Integrand size = 15, antiderivative size = 58 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {(c+d x)^{11}}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^{11}}{132 (b c-a d)^2 (a+b x)^{11}} \]
Leaf count is larger than twice the leaf count of optimal. \(684\) vs. \(2(58)=116\).
Time = 0.17 (sec) , antiderivative size = 684, normalized size of antiderivative = 11.79 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {a^{10} d^{10}+2 a^9 b d^9 (c+6 d x)+3 a^8 b^2 d^8 \left (c^2+8 c d x+22 d^2 x^2\right )+4 a^7 b^3 d^7 \left (c^3+9 c^2 d x+33 c d^2 x^2+55 d^3 x^3\right )+a^6 b^4 d^6 \left (5 c^4+48 c^3 d x+198 c^2 d^2 x^2+440 c d^3 x^3+495 d^4 x^4\right )+6 a^5 b^5 d^5 \left (c^5+10 c^4 d x+44 c^3 d^2 x^2+110 c^2 d^3 x^3+165 c d^4 x^4+132 d^5 x^5\right )+a^4 b^6 d^4 \left (7 c^6+72 c^5 d x+330 c^4 d^2 x^2+880 c^3 d^3 x^3+1485 c^2 d^4 x^4+1584 c d^5 x^5+924 d^6 x^6\right )+4 a^3 b^7 d^3 \left (2 c^7+21 c^6 d x+99 c^5 d^2 x^2+275 c^4 d^3 x^3+495 c^3 d^4 x^4+594 c^2 d^5 x^5+462 c d^6 x^6+198 d^7 x^7\right )+3 a^2 b^8 d^2 \left (3 c^8+32 c^7 d x+154 c^6 d^2 x^2+440 c^5 d^3 x^3+825 c^4 d^4 x^4+1056 c^3 d^5 x^5+924 c^2 d^6 x^6+528 c d^7 x^7+165 d^8 x^8\right )+2 a b^9 d \left (5 c^9+54 c^8 d x+264 c^7 d^2 x^2+770 c^6 d^3 x^3+1485 c^5 d^4 x^4+1980 c^4 d^5 x^5+1848 c^3 d^6 x^6+1188 c^2 d^7 x^7+495 c d^8 x^8+110 d^9 x^9\right )+b^{10} \left (11 c^{10}+120 c^9 d x+594 c^8 d^2 x^2+1760 c^7 d^3 x^3+3465 c^6 d^4 x^4+4752 c^5 d^5 x^5+4620 c^4 d^6 x^6+3168 c^3 d^7 x^7+1485 c^2 d^8 x^8+440 c d^9 x^9+66 d^{10} x^{10}\right )}{132 b^{11} (a+b x)^{12}} \]
-1/132*(a^10*d^10 + 2*a^9*b*d^9*(c + 6*d*x) + 3*a^8*b^2*d^8*(c^2 + 8*c*d*x + 22*d^2*x^2) + 4*a^7*b^3*d^7*(c^3 + 9*c^2*d*x + 33*c*d^2*x^2 + 55*d^3*x^ 3) + a^6*b^4*d^6*(5*c^4 + 48*c^3*d*x + 198*c^2*d^2*x^2 + 440*c*d^3*x^3 + 4 95*d^4*x^4) + 6*a^5*b^5*d^5*(c^5 + 10*c^4*d*x + 44*c^3*d^2*x^2 + 110*c^2*d ^3*x^3 + 165*c*d^4*x^4 + 132*d^5*x^5) + a^4*b^6*d^4*(7*c^6 + 72*c^5*d*x + 330*c^4*d^2*x^2 + 880*c^3*d^3*x^3 + 1485*c^2*d^4*x^4 + 1584*c*d^5*x^5 + 92 4*d^6*x^6) + 4*a^3*b^7*d^3*(2*c^7 + 21*c^6*d*x + 99*c^5*d^2*x^2 + 275*c^4* d^3*x^3 + 495*c^3*d^4*x^4 + 594*c^2*d^5*x^5 + 462*c*d^6*x^6 + 198*d^7*x^7) + 3*a^2*b^8*d^2*(3*c^8 + 32*c^7*d*x + 154*c^6*d^2*x^2 + 440*c^5*d^3*x^3 + 825*c^4*d^4*x^4 + 1056*c^3*d^5*x^5 + 924*c^2*d^6*x^6 + 528*c*d^7*x^7 + 16 5*d^8*x^8) + 2*a*b^9*d*(5*c^9 + 54*c^8*d*x + 264*c^7*d^2*x^2 + 770*c^6*d^3 *x^3 + 1485*c^5*d^4*x^4 + 1980*c^4*d^5*x^5 + 1848*c^3*d^6*x^6 + 1188*c^2*d ^7*x^7 + 495*c*d^8*x^8 + 110*d^9*x^9) + b^10*(11*c^10 + 120*c^9*d*x + 594* c^8*d^2*x^2 + 1760*c^7*d^3*x^3 + 3465*c^6*d^4*x^4 + 4752*c^5*d^5*x^5 + 462 0*c^4*d^6*x^6 + 3168*c^3*d^7*x^7 + 1485*c^2*d^8*x^8 + 440*c*d^9*x^9 + 66*d ^10*x^10))/(b^11*(a + b*x)^12)
Time = 0.16 (sec) , antiderivative size = 58, 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 = {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.
| \(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {d \int \frac {(c+d x)^{10}}{(a+b x)^{12}}dx}{12 (b c-a d)}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {d (c+d x)^{11}}{132 (a+b x)^{11} (b c-a d)^2}-\frac {(c+d x)^{11}}{12 (a+b x)^{12} (b c-a d)}\) |
-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)
3.14.24.3.1 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. \(2(54)=108\).
Time = 0.23 (sec) , antiderivative size = 831, normalized size of antiderivative = 14.33
| method | result | size |
| risch | \(\frac {-\frac {d^{10} x^{10}}{2 b}-\frac {5 d^{9} \left (a d +2 b c \right ) x^{9}}{3 b^{2}}-\frac {15 d^{8} \left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) x^{8}}{4 b^{3}}-\frac {6 d^{7} \left (a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) x^{7}}{b^{4}}-\frac {7 d^{6} \left (a^{4} d^{4}+2 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +5 b^{4} c^{4}\right ) x^{6}}{b^{5}}-\frac {6 d^{5} \left (a^{5} d^{5}+2 a^{4} b c \,d^{4}+3 a^{3} b^{2} c^{2} d^{3}+4 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +6 b^{5} c^{5}\right ) x^{5}}{b^{6}}-\frac {15 d^{4} \left (a^{6} d^{6}+2 a^{5} b c \,d^{5}+3 a^{4} b^{2} c^{2} d^{4}+4 a^{3} b^{3} c^{3} d^{3}+5 a^{2} b^{4} c^{4} d^{2}+6 a \,b^{5} c^{5} d +7 b^{6} c^{6}\right ) x^{4}}{4 b^{7}}-\frac {5 d^{3} \left (a^{7} d^{7}+2 a^{6} b c \,d^{6}+3 a^{5} b^{2} c^{2} d^{5}+4 a^{4} b^{3} c^{3} d^{4}+5 a^{3} b^{4} c^{4} d^{3}+6 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +8 b^{7} c^{7}\right ) x^{3}}{3 b^{8}}-\frac {d^{2} \left (a^{8} d^{8}+2 a^{7} b c \,d^{7}+3 a^{6} b^{2} c^{2} d^{6}+4 a^{5} b^{3} c^{3} d^{5}+5 a^{4} b^{4} c^{4} d^{4}+6 a^{3} b^{5} c^{5} d^{3}+7 a^{2} b^{6} c^{6} d^{2}+8 a \,b^{7} c^{7} d +9 b^{8} c^{8}\right ) x^{2}}{2 b^{9}}-\frac {d \left (a^{9} d^{9}+2 a^{8} b c \,d^{8}+3 a^{7} b^{2} c^{2} d^{7}+4 a^{6} b^{3} c^{3} d^{6}+5 a^{5} b^{4} c^{4} d^{5}+6 a^{4} b^{5} c^{5} d^{4}+7 a^{3} b^{6} c^{6} d^{3}+8 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d +10 b^{9} c^{9}\right ) x}{11 b^{10}}-\frac {a^{10} d^{10}+2 a^{9} b c \,d^{9}+3 a^{8} b^{2} c^{2} d^{8}+4 a^{7} b^{3} c^{3} d^{7}+5 a^{6} b^{4} c^{4} d^{6}+6 a^{5} b^{5} c^{5} d^{5}+7 a^{4} b^{6} c^{6} d^{4}+8 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+10 a \,b^{9} c^{9} d +11 b^{10} c^{10}}{132 b^{11}}}{\left (b x +a \right )^{12}}\) | \(831\) |
| default | \(\frac {10 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{11 b^{11} \left (b x +a \right )^{11}}+\frac {10 d^{9} \left (a d -b c \right )}{3 b^{11} \left (b x +a \right )^{3}}+\frac {40 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 )}{3 b^{11} \left (b x +a \right )^{9}}-\frac {35 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 )^{6}}-\frac {105 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 )}{4 b^{11} \left (b x +a \right )^{8}}-\frac {45 d^{8} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 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}}{12 b^{11} \left (b x +a \right )^{12}}+\frac {36 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 )}{b^{11} \left (b x +a \right )^{7}}-\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 +b^{8} c^{8}\right )}{2 b^{11} \left (b x +a \right )^{10}}-\frac {d^{10}}{2 b^{11} \left (b x +a \right )^{2}}+\frac {24 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 )^{5}}\) | \(867\) |
| norman | \(\frac {-\frac {d^{10} x^{10}}{2 b}+\frac {5 \left (-a b \,d^{10}-2 b^{2} c \,d^{9}\right ) x^{9}}{3 b^{3}}+\frac {15 \left (-a^{2} b \,d^{10}-2 a \,b^{2} c \,d^{9}-3 b^{3} c^{2} d^{8}\right ) x^{8}}{4 b^{4}}+\frac {6 \left (-a^{3} b \,d^{10}-2 a^{2} b^{2} c \,d^{9}-3 a \,b^{3} c^{2} d^{8}-4 b^{4} c^{3} d^{7}\right ) x^{7}}{b^{5}}+\frac {7 \left (-a^{4} b \,d^{10}-2 a^{3} b^{2} c \,d^{9}-3 a^{2} b^{3} c^{2} d^{8}-4 a \,b^{4} c^{3} d^{7}-5 b^{5} c^{4} d^{6}\right ) x^{6}}{b^{6}}+\frac {6 \left (-a^{5} b \,d^{10}-2 a^{4} b^{2} c \,d^{9}-3 a^{3} b^{3} c^{2} d^{8}-4 a^{2} b^{4} c^{3} d^{7}-5 a \,b^{5} c^{4} d^{6}-6 b^{6} c^{5} d^{5}\right ) x^{5}}{b^{7}}+\frac {15 \left (-a^{6} b \,d^{10}-2 a^{5} b^{2} c \,d^{9}-3 a^{4} b^{3} c^{2} d^{8}-4 a^{3} b^{4} c^{3} d^{7}-5 a^{2} b^{5} c^{4} d^{6}-6 a \,b^{6} c^{5} d^{5}-7 b^{7} c^{6} d^{4}\right ) x^{4}}{4 b^{8}}+\frac {5 \left (-a^{7} b \,d^{10}-2 a^{6} b^{2} c \,d^{9}-3 a^{5} b^{3} c^{2} d^{8}-4 a^{4} b^{4} c^{3} d^{7}-5 a^{3} b^{5} c^{4} d^{6}-6 a^{2} b^{6} c^{5} d^{5}-7 a \,b^{7} c^{6} d^{4}-8 b^{8} c^{7} d^{3}\right ) x^{3}}{3 b^{9}}+\frac {\left (-a^{8} b \,d^{10}-2 a^{7} b^{2} c \,d^{9}-3 a^{6} b^{3} c^{2} d^{8}-4 a^{5} b^{4} c^{3} d^{7}-5 a^{4} b^{5} c^{4} d^{6}-6 a^{3} b^{6} c^{5} d^{5}-7 a^{2} b^{7} c^{6} d^{4}-8 a \,b^{8} c^{7} d^{3}-9 b^{9} c^{8} d^{2}\right ) x^{2}}{2 b^{10}}+\frac {\left (-a^{9} b \,d^{10}-2 a^{8} b^{2} c \,d^{9}-3 a^{7} b^{3} c^{2} d^{8}-4 a^{6} b^{4} c^{3} d^{7}-5 a^{5} b^{5} c^{4} d^{6}-6 a^{4} b^{6} c^{5} d^{5}-7 a^{3} b^{7} c^{6} d^{4}-8 a^{2} b^{8} c^{7} d^{3}-9 a \,b^{9} c^{8} d^{2}-10 b^{10} c^{9} d \right ) x}{11 b^{11}}+\frac {-a^{10} b \,d^{10}-2 a^{9} b^{2} c \,d^{9}-3 a^{8} b^{3} c^{2} d^{8}-4 a^{7} b^{4} c^{3} d^{7}-5 a^{6} b^{5} c^{4} d^{6}-6 a^{5} b^{6} c^{5} d^{5}-7 a^{4} b^{7} c^{6} d^{4}-8 a^{3} b^{8} c^{7} d^{3}-9 a^{2} b^{9} c^{8} d^{2}-10 a \,b^{10} c^{9} d -11 b^{11} c^{10}}{132 b^{12}}}{\left (b x +a \right )^{12}}\) | \(889\) |
| gosper | \(-\frac {66 x^{10} d^{10} b^{10}+220 x^{9} a \,b^{9} d^{10}+440 x^{9} b^{10} c \,d^{9}+495 x^{8} a^{2} b^{8} d^{10}+990 x^{8} a \,b^{9} c \,d^{9}+1485 x^{8} b^{10} c^{2} d^{8}+792 x^{7} a^{3} b^{7} d^{10}+1584 x^{7} a^{2} b^{8} c \,d^{9}+2376 x^{7} a \,b^{9} c^{2} d^{8}+3168 x^{7} b^{10} c^{3} d^{7}+924 x^{6} a^{4} b^{6} d^{10}+1848 x^{6} a^{3} b^{7} c \,d^{9}+2772 x^{6} a^{2} b^{8} c^{2} d^{8}+3696 x^{6} a \,b^{9} c^{3} d^{7}+4620 x^{6} b^{10} c^{4} d^{6}+792 x^{5} a^{5} b^{5} d^{10}+1584 x^{5} a^{4} b^{6} c \,d^{9}+2376 x^{5} a^{3} b^{7} c^{2} d^{8}+3168 x^{5} a^{2} b^{8} c^{3} d^{7}+3960 x^{5} a \,b^{9} c^{4} d^{6}+4752 x^{5} b^{10} c^{5} d^{5}+495 x^{4} a^{6} b^{4} d^{10}+990 x^{4} a^{5} b^{5} c \,d^{9}+1485 x^{4} a^{4} b^{6} c^{2} d^{8}+1980 x^{4} a^{3} b^{7} c^{3} d^{7}+2475 x^{4} a^{2} b^{8} c^{4} d^{6}+2970 x^{4} a \,b^{9} c^{5} d^{5}+3465 x^{4} b^{10} c^{6} d^{4}+220 x^{3} a^{7} b^{3} d^{10}+440 x^{3} a^{6} b^{4} c \,d^{9}+660 x^{3} a^{5} b^{5} c^{2} d^{8}+880 x^{3} a^{4} b^{6} c^{3} d^{7}+1100 x^{3} a^{3} b^{7} c^{4} d^{6}+1320 x^{3} a^{2} b^{8} c^{5} d^{5}+1540 x^{3} a \,b^{9} c^{6} d^{4}+1760 x^{3} b^{10} c^{7} d^{3}+66 x^{2} a^{8} b^{2} d^{10}+132 x^{2} a^{7} b^{3} c \,d^{9}+198 x^{2} a^{6} b^{4} c^{2} d^{8}+264 x^{2} a^{5} b^{5} c^{3} d^{7}+330 x^{2} a^{4} b^{6} c^{4} d^{6}+396 x^{2} a^{3} b^{7} c^{5} d^{5}+462 x^{2} a^{2} b^{8} c^{6} d^{4}+528 x^{2} a \,b^{9} c^{7} d^{3}+594 x^{2} b^{10} c^{8} d^{2}+12 x \,a^{9} b \,d^{10}+24 x \,a^{8} b^{2} c \,d^{9}+36 x \,a^{7} b^{3} c^{2} d^{8}+48 x \,a^{6} b^{4} c^{3} d^{7}+60 x \,a^{5} b^{5} c^{4} d^{6}+72 x \,a^{4} b^{6} c^{5} d^{5}+84 x \,a^{3} b^{7} c^{6} d^{4}+96 x \,a^{2} b^{8} c^{7} d^{3}+108 x a \,b^{9} c^{8} d^{2}+120 x \,b^{10} c^{9} d +a^{10} d^{10}+2 a^{9} b c \,d^{9}+3 a^{8} b^{2} c^{2} d^{8}+4 a^{7} b^{3} c^{3} d^{7}+5 a^{6} b^{4} c^{4} d^{6}+6 a^{5} b^{5} c^{5} d^{5}+7 a^{4} b^{6} c^{6} d^{4}+8 a^{3} b^{7} c^{7} d^{3}+9 a^{2} b^{8} c^{8} d^{2}+10 a \,b^{9} c^{9} d +11 b^{10} c^{10}}{132 b^{11} \left (b x +a \right )^{12}}\) | \(962\) |
| parallelrisch | \(\frac {-66 d^{10} x^{10} b^{11}-220 a \,b^{10} d^{10} x^{9}-440 b^{11} c \,d^{9} x^{9}-495 a^{2} b^{9} d^{10} x^{8}-990 a \,b^{10} c \,d^{9} x^{8}-1485 b^{11} c^{2} d^{8} x^{8}-792 a^{3} b^{8} d^{10} x^{7}-1584 a^{2} b^{9} c \,d^{9} x^{7}-2376 a \,b^{10} c^{2} d^{8} x^{7}-3168 b^{11} c^{3} d^{7} x^{7}-924 a^{4} b^{7} d^{10} x^{6}-1848 a^{3} b^{8} c \,d^{9} x^{6}-2772 a^{2} b^{9} c^{2} d^{8} x^{6}-3696 a \,b^{10} c^{3} d^{7} x^{6}-4620 b^{11} c^{4} d^{6} x^{6}-792 a^{5} b^{6} d^{10} x^{5}-1584 a^{4} b^{7} c \,d^{9} x^{5}-2376 a^{3} b^{8} c^{2} d^{8} x^{5}-3168 a^{2} b^{9} c^{3} d^{7} x^{5}-3960 a \,b^{10} c^{4} d^{6} x^{5}-4752 b^{11} c^{5} d^{5} x^{5}-495 a^{6} b^{5} d^{10} x^{4}-990 a^{5} b^{6} c \,d^{9} x^{4}-1485 a^{4} b^{7} c^{2} d^{8} x^{4}-1980 a^{3} b^{8} c^{3} d^{7} x^{4}-2475 a^{2} b^{9} c^{4} d^{6} x^{4}-2970 a \,b^{10} c^{5} d^{5} x^{4}-3465 b^{11} c^{6} d^{4} x^{4}-220 a^{7} b^{4} d^{10} x^{3}-440 a^{6} b^{5} c \,d^{9} x^{3}-660 a^{5} b^{6} c^{2} d^{8} x^{3}-880 a^{4} b^{7} c^{3} d^{7} x^{3}-1100 a^{3} b^{8} c^{4} d^{6} x^{3}-1320 a^{2} b^{9} c^{5} d^{5} x^{3}-1540 a \,b^{10} c^{6} d^{4} x^{3}-1760 b^{11} c^{7} d^{3} x^{3}-66 a^{8} b^{3} d^{10} x^{2}-132 a^{7} b^{4} c \,d^{9} x^{2}-198 a^{6} b^{5} c^{2} d^{8} x^{2}-264 a^{5} b^{6} c^{3} d^{7} x^{2}-330 a^{4} b^{7} c^{4} d^{6} x^{2}-396 a^{3} b^{8} c^{5} d^{5} x^{2}-462 a^{2} b^{9} c^{6} d^{4} x^{2}-528 a \,b^{10} c^{7} d^{3} x^{2}-594 b^{11} c^{8} d^{2} x^{2}-12 a^{9} b^{2} d^{10} x -24 a^{8} b^{3} c \,d^{9} x -36 a^{7} b^{4} c^{2} d^{8} x -48 a^{6} b^{5} c^{3} d^{7} x -60 a^{5} b^{6} c^{4} d^{6} x -72 a^{4} b^{7} c^{5} d^{5} x -84 a^{3} b^{8} c^{6} d^{4} x -96 a^{2} b^{9} c^{7} d^{3} x -108 a \,b^{10} c^{8} d^{2} x -120 b^{11} c^{9} d x -a^{10} b \,d^{10}-2 a^{9} b^{2} c \,d^{9}-3 a^{8} b^{3} c^{2} d^{8}-4 a^{7} b^{4} c^{3} d^{7}-5 a^{6} b^{5} c^{4} d^{6}-6 a^{5} b^{6} c^{5} d^{5}-7 a^{4} b^{7} c^{6} d^{4}-8 a^{3} b^{8} c^{7} d^{3}-9 a^{2} b^{9} c^{8} d^{2}-10 a \,b^{10} c^{9} d -11 b^{11} c^{10}}{132 b^{12} \left (b x +a \right )^{12}}\) | \(968\) |
(-1/2/b*d^10*x^10-5/3/b^2*d^9*(a*d+2*b*c)*x^9-15/4/b^3*d^8*(a^2*d^2+2*a*b* c*d+3*b^2*c^2)*x^8-6/b^4*d^7*(a^3*d^3+2*a^2*b*c*d^2+3*a*b^2*c^2*d+4*b^3*c^ 3)*x^7-7/b^5*d^6*(a^4*d^4+2*a^3*b*c*d^3+3*a^2*b^2*c^2*d^2+4*a*b^3*c^3*d+5* b^4*c^4)*x^6-6/b^6*d^5*(a^5*d^5+2*a^4*b*c*d^4+3*a^3*b^2*c^2*d^3+4*a^2*b^3* c^3*d^2+5*a*b^4*c^4*d+6*b^5*c^5)*x^5-15/4/b^7*d^4*(a^6*d^6+2*a^5*b*c*d^5+3 *a^4*b^2*c^2*d^4+4*a^3*b^3*c^3*d^3+5*a^2*b^4*c^4*d^2+6*a*b^5*c^5*d+7*b^6*c ^6)*x^4-5/3/b^8*d^3*(a^7*d^7+2*a^6*b*c*d^6+3*a^5*b^2*c^2*d^5+4*a^4*b^3*c^3 *d^4+5*a^3*b^4*c^4*d^3+6*a^2*b^5*c^5*d^2+7*a*b^6*c^6*d+8*b^7*c^7)*x^3-1/2/ b^9*d^2*(a^8*d^8+2*a^7*b*c*d^7+3*a^6*b^2*c^2*d^6+4*a^5*b^3*c^3*d^5+5*a^4*b ^4*c^4*d^4+6*a^3*b^5*c^5*d^3+7*a^2*b^6*c^6*d^2+8*a*b^7*c^7*d+9*b^8*c^8)*x^ 2-1/11/b^10*d*(a^9*d^9+2*a^8*b*c*d^8+3*a^7*b^2*c^2*d^7+4*a^6*b^3*c^3*d^6+5 *a^5*b^4*c^4*d^5+6*a^4*b^5*c^5*d^4+7*a^3*b^6*c^6*d^3+8*a^2*b^7*c^7*d^2+9*a *b^8*c^8*d+10*b^9*c^9)*x-1/132/b^11*(a^10*d^10+2*a^9*b*c*d^9+3*a^8*b^2*c^2 *d^8+4*a^7*b^3*c^3*d^7+5*a^6*b^4*c^4*d^6+6*a^5*b^5*c^5*d^5+7*a^4*b^6*c^6*d ^4+8*a^3*b^7*c^7*d^3+9*a^2*b^8*c^8*d^2+10*a*b^9*c^9*d+11*b^10*c^10))/(b*x+ a)^12
Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (54) = 108\).
Time = 0.24 (sec) , antiderivative size = 986, normalized size of antiderivative = 17.00 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {66 \, b^{10} d^{10} x^{10} + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10} + 220 \, {\left (2 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 495 \, {\left (3 \, b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 792 \, {\left (4 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 2 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 924 \, {\left (5 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 3 \, a^{2} b^{8} c^{2} d^{8} + 2 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 792 \, {\left (6 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 4 \, a^{2} b^{8} c^{3} d^{7} + 3 \, a^{3} b^{7} c^{2} d^{8} + 2 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 495 \, {\left (7 \, b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 4 \, a^{3} b^{7} c^{3} d^{7} + 3 \, a^{4} b^{6} c^{2} d^{8} + 2 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 220 \, {\left (8 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 6 \, a^{2} b^{8} c^{5} d^{5} + 5 \, a^{3} b^{7} c^{4} d^{6} + 4 \, a^{4} b^{6} c^{3} d^{7} + 3 \, a^{5} b^{5} c^{2} d^{8} + 2 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 66 \, {\left (9 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 7 \, a^{2} b^{8} c^{6} d^{4} + 6 \, a^{3} b^{7} c^{5} d^{5} + 5 \, a^{4} b^{6} c^{4} d^{6} + 4 \, a^{5} b^{5} c^{3} d^{7} + 3 \, a^{6} b^{4} c^{2} d^{8} + 2 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 12 \, {\left (10 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 8 \, a^{2} b^{8} c^{7} d^{3} + 7 \, a^{3} b^{7} c^{6} d^{4} + 6 \, a^{4} b^{6} c^{5} d^{5} + 5 \, a^{5} b^{5} c^{4} d^{6} + 4 \, a^{6} b^{4} c^{3} d^{7} + 3 \, a^{7} b^{3} c^{2} d^{8} + 2 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{132 \, {\left (b^{23} x^{12} + 12 \, a b^{22} x^{11} + 66 \, a^{2} b^{21} x^{10} + 220 \, a^{3} b^{20} x^{9} + 495 \, a^{4} b^{19} x^{8} + 792 \, a^{5} b^{18} x^{7} + 924 \, a^{6} b^{17} x^{6} + 792 \, a^{7} b^{16} x^{5} + 495 \, a^{8} b^{15} x^{4} + 220 \, a^{9} b^{14} x^{3} + 66 \, a^{10} b^{13} x^{2} + 12 \, a^{11} b^{12} x + a^{12} b^{11}\right )}} \]
-1/132*(66*b^10*d^10*x^10 + 11*b^10*c^10 + 10*a*b^9*c^9*d + 9*a^2*b^8*c^8* d^2 + 8*a^3*b^7*c^7*d^3 + 7*a^4*b^6*c^6*d^4 + 6*a^5*b^5*c^5*d^5 + 5*a^6*b^ 4*c^4*d^6 + 4*a^7*b^3*c^3*d^7 + 3*a^8*b^2*c^2*d^8 + 2*a^9*b*c*d^9 + a^10*d ^10 + 220*(2*b^10*c*d^9 + a*b^9*d^10)*x^9 + 495*(3*b^10*c^2*d^8 + 2*a*b^9* c*d^9 + a^2*b^8*d^10)*x^8 + 792*(4*b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 + 2*a^2* b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 924*(5*b^10*c^4*d^6 + 4*a*b^9*c^3*d^7 + 3* a^2*b^8*c^2*d^8 + 2*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 792*(6*b^10*c^5*d^ 5 + 5*a*b^9*c^4*d^6 + 4*a^2*b^8*c^3*d^7 + 3*a^3*b^7*c^2*d^8 + 2*a^4*b^6*c* d^9 + a^5*b^5*d^10)*x^5 + 495*(7*b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 + 5*a^2*b^ 8*c^4*d^6 + 4*a^3*b^7*c^3*d^7 + 3*a^4*b^6*c^2*d^8 + 2*a^5*b^5*c*d^9 + a^6* b^4*d^10)*x^4 + 220*(8*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 6*a^2*b^8*c^5*d^5 + 5*a^3*b^7*c^4*d^6 + 4*a^4*b^6*c^3*d^7 + 3*a^5*b^5*c^2*d^8 + 2*a^6*b^4*c* d^9 + a^7*b^3*d^10)*x^3 + 66*(9*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 7*a^2*b^8 *c^6*d^4 + 6*a^3*b^7*c^5*d^5 + 5*a^4*b^6*c^4*d^6 + 4*a^5*b^5*c^3*d^7 + 3*a ^6*b^4*c^2*d^8 + 2*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 12*(10*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 8*a^2*b^8*c^7*d^3 + 7*a^3*b^7*c^6*d^4 + 6*a^4*b^6*c^5*d ^5 + 5*a^5*b^5*c^4*d^6 + 4*a^6*b^4*c^3*d^7 + 3*a^7*b^3*c^2*d^8 + 2*a^8*b^2 *c*d^9 + a^9*b*d^10)*x)/(b^23*x^12 + 12*a*b^22*x^11 + 66*a^2*b^21*x^10 + 2 20*a^3*b^20*x^9 + 495*a^4*b^19*x^8 + 792*a^5*b^18*x^7 + 924*a^6*b^17*x^6 + 792*a^7*b^16*x^5 + 495*a^8*b^15*x^4 + 220*a^9*b^14*x^3 + 66*a^10*b^13*...
Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (54) = 108\).
Time = 0.26 (sec) , antiderivative size = 986, normalized size of antiderivative = 17.00 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {66 \, b^{10} d^{10} x^{10} + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10} + 220 \, {\left (2 \, b^{10} c d^{9} + a b^{9} d^{10}\right )} x^{9} + 495 \, {\left (3 \, b^{10} c^{2} d^{8} + 2 \, a b^{9} c d^{9} + a^{2} b^{8} d^{10}\right )} x^{8} + 792 \, {\left (4 \, b^{10} c^{3} d^{7} + 3 \, a b^{9} c^{2} d^{8} + 2 \, a^{2} b^{8} c d^{9} + a^{3} b^{7} d^{10}\right )} x^{7} + 924 \, {\left (5 \, b^{10} c^{4} d^{6} + 4 \, a b^{9} c^{3} d^{7} + 3 \, a^{2} b^{8} c^{2} d^{8} + 2 \, a^{3} b^{7} c d^{9} + a^{4} b^{6} d^{10}\right )} x^{6} + 792 \, {\left (6 \, b^{10} c^{5} d^{5} + 5 \, a b^{9} c^{4} d^{6} + 4 \, a^{2} b^{8} c^{3} d^{7} + 3 \, a^{3} b^{7} c^{2} d^{8} + 2 \, a^{4} b^{6} c d^{9} + a^{5} b^{5} d^{10}\right )} x^{5} + 495 \, {\left (7 \, b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} + 5 \, a^{2} b^{8} c^{4} d^{6} + 4 \, a^{3} b^{7} c^{3} d^{7} + 3 \, a^{4} b^{6} c^{2} d^{8} + 2 \, a^{5} b^{5} c d^{9} + a^{6} b^{4} d^{10}\right )} x^{4} + 220 \, {\left (8 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 6 \, a^{2} b^{8} c^{5} d^{5} + 5 \, a^{3} b^{7} c^{4} d^{6} + 4 \, a^{4} b^{6} c^{3} d^{7} + 3 \, a^{5} b^{5} c^{2} d^{8} + 2 \, a^{6} b^{4} c d^{9} + a^{7} b^{3} d^{10}\right )} x^{3} + 66 \, {\left (9 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 7 \, a^{2} b^{8} c^{6} d^{4} + 6 \, a^{3} b^{7} c^{5} d^{5} + 5 \, a^{4} b^{6} c^{4} d^{6} + 4 \, a^{5} b^{5} c^{3} d^{7} + 3 \, a^{6} b^{4} c^{2} d^{8} + 2 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 12 \, {\left (10 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 8 \, a^{2} b^{8} c^{7} d^{3} + 7 \, a^{3} b^{7} c^{6} d^{4} + 6 \, a^{4} b^{6} c^{5} d^{5} + 5 \, a^{5} b^{5} c^{4} d^{6} + 4 \, a^{6} b^{4} c^{3} d^{7} + 3 \, a^{7} b^{3} c^{2} d^{8} + 2 \, a^{8} b^{2} c d^{9} + a^{9} b d^{10}\right )} x}{132 \, {\left (b^{23} x^{12} + 12 \, a b^{22} x^{11} + 66 \, a^{2} b^{21} x^{10} + 220 \, a^{3} b^{20} x^{9} + 495 \, a^{4} b^{19} x^{8} + 792 \, a^{5} b^{18} x^{7} + 924 \, a^{6} b^{17} x^{6} + 792 \, a^{7} b^{16} x^{5} + 495 \, a^{8} b^{15} x^{4} + 220 \, a^{9} b^{14} x^{3} + 66 \, a^{10} b^{13} x^{2} + 12 \, a^{11} b^{12} x + a^{12} b^{11}\right )}} \]
-1/132*(66*b^10*d^10*x^10 + 11*b^10*c^10 + 10*a*b^9*c^9*d + 9*a^2*b^8*c^8* d^2 + 8*a^3*b^7*c^7*d^3 + 7*a^4*b^6*c^6*d^4 + 6*a^5*b^5*c^5*d^5 + 5*a^6*b^ 4*c^4*d^6 + 4*a^7*b^3*c^3*d^7 + 3*a^8*b^2*c^2*d^8 + 2*a^9*b*c*d^9 + a^10*d ^10 + 220*(2*b^10*c*d^9 + a*b^9*d^10)*x^9 + 495*(3*b^10*c^2*d^8 + 2*a*b^9* c*d^9 + a^2*b^8*d^10)*x^8 + 792*(4*b^10*c^3*d^7 + 3*a*b^9*c^2*d^8 + 2*a^2* b^8*c*d^9 + a^3*b^7*d^10)*x^7 + 924*(5*b^10*c^4*d^6 + 4*a*b^9*c^3*d^7 + 3* a^2*b^8*c^2*d^8 + 2*a^3*b^7*c*d^9 + a^4*b^6*d^10)*x^6 + 792*(6*b^10*c^5*d^ 5 + 5*a*b^9*c^4*d^6 + 4*a^2*b^8*c^3*d^7 + 3*a^3*b^7*c^2*d^8 + 2*a^4*b^6*c* d^9 + a^5*b^5*d^10)*x^5 + 495*(7*b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 + 5*a^2*b^ 8*c^4*d^6 + 4*a^3*b^7*c^3*d^7 + 3*a^4*b^6*c^2*d^8 + 2*a^5*b^5*c*d^9 + a^6* b^4*d^10)*x^4 + 220*(8*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 6*a^2*b^8*c^5*d^5 + 5*a^3*b^7*c^4*d^6 + 4*a^4*b^6*c^3*d^7 + 3*a^5*b^5*c^2*d^8 + 2*a^6*b^4*c* d^9 + a^7*b^3*d^10)*x^3 + 66*(9*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 7*a^2*b^8 *c^6*d^4 + 6*a^3*b^7*c^5*d^5 + 5*a^4*b^6*c^4*d^6 + 4*a^5*b^5*c^3*d^7 + 3*a ^6*b^4*c^2*d^8 + 2*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 12*(10*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 8*a^2*b^8*c^7*d^3 + 7*a^3*b^7*c^6*d^4 + 6*a^4*b^6*c^5*d ^5 + 5*a^5*b^5*c^4*d^6 + 4*a^6*b^4*c^3*d^7 + 3*a^7*b^3*c^2*d^8 + 2*a^8*b^2 *c*d^9 + a^9*b*d^10)*x)/(b^23*x^12 + 12*a*b^22*x^11 + 66*a^2*b^21*x^10 + 2 20*a^3*b^20*x^9 + 495*a^4*b^19*x^8 + 792*a^5*b^18*x^7 + 924*a^6*b^17*x^6 + 792*a^7*b^16*x^5 + 495*a^8*b^15*x^4 + 220*a^9*b^14*x^3 + 66*a^10*b^13*...
Leaf count of result is larger than twice the leaf count of optimal. 961 vs. \(2 (54) = 108\).
Time = 0.34 (sec) , antiderivative size = 961, normalized size of antiderivative = 16.57 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=-\frac {66 \, b^{10} d^{10} x^{10} + 440 \, b^{10} c d^{9} x^{9} + 220 \, a b^{9} d^{10} x^{9} + 1485 \, b^{10} c^{2} d^{8} x^{8} + 990 \, a b^{9} c d^{9} x^{8} + 495 \, a^{2} b^{8} d^{10} x^{8} + 3168 \, b^{10} c^{3} d^{7} x^{7} + 2376 \, a b^{9} c^{2} d^{8} x^{7} + 1584 \, a^{2} b^{8} c d^{9} x^{7} + 792 \, a^{3} b^{7} d^{10} x^{7} + 4620 \, b^{10} c^{4} d^{6} x^{6} + 3696 \, a b^{9} c^{3} d^{7} x^{6} + 2772 \, a^{2} b^{8} c^{2} d^{8} x^{6} + 1848 \, a^{3} b^{7} c d^{9} x^{6} + 924 \, a^{4} b^{6} d^{10} x^{6} + 4752 \, b^{10} c^{5} d^{5} x^{5} + 3960 \, a b^{9} c^{4} d^{6} x^{5} + 3168 \, a^{2} b^{8} c^{3} d^{7} x^{5} + 2376 \, a^{3} b^{7} c^{2} d^{8} x^{5} + 1584 \, a^{4} b^{6} c d^{9} x^{5} + 792 \, a^{5} b^{5} d^{10} x^{5} + 3465 \, b^{10} c^{6} d^{4} x^{4} + 2970 \, a b^{9} c^{5} d^{5} x^{4} + 2475 \, a^{2} b^{8} c^{4} d^{6} x^{4} + 1980 \, a^{3} b^{7} c^{3} d^{7} x^{4} + 1485 \, a^{4} b^{6} c^{2} d^{8} x^{4} + 990 \, a^{5} b^{5} c d^{9} x^{4} + 495 \, a^{6} b^{4} d^{10} x^{4} + 1760 \, b^{10} c^{7} d^{3} x^{3} + 1540 \, a b^{9} c^{6} d^{4} x^{3} + 1320 \, a^{2} b^{8} c^{5} d^{5} x^{3} + 1100 \, a^{3} b^{7} c^{4} d^{6} x^{3} + 880 \, a^{4} b^{6} c^{3} d^{7} x^{3} + 660 \, a^{5} b^{5} c^{2} d^{8} x^{3} + 440 \, a^{6} b^{4} c d^{9} x^{3} + 220 \, a^{7} b^{3} d^{10} x^{3} + 594 \, b^{10} c^{8} d^{2} x^{2} + 528 \, a b^{9} c^{7} d^{3} x^{2} + 462 \, a^{2} b^{8} c^{6} d^{4} x^{2} + 396 \, a^{3} b^{7} c^{5} d^{5} x^{2} + 330 \, a^{4} b^{6} c^{4} d^{6} x^{2} + 264 \, a^{5} b^{5} c^{3} d^{7} x^{2} + 198 \, a^{6} b^{4} c^{2} d^{8} x^{2} + 132 \, a^{7} b^{3} c d^{9} x^{2} + 66 \, a^{8} b^{2} d^{10} x^{2} + 120 \, b^{10} c^{9} d x + 108 \, a b^{9} c^{8} d^{2} x + 96 \, a^{2} b^{8} c^{7} d^{3} x + 84 \, a^{3} b^{7} c^{6} d^{4} x + 72 \, a^{4} b^{6} c^{5} d^{5} x + 60 \, a^{5} b^{5} c^{4} d^{6} x + 48 \, a^{6} b^{4} c^{3} d^{7} x + 36 \, a^{7} b^{3} c^{2} d^{8} x + 24 \, a^{8} b^{2} c d^{9} x + 12 \, a^{9} b d^{10} x + 11 \, b^{10} c^{10} + 10 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 8 \, a^{3} b^{7} c^{7} d^{3} + 7 \, a^{4} b^{6} c^{6} d^{4} + 6 \, a^{5} b^{5} c^{5} d^{5} + 5 \, a^{6} b^{4} c^{4} d^{6} + 4 \, a^{7} b^{3} c^{3} d^{7} + 3 \, a^{8} b^{2} c^{2} d^{8} + 2 \, a^{9} b c d^{9} + a^{10} d^{10}}{132 \, {\left (b x + a\right )}^{12} b^{11}} \]
-1/132*(66*b^10*d^10*x^10 + 440*b^10*c*d^9*x^9 + 220*a*b^9*d^10*x^9 + 1485 *b^10*c^2*d^8*x^8 + 990*a*b^9*c*d^9*x^8 + 495*a^2*b^8*d^10*x^8 + 3168*b^10 *c^3*d^7*x^7 + 2376*a*b^9*c^2*d^8*x^7 + 1584*a^2*b^8*c*d^9*x^7 + 792*a^3*b ^7*d^10*x^7 + 4620*b^10*c^4*d^6*x^6 + 3696*a*b^9*c^3*d^7*x^6 + 2772*a^2*b^ 8*c^2*d^8*x^6 + 1848*a^3*b^7*c*d^9*x^6 + 924*a^4*b^6*d^10*x^6 + 4752*b^10* c^5*d^5*x^5 + 3960*a*b^9*c^4*d^6*x^5 + 3168*a^2*b^8*c^3*d^7*x^5 + 2376*a^3 *b^7*c^2*d^8*x^5 + 1584*a^4*b^6*c*d^9*x^5 + 792*a^5*b^5*d^10*x^5 + 3465*b^ 10*c^6*d^4*x^4 + 2970*a*b^9*c^5*d^5*x^4 + 2475*a^2*b^8*c^4*d^6*x^4 + 1980* a^3*b^7*c^3*d^7*x^4 + 1485*a^4*b^6*c^2*d^8*x^4 + 990*a^5*b^5*c*d^9*x^4 + 4 95*a^6*b^4*d^10*x^4 + 1760*b^10*c^7*d^3*x^3 + 1540*a*b^9*c^6*d^4*x^3 + 132 0*a^2*b^8*c^5*d^5*x^3 + 1100*a^3*b^7*c^4*d^6*x^3 + 880*a^4*b^6*c^3*d^7*x^3 + 660*a^5*b^5*c^2*d^8*x^3 + 440*a^6*b^4*c*d^9*x^3 + 220*a^7*b^3*d^10*x^3 + 594*b^10*c^8*d^2*x^2 + 528*a*b^9*c^7*d^3*x^2 + 462*a^2*b^8*c^6*d^4*x^2 + 396*a^3*b^7*c^5*d^5*x^2 + 330*a^4*b^6*c^4*d^6*x^2 + 264*a^5*b^5*c^3*d^7*x ^2 + 198*a^6*b^4*c^2*d^8*x^2 + 132*a^7*b^3*c*d^9*x^2 + 66*a^8*b^2*d^10*x^2 + 120*b^10*c^9*d*x + 108*a*b^9*c^8*d^2*x + 96*a^2*b^8*c^7*d^3*x + 84*a^3* b^7*c^6*d^4*x + 72*a^4*b^6*c^5*d^5*x + 60*a^5*b^5*c^4*d^6*x + 48*a^6*b^4*c ^3*d^7*x + 36*a^7*b^3*c^2*d^8*x + 24*a^8*b^2*c*d^9*x + 12*a^9*b*d^10*x + 1 1*b^10*c^10 + 10*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 8*a^3*b^7*c^7*d^3 + 7*a ^4*b^6*c^6*d^4 + 6*a^5*b^5*c^5*d^5 + 5*a^6*b^4*c^4*d^6 + 4*a^7*b^3*c^3*...
Time = 0.42 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {(c+d x)^{10}}{(a+b x)^{13}} \, dx=\frac {{\left (c+d\,x\right )}^{11}\,\left (12\,a\,d-11\,b\,c+b\,d\,x\right )}{132\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^{12}} \]