Integrand size = 15, antiderivative size = 258 \[ \int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx=\frac {210 d^4 (b c-a d)^6 x}{b^{10}}-\frac {(b c-a d)^{10}}{3 b^{11} (a+b x)^3}-\frac {5 d (b c-a d)^9}{b^{11} (a+b x)^2}-\frac {45 d^2 (b c-a d)^8}{b^{11} (a+b x)}+\frac {126 d^5 (b c-a d)^5 (a+b x)^2}{b^{11}}+\frac {70 d^6 (b c-a d)^4 (a+b x)^3}{b^{11}}+\frac {30 d^7 (b c-a d)^3 (a+b x)^4}{b^{11}}+\frac {9 d^8 (b c-a d)^2 (a+b x)^5}{b^{11}}+\frac {5 d^9 (b c-a d) (a+b x)^6}{3 b^{11}}+\frac {d^{10} (a+b x)^7}{7 b^{11}}+\frac {120 d^3 (b c-a d)^7 \log (a+b x)}{b^{11}} \]
210*d^4*(-a*d+b*c)^6*x/b^10-1/3*(-a*d+b*c)^10/b^11/(b*x+a)^3-5*d*(-a*d+b*c )^9/b^11/(b*x+a)^2-45*d^2*(-a*d+b*c)^8/b^11/(b*x+a)+126*d^5*(-a*d+b*c)^5*( b*x+a)^2/b^11+70*d^6*(-a*d+b*c)^4*(b*x+a)^3/b^11+30*d^7*(-a*d+b*c)^3*(b*x+ a)^4/b^11+9*d^8*(-a*d+b*c)^2*(b*x+a)^5/b^11+5/3*d^9*(-a*d+b*c)*(b*x+a)^6/b ^11+1/7*d^10*(b*x+a)^7/b^11+120*d^3*(-a*d+b*c)^7*ln(b*x+a)/b^11
Time = 0.10 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.66 \[ \int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx=\frac {21 b d^4 \left (210 b^6 c^6-1008 a b^5 c^5 d+2100 a^2 b^4 c^4 d^2-2400 a^3 b^3 c^3 d^3+1575 a^4 b^2 c^2 d^4-560 a^5 b c d^5+84 a^6 d^6\right ) x+21 b^2 d^5 \left (126 b^5 c^5-420 a b^4 c^4 d+600 a^2 b^3 c^3 d^2-450 a^3 b^2 c^2 d^3+175 a^4 b c d^4-28 a^5 d^5\right ) x^2+35 b^3 d^6 \left (42 b^4 c^4-96 a b^3 c^3 d+90 a^2 b^2 c^2 d^2-40 a^3 b c d^3+7 a^4 d^4\right ) x^3+105 b^4 d^7 \left (6 b^3 c^3-9 a b^2 c^2 d+5 a^2 b c d^2-a^3 d^3\right ) x^4+21 b^5 d^8 \left (9 b^2 c^2-8 a b c d+2 a^2 d^2\right ) x^5+7 b^6 d^9 (5 b c-2 a d) x^6+3 b^7 d^{10} x^7-\frac {7 (b c-a d)^{10}}{(a+b x)^3}+\frac {105 d (-b c+a d)^9}{(a+b x)^2}-\frac {945 d^2 (b c-a d)^8}{a+b x}+2520 d^3 (b c-a d)^7 \log (a+b x)}{21 b^{11}} \]
(21*b*d^4*(210*b^6*c^6 - 1008*a*b^5*c^5*d + 2100*a^2*b^4*c^4*d^2 - 2400*a^ 3*b^3*c^3*d^3 + 1575*a^4*b^2*c^2*d^4 - 560*a^5*b*c*d^5 + 84*a^6*d^6)*x + 2 1*b^2*d^5*(126*b^5*c^5 - 420*a*b^4*c^4*d + 600*a^2*b^3*c^3*d^2 - 450*a^3*b ^2*c^2*d^3 + 175*a^4*b*c*d^4 - 28*a^5*d^5)*x^2 + 35*b^3*d^6*(42*b^4*c^4 - 96*a*b^3*c^3*d + 90*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 7*a^4*d^4)*x^3 + 10 5*b^4*d^7*(6*b^3*c^3 - 9*a*b^2*c^2*d + 5*a^2*b*c*d^2 - a^3*d^3)*x^4 + 21*b ^5*d^8*(9*b^2*c^2 - 8*a*b*c*d + 2*a^2*d^2)*x^5 + 7*b^6*d^9*(5*b*c - 2*a*d) *x^6 + 3*b^7*d^10*x^7 - (7*(b*c - a*d)^10)/(a + b*x)^3 + (105*d*(-(b*c) + a*d)^9)/(a + b*x)^2 - (945*d^2*(b*c - a*d)^8)/(a + b*x) + 2520*d^3*(b*c - a*d)^7*Log[a + b*x])/(21*b^11)
Time = 0.60 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {10 d^9 (a+b x)^5 (b c-a d)}{b^{10}}+\frac {45 d^8 (a+b x)^4 (b c-a d)^2}{b^{10}}+\frac {120 d^7 (a+b x)^3 (b c-a d)^3}{b^{10}}+\frac {210 d^6 (a+b x)^2 (b c-a d)^4}{b^{10}}+\frac {252 d^5 (a+b x) (b c-a d)^5}{b^{10}}+\frac {210 d^4 (b c-a d)^6}{b^{10}}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^2}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^3}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^4}+\frac {d^{10} (a+b x)^6}{b^{10}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 d^9 (a+b x)^6 (b c-a d)}{3 b^{11}}+\frac {9 d^8 (a+b x)^5 (b c-a d)^2}{b^{11}}+\frac {30 d^7 (a+b x)^4 (b c-a d)^3}{b^{11}}+\frac {70 d^6 (a+b x)^3 (b c-a d)^4}{b^{11}}+\frac {126 d^5 (a+b x)^2 (b c-a d)^5}{b^{11}}+\frac {120 d^3 (b c-a d)^7 \log (a+b x)}{b^{11}}-\frac {45 d^2 (b c-a d)^8}{b^{11} (a+b x)}-\frac {5 d (b c-a d)^9}{b^{11} (a+b x)^2}-\frac {(b c-a d)^{10}}{3 b^{11} (a+b x)^3}+\frac {d^{10} (a+b x)^7}{7 b^{11}}+\frac {210 d^4 x (b c-a d)^6}{b^{10}}\) |
(210*d^4*(b*c - a*d)^6*x)/b^10 - (b*c - a*d)^10/(3*b^11*(a + b*x)^3) - (5* d*(b*c - a*d)^9)/(b^11*(a + b*x)^2) - (45*d^2*(b*c - a*d)^8)/(b^11*(a + b* x)) + (126*d^5*(b*c - a*d)^5*(a + b*x)^2)/b^11 + (70*d^6*(b*c - a*d)^4*(a + b*x)^3)/b^11 + (30*d^7*(b*c - a*d)^3*(a + b*x)^4)/b^11 + (9*d^8*(b*c - a *d)^2*(a + b*x)^5)/b^11 + (5*d^9*(b*c - a*d)*(a + b*x)^6)/(3*b^11) + (d^10 *(a + b*x)^7)/(7*b^11) + (120*d^3*(b*c - a*d)^7*Log[a + b*x])/b^11
3.14.15.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(840\) vs. \(2(252)=504\).
Time = 0.22 (sec) , antiderivative size = 841, normalized size of antiderivative = 3.26
| method | result | size |
| norman | \(\frac {-\frac {660 a^{10} d^{10}-4620 a^{9} b c \,d^{9}+13860 a^{8} b^{2} c^{2} d^{8}-23100 a^{7} b^{3} c^{3} d^{7}+23100 a^{6} b^{4} c^{4} d^{6}-13860 a^{5} b^{5} c^{5} d^{5}+4620 a^{4} b^{6} c^{6} d^{4}-660 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +b^{10} c^{10}}{3 b^{11}}+\frac {d^{10} x^{10}}{7 b}-\frac {3 \left (120 a^{8} d^{10}-840 a^{7} b c \,d^{9}+2520 a^{6} b^{2} c^{2} d^{8}-4200 a^{5} b^{3} c^{3} d^{7}+4200 a^{4} b^{4} c^{4} d^{6}-2520 a^{3} b^{5} c^{5} d^{5}+840 a^{2} b^{6} c^{6} d^{4}-120 a \,b^{7} c^{7} d^{3}+15 b^{8} c^{8} d^{2}\right ) x^{2}}{b^{9}}-\frac {\left (540 a^{9} d^{10}-3780 a^{8} b c \,d^{9}+11340 a^{7} b^{2} c^{2} d^{8}-18900 a^{6} b^{3} c^{3} d^{7}+18900 a^{5} b^{4} c^{4} d^{6}-11340 a^{4} b^{5} c^{5} d^{5}+3780 a^{3} b^{6} c^{6} d^{4}-540 a^{2} b^{7} c^{7} d^{3}+45 a \,b^{8} c^{8} d^{2}+5 b^{9} c^{9} d \right ) x}{b^{10}}+\frac {30 d^{4} \left (a^{6} d^{6}-7 a^{5} b c \,d^{5}+21 a^{4} b^{2} c^{2} d^{4}-35 a^{3} b^{3} c^{3} d^{3}+35 a^{2} b^{4} c^{4} d^{2}-21 a \,b^{5} c^{5} d +7 b^{6} c^{6}\right ) x^{4}}{b^{7}}-\frac {6 d^{5} \left (a^{5} d^{5}-7 a^{4} b c \,d^{4}+21 a^{3} b^{2} c^{2} d^{3}-35 a^{2} b^{3} c^{3} d^{2}+35 a \,b^{4} c^{4} d -21 b^{5} c^{5}\right ) x^{5}}{b^{6}}+\frac {2 d^{6} \left (a^{4} d^{4}-7 a^{3} b c \,d^{3}+21 a^{2} b^{2} c^{2} d^{2}-35 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right ) x^{6}}{b^{5}}-\frac {6 d^{7} \left (a^{3} d^{3}-7 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d -35 b^{3} c^{3}\right ) x^{7}}{7 b^{4}}+\frac {3 d^{8} \left (a^{2} d^{2}-7 a b c d +21 b^{2} c^{2}\right ) x^{8}}{7 b^{3}}-\frac {5 d^{9} \left (a d -7 b c \right ) x^{9}}{21 b^{2}}}{\left (b x +a \right )^{3}}-\frac {120 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right ) \ln \left (b x +a \right )}{b^{11}}\) | \(841\) |
| default | \(\frac {d^{4} \left (9 b^{6} c^{2} d^{4} x^{5}-1008 a \,b^{5} c^{5} d x +210 b^{6} c^{6} x +84 a^{6} d^{6} x +\frac {1}{7} d^{6} x^{7} b^{6}-160 a \,b^{5} c^{3} d^{3} x^{3}+175 a^{4} b^{2} c \,d^{5} x^{2}-450 a^{3} b^{3} c^{2} d^{4} x^{2}+600 a^{2} b^{4} c^{3} d^{3} x^{2}-420 a \,b^{5} c^{4} d^{2} x^{2}-560 a^{5} b c \,d^{5} x +1575 a^{4} b^{2} c^{2} d^{4} x -2400 a^{3} b^{3} c^{3} d^{3} x +2100 a^{2} b^{4} c^{4} d^{2} x +30 b^{6} c^{3} d^{3} x^{4}+\frac {35}{3} a^{4} b^{2} d^{6} x^{3}+70 b^{6} c^{4} d^{2} x^{3}-28 a^{5} b \,d^{6} x^{2}+126 b^{6} c^{5} d \,x^{2}-\frac {2}{3} a \,b^{5} d^{6} x^{6}+\frac {5}{3} b^{6} c \,d^{5} x^{6}+2 a^{2} b^{4} d^{6} x^{5}-5 a^{3} b^{3} d^{6} x^{4}+25 a^{2} b^{4} c \,d^{5} x^{4}-45 a \,b^{5} c^{2} d^{4} x^{4}-\frac {200}{3} a^{3} b^{3} c \,d^{5} x^{3}+150 a^{2} b^{4} c^{2} d^{4} x^{3}-8 a \,b^{5} c \,d^{5} x^{5}\right )}{b^{10}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{3 b^{11} \left (b x +a \right )^{3}}-\frac {120 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right ) \ln \left (b x +a \right )}{b^{11}}+\frac {5 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{b^{11} \left (b x +a \right )^{2}}-\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{b^{11} \left (b x +a \right )}\) | \(896\) |
| risch | \(\frac {9 d^{8} c^{2} x^{5}}{b^{4}}+\frac {210 d^{4} c^{6} x}{b^{4}}+\frac {84 d^{10} a^{6} x}{b^{10}}+\frac {30 d^{7} c^{3} x^{4}}{b^{4}}+\frac {35 d^{10} a^{4} x^{3}}{3 b^{8}}+\frac {70 d^{6} c^{4} x^{3}}{b^{4}}-\frac {28 d^{10} a^{5} x^{2}}{b^{9}}+\frac {126 d^{5} c^{5} x^{2}}{b^{4}}-\frac {2 d^{10} a \,x^{6}}{3 b^{5}}+\frac {5 d^{9} c \,x^{6}}{3 b^{4}}+\frac {2 d^{10} a^{2} x^{5}}{b^{6}}-\frac {5 d^{10} a^{3} x^{4}}{b^{7}}-\frac {120 d^{10} \ln \left (b x +a \right ) a^{7}}{b^{11}}+\frac {120 d^{3} \ln \left (b x +a \right ) c^{7}}{b^{4}}-\frac {1008 d^{5} a \,c^{5} x}{b^{5}}-\frac {160 d^{7} a \,c^{3} x^{3}}{b^{5}}+\frac {840 d^{9} \ln \left (b x +a \right ) a^{6} c}{b^{10}}-\frac {2520 d^{8} \ln \left (b x +a \right ) a^{5} c^{2}}{b^{9}}+\frac {4200 d^{7} \ln \left (b x +a \right ) a^{4} c^{3}}{b^{8}}-\frac {4200 d^{6} \ln \left (b x +a \right ) a^{3} c^{4}}{b^{7}}+\frac {2520 d^{5} \ln \left (b x +a \right ) a^{2} c^{5}}{b^{6}}-\frac {840 d^{4} \ln \left (b x +a \right ) a \,c^{6}}{b^{5}}+\frac {175 d^{9} a^{4} c \,x^{2}}{b^{8}}-\frac {450 d^{8} a^{3} c^{2} x^{2}}{b^{7}}+\frac {600 d^{7} a^{2} c^{3} x^{2}}{b^{6}}-\frac {420 d^{6} a \,c^{4} x^{2}}{b^{5}}-\frac {560 d^{9} a^{5} c x}{b^{9}}+\frac {1575 d^{8} a^{4} c^{2} x}{b^{8}}-\frac {2400 d^{7} a^{3} c^{3} x}{b^{7}}+\frac {2100 d^{6} a^{2} c^{4} x}{b^{6}}+\frac {25 d^{9} a^{2} c \,x^{4}}{b^{6}}-\frac {45 d^{8} a \,c^{2} x^{4}}{b^{5}}-\frac {200 d^{9} a^{3} c \,x^{3}}{3 b^{7}}+\frac {150 d^{8} a^{2} c^{2} x^{3}}{b^{6}}-\frac {8 d^{9} a c \,x^{5}}{b^{5}}+\frac {d^{10} x^{7}}{7 b^{4}}+\frac {\left (-45 a^{8} b \,d^{10}+360 a^{7} b^{2} c \,d^{9}-1260 a^{6} b^{3} c^{2} d^{8}+2520 a^{5} b^{4} c^{3} d^{7}-3150 a^{4} b^{5} c^{4} d^{6}+2520 a^{3} b^{6} c^{5} d^{5}-1260 a^{2} b^{7} c^{6} d^{4}+360 a \,b^{8} c^{7} d^{3}-45 b^{9} c^{8} d^{2}\right ) x^{2}-5 d \left (17 a^{9} d^{9}-135 a^{8} b c \,d^{8}+468 a^{7} b^{2} c^{2} d^{7}-924 a^{6} b^{3} c^{3} d^{6}+1134 a^{5} b^{4} c^{4} d^{5}-882 a^{4} b^{5} c^{5} d^{4}+420 a^{3} b^{6} c^{6} d^{3}-108 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d +b^{9} c^{9}\right ) x -\frac {121 a^{10} d^{10}-955 a^{9} b c \,d^{9}+3285 a^{8} b^{2} c^{2} d^{8}-6420 a^{7} b^{3} c^{3} d^{7}+7770 a^{6} b^{4} c^{4} d^{6}-5922 a^{5} b^{5} c^{5} d^{5}+2730 a^{4} b^{6} c^{6} d^{4}-660 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}+5 a \,b^{9} c^{9} d +b^{10} c^{10}}{3 b}}{b^{10} \left (b x +a \right )^{3}}\) | \(942\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1499\) |
(-1/3*(660*a^10*d^10-4620*a^9*b*c*d^9+13860*a^8*b^2*c^2*d^8-23100*a^7*b^3* c^3*d^7+23100*a^6*b^4*c^4*d^6-13860*a^5*b^5*c^5*d^5+4620*a^4*b^6*c^6*d^4-6 60*a^3*b^7*c^7*d^3+45*a^2*b^8*c^8*d^2+5*a*b^9*c^9*d+b^10*c^10)/b^11+1/7/b* d^10*x^10-3*(120*a^8*d^10-840*a^7*b*c*d^9+2520*a^6*b^2*c^2*d^8-4200*a^5*b^ 3*c^3*d^7+4200*a^4*b^4*c^4*d^6-2520*a^3*b^5*c^5*d^5+840*a^2*b^6*c^6*d^4-12 0*a*b^7*c^7*d^3+15*b^8*c^8*d^2)/b^9*x^2-(540*a^9*d^10-3780*a^8*b*c*d^9+113 40*a^7*b^2*c^2*d^8-18900*a^6*b^3*c^3*d^7+18900*a^5*b^4*c^4*d^6-11340*a^4*b ^5*c^5*d^5+3780*a^3*b^6*c^6*d^4-540*a^2*b^7*c^7*d^3+45*a*b^8*c^8*d^2+5*b^9 *c^9*d)/b^10*x+30*d^4*(a^6*d^6-7*a^5*b*c*d^5+21*a^4*b^2*c^2*d^4-35*a^3*b^3 *c^3*d^3+35*a^2*b^4*c^4*d^2-21*a*b^5*c^5*d+7*b^6*c^6)/b^7*x^4-6*d^5*(a^5*d ^5-7*a^4*b*c*d^4+21*a^3*b^2*c^2*d^3-35*a^2*b^3*c^3*d^2+35*a*b^4*c^4*d-21*b ^5*c^5)/b^6*x^5+2*d^6*(a^4*d^4-7*a^3*b*c*d^3+21*a^2*b^2*c^2*d^2-35*a*b^3*c ^3*d+35*b^4*c^4)/b^5*x^6-6/7*d^7*(a^3*d^3-7*a^2*b*c*d^2+21*a*b^2*c^2*d-35* b^3*c^3)/b^4*x^7+3/7*d^8*(a^2*d^2-7*a*b*c*d+21*b^2*c^2)/b^3*x^8-5/21*d^9*( a*d-7*b*c)/b^2*x^9)/(b*x+a)^3-120/b^11*d^3*(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)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 1316 vs. \(2 (252) = 504\).
Time = 0.23 (sec) , antiderivative size = 1316, normalized size of antiderivative = 5.10 \[ \int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx=\text {Too large to display} \]
1/21*(3*b^10*d^10*x^10 - 7*b^10*c^10 - 35*a*b^9*c^9*d - 315*a^2*b^8*c^8*d^ 2 + 4620*a^3*b^7*c^7*d^3 - 19110*a^4*b^6*c^6*d^4 + 41454*a^5*b^5*c^5*d^5 - 54390*a^6*b^4*c^4*d^6 + 44940*a^7*b^3*c^3*d^7 - 22995*a^8*b^2*c^2*d^8 + 6 685*a^9*b*c*d^9 - 847*a^10*d^10 + 5*(7*b^10*c*d^9 - a*b^9*d^10)*x^9 + 9*(2 1*b^10*c^2*d^8 - 7*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 18*(35*b^10*c^3*d^7 - 21*a*b^9*c^2*d^8 + 7*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + 42*(35*b^10*c^4* d^6 - 35*a*b^9*c^3*d^7 + 21*a^2*b^8*c^2*d^8 - 7*a^3*b^7*c*d^9 + a^4*b^6*d^ 10)*x^6 + 126*(21*b^10*c^5*d^5 - 35*a*b^9*c^4*d^6 + 35*a^2*b^8*c^3*d^7 - 2 1*a^3*b^7*c^2*d^8 + 7*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 630*(7*b^10*c^6* d^4 - 21*a*b^9*c^5*d^5 + 35*a^2*b^8*c^4*d^6 - 35*a^3*b^7*c^3*d^7 + 21*a^4* b^6*c^2*d^8 - 7*a^5*b^5*c*d^9 + a^6*b^4*d^10)*x^4 + 7*(1890*a*b^9*c^6*d^4 - 7938*a^2*b^8*c^5*d^5 + 15330*a^3*b^7*c^4*d^6 - 16680*a^4*b^6*c^3*d^7 + 1 0575*a^5*b^5*c^2*d^8 - 3665*a^6*b^4*c*d^9 + 539*a^7*b^3*d^10)*x^3 - 21*(45 *b^10*c^8*d^2 - 360*a*b^9*c^7*d^3 + 630*a^2*b^8*c^6*d^4 + 378*a^3*b^7*c^5* d^5 - 2730*a^4*b^6*c^4*d^6 + 4080*a^5*b^5*c^3*d^7 - 3015*a^6*b^4*c^2*d^8 + 1145*a^7*b^3*c*d^9 - 179*a^8*b^2*d^10)*x^2 - 21*(5*b^10*c^9*d + 45*a*b^9* c^8*d^2 - 540*a^2*b^8*c^7*d^3 + 1890*a^3*b^7*c^6*d^4 - 3402*a^4*b^6*c^5*d^ 5 + 3570*a^5*b^5*c^4*d^6 - 2220*a^6*b^4*c^3*d^7 + 765*a^7*b^3*c^2*d^8 - 11 5*a^8*b^2*c*d^9 + a^9*b*d^10)*x + 2520*(a^3*b^7*c^7*d^3 - 7*a^4*b^6*c^6*d^ 4 + 21*a^5*b^5*c^5*d^5 - 35*a^6*b^4*c^4*d^6 + 35*a^7*b^3*c^3*d^7 - 21*a...
Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 891 vs. \(2 (252) = 504\).
Time = 0.26 (sec) , antiderivative size = 891, normalized size of antiderivative = 3.45 \[ \int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx=-\frac {b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 660 \, a^{3} b^{7} c^{7} d^{3} + 2730 \, a^{4} b^{6} c^{6} d^{4} - 5922 \, a^{5} b^{5} c^{5} d^{5} + 7770 \, a^{6} b^{4} c^{4} d^{6} - 6420 \, a^{7} b^{3} c^{3} d^{7} + 3285 \, a^{8} b^{2} c^{2} d^{8} - 955 \, a^{9} b c d^{9} + 121 \, a^{10} d^{10} + 135 \, {\left (b^{10} c^{8} d^{2} - 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} - 56 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} - 56 \, a^{5} b^{5} c^{3} d^{7} + 28 \, a^{6} b^{4} c^{2} d^{8} - 8 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 15 \, {\left (b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} - 108 \, a^{2} b^{8} c^{7} d^{3} + 420 \, a^{3} b^{7} c^{6} d^{4} - 882 \, a^{4} b^{6} c^{5} d^{5} + 1134 \, a^{5} b^{5} c^{4} d^{6} - 924 \, a^{6} b^{4} c^{3} d^{7} + 468 \, a^{7} b^{3} c^{2} d^{8} - 135 \, a^{8} b^{2} c d^{9} + 17 \, a^{9} b d^{10}\right )} x}{3 \, {\left (b^{14} x^{3} + 3 \, a b^{13} x^{2} + 3 \, a^{2} b^{12} x + a^{3} b^{11}\right )}} + \frac {3 \, b^{6} d^{10} x^{7} + 7 \, {\left (5 \, b^{6} c d^{9} - 2 \, a b^{5} d^{10}\right )} x^{6} + 21 \, {\left (9 \, b^{6} c^{2} d^{8} - 8 \, a b^{5} c d^{9} + 2 \, a^{2} b^{4} d^{10}\right )} x^{5} + 105 \, {\left (6 \, b^{6} c^{3} d^{7} - 9 \, a b^{5} c^{2} d^{8} + 5 \, a^{2} b^{4} c d^{9} - a^{3} b^{3} d^{10}\right )} x^{4} + 35 \, {\left (42 \, b^{6} c^{4} d^{6} - 96 \, a b^{5} c^{3} d^{7} + 90 \, a^{2} b^{4} c^{2} d^{8} - 40 \, a^{3} b^{3} c d^{9} + 7 \, a^{4} b^{2} d^{10}\right )} x^{3} + 21 \, {\left (126 \, b^{6} c^{5} d^{5} - 420 \, a b^{5} c^{4} d^{6} + 600 \, a^{2} b^{4} c^{3} d^{7} - 450 \, a^{3} b^{3} c^{2} d^{8} + 175 \, a^{4} b^{2} c d^{9} - 28 \, a^{5} b d^{10}\right )} x^{2} + 21 \, {\left (210 \, b^{6} c^{6} d^{4} - 1008 \, a b^{5} c^{5} d^{5} + 2100 \, a^{2} b^{4} c^{4} d^{6} - 2400 \, a^{3} b^{3} c^{3} d^{7} + 1575 \, a^{4} b^{2} c^{2} d^{8} - 560 \, a^{5} b c d^{9} + 84 \, a^{6} d^{10}\right )} x}{21 \, b^{10}} + \frac {120 \, {\left (b^{7} c^{7} d^{3} - 7 \, a b^{6} c^{6} d^{4} + 21 \, a^{2} b^{5} c^{5} d^{5} - 35 \, a^{3} b^{4} c^{4} d^{6} + 35 \, a^{4} b^{3} c^{3} d^{7} - 21 \, a^{5} b^{2} c^{2} d^{8} + 7 \, a^{6} b c d^{9} - a^{7} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \]
-1/3*(b^10*c^10 + 5*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^2 - 660*a^3*b^7*c^7*d^3 + 2730*a^4*b^6*c^6*d^4 - 5922*a^5*b^5*c^5*d^5 + 7770*a^6*b^4*c^4*d^6 - 64 20*a^7*b^3*c^3*d^7 + 3285*a^8*b^2*c^2*d^8 - 955*a^9*b*c*d^9 + 121*a^10*d^1 0 + 135*(b^10*c^8*d^2 - 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 - 56*a^3*b^7* c^5*d^5 + 70*a^4*b^6*c^4*d^6 - 56*a^5*b^5*c^3*d^7 + 28*a^6*b^4*c^2*d^8 - 8 *a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 15*(b^10*c^9*d + 9*a*b^9*c^8*d^2 - 10 8*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 - 882*a^4*b^6*c^5*d^5 + 1134*a^5*b ^5*c^4*d^6 - 924*a^6*b^4*c^3*d^7 + 468*a^7*b^3*c^2*d^8 - 135*a^8*b^2*c*d^9 + 17*a^9*b*d^10)*x)/(b^14*x^3 + 3*a*b^13*x^2 + 3*a^2*b^12*x + a^3*b^11) + 1/21*(3*b^6*d^10*x^7 + 7*(5*b^6*c*d^9 - 2*a*b^5*d^10)*x^6 + 21*(9*b^6*c^2 *d^8 - 8*a*b^5*c*d^9 + 2*a^2*b^4*d^10)*x^5 + 105*(6*b^6*c^3*d^7 - 9*a*b^5* c^2*d^8 + 5*a^2*b^4*c*d^9 - a^3*b^3*d^10)*x^4 + 35*(42*b^6*c^4*d^6 - 96*a* b^5*c^3*d^7 + 90*a^2*b^4*c^2*d^8 - 40*a^3*b^3*c*d^9 + 7*a^4*b^2*d^10)*x^3 + 21*(126*b^6*c^5*d^5 - 420*a*b^5*c^4*d^6 + 600*a^2*b^4*c^3*d^7 - 450*a^3* b^3*c^2*d^8 + 175*a^4*b^2*c*d^9 - 28*a^5*b*d^10)*x^2 + 21*(210*b^6*c^6*d^4 - 1008*a*b^5*c^5*d^5 + 2100*a^2*b^4*c^4*d^6 - 2400*a^3*b^3*c^3*d^7 + 1575 *a^4*b^2*c^2*d^8 - 560*a^5*b*c*d^9 + 84*a^6*d^10)*x)/b^10 + 120*(b^7*c^7*d ^3 - 7*a*b^6*c^6*d^4 + 21*a^2*b^5*c^5*d^5 - 35*a^3*b^4*c^4*d^6 + 35*a^4*b^ 3*c^3*d^7 - 21*a^5*b^2*c^2*d^8 + 7*a^6*b*c*d^9 - a^7*d^10)*log(b*x + a)/b^ 11
Leaf count of result is larger than twice the leaf count of optimal. 907 vs. \(2 (252) = 504\).
Time = 0.40 (sec) , antiderivative size = 907, normalized size of antiderivative = 3.52 \[ \int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx=\frac {120 \, {\left (b^{7} c^{7} d^{3} - 7 \, a b^{6} c^{6} d^{4} + 21 \, a^{2} b^{5} c^{5} d^{5} - 35 \, a^{3} b^{4} c^{4} d^{6} + 35 \, a^{4} b^{3} c^{3} d^{7} - 21 \, a^{5} b^{2} c^{2} d^{8} + 7 \, a^{6} b c d^{9} - a^{7} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {b^{10} c^{10} + 5 \, a b^{9} c^{9} d + 45 \, a^{2} b^{8} c^{8} d^{2} - 660 \, a^{3} b^{7} c^{7} d^{3} + 2730 \, a^{4} b^{6} c^{6} d^{4} - 5922 \, a^{5} b^{5} c^{5} d^{5} + 7770 \, a^{6} b^{4} c^{4} d^{6} - 6420 \, a^{7} b^{3} c^{3} d^{7} + 3285 \, a^{8} b^{2} c^{2} d^{8} - 955 \, a^{9} b c d^{9} + 121 \, a^{10} d^{10} + 135 \, {\left (b^{10} c^{8} d^{2} - 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} - 56 \, a^{3} b^{7} c^{5} d^{5} + 70 \, a^{4} b^{6} c^{4} d^{6} - 56 \, a^{5} b^{5} c^{3} d^{7} + 28 \, a^{6} b^{4} c^{2} d^{8} - 8 \, a^{7} b^{3} c d^{9} + a^{8} b^{2} d^{10}\right )} x^{2} + 15 \, {\left (b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} - 108 \, a^{2} b^{8} c^{7} d^{3} + 420 \, a^{3} b^{7} c^{6} d^{4} - 882 \, a^{4} b^{6} c^{5} d^{5} + 1134 \, a^{5} b^{5} c^{4} d^{6} - 924 \, a^{6} b^{4} c^{3} d^{7} + 468 \, a^{7} b^{3} c^{2} d^{8} - 135 \, a^{8} b^{2} c d^{9} + 17 \, a^{9} b d^{10}\right )} x}{3 \, {\left (b x + a\right )}^{3} b^{11}} + \frac {3 \, b^{24} d^{10} x^{7} + 35 \, b^{24} c d^{9} x^{6} - 14 \, a b^{23} d^{10} x^{6} + 189 \, b^{24} c^{2} d^{8} x^{5} - 168 \, a b^{23} c d^{9} x^{5} + 42 \, a^{2} b^{22} d^{10} x^{5} + 630 \, b^{24} c^{3} d^{7} x^{4} - 945 \, a b^{23} c^{2} d^{8} x^{4} + 525 \, a^{2} b^{22} c d^{9} x^{4} - 105 \, a^{3} b^{21} d^{10} x^{4} + 1470 \, b^{24} c^{4} d^{6} x^{3} - 3360 \, a b^{23} c^{3} d^{7} x^{3} + 3150 \, a^{2} b^{22} c^{2} d^{8} x^{3} - 1400 \, a^{3} b^{21} c d^{9} x^{3} + 245 \, a^{4} b^{20} d^{10} x^{3} + 2646 \, b^{24} c^{5} d^{5} x^{2} - 8820 \, a b^{23} c^{4} d^{6} x^{2} + 12600 \, a^{2} b^{22} c^{3} d^{7} x^{2} - 9450 \, a^{3} b^{21} c^{2} d^{8} x^{2} + 3675 \, a^{4} b^{20} c d^{9} x^{2} - 588 \, a^{5} b^{19} d^{10} x^{2} + 4410 \, b^{24} c^{6} d^{4} x - 21168 \, a b^{23} c^{5} d^{5} x + 44100 \, a^{2} b^{22} c^{4} d^{6} x - 50400 \, a^{3} b^{21} c^{3} d^{7} x + 33075 \, a^{4} b^{20} c^{2} d^{8} x - 11760 \, a^{5} b^{19} c d^{9} x + 1764 \, a^{6} b^{18} d^{10} x}{21 \, b^{28}} \]
120*(b^7*c^7*d^3 - 7*a*b^6*c^6*d^4 + 21*a^2*b^5*c^5*d^5 - 35*a^3*b^4*c^4*d ^6 + 35*a^4*b^3*c^3*d^7 - 21*a^5*b^2*c^2*d^8 + 7*a^6*b*c*d^9 - a^7*d^10)*l og(abs(b*x + a))/b^11 - 1/3*(b^10*c^10 + 5*a*b^9*c^9*d + 45*a^2*b^8*c^8*d^ 2 - 660*a^3*b^7*c^7*d^3 + 2730*a^4*b^6*c^6*d^4 - 5922*a^5*b^5*c^5*d^5 + 77 70*a^6*b^4*c^4*d^6 - 6420*a^7*b^3*c^3*d^7 + 3285*a^8*b^2*c^2*d^8 - 955*a^9 *b*c*d^9 + 121*a^10*d^10 + 135*(b^10*c^8*d^2 - 8*a*b^9*c^7*d^3 + 28*a^2*b^ 8*c^6*d^4 - 56*a^3*b^7*c^5*d^5 + 70*a^4*b^6*c^4*d^6 - 56*a^5*b^5*c^3*d^7 + 28*a^6*b^4*c^2*d^8 - 8*a^7*b^3*c*d^9 + a^8*b^2*d^10)*x^2 + 15*(b^10*c^9*d + 9*a*b^9*c^8*d^2 - 108*a^2*b^8*c^7*d^3 + 420*a^3*b^7*c^6*d^4 - 882*a^4*b ^6*c^5*d^5 + 1134*a^5*b^5*c^4*d^6 - 924*a^6*b^4*c^3*d^7 + 468*a^7*b^3*c^2* d^8 - 135*a^8*b^2*c*d^9 + 17*a^9*b*d^10)*x)/((b*x + a)^3*b^11) + 1/21*(3*b ^24*d^10*x^7 + 35*b^24*c*d^9*x^6 - 14*a*b^23*d^10*x^6 + 189*b^24*c^2*d^8*x ^5 - 168*a*b^23*c*d^9*x^5 + 42*a^2*b^22*d^10*x^5 + 630*b^24*c^3*d^7*x^4 - 945*a*b^23*c^2*d^8*x^4 + 525*a^2*b^22*c*d^9*x^4 - 105*a^3*b^21*d^10*x^4 + 1470*b^24*c^4*d^6*x^3 - 3360*a*b^23*c^3*d^7*x^3 + 3150*a^2*b^22*c^2*d^8*x^ 3 - 1400*a^3*b^21*c*d^9*x^3 + 245*a^4*b^20*d^10*x^3 + 2646*b^24*c^5*d^5*x^ 2 - 8820*a*b^23*c^4*d^6*x^2 + 12600*a^2*b^22*c^3*d^7*x^2 - 9450*a^3*b^21*c ^2*d^8*x^2 + 3675*a^4*b^20*c*d^9*x^2 - 588*a^5*b^19*d^10*x^2 + 4410*b^24*c ^6*d^4*x - 21168*a*b^23*c^5*d^5*x + 44100*a^2*b^22*c^4*d^6*x - 50400*a^3*b ^21*c^3*d^7*x + 33075*a^4*b^20*c^2*d^8*x - 11760*a^5*b^19*c*d^9*x + 176...
Time = 0.42 (sec) , antiderivative size = 2219, normalized size of antiderivative = 8.60 \[ \int \frac {(c+d x)^{10}}{(a+b x)^4} \, dx=\text {Too large to display} \]
x^3*((4*a*((4*a*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/ b^6 + (45*c^2*d^8)/b^4))/b + (4*a^3*d^10)/b^7 - (120*c^3*d^7)/b^4 - (6*a^2 *((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^2))/(3*b) - (a^4*d^10)/(3*b^8) + (70 *c^4*d^6)/b^4 + (4*a^3*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/(3*b^3) - (2*a^2 *((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d ^8)/b^4))/b^2) - x^6*((2*a*d^10)/(3*b^5) - (5*c*d^9)/(3*b^4)) - x^4*((a*(( 4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8) /b^4))/b + (a^3*d^10)/b^7 - (30*c^3*d^7)/b^4 - (3*a^2*((4*a*d^10)/b^5 - (1 0*c*d^9)/b^4))/(2*b^2)) + x^5*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/(5* b) - (6*a^2*d^10)/(5*b^6) + (9*c^2*d^8)/b^4) - x*((4*a*((252*c^5*d^5)/b^4 - (4*a*((4*a*((4*a*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^1 0)/b^6 + (45*c^2*d^8)/b^4))/b + (4*a^3*d^10)/b^7 - (120*c^3*d^7)/b^4 - (6* a^2*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^2))/b - (a^4*d^10)/b^8 + (210*c^4 *d^6)/b^4 + (4*a^3*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^3 - (6*a^2*((4*a*( (4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4) )/b^2))/b + (a^4*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^4 + (6*a^2*((4*a*((4 *a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/ b^4))/b + (4*a^3*d^10)/b^7 - (120*c^3*d^7)/b^4 - (6*a^2*((4*a*d^10)/b^5 - (10*c*d^9)/b^4))/b^2))/b^2 - (4*a^3*((4*a*((4*a*d^10)/b^5 - (10*c*d^9)/b^4 ))/b - (6*a^2*d^10)/b^6 + (45*c^2*d^8)/b^4))/b^3))/b - (210*c^6*d^4)/b^...