac {d^{2} \left (a^{8} d^{8}+3 a^{7} b c \,d^{7}+6 a^{6} b^{2} c^{2} d^{6}+10 a^{5} b^{3} c^{3} d^{5}+15 a^{4} b^{4} c^{4} d^{4}+21 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}+36 a \,b^{7} c^{7} d +45 b^{8} c^{8}\right ) x^{2}}{11 b^{9}}-\frac {d \left (a^{9} d^{9}+3 a^{8} b c \,d^{8}+6 a^{7} b^{2} c^{2} d^{7}+10 a^{6} b^{3} c^{3} d^{6}+15 a^{5} b^{4} c^{4} d^{5}+21 a^{4} b^{5} c^{5} d^{4}+28 a^{3} b^{6} c^{6} d^{3}+36 a^{2} b^{7} c^{7} d^{2}+45 a \,b^{8} c^{8} d +55 b^{9} c^{9}\right ) x}{66 b^{10}}-\frac {a^{10} d^{10}+3 a^{9} b c \,d^{9}+6 a^{8} b^{2} c^{2} d^{8}+10 a^{7} b^{3} c^{3} d^{7}+15 a^{6} b^{4} c^{4} d^{6}+21 a^{5} b^{5} c^{5} d^{5}+28 a^{4} b^{6} c^{6} d^{4}+36 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+55 a \,b^{9} c^{9} d +66 b^{10} c^{10}}{858 b^{11}}}{\left (b x +a \right )^{13}}\) | \(831\) |
| default | \(-\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 +b^{8} c^{8}\right )}{11 b^{11} \left (b x +a \right )^{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}}{13 b^{11} \left (b x +a \right )^{13}}-\frac {d^{10}}{3 b^{11} \left (b x +a \right )^{3}}-\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 +b^{6} c^{6}\right )}{3 b^{11} \left (b x +a \right )^{9}}+\frac {20 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 )^{6}}+\frac {63 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{2 b^{11} \left (b x +a \right )^{8}}+\frac {5 d^{9} \left (a d -b c \right )}{2 b^{11} \left (b x +a \right )^{4}}+\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 -b^{9} c^{9}\right )}{6 b^{11} \left (b x +a \right )^{12}}-\frac {30 d^{6} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{b^{11} \left (b x +a \right )^{7}}+\frac {12 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 )^{10}}-\frac {9 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{11} \left (b x +a \right )^{5}}\) | \(867\) |
| norman | \(\frac {-\frac {d^{10} x^{10}}{3 b}+\frac {5 \left (-a \,b^{2} d^{10}-3 b^{3} c \,d^{9}\right ) x^{9}}{6 b^{4}}+\frac {3 \left (-a^{2} b^{2} d^{10}-3 a \,b^{3} c \,d^{9}-6 b^{4} c^{2} d^{8}\right ) x^{8}}{2 b^{5}}+\frac {2 \left (-a^{3} b^{2} d^{10}-3 a^{2} b^{3} c \,d^{9}-6 a \,b^{4} c^{2} d^{8}-10 b^{5} c^{3} d^{7}\right ) x^{7}}{b^{6}}+\frac {2 \left (-a^{4} b^{2} d^{10}-3 a^{3} b^{3} c \,d^{9}-6 a^{2} b^{4} c^{2} d^{8}-10 a \,b^{5} c^{3} d^{7}-15 b^{6} c^{4} d^{6}\right ) x^{6}}{b^{7}}+\frac {3 \left (-a^{5} b^{2} d^{10}-3 a^{4} b^{3} c \,d^{9}-6 a^{3} b^{4} c^{2} d^{8}-10 a^{2} b^{5} c^{3} d^{7}-15 a \,b^{6} c^{4} d^{6}-21 b^{7} c^{5} d^{5}\right ) x^{5}}{2 b^{8}}+\frac {5 \left (-a^{6} b^{2} d^{10}-3 a^{5} b^{3} c \,d^{9}-6 a^{4} b^{4} c^{2} d^{8}-10 a^{3} b^{5} c^{3} d^{7}-15 a^{2} b^{6} c^{4} d^{6}-21 a \,b^{7} c^{5} d^{5}-28 b^{8} c^{6} d^{4}\right ) x^{4}}{6 b^{9}}+\frac {\left (-a^{7} b^{2} d^{10}-3 a^{6} b^{3} c \,d^{9}-6 a^{5} b^{4} c^{2} d^{8}-10 a^{4} b^{5} c^{3} d^{7}-15 a^{3} b^{6} c^{4} d^{6}-21 a^{2} b^{7} c^{5} d^{5}-28 a \,b^{8} c^{6} d^{4}-36 b^{9} c^{7} d^{3}\right ) x^{3}}{3 b^{10}}+\frac {\left (-a^{8} b^{2} d^{10}-3 a^{7} b^{3} c \,d^{9}-6 a^{6} b^{4} c^{2} d^{8}-10 a^{5} b^{5} c^{3} d^{7}-15 a^{4} b^{6} c^{4} d^{6}-21 a^{3} b^{7} c^{5} d^{5}-28 a^{2} b^{8} c^{6} d^{4}-36 a \,b^{9} c^{7} d^{3}-45 b^{10} c^{8} d^{2}\right ) x^{2}}{11 b^{11}}+\frac {\left (-a^{9} b^{2} d^{10}-3 a^{8} b^{3} c \,d^{9}-6 a^{7} b^{4} c^{2} d^{8}-10 a^{6} b^{5} c^{3} d^{7}-15 a^{5} b^{6} c^{4} d^{6}-21 a^{4} b^{7} c^{5} d^{5}-28 a^{3} b^{8} c^{6} d^{4}-36 a^{2} b^{9} c^{7} d^{3}-45 a \,b^{10} c^{8} d^{2}-55 b^{11} c^{9} d \right ) x}{66 b^{12}}+\frac {-a^{10} b^{2} d^{10}-3 a^{9} b^{3} c \,d^{9}-6 a^{8} b^{4} c^{2} d^{8}-10 a^{7} b^{5} c^{3} d^{7}-15 a^{6} b^{6} c^{4} d^{6}-21 a^{5} b^{7} c^{5} d^{5}-28 a^{4} b^{8} c^{6} d^{4}-36 a^{3} b^{9} c^{7} d^{3}-45 a^{2} b^{10} c^{8} d^{2}-55 a \,b^{11} c^{9} d -66 b^{12} c^{10}}{858 b^{13}}}{\left (b x +a \right )^{13}}\) | \(909\) |
| gosper | \(-\frac {286 x^{10} d^{10} b^{10}+715 x^{9} a \,b^{9} d^{10}+2145 x^{9} b^{10} c \,d^{9}+1287 x^{8} a^{2} b^{8} d^{10}+3861 x^{8} a \,b^{9} c \,d^{9}+7722 x^{8} b^{10} c^{2} d^{8}+1716 x^{7} a^{3} b^{7} d^{10}+5148 x^{7} a^{2} b^{8} c \,d^{9}+10296 x^{7} a \,b^{9} c^{2} d^{8}+17160 x^{7} b^{10} c^{3} d^{7}+1716 x^{6} a^{4} b^{6} d^{10}+5148 x^{6} a^{3} b^{7} c \,d^{9}+10296 x^{6} a^{2} b^{8} c^{2} d^{8}+17160 x^{6} a \,b^{9} c^{3} d^{7}+25740 x^{6} b^{10} c^{4} d^{6}+1287 x^{5} a^{5} b^{5} d^{10}+3861 x^{5} a^{4} b^{6} c \,d^{9}+7722 x^{5} a^{3} b^{7} c^{2} d^{8}+12870 x^{5} a^{2} b^{8} c^{3} d^{7}+19305 x^{5} a \,b^{9} c^{4} d^{6}+27027 x^{5} b^{10} c^{5} d^{5}+715 x^{4} a^{6} b^{4} d^{10}+2145 x^{4} a^{5} b^{5} c \,d^{9}+4290 x^{4} a^{4} b^{6} c^{2} d^{8}+7150 x^{4} a^{3} b^{7} c^{3} d^{7}+10725 x^{4} a^{2} b^{8} c^{4} d^{6}+15015 x^{4} a \,b^{9} c^{5} d^{5}+20020 x^{4} b^{10} c^{6} d^{4}+286 x^{3} a^{7} b^{3} d^{10}+858 x^{3} a^{6} b^{4} c \,d^{9}+1716 x^{3} a^{5} b^{5} c^{2} d^{8}+2860 x^{3} a^{4} b^{6} c^{3} d^{7}+4290 x^{3} a^{3} b^{7} c^{4} d^{6}+6006 x^{3} a^{2} b^{8} c^{5} d^{5}+8008 x^{3} a \,b^{9} c^{6} d^{4}+10296 x^{3} b^{10} c^{7} d^{3}+78 x^{2} a^{8} b^{2} d^{10}+234 x^{2} a^{7} b^{3} c \,d^{9}+468 x^{2} a^{6} b^{4} c^{2} d^{8}+780 x^{2} a^{5} b^{5} c^{3} d^{7}+1170 x^{2} a^{4} b^{6} c^{4} d^{6}+1638 x^{2} a^{3} b^{7} c^{5} d^{5}+2184 x^{2} a^{2} b^{8} c^{6} d^{4}+2808 x^{2} a \,b^{9} c^{7} d^{3}+3510 x^{2} b^{10} c^{8} d^{2}+13 x \,a^{9} b \,d^{10}+39 x \,a^{8} b^{2} c \,d^{9}+78 x \,a^{7} b^{3} c^{2} d^{8}+130 x \,a^{6} b^{4} c^{3} d^{7}+195 x \,a^{5} b^{5} c^{4} d^{6}+273 x \,a^{4} b^{6} c^{5} d^{5}+364 x \,a^{3} b^{7} c^{6} d^{4}+468 x \,a^{2} b^{8} c^{7} d^{3}+585 x a \,b^{9} c^{8} d^{2}+715 x \,b^{10} c^{9} d +a^{10} d^{10}+3 a^{9} b c \,d^{9}+6 a^{8} b^{2} c^{2} d^{8}+10 a^{7} b^{3} c^{3} d^{7}+15 a^{6} b^{4} c^{4} d^{6}+21 a^{5} b^{5} c^{5} d^{5}+28 a^{4} b^{6} c^{6} d^{4}+36 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+55 a \,b^{9} c^{9} d +66 b^{10} c^{10}}{858 b^{11} \left (b x +a \right )^{13}}\) | \(962\) |
| parallelrisch | \(\frac {-286 d^{10} x^{10} b^{12}-715 a \,b^{11} d^{10} x^{9}-2145 b^{12} c \,d^{9} x^{9}-1287 a^{2} b^{10} d^{10} x^{8}-3861 a \,b^{11} c \,d^{9} x^{8}-7722 b^{12} c^{2} d^{8} x^{8}-1716 a^{3} b^{9} d^{10} x^{7}-5148 a^{2} b^{10} c \,d^{9} x^{7}-10296 a \,b^{11} c^{2} d^{8} x^{7}-17160 b^{12} c^{3} d^{7} x^{7}-1716 a^{4} b^{8} d^{10} x^{6}-5148 a^{3} b^{9} c \,d^{9} x^{6}-10296 a^{2} b^{10} c^{2} d^{8} x^{6}-17160 a \,b^{11} c^{3} d^{7} x^{6}-25740 b^{12} c^{4} d^{6} x^{6}-1287 a^{5} b^{7} d^{10} x^{5}-3861 a^{4} b^{8} c \,d^{9} x^{5}-7722 a^{3} b^{9} c^{2} d^{8} x^{5}-12870 a^{2} b^{10} c^{3} d^{7} x^{5}-19305 a \,b^{11} c^{4} d^{6} x^{5}-27027 b^{12} c^{5} d^{5} x^{5}-715 a^{6} b^{6} d^{10} x^{4}-2145 a^{5} b^{7} c \,d^{9} x^{4}-4290 a^{4} b^{8} c^{2} d^{8} x^{4}-7150 a^{3} b^{9} c^{3} d^{7} x^{4}-10725 a^{2} b^{10} c^{4} d^{6} x^{4}-15015 a \,b^{11} c^{5} d^{5} x^{4}-20020 b^{12} c^{6} d^{4} x^{4}-286 a^{7} b^{5} d^{10} x^{3}-858 a^{6} b^{6} c \,d^{9} x^{3}-1716 a^{5} b^{7} c^{2} d^{8} x^{3}-2860 a^{4} b^{8} c^{3} d^{7} x^{3}-4290 a^{3} b^{9} c^{4} d^{6} x^{3}-6006 a^{2} b^{10} c^{5} d^{5} x^{3}-8008 a \,b^{11} c^{6} d^{4} x^{3}-10296 b^{12} c^{7} d^{3} x^{3}-78 a^{8} b^{4} d^{10} x^{2}-234 a^{7} b^{5} c \,d^{9} x^{2}-468 a^{6} b^{6} c^{2} d^{8} x^{2}-780 a^{5} b^{7} c^{3} d^{7} x^{2}-1170 a^{4} b^{8} c^{4} d^{6} x^{2}-1638 a^{3} b^{9} c^{5} d^{5} x^{2}-2184 a^{2} b^{10} c^{6} d^{4} x^{2}-2808 a \,b^{11} c^{7} d^{3} x^{2}-3510 b^{12} c^{8} d^{2} x^{2}-13 a^{9} b^{3} d^{10} x -39 a^{8} b^{4} c \,d^{9} x -78 a^{7} b^{5} c^{2} d^{8} x -130 a^{6} b^{6} c^{3} d^{7} x -195 a^{5} b^{7} c^{4} d^{6} x -273 a^{4} b^{8} c^{5} d^{5} x -364 a^{3} b^{9} c^{6} d^{4} x -468 a^{2} b^{10} c^{7} d^{3} x -585 a \,b^{11} c^{8} d^{2} x -715 b^{12} c^{9} d x -a^{10} b^{2} d^{10}-3 a^{9} b^{3} c \,d^{9}-6 a^{8} b^{4} c^{2} d^{8}-10 a^{7} b^{5} c^{3} d^{7}-15 a^{6} b^{6} c^{4} d^{6}-21 a^{5} b^{7} c^{5} d^{5}-28 a^{4} b^{8} c^{6} d^{4}-36 a^{3} b^{9} c^{7} d^{3}-45 a^{2} b^{10} c^{8} d^{2}-55 a \,b^{11} c^{9} d -66 b^{12} c^{10}}{858 b^{13} \left (b x +a \right )^{13}}\) | \(970\) |
(-1/3/b*d^10*x^10-5/6/b^2*d^9*(a*d+3*b*c)*x^9-3/2/b^3*d^8*(a^2*d^2+3*a*b*c *d+6*b^2*c^2)*x^8-2/b^4*d^7*(a^3*d^3+3*a^2*b*c*d^2+6*a*b^2*c^2*d+10*b^3*c^ 3)*x^7-2/b^5*d^6*(a^4*d^4+3*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2+10*a*b^3*c^3*d+1 5*b^4*c^4)*x^6-3/2/b^6*d^5*(a^5*d^5+3*a^4*b*c*d^4+6*a^3*b^2*c^2*d^3+10*a^2 *b^3*c^3*d^2+15*a*b^4*c^4*d+21*b^5*c^5)*x^5-5/6/b^7*d^4*(a^6*d^6+3*a^5*b*c *d^5+6*a^4*b^2*c^2*d^4+10*a^3*b^3*c^3*d^3+15*a^2*b^4*c^4*d^2+21*a*b^5*c^5* d+28*b^6*c^6)*x^4-1/3/b^8*d^3*(a^7*d^7+3*a^6*b*c*d^6+6*a^5*b^2*c^2*d^5+10* a^4*b^3*c^3*d^4+15*a^3*b^4*c^4*d^3+21*a^2*b^5*c^5*d^2+28*a*b^6*c^6*d+36*b^ 7*c^7)*x^3-1/11/b^9*d^2*(a^8*d^8+3*a^7*b*c*d^7+6*a^6*b^2*c^2*d^6+10*a^5*b^ 3*c^3*d^5+15*a^4*b^4*c^4*d^4+21*a^3*b^5*c^5*d^3+28*a^2*b^6*c^6*d^2+36*a*b^ 7*c^7*d+45*b^8*c^8)*x^2-1/66/b^10*d*(a^9*d^9+3*a^8*b*c*d^8+6*a^7*b^2*c^2*d ^7+10*a^6*b^3*c^3*d^6+15*a^5*b^4*c^4*d^5+21*a^4*b^5*c^5*d^4+28*a^3*b^6*c^6 *d^3+36*a^2*b^7*c^7*d^2+45*a*b^8*c^8*d+55*b^9*c^9)*x-1/858/b^11*(a^10*d^10 +3*a^9*b*c*d^9+6*a^8*b^2*c^2*d^8+10*a^7*b^3*c^3*d^7+15*a^6*b^4*c^4*d^6+21* a^5*b^5*c^5*d^5+28*a^4*b^6*c^6*d^4+36*a^3*b^7*c^7*d^3+45*a^2*b^8*c^8*d^2+5 5*a*b^9*c^9*d+66*b^10*c^10))/(b*x+a)^13
Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (83) = 166\).
Time = 0.24 (sec) , antiderivative size = 997, normalized size of antiderivative = 11.20 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=-\frac {286 \, b^{10} d^{10} x^{10} + 66 \, b^{10} c^{10} + 55 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 36 \, a^{3} b^{7} c^{7} d^{3} + 28 \, a^{4} b^{6} c^{6} d^{4} + 21 \, a^{5} b^{5} c^{5} d^{5} + 15 \, a^{6} b^{4} c^{4} d^{6} + 10 \, a^{7} b^{3} c^{3} d^{7} + 6 \, a^{8} b^{2} c^{2} d^{8} + 3 \, a^{9} b c d^{9} + a^{10} d^{10} + 715 \, {\left (3 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 1287 \, {\left (6 \, b^{10} c^{2} d^{8} + 3 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 1716 \, {\left (10 \, b^{10} c^{3} d^{7} + 6 \, a b^{9} c^{2} d^{8} + 3 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 1716 \, {\left (15 \, b^{10} c^{4} d^{6} + 10 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} + 3 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 1287 \, {\left (21 \, b^{10} c^{5} d^{5} + 15 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} + 6 \, a^{3} b^{7} c^{2} d^{8} + 3 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 715 \, {\left (28 \, b^{10} c^{6} d^{4} + 21 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} + 10 \, a^{3} b^{7} c^{3} d^{7} + 6 \, a^{4} b^{6} c^{2} d^{8} + 3 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 286 \, {\left (36 \, b^{10} c^{7} d^{3} + 28 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 15 \, a^{3} b^{7} c^{4} d^{6} + 10 \, a^{4} b^{6} c^{3} d^{7} + 6 \, a^{5} b^{5} c^{2} d^{8} + 3 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 78 \, {\left (45 \, b^{10} c^{8} d^{2} + 36 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 21 \, a^{3} b^{7} c^{5} d^{5} + 15 \, a^{4} b^{6} c^{4} d^{6} + 10 \, a^{5} b^{5} c^{3} d^{7} + 6 \, a^{6} b^{4} c^{2} d^{8} + 3 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 13 \, {\left (55 \, b^{10} c^{9} d + 45 \, a b^{9} c^{8} d^{2} + 36 \, a^{2} b^{8} c^{7} d^{3} + 28 \, a^{3} b^{7} c^{6} d^{4} + 21 \, a^{4} b^{6} c^{5} d^{5} + 15 \, a^{5} b^{5} c^{4} d^{6} + 10 \, a^{6} b^{4} c^{3} d^{7} + 6 \, a^{7} b^{3} c^{2} d^{8} + 3 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{858 \, {\left (b^{24} x^{13} + 13 \, a b^{23} x^{12} + 78 \, a^{2} b^{22} x^{11} + 286 \, a^{3} b^{21} x^{10} + 715 \, a^{4} b^{20} x^{9} + 1287 \, a^{5} b^{19} x^{8} + 1716 \, a^{6} b^{18} x^{7} + 1716 \, a^{7} b^{17} x^{6} + 1287 \, a^{8} b^{16} x^{5} + 715 \, a^{9} b^{15} x^{4} + 286 \, a^{10} b^{14} x^{3} + 78 \, a^{11} b^{13} x^{2} + 13 \, a^{12} b^{12} x + a^{13} b^{11}\right )}} \]
-1/858*(286*b^10*d^10*x^10 + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8*c^ 8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 + 15* a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 + 3*a^9*b*c*d^9 + a^10*d^10 + 715*(3*b^10*c*d^9 + a*b^9*d^10)*x^9 + 1287*(6*b^10*c^2*d^8 + 3*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 1716*(10*b^10*c^3*d^7 + 6*a*b^9*c^2*d^ 8 + 3*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 1716*(15*b^10*c^4*d^6 + 10*a*b^9 *c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 3*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 1287* (21*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d ^8 + 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 715*(28*b^10*c^6*d^4 + 21*a*b^9 *c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 6*a^4*b^6*c^2*d^8 + 3 *a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 286*(36*b^10*c^7*d^3 + 28*a*b^9*c^6*d ^4 + 21*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7 + 6*a^5* b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 78*(45*b^10*c^8*d^2 + 36*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 21*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^ 4*d^6 + 10*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 + 3*a^7*b^3*c*d^9 + a^8*b^2 *d^10)*x^2 + 13*(55*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 + 2 8*a^3*b^7*c^6*d^4 + 21*a^4*b^6*c^5*d^5 + 15*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c ^3*d^7 + 6*a^7*b^3*c^2*d^8 + 3*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^24*x^13 + 13*a*b^23*x^12 + 78*a^2*b^22*x^11 + 286*a^3*b^21*x^10 + 715*a^4*b^20*x^9 + 1287*a^5*b^19*x^8 + 1716*a^6*b^18*x^7 + 1716*a^7*b^17*x^6 + 1287*a^8*...
Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (83) = 166\).
Time = 0.25 (sec) , antiderivative size = 997, normalized size of antiderivative = 11.20 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=-\frac {286 \, b^{10} d^{10} x^{10} + 66 \, b^{10} c^{10} + 55 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 36 \, a^{3} b^{7} c^{7} d^{3} + 28 \, a^{4} b^{6} c^{6} d^{4} + 21 \, a^{5} b^{5} c^{5} d^{5} + 15 \, a^{6} b^{4} c^{4} d^{6} + 10 \, a^{7} b^{3} c^{3} d^{7} + 6 \, a^{8} b^{2} c^{2} d^{8} + 3 \, a^{9} b c d^{9} + a^{10} d^{10} + 715 \, {\left (3 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 1287 \, {\left (6 \, b^{10} c^{2} d^{8} + 3 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 1716 \, {\left (10 \, b^{10} c^{3} d^{7} + 6 \, a b^{9} c^{2} d^{8} + 3 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 1716 \, {\left (15 \, b^{10} c^{4} d^{6} + 10 \, a b^{9} c^{3} d^{7} + 6 \, a^{2} b^{8} c^{2} d^{8} + 3 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 1287 \, {\left (21 \, b^{10} c^{5} d^{5} + 15 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} + 6 \, a^{3} b^{7} c^{2} d^{8} + 3 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 715 \, {\left (28 \, b^{10} c^{6} d^{4} + 21 \, a b^{9} c^{5} d^{5} + 15 \, a^{2} b^{8} c^{4} d^{6} + 10 \, a^{3} b^{7} c^{3} d^{7} + 6 \, a^{4} b^{6} c^{2} d^{8} + 3 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 286 \, {\left (36 \, b^{10} c^{7} d^{3} + 28 \, a b^{9} c^{6} d^{4} + 21 \, a^{2} b^{8} c^{5} d^{5} + 15 \, a^{3} b^{7} c^{4} d^{6} + 10 \, a^{4} b^{6} c^{3} d^{7} + 6 \, a^{5} b^{5} c^{2} d^{8} + 3 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 78 \, {\left (45 \, b^{10} c^{8} d^{2} + 36 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 21 \, a^{3} b^{7} c^{5} d^{5} + 15 \, a^{4} b^{6} c^{4} d^{6} + 10 \, a^{5} b^{5} c^{3} d^{7} + 6 \, a^{6} b^{4} c^{2} d^{8} + 3 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 13 \, {\left (55 \, b^{10} c^{9} d + 45 \, a b^{9} c^{8} d^{2} + 36 \, a^{2} b^{8} c^{7} d^{3} + 28 \, a^{3} b^{7} c^{6} d^{4} + 21 \, a^{4} b^{6} c^{5} d^{5} + 15 \, a^{5} b^{5} c^{4} d^{6} + 10 \, a^{6} b^{4} c^{3} d^{7} + 6 \, a^{7} b^{3} c^{2} d^{8} + 3 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{858 \, {\left (b^{24} x^{13} + 13 \, a b^{23} x^{12} + 78 \, a^{2} b^{22} x^{11} + 286 \, a^{3} b^{21} x^{10} + 715 \, a^{4} b^{20} x^{9} + 1287 \, a^{5} b^{19} x^{8} + 1716 \, a^{6} b^{18} x^{7} + 1716 \, a^{7} b^{17} x^{6} + 1287 \, a^{8} b^{16} x^{5} + 715 \, a^{9} b^{15} x^{4} + 286 \, a^{10} b^{14} x^{3} + 78 \, a^{11} b^{13} x^{2} + 13 \, a^{12} b^{12} x + a^{13} b^{11}\right )}} \]
-1/858*(286*b^10*d^10*x^10 + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8*c^ 8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 + 15* a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 + 3*a^9*b*c*d^9 + a^10*d^10 + 715*(3*b^10*c*d^9 + a*b^9*d^10)*x^9 + 1287*(6*b^10*c^2*d^8 + 3*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 1716*(10*b^10*c^3*d^7 + 6*a*b^9*c^2*d^ 8 + 3*a^2*b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 1716*(15*b^10*c^4*d^6 + 10*a*b^9 *c^3*d^7 + 6*a^2*b^8*c^2*d^8 + 3*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 1287* (21*b^10*c^5*d^5 + 15*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 + 6*a^3*b^7*c^2*d ^8 + 3*a^4*b^6*c*d^9 + a^5*b^5*d^10)*x^5 + 715*(28*b^10*c^6*d^4 + 21*a*b^9 *c^5*d^5 + 15*a^2*b^8*c^4*d^6 + 10*a^3*b^7*c^3*d^7 + 6*a^4*b^6*c^2*d^8 + 3 *a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 286*(36*b^10*c^7*d^3 + 28*a*b^9*c^6*d ^4 + 21*a^2*b^8*c^5*d^5 + 15*a^3*b^7*c^4*d^6 + 10*a^4*b^6*c^3*d^7 + 6*a^5* b^5*c^2*d^8 + 3*a^6*b^4*c*d^9 + a^7*b^3*d^10)*x^3 + 78*(45*b^10*c^8*d^2 + 36*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 21*a^3*b^7*c^5*d^5 + 15*a^4*b^6*c^ 4*d^6 + 10*a^5*b^5*c^3*d^7 + 6*a^6*b^4*c^2*d^8 + 3*a^7*b^3*c*d^9 + a^8*b^2 *d^10)*x^2 + 13*(55*b^10*c^9*d + 45*a*b^9*c^8*d^2 + 36*a^2*b^8*c^7*d^3 + 2 8*a^3*b^7*c^6*d^4 + 21*a^4*b^6*c^5*d^5 + 15*a^5*b^5*c^4*d^6 + 10*a^6*b^4*c ^3*d^7 + 6*a^7*b^3*c^2*d^8 + 3*a^8*b^2*c*d^9 + a^9*b*d^10)*x)/(b^24*x^13 + 13*a*b^23*x^12 + 78*a^2*b^22*x^11 + 286*a^3*b^21*x^10 + 715*a^4*b^20*x^9 + 1287*a^5*b^19*x^8 + 1716*a^6*b^18*x^7 + 1716*a^7*b^17*x^6 + 1287*a^8*...
Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (83) = 166\).
Time = 0.29 (sec) , antiderivative size = 961, normalized size of antiderivative = 10.80 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=-\frac {286 \, b^{10} d^{10} x^{10} + 2145 \, b^{10} c d^{9} x^{9} + 715 \, a b^{9} d^{10} x^{9} + 7722 \, b^{10} c^{2} d^{8} x^{8} + 3861 \, a b^{9} c d^{9} x^{8} + 1287 \, a^{2} b^{8} d^{10} x^{8} + 17160 \, b^{10} c^{3} d^{7} x^{7} + 10296 \, a b^{9} c^{2} d^{8} x^{7} + 5148 \, a^{2} b^{8} c d^{9} x^{7} + 1716 \, a^{3} b^{7} d^{10} x^{7} + 25740 \, b^{10} c^{4} d^{6} x^{6} + 17160 \, a b^{9} c^{3} d^{7} x^{6} + 10296 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 5148 \, a^{3} b^{7} c d^{9} x^{6} + 1716 \, a^{4} b^{6} d^{10} x^{6} + 27027 \, b^{10} c^{5} d^{5} x^{5} + 19305 \, a b^{9} c^{4} d^{6} x^{5} + 12870 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 7722 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 3861 \, a^{4} b^{6} c d^{9} x^{5} + 1287 \, a^{5} b^{5} d^{10} x^{5} + 20020 \, b^{10} c^{6} d^{4} x^{4} + 15015 \, a b^{9} c^{5} d^{5} x^{4} + 10725 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 7150 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 4290 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 2145 \, a^{5} b^{5} c d^{9} x^{4} + 715 \, a^{6} b^{4} d^{10} x^{4} + 10296 \, b^{10} c^{7} d^{3} x^{3} + 8008 \, a b^{9} c^{6} d^{4} x^{3} + 6006 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 4290 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 2860 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 1716 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 858 \, a^{6} b^{4} c d^{9} x^{3} + 286 \, a^{7} b^{3} d^{10} x^{3} + 3510 \, b^{10} c^{8} d^{2} x^{2} + 2808 \, a b^{9} c^{7} d^{3} x^{2} + 2184 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 1638 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 1170 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 780 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 468 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 234 \, a^{7} b^{3} c d^{9} x^{2} + 78 \, a^{8} b^{2} d^{10} x^{2} + 715 \, b^{10} c^{9} d x + 585 \, a b^{9} c^{8} d^{2} x + 468 \, a^{2} b^{8} c^{7} d^{3} x + 364 \, a^{3} b^{7} c^{6} d^{4} x + 273 \, a^{4} b^{6} c^{5} d^{5} x + 195 \, a^{5} b^{5} c^{4} d^{6} x + 130 \, a^{6} b^{4} c^{3} d^{7} x + 78 \, a^{7} b^{3} c^{2} d^{8} x + 39 \, a^{8} b^{2} c d^{9} x + 13 \, a^{9} b d^{10} x + 66 \, b^{10} c^{10} + 55 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} + 36 \, a^{3} b^{7} c^{7} d^{3} + 28 \, a^{4} b^{6} c^{6} d^{4} + 21 \, a^{5} b^{5} c^{5} d^{5} + 15 \, a^{6} b^{4} c^{4} d^{6} + 10 \, a^{7} b^{3} c^{3} d^{7} + 6 \, a^{8} b^{2} c^{2} d^{8} + 3 \, a^{9} b c d^{9} + a^{10} d^{10}}{858 \, {\left (b x + a\right )}^{13} b^{11}} \]
-1/858*(286*b^10*d^10*x^10 + 2145*b^10*c*d^9*x^9 + 715*a*b^9*d^10*x^9 + 77 22*b^10*c^2*d^8*x^8 + 3861*a*b^9*c*d^9*x^8 + 1287*a^2*b^8*d^10*x^8 + 17160 *b^10*c^3*d^7*x^7 + 10296*a*b^9*c^2*d^8*x^7 + 5148*a^2*b^8*c*d^9*x^7 + 171 6*a^3*b^7*d^10*x^7 + 25740*b^10*c^4*d^6*x^6 + 17160*a*b^9*c^3*d^7*x^6 + 10 296*a^2*b^8*c^2*d^8*x^6 + 5148*a^3*b^7*c*d^9*x^6 + 1716*a^4*b^6*d^10*x^6 + 27027*b^10*c^5*d^5*x^5 + 19305*a*b^9*c^4*d^6*x^5 + 12870*a^2*b^8*c^3*d^7* x^5 + 7722*a^3*b^7*c^2*d^8*x^5 + 3861*a^4*b^6*c*d^9*x^5 + 1287*a^5*b^5*d^1 0*x^5 + 20020*b^10*c^6*d^4*x^4 + 15015*a*b^9*c^5*d^5*x^4 + 10725*a^2*b^8*c ^4*d^6*x^4 + 7150*a^3*b^7*c^3*d^7*x^4 + 4290*a^4*b^6*c^2*d^8*x^4 + 2145*a^ 5*b^5*c*d^9*x^4 + 715*a^6*b^4*d^10*x^4 + 10296*b^10*c^7*d^3*x^3 + 8008*a*b ^9*c^6*d^4*x^3 + 6006*a^2*b^8*c^5*d^5*x^3 + 4290*a^3*b^7*c^4*d^6*x^3 + 286 0*a^4*b^6*c^3*d^7*x^3 + 1716*a^5*b^5*c^2*d^8*x^3 + 858*a^6*b^4*c*d^9*x^3 + 286*a^7*b^3*d^10*x^3 + 3510*b^10*c^8*d^2*x^2 + 2808*a*b^9*c^7*d^3*x^2 + 2 184*a^2*b^8*c^6*d^4*x^2 + 1638*a^3*b^7*c^5*d^5*x^2 + 1170*a^4*b^6*c^4*d^6* x^2 + 780*a^5*b^5*c^3*d^7*x^2 + 468*a^6*b^4*c^2*d^8*x^2 + 234*a^7*b^3*c*d^ 9*x^2 + 78*a^8*b^2*d^10*x^2 + 715*b^10*c^9*d*x + 585*a*b^9*c^8*d^2*x + 468 *a^2*b^8*c^7*d^3*x + 364*a^3*b^7*c^6*d^4*x + 273*a^4*b^6*c^5*d^5*x + 195*a ^5*b^5*c^4*d^6*x + 130*a^6*b^4*c^3*d^7*x + 78*a^7*b^3*c^2*d^8*x + 39*a^8*b ^2*c*d^9*x + 13*a^9*b*d^10*x + 66*b^10*c^10 + 55*a*b^9*c^9*d + 45*a^2*b^8* c^8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^6*d^4 + 21*a^5*b^5*c^5*d^5 ...
Time = 0.50 (sec) , antiderivative size = 1098, normalized size of antiderivative = 12.34 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{14}} \, dx=-\frac {a^{10}\,d^{10}+3\,a^9\,b\,c\,d^9+13\,a^9\,b\,d^{10}\,x+6\,a^8\,b^2\,c^2\,d^8+39\,a^8\,b^2\,c\,d^9\,x+78\,a^8\,b^2\,d^{10}\,x^2+10\,a^7\,b^3\,c^3\,d^7+78\,a^7\,b^3\,c^2\,d^8\,x+234\,a^7\,b^3\,c\,d^9\,x^2+286\,a^7\,b^3\,d^{10}\,x^3+15\,a^6\,b^4\,c^4\,d^6+130\,a^6\,b^4\,c^3\,d^7\,x+468\,a^6\,b^4\,c^2\,d^8\,x^2+858\,a^6\,b^4\,c\,d^9\,x^3+715\,a^6\,b^4\,d^{10}\,x^4+21\,a^5\,b^5\,c^5\,d^5+195\,a^5\,b^5\,c^4\,d^6\,x+780\,a^5\,b^5\,c^3\,d^7\,x^2+1716\,a^5\,b^5\,c^2\,d^8\,x^3+2145\,a^5\,b^5\,c\,d^9\,x^4+1287\,a^5\,b^5\,d^{10}\,x^5+28\,a^4\,b^6\,c^6\,d^4+273\,a^4\,b^6\,c^5\,d^5\,x+1170\,a^4\,b^6\,c^4\,d^6\,x^2+2860\,a^4\,b^6\,c^3\,d^7\,x^3+4290\,a^4\,b^6\,c^2\,d^8\,x^4+3861\,a^4\,b^6\,c\,d^9\,x^5+1716\,a^4\,b^6\,d^{10}\,x^6+36\,a^3\,b^7\,c^7\,d^3+364\,a^3\,b^7\,c^6\,d^4\,x+1638\,a^3\,b^7\,c^5\,d^5\,x^2+4290\,a^3\,b^7\,c^4\,d^6\,x^3+7150\,a^3\,b^7\,c^3\,d^7\,x^4+7722\,a^3\,b^7\,c^2\,d^8\,x^5+5148\,a^3\,b^7\,c\,d^9\,x^6+1716\,a^3\,b^7\,d^{10}\,x^7+45\,a^2\,b^8\,c^8\,d^2+468\,a^2\,b^8\,c^7\,d^3\,x+2184\,a^2\,b^8\,c^6\,d^4\,x^2+6006\,a^2\,b^8\,c^5\,d^5\,x^3+10725\,a^2\,b^8\,c^4\,d^6\,x^4+12870\,a^2\,b^8\,c^3\,d^7\,x^5+10296\,a^2\,b^8\,c^2\,d^8\,x^6+5148\,a^2\,b^8\,c\,d^9\,x^7+1287\,a^2\,b^8\,d^{10}\,x^8+55\,a\,b^9\,c^9\,d+585\,a\,b^9\,c^8\,d^2\,x+2808\,a\,b^9\,c^7\,d^3\,x^2+8008\,a\,b^9\,c^6\,d^4\,x^3+15015\,a\,b^9\,c^5\,d^5\,x^4+19305\,a\,b^9\,c^4\,d^6\,x^5+17160\,a\,b^9\,c^3\,d^7\,x^6+10296\,a\,b^9\,c^2\,d^8\,x^7+3861\,a\,b^9\,c\,d^9\,x^8+715\,a\,b^9\,d^{10}\,x^9+66\,b^{10}\,c^{10}+715\,b^{10}\,c^9\,d\,x+3510\,b^{10}\,c^8\,d^2\,x^2+10296\,b^{10}\,c^7\,d^3\,x^3+20020\,b^{10}\,c^6\,d^4\,x^4+27027\,b^{10}\,c^5\,d^5\,x^5+25740\,b^{10}\,c^4\,d^6\,x^6+17160\,b^{10}\,c^3\,d^7\,x^7+7722\,b^{10}\,c^2\,d^8\,x^8+2145\,b^{10}\,c\,d^9\,x^9+286\,b^{10}\,d^{10}\,x^{10}}{858\,a^{13}\,b^{11}+11154\,a^{12}\,b^{12}\,x+66924\,a^{11}\,b^{13}\,x^2+245388\,a^{10}\,b^{14}\,x^3+613470\,a^9\,b^{15}\,x^4+1104246\,a^8\,b^{16}\,x^5+1472328\,a^7\,b^{17}\,x^6+1472328\,a^6\,b^{18}\,x^7+1104246\,a^5\,b^{19}\,x^8+613470\,a^4\,b^{20}\,x^9+245388\,a^3\,b^{21}\,x^{10}+66924\,a^2\,b^{22}\,x^{11}+11154\,a\,b^{23}\,x^{12}+858\,b^{24}\,x^{13}} \]
-(a^10*d^10 + 66*b^10*c^10 + 286*b^10*d^10*x^10 + 715*a*b^9*d^10*x^9 + 214 5*b^10*c*d^9*x^9 + 45*a^2*b^8*c^8*d^2 + 36*a^3*b^7*c^7*d^3 + 28*a^4*b^6*c^ 6*d^4 + 21*a^5*b^5*c^5*d^5 + 15*a^6*b^4*c^4*d^6 + 10*a^7*b^3*c^3*d^7 + 6*a ^8*b^2*c^2*d^8 + 78*a^8*b^2*d^10*x^2 + 286*a^7*b^3*d^10*x^3 + 715*a^6*b^4* d^10*x^4 + 1287*a^5*b^5*d^10*x^5 + 1716*a^4*b^6*d^10*x^6 + 1716*a^3*b^7*d^ 10*x^7 + 1287*a^2*b^8*d^10*x^8 + 3510*b^10*c^8*d^2*x^2 + 10296*b^10*c^7*d^ 3*x^3 + 20020*b^10*c^6*d^4*x^4 + 27027*b^10*c^5*d^5*x^5 + 25740*b^10*c^4*d ^6*x^6 + 17160*b^10*c^3*d^7*x^7 + 7722*b^10*c^2*d^8*x^8 + 55*a*b^9*c^9*d + 3*a^9*b*c*d^9 + 13*a^9*b*d^10*x + 715*b^10*c^9*d*x + 2184*a^2*b^8*c^6*d^4 *x^2 + 1638*a^3*b^7*c^5*d^5*x^2 + 1170*a^4*b^6*c^4*d^6*x^2 + 780*a^5*b^5*c ^3*d^7*x^2 + 468*a^6*b^4*c^2*d^8*x^2 + 6006*a^2*b^8*c^5*d^5*x^3 + 4290*a^3 *b^7*c^4*d^6*x^3 + 2860*a^4*b^6*c^3*d^7*x^3 + 1716*a^5*b^5*c^2*d^8*x^3 + 1 0725*a^2*b^8*c^4*d^6*x^4 + 7150*a^3*b^7*c^3*d^7*x^4 + 4290*a^4*b^6*c^2*d^8 *x^4 + 12870*a^2*b^8*c^3*d^7*x^5 + 7722*a^3*b^7*c^2*d^8*x^5 + 10296*a^2*b^ 8*c^2*d^8*x^6 + 585*a*b^9*c^8*d^2*x + 39*a^8*b^2*c*d^9*x + 3861*a*b^9*c*d^ 9*x^8 + 468*a^2*b^8*c^7*d^3*x + 364*a^3*b^7*c^6*d^4*x + 273*a^4*b^6*c^5*d^ 5*x + 195*a^5*b^5*c^4*d^6*x + 130*a^6*b^4*c^3*d^7*x + 78*a^7*b^3*c^2*d^8*x + 2808*a*b^9*c^7*d^3*x^2 + 234*a^7*b^3*c*d^9*x^2 + 8008*a*b^9*c^6*d^4*x^3 + 858*a^6*b^4*c*d^9*x^3 + 15015*a*b^9*c^5*d^5*x^4 + 2145*a^5*b^5*c*d^9*x^ 4 + 19305*a*b^9*c^4*d^6*x^5 + 3861*a^4*b^6*c*d^9*x^5 + 17160*a*b^9*c^3*...