Integrand size = 15, antiderivative size = 262 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\frac {120 d^7 (b c-a d)^3 x}{b^{10}}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}+\frac {45 d^8 (b c-a d)^2 (a+b x)^2}{2 b^{11}}+\frac {10 d^9 (b c-a d) (a+b x)^3}{3 b^{11}}+\frac {d^{10} (a+b x)^4}{4 b^{11}}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}} \]
120*d^7*(-a*d+b*c)^3*x/b^10-1/6*(-a*d+b*c)^10/b^11/(b*x+a)^6-2*d*(-a*d+b*c )^9/b^11/(b*x+a)^5-45/4*d^2*(-a*d+b*c)^8/b^11/(b*x+a)^4-40*d^3*(-a*d+b*c)^ 7/b^11/(b*x+a)^3-105*d^4*(-a*d+b*c)^6/b^11/(b*x+a)^2-252*d^5*(-a*d+b*c)^5/ b^11/(b*x+a)+45/2*d^8*(-a*d+b*c)^2*(b*x+a)^2/b^11+10/3*d^9*(-a*d+b*c)*(b*x +a)^3/b^11+1/4*d^10*(b*x+a)^4/b^11+210*d^6*(-a*d+b*c)^4*ln(b*x+a)/b^11
Time = 0.13 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.01 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\frac {12 b d^7 \left (120 b^3 c^3-315 a b^2 c^2 d+280 a^2 b c d^2-84 a^3 d^3\right ) x+6 b^2 d^8 \left (45 b^2 c^2-70 a b c d+28 a^2 d^2\right ) x^2+4 b^3 d^9 (10 b c-7 a d) x^3+3 b^4 d^{10} x^4-\frac {2 (b c-a d)^{10}}{(a+b x)^6}+\frac {24 d (-b c+a d)^9}{(a+b x)^5}-\frac {135 d^2 (b c-a d)^8}{(a+b x)^4}+\frac {480 d^3 (-b c+a d)^7}{(a+b x)^3}-\frac {1260 d^4 (b c-a d)^6}{(a+b x)^2}+\frac {3024 d^5 (-b c+a d)^5}{a+b x}+2520 d^6 (b c-a d)^4 \log (a+b x)}{12 b^{11}} \]
(12*b*d^7*(120*b^3*c^3 - 315*a*b^2*c^2*d + 280*a^2*b*c*d^2 - 84*a^3*d^3)*x + 6*b^2*d^8*(45*b^2*c^2 - 70*a*b*c*d + 28*a^2*d^2)*x^2 + 4*b^3*d^9*(10*b* c - 7*a*d)*x^3 + 3*b^4*d^10*x^4 - (2*(b*c - a*d)^10)/(a + b*x)^6 + (24*d*( -(b*c) + a*d)^9)/(a + b*x)^5 - (135*d^2*(b*c - a*d)^8)/(a + b*x)^4 + (480* d^3*(-(b*c) + a*d)^7)/(a + b*x)^3 - (1260*d^4*(b*c - a*d)^6)/(a + b*x)^2 + (3024*d^5*(-(b*c) + a*d)^5)/(a + b*x) + 2520*d^6*(b*c - a*d)^4*Log[a + b* x])/(12*b^11)
Time = 0.58 (sec) , antiderivative size = 262, 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.
\(\displaystyle \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {10 d^9 (a+b x)^2 (b c-a d)}{b^{10}}+\frac {45 d^8 (a+b x) (b c-a d)^2}{b^{10}}+\frac {120 d^7 (b c-a d)^3}{b^{10}}+\frac {210 d^6 (b c-a d)^4}{b^{10} (a+b x)}+\frac {252 d^5 (b c-a d)^5}{b^{10} (a+b x)^2}+\frac {210 d^4 (b c-a d)^6}{b^{10} (a+b x)^3}+\frac {120 d^3 (b c-a d)^7}{b^{10} (a+b x)^4}+\frac {45 d^2 (b c-a d)^8}{b^{10} (a+b x)^5}+\frac {10 d (b c-a d)^9}{b^{10} (a+b x)^6}+\frac {(b c-a d)^{10}}{b^{10} (a+b x)^7}+\frac {d^{10} (a+b x)^3}{b^{10}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {10 d^9 (a+b x)^3 (b c-a d)}{3 b^{11}}+\frac {45 d^8 (a+b x)^2 (b c-a d)^2}{2 b^{11}}+\frac {210 d^6 (b c-a d)^4 \log (a+b x)}{b^{11}}-\frac {252 d^5 (b c-a d)^5}{b^{11} (a+b x)}-\frac {105 d^4 (b c-a d)^6}{b^{11} (a+b x)^2}-\frac {40 d^3 (b c-a d)^7}{b^{11} (a+b x)^3}-\frac {45 d^2 (b c-a d)^8}{4 b^{11} (a+b x)^4}-\frac {2 d (b c-a d)^9}{b^{11} (a+b x)^5}-\frac {(b c-a d)^{10}}{6 b^{11} (a+b x)^6}+\frac {d^{10} (a+b x)^4}{4 b^{11}}+\frac {120 d^7 x (b c-a d)^3}{b^{10}}\) |
(120*d^7*(b*c - a*d)^3*x)/b^10 - (b*c - a*d)^10/(6*b^11*(a + b*x)^6) - (2* d*(b*c - a*d)^9)/(b^11*(a + b*x)^5) - (45*d^2*(b*c - a*d)^8)/(4*b^11*(a + b*x)^4) - (40*d^3*(b*c - a*d)^7)/(b^11*(a + b*x)^3) - (105*d^4*(b*c - a*d) ^6)/(b^11*(a + b*x)^2) - (252*d^5*(b*c - a*d)^5)/(b^11*(a + b*x)) + (45*d^ 8*(b*c - a*d)^2*(a + b*x)^2)/(2*b^11) + (10*d^9*(b*c - a*d)*(a + b*x)^3)/( 3*b^11) + (d^10*(a + b*x)^4)/(4*b^11) + (210*d^6*(b*c - a*d)^4*Log[a + b*x ])/b^11
3.14.18.3.1 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. \(849\) vs. \(2(252)=504\).
Time = 0.24 (sec) , antiderivative size = 850, normalized size of antiderivative = 3.24
method | result | size |
norman | \(\frac {\frac {6174 a^{10} d^{10}-24696 a^{9} b c \,d^{9}+37044 a^{8} b^{2} c^{2} d^{8}-24696 a^{7} b^{3} c^{3} d^{7}+6174 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-84 a^{4} b^{6} c^{6} d^{4}-24 a^{3} b^{7} c^{7} d^{3}-9 a^{2} b^{8} c^{8} d^{2}-4 a \,b^{9} c^{9} d -2 b^{10} c^{10}}{12 b^{11}}+\frac {d^{10} x^{10}}{4 b}+\frac {6 \left (210 a^{5} d^{10}-840 a^{4} b c \,d^{9}+1260 a^{3} b^{2} c^{2} d^{8}-840 a^{2} b^{3} c^{3} d^{7}+210 a \,b^{4} c^{4} d^{6}-42 b^{5} c^{5} d^{5}\right ) x^{5}}{b^{6}}+\frac {15 \left (315 a^{6} d^{10}-1260 a^{5} b c \,d^{9}+1890 a^{4} b^{2} c^{2} d^{8}-1260 a^{3} b^{3} c^{3} d^{7}+315 a^{2} b^{4} c^{4} d^{6}-42 a \,b^{5} c^{5} d^{5}-7 b^{6} c^{6} d^{4}\right ) x^{4}}{b^{7}}+\frac {20 \left (385 a^{7} d^{10}-1540 a^{6} b c \,d^{9}+2310 a^{5} b^{2} c^{2} d^{8}-1540 a^{4} b^{3} c^{3} d^{7}+385 a^{3} b^{4} c^{4} d^{6}-42 a^{2} b^{5} c^{5} d^{5}-7 a \,b^{6} c^{6} d^{4}-2 b^{7} c^{7} d^{3}\right ) x^{3}}{b^{8}}+\frac {15 \left (1750 a^{8} d^{10}-7000 a^{7} b c \,d^{9}+10500 a^{6} b^{2} c^{2} d^{8}-7000 a^{5} b^{3} c^{3} d^{7}+1750 a^{4} b^{4} c^{4} d^{6}-168 a^{3} b^{5} c^{5} d^{5}-28 a^{2} b^{6} c^{6} d^{4}-8 a \,b^{7} c^{7} d^{3}-3 b^{8} c^{8} d^{2}\right ) x^{2}}{4 b^{9}}+\frac {\left (5754 a^{9} d^{10}-23016 a^{8} b c \,d^{9}+34524 a^{7} b^{2} c^{2} d^{8}-23016 a^{6} b^{3} c^{3} d^{7}+5754 a^{5} b^{4} c^{4} d^{6}-504 a^{4} b^{5} c^{5} d^{5}-84 a^{3} b^{6} c^{6} d^{4}-24 a^{2} b^{7} c^{7} d^{3}-9 a \,b^{8} c^{8} d^{2}-4 b^{9} c^{9} d \right ) x}{2 b^{10}}-\frac {30 d^{7} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{7}}{b^{4}}+\frac {15 d^{8} \left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) x^{8}}{4 b^{3}}-\frac {5 d^{9} \left (a d -4 b c \right ) x^{9}}{6 b^{2}}}{\left (b x +a \right )^{6}}+\frac {210 d^{6} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{b^{11}}\) | \(850\) |
default | \(-\frac {d^{7} \left (-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {7}{3} x^{3} a \,b^{2} d^{3}-\frac {10}{3} x^{3} b^{3} c \,d^{2}-14 x^{2} a^{2} b \,d^{3}+35 x^{2} a \,b^{2} c \,d^{2}-\frac {45}{2} x^{2} b^{3} c^{2} d +84 a^{3} d^{3} x -280 a^{2} b c \,d^{2} x +315 a \,b^{2} c^{2} d x -120 b^{3} c^{3} x \right )}{b^{10}}+\frac {40 d^{3} \left (a^{7} d^{7}-7 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}+35 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -b^{7} c^{7}\right )}{b^{11} \left (b x +a \right )^{3}}+\frac {210 d^{6} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) \ln \left (b x +a \right )}{b^{11}}-\frac {a^{10} d^{10}-10 a^{9} b c \,d^{9}+45 a^{8} b^{2} c^{2} d^{8}-120 a^{7} b^{3} c^{3} d^{7}+210 a^{6} b^{4} c^{4} d^{6}-252 a^{5} b^{5} c^{5} d^{5}+210 a^{4} b^{6} c^{6} d^{4}-120 a^{3} b^{7} c^{7} d^{3}+45 a^{2} b^{8} c^{8} d^{2}-10 a \,b^{9} c^{9} d +b^{10} c^{10}}{6 b^{11} \left (b x +a \right )^{6}}-\frac {45 d^{2} \left (a^{8} d^{8}-8 a^{7} b c \,d^{7}+28 a^{6} b^{2} c^{2} d^{6}-56 a^{5} b^{3} c^{3} d^{5}+70 a^{4} b^{4} c^{4} d^{4}-56 a^{3} b^{5} c^{5} d^{3}+28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d +b^{8} c^{8}\right )}{4 b^{11} \left (b x +a \right )^{4}}-\frac {105 d^{4} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{b^{11} \left (b x +a \right )^{2}}+\frac {2 d \left (a^{9} d^{9}-9 a^{8} b c \,d^{8}+36 a^{7} b^{2} c^{2} d^{7}-84 a^{6} b^{3} c^{3} d^{6}+126 a^{5} b^{4} c^{4} d^{5}-126 a^{4} b^{5} c^{5} d^{4}+84 a^{3} b^{6} c^{6} d^{3}-36 a^{2} b^{7} c^{7} d^{2}+9 a \,b^{8} c^{8} d -b^{9} c^{9}\right )}{b^{11} \left (b x +a \right )^{5}}+\frac {252 d^{5} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{11} \left (b x +a \right )}\) | \(862\) |
risch | \(\frac {d^{10} x^{4}}{4 b^{7}}-\frac {7 d^{10} x^{3} a}{3 b^{8}}+\frac {10 d^{9} x^{3} c}{3 b^{7}}+\frac {14 d^{10} x^{2} a^{2}}{b^{9}}-\frac {35 d^{9} x^{2} a c}{b^{8}}+\frac {45 d^{8} x^{2} c^{2}}{2 b^{7}}-\frac {84 d^{10} a^{3} x}{b^{10}}+\frac {280 d^{9} a^{2} c x}{b^{9}}-\frac {315 d^{8} a \,c^{2} x}{b^{8}}+\frac {120 d^{7} c^{3} x}{b^{7}}+\frac {\left (252 a^{5} b^{4} d^{10}-1260 a^{4} b^{5} c \,d^{9}+2520 a^{3} b^{6} c^{2} d^{8}-2520 a^{2} b^{7} c^{3} d^{7}+1260 a \,b^{8} c^{4} d^{6}-252 b^{9} c^{5} d^{5}\right ) x^{5}+105 b^{3} d^{4} \left (11 a^{6} d^{6}-54 a^{5} b c \,d^{5}+105 a^{4} b^{2} c^{2} d^{4}-100 a^{3} b^{3} c^{3} d^{3}+45 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d -b^{6} c^{6}\right ) x^{4}+20 b^{2} d^{3} \left (107 a^{7} d^{7}-518 a^{6} b c \,d^{6}+987 a^{5} b^{2} c^{2} d^{5}-910 a^{4} b^{3} c^{3} d^{4}+385 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -2 b^{7} c^{7}\right ) x^{3}+\frac {15 b \,d^{2} \left (533 a^{8} d^{8}-2552 a^{7} b c \,d^{7}+4788 a^{6} b^{2} c^{2} d^{6}-4312 a^{5} b^{3} c^{3} d^{5}+1750 a^{4} b^{4} c^{4} d^{4}-168 a^{3} b^{5} c^{5} d^{3}-28 a^{2} b^{6} c^{6} d^{2}-8 a \,b^{7} c^{7} d -3 b^{8} c^{8}\right ) x^{2}}{4}+\frac {d \left (1879 a^{9} d^{9}-8916 a^{8} b c \,d^{8}+16524 a^{7} b^{2} c^{2} d^{7}-14616 a^{6} b^{3} c^{3} d^{6}+5754 a^{5} b^{4} c^{4} d^{5}-504 a^{4} b^{5} c^{5} d^{4}-84 a^{3} b^{6} c^{6} d^{3}-24 a^{2} b^{7} c^{7} d^{2}-9 a \,b^{8} c^{8} d -4 b^{9} c^{9}\right ) x}{2}+\frac {2131 a^{10} d^{10}-10036 a^{9} b c \,d^{9}+18414 a^{8} b^{2} c^{2} d^{8}-16056 a^{7} b^{3} c^{3} d^{7}+6174 a^{6} b^{4} c^{4} d^{6}-504 a^{5} b^{5} c^{5} d^{5}-84 a^{4} b^{6} c^{6} d^{4}-24 a^{3} b^{7} c^{7} d^{3}-9 a^{2} b^{8} c^{8} d^{2}-4 a \,b^{9} c^{9} d -2 b^{10} c^{10}}{12 b}}{b^{10} \left (b x +a \right )^{6}}+\frac {210 d^{10} \ln \left (b x +a \right ) a^{4}}{b^{11}}-\frac {840 d^{9} \ln \left (b x +a \right ) a^{3} c}{b^{10}}+\frac {1260 d^{8} \ln \left (b x +a \right ) a^{2} c^{2}}{b^{9}}-\frac {840 d^{7} \ln \left (b x +a \right ) a \,c^{3}}{b^{8}}+\frac {210 d^{6} \ln \left (b x +a \right ) c^{4}}{b^{7}}\) | \(884\) |
parallelrisch | \(\text {Expression too large to display}\) | \(1619\) |
(1/12*(6174*a^10*d^10-24696*a^9*b*c*d^9+37044*a^8*b^2*c^2*d^8-24696*a^7*b^ 3*c^3*d^7+6174*a^6*b^4*c^4*d^6-504*a^5*b^5*c^5*d^5-84*a^4*b^6*c^6*d^4-24*a ^3*b^7*c^7*d^3-9*a^2*b^8*c^8*d^2-4*a*b^9*c^9*d-2*b^10*c^10)/b^11+1/4/b*d^1 0*x^10+6*(210*a^5*d^10-840*a^4*b*c*d^9+1260*a^3*b^2*c^2*d^8-840*a^2*b^3*c^ 3*d^7+210*a*b^4*c^4*d^6-42*b^5*c^5*d^5)/b^6*x^5+15*(315*a^6*d^10-1260*a^5* b*c*d^9+1890*a^4*b^2*c^2*d^8-1260*a^3*b^3*c^3*d^7+315*a^2*b^4*c^4*d^6-42*a *b^5*c^5*d^5-7*b^6*c^6*d^4)/b^7*x^4+20*(385*a^7*d^10-1540*a^6*b*c*d^9+2310 *a^5*b^2*c^2*d^8-1540*a^4*b^3*c^3*d^7+385*a^3*b^4*c^4*d^6-42*a^2*b^5*c^5*d ^5-7*a*b^6*c^6*d^4-2*b^7*c^7*d^3)/b^8*x^3+15/4*(1750*a^8*d^10-7000*a^7*b*c *d^9+10500*a^6*b^2*c^2*d^8-7000*a^5*b^3*c^3*d^7+1750*a^4*b^4*c^4*d^6-168*a ^3*b^5*c^5*d^5-28*a^2*b^6*c^6*d^4-8*a*b^7*c^7*d^3-3*b^8*c^8*d^2)/b^9*x^2+1 /2*(5754*a^9*d^10-23016*a^8*b*c*d^9+34524*a^7*b^2*c^2*d^8-23016*a^6*b^3*c^ 3*d^7+5754*a^5*b^4*c^4*d^6-504*a^4*b^5*c^5*d^5-84*a^3*b^6*c^6*d^4-24*a^2*b ^7*c^7*d^3-9*a*b^8*c^8*d^2-4*b^9*c^9*d)/b^10*x-30*d^7*(a^3*d^3-4*a^2*b*c*d ^2+6*a*b^2*c^2*d-4*b^3*c^3)/b^4*x^7+15/4*d^8*(a^2*d^2-4*a*b*c*d+6*b^2*c^2) /b^3*x^8-5/6*d^9*(a*d-4*b*c)/b^2*x^9)/(b*x+a)^6+210/b^11*d^6*(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)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 1386 vs. \(2 (252) = 504\).
Time = 0.24 (sec) , antiderivative size = 1386, normalized size of antiderivative = 5.29 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\text {Too large to display} \]
1/12*(3*b^10*d^10*x^10 - 2*b^10*c^10 - 4*a*b^9*c^9*d - 9*a^2*b^8*c^8*d^2 - 24*a^3*b^7*c^7*d^3 - 84*a^4*b^6*c^6*d^4 - 504*a^5*b^5*c^5*d^5 + 6174*a^6* b^4*c^4*d^6 - 16056*a^7*b^3*c^3*d^7 + 18414*a^8*b^2*c^2*d^8 - 10036*a^9*b* c*d^9 + 2131*a^10*d^10 + 10*(4*b^10*c*d^9 - a*b^9*d^10)*x^9 + 45*(6*b^10*c ^2*d^8 - 4*a*b^9*c*d^9 + a^2*b^8*d^10)*x^8 + 360*(4*b^10*c^3*d^7 - 6*a*b^9 *c^2*d^8 + 4*a^2*b^8*c*d^9 - a^3*b^7*d^10)*x^7 + (8640*a*b^9*c^3*d^7 - 186 30*a^2*b^8*c^2*d^8 + 14660*a^3*b^7*c*d^9 - 4043*a^4*b^6*d^10)*x^6 - 6*(504 *b^10*c^5*d^5 - 2520*a*b^9*c^4*d^6 + 1440*a^2*b^8*c^3*d^7 + 3510*a^3*b^7*c ^2*d^8 - 4580*a^4*b^6*c*d^9 + 1523*a^5*b^5*d^10)*x^5 - 15*(84*b^10*c^6*d^4 + 504*a*b^9*c^5*d^5 - 3780*a^2*b^8*c^4*d^6 + 6480*a^3*b^7*c^3*d^7 - 4050* a^4*b^6*c^2*d^8 + 460*a^5*b^5*c*d^9 + 263*a^6*b^4*d^10)*x^4 - 20*(24*b^10* c^7*d^3 + 84*a*b^9*c^6*d^4 + 504*a^2*b^8*c^5*d^5 - 4620*a^3*b^7*c^4*d^6 + 9840*a^4*b^6*c^3*d^7 - 9090*a^5*b^5*c^2*d^8 + 3820*a^6*b^4*c*d^9 - 577*a^7 *b^3*d^10)*x^3 - 15*(9*b^10*c^8*d^2 + 24*a*b^9*c^7*d^3 + 84*a^2*b^8*c^6*d^ 4 + 504*a^3*b^7*c^5*d^5 - 5250*a^4*b^6*c^4*d^6 + 12360*a^5*b^5*c^3*d^7 - 1 2870*a^6*b^4*c^2*d^8 + 6340*a^7*b^3*c*d^9 - 1207*a^8*b^2*d^10)*x^2 - 6*(4* b^10*c^9*d + 9*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 5 04*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14376*a^6*b^4*c^3*d^7 - 15894* a^7*b^3*c^2*d^8 + 8356*a^8*b^2*c*d^9 - 1711*a^9*b*d^10)*x + 2520*(a^6*b^4* c^4*d^6 - 4*a^7*b^3*c^3*d^7 + 6*a^8*b^2*c^2*d^8 - 4*a^9*b*c*d^9 + a^10*...
Timed out. \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (252) = 504\).
Time = 0.26 (sec) , antiderivative size = 925, normalized size of antiderivative = 3.53 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=-\frac {2 \, b^{10} c^{10} + 4 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 24 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} - 6174 \, a^{6} b^{4} c^{4} d^{6} + 16056 \, a^{7} b^{3} c^{3} d^{7} - 18414 \, a^{8} b^{2} c^{2} d^{8} + 10036 \, a^{9} b c d^{9} - 2131 \, a^{10} d^{10} + 3024 \, {\left (b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}\right )} x^{5} + 1260 \, {\left (b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} - 45 \, a^{2} b^{8} c^{4} d^{6} + 100 \, a^{3} b^{7} c^{3} d^{7} - 105 \, a^{4} b^{6} c^{2} d^{8} + 54 \, a^{5} b^{5} c d^{9} - 11 \, a^{6} b^{4} d^{10}\right )} x^{4} + 240 \, {\left (2 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 42 \, a^{2} b^{8} c^{5} d^{5} - 385 \, a^{3} b^{7} c^{4} d^{6} + 910 \, a^{4} b^{6} c^{3} d^{7} - 987 \, a^{5} b^{5} c^{2} d^{8} + 518 \, a^{6} b^{4} c d^{9} - 107 \, a^{7} b^{3} d^{10}\right )} x^{3} + 45 \, {\left (3 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} - 1750 \, a^{4} b^{6} c^{4} d^{6} + 4312 \, a^{5} b^{5} c^{3} d^{7} - 4788 \, a^{6} b^{4} c^{2} d^{8} + 2552 \, a^{7} b^{3} c d^{9} - 533 \, a^{8} b^{2} d^{10}\right )} x^{2} + 6 \, {\left (4 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 24 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} - 5754 \, a^{5} b^{5} c^{4} d^{6} + 14616 \, a^{6} b^{4} c^{3} d^{7} - 16524 \, a^{7} b^{3} c^{2} d^{8} + 8916 \, a^{8} b^{2} c d^{9} - 1879 \, a^{9} b d^{10}\right )} x}{12 \, {\left (b^{17} x^{6} + 6 \, a b^{16} x^{5} + 15 \, a^{2} b^{15} x^{4} + 20 \, a^{3} b^{14} x^{3} + 15 \, a^{4} b^{13} x^{2} + 6 \, a^{5} b^{12} x + a^{6} b^{11}\right )}} + \frac {3 \, b^{3} d^{10} x^{4} + 4 \, {\left (10 \, b^{3} c d^{9} - 7 \, a b^{2} d^{10}\right )} x^{3} + 6 \, {\left (45 \, b^{3} c^{2} d^{8} - 70 \, a b^{2} c d^{9} + 28 \, a^{2} b d^{10}\right )} x^{2} + 12 \, {\left (120 \, b^{3} c^{3} d^{7} - 315 \, a b^{2} c^{2} d^{8} + 280 \, a^{2} b c d^{9} - 84 \, a^{3} d^{10}\right )} x}{12 \, b^{10}} + \frac {210 \, {\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} \log \left (b x + a\right )}{b^{11}} \]
-1/12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^ 3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^5 - 6174*a^6*b^4*c^4*d^6 + 1605 6*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 10036*a^9*b*c*d^9 - 2131*a^10* d^10 + 3024*(b^10*c^5*d^5 - 5*a*b^9*c^4*d^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3* b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d^10)*x^5 + 1260*(b^10*c^6*d^4 + 6 *a*b^9*c^5*d^5 - 45*a^2*b^8*c^4*d^6 + 100*a^3*b^7*c^3*d^7 - 105*a^4*b^6*c^ 2*d^8 + 54*a^5*b^5*c*d^9 - 11*a^6*b^4*d^10)*x^4 + 240*(2*b^10*c^7*d^3 + 7* a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d^5 - 385*a^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3 *d^7 - 987*a^5*b^5*c^2*d^8 + 518*a^6*b^4*c*d^9 - 107*a^7*b^3*d^10)*x^3 + 4 5*(3*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 28*a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5 *d^5 - 1750*a^4*b^6*c^4*d^6 + 4312*a^5*b^5*c^3*d^7 - 4788*a^6*b^4*c^2*d^8 + 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)*x^2 + 6*(4*b^10*c^9*d + 9*a*b^9*c ^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7*c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5 754*a^5*b^5*c^4*d^6 + 14616*a^6*b^4*c^3*d^7 - 16524*a^7*b^3*c^2*d^8 + 8916 *a^8*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/(b^17*x^6 + 6*a*b^16*x^5 + 15*a^2*b^1 5*x^4 + 20*a^3*b^14*x^3 + 15*a^4*b^13*x^2 + 6*a^5*b^12*x + a^6*b^11) + 1/1 2*(3*b^3*d^10*x^4 + 4*(10*b^3*c*d^9 - 7*a*b^2*d^10)*x^3 + 6*(45*b^3*c^2*d^ 8 - 70*a*b^2*c*d^9 + 28*a^2*b*d^10)*x^2 + 12*(120*b^3*c^3*d^7 - 315*a*b^2* c^2*d^8 + 280*a^2*b*c*d^9 - 84*a^3*d^10)*x)/b^10 + 210*(b^4*c^4*d^6 - 4*a* b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a^4*d^10)*log(b*x + a...
Leaf count of result is larger than twice the leaf count of optimal. 878 vs. \(2 (252) = 504\).
Time = 0.37 (sec) , antiderivative size = 878, normalized size of antiderivative = 3.35 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=\frac {210 \, {\left (b^{4} c^{4} d^{6} - 4 \, a b^{3} c^{3} d^{7} + 6 \, a^{2} b^{2} c^{2} d^{8} - 4 \, a^{3} b c d^{9} + a^{4} d^{10}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{11}} - \frac {2 \, b^{10} c^{10} + 4 \, a b^{9} c^{9} d + 9 \, a^{2} b^{8} c^{8} d^{2} + 24 \, a^{3} b^{7} c^{7} d^{3} + 84 \, a^{4} b^{6} c^{6} d^{4} + 504 \, a^{5} b^{5} c^{5} d^{5} - 6174 \, a^{6} b^{4} c^{4} d^{6} + 16056 \, a^{7} b^{3} c^{3} d^{7} - 18414 \, a^{8} b^{2} c^{2} d^{8} + 10036 \, a^{9} b c d^{9} - 2131 \, a^{10} d^{10} + 3024 \, {\left (b^{10} c^{5} d^{5} - 5 \, a b^{9} c^{4} d^{6} + 10 \, a^{2} b^{8} c^{3} d^{7} - 10 \, a^{3} b^{7} c^{2} d^{8} + 5 \, a^{4} b^{6} c d^{9} - a^{5} b^{5} d^{10}\right )} x^{5} + 1260 \, {\left (b^{10} c^{6} d^{4} + 6 \, a b^{9} c^{5} d^{5} - 45 \, a^{2} b^{8} c^{4} d^{6} + 100 \, a^{3} b^{7} c^{3} d^{7} - 105 \, a^{4} b^{6} c^{2} d^{8} + 54 \, a^{5} b^{5} c d^{9} - 11 \, a^{6} b^{4} d^{10}\right )} x^{4} + 240 \, {\left (2 \, b^{10} c^{7} d^{3} + 7 \, a b^{9} c^{6} d^{4} + 42 \, a^{2} b^{8} c^{5} d^{5} - 385 \, a^{3} b^{7} c^{4} d^{6} + 910 \, a^{4} b^{6} c^{3} d^{7} - 987 \, a^{5} b^{5} c^{2} d^{8} + 518 \, a^{6} b^{4} c d^{9} - 107 \, a^{7} b^{3} d^{10}\right )} x^{3} + 45 \, {\left (3 \, b^{10} c^{8} d^{2} + 8 \, a b^{9} c^{7} d^{3} + 28 \, a^{2} b^{8} c^{6} d^{4} + 168 \, a^{3} b^{7} c^{5} d^{5} - 1750 \, a^{4} b^{6} c^{4} d^{6} + 4312 \, a^{5} b^{5} c^{3} d^{7} - 4788 \, a^{6} b^{4} c^{2} d^{8} + 2552 \, a^{7} b^{3} c d^{9} - 533 \, a^{8} b^{2} d^{10}\right )} x^{2} + 6 \, {\left (4 \, b^{10} c^{9} d + 9 \, a b^{9} c^{8} d^{2} + 24 \, a^{2} b^{8} c^{7} d^{3} + 84 \, a^{3} b^{7} c^{6} d^{4} + 504 \, a^{4} b^{6} c^{5} d^{5} - 5754 \, a^{5} b^{5} c^{4} d^{6} + 14616 \, a^{6} b^{4} c^{3} d^{7} - 16524 \, a^{7} b^{3} c^{2} d^{8} + 8916 \, a^{8} b^{2} c d^{9} - 1879 \, a^{9} b d^{10}\right )} x}{12 \, {\left (b x + a\right )}^{6} b^{11}} + \frac {3 \, b^{21} d^{10} x^{4} + 40 \, b^{21} c d^{9} x^{3} - 28 \, a b^{20} d^{10} x^{3} + 270 \, b^{21} c^{2} d^{8} x^{2} - 420 \, a b^{20} c d^{9} x^{2} + 168 \, a^{2} b^{19} d^{10} x^{2} + 1440 \, b^{21} c^{3} d^{7} x - 3780 \, a b^{20} c^{2} d^{8} x + 3360 \, a^{2} b^{19} c d^{9} x - 1008 \, a^{3} b^{18} d^{10} x}{12 \, b^{28}} \]
210*(b^4*c^4*d^6 - 4*a*b^3*c^3*d^7 + 6*a^2*b^2*c^2*d^8 - 4*a^3*b*c*d^9 + a ^4*d^10)*log(abs(b*x + a))/b^11 - 1/12*(2*b^10*c^10 + 4*a*b^9*c^9*d + 9*a^ 2*b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5* d^5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 10036*a^9*b*c*d^9 - 2131*a^10*d^10 + 3024*(b^10*c^5*d^5 - 5*a*b^9*c^4*d ^6 + 10*a^2*b^8*c^3*d^7 - 10*a^3*b^7*c^2*d^8 + 5*a^4*b^6*c*d^9 - a^5*b^5*d ^10)*x^5 + 1260*(b^10*c^6*d^4 + 6*a*b^9*c^5*d^5 - 45*a^2*b^8*c^4*d^6 + 100 *a^3*b^7*c^3*d^7 - 105*a^4*b^6*c^2*d^8 + 54*a^5*b^5*c*d^9 - 11*a^6*b^4*d^1 0)*x^4 + 240*(2*b^10*c^7*d^3 + 7*a*b^9*c^6*d^4 + 42*a^2*b^8*c^5*d^5 - 385* a^3*b^7*c^4*d^6 + 910*a^4*b^6*c^3*d^7 - 987*a^5*b^5*c^2*d^8 + 518*a^6*b^4* c*d^9 - 107*a^7*b^3*d^10)*x^3 + 45*(3*b^10*c^8*d^2 + 8*a*b^9*c^7*d^3 + 28* a^2*b^8*c^6*d^4 + 168*a^3*b^7*c^5*d^5 - 1750*a^4*b^6*c^4*d^6 + 4312*a^5*b^ 5*c^3*d^7 - 4788*a^6*b^4*c^2*d^8 + 2552*a^7*b^3*c*d^9 - 533*a^8*b^2*d^10)* x^2 + 6*(4*b^10*c^9*d + 9*a*b^9*c^8*d^2 + 24*a^2*b^8*c^7*d^3 + 84*a^3*b^7* c^6*d^4 + 504*a^4*b^6*c^5*d^5 - 5754*a^5*b^5*c^4*d^6 + 14616*a^6*b^4*c^3*d ^7 - 16524*a^7*b^3*c^2*d^8 + 8916*a^8*b^2*c*d^9 - 1879*a^9*b*d^10)*x)/((b* x + a)^6*b^11) + 1/12*(3*b^21*d^10*x^4 + 40*b^21*c*d^9*x^3 - 28*a*b^20*d^1 0*x^3 + 270*b^21*c^2*d^8*x^2 - 420*a*b^20*c*d^9*x^2 + 168*a^2*b^19*d^10*x^ 2 + 1440*b^21*c^3*d^7*x - 3780*a*b^20*c^2*d^8*x + 3360*a^2*b^19*c*d^9*x - 1008*a^3*b^18*d^10*x)/b^28
Time = 0.62 (sec) , antiderivative size = 997, normalized size of antiderivative = 3.81 \[ \int \frac {(c+d x)^{10}}{(a+b x)^7} \, dx=x^2\,\left (\frac {7\,a\,\left (\frac {7\,a\,d^{10}}{b^8}-\frac {10\,c\,d^9}{b^7}\right )}{2\,b}-\frac {21\,a^2\,d^{10}}{2\,b^9}+\frac {45\,c^2\,d^8}{2\,b^7}\right )-\frac {x^4\,\left (-1155\,a^6\,b^3\,d^{10}+5670\,a^5\,b^4\,c\,d^9-11025\,a^4\,b^5\,c^2\,d^8+10500\,a^3\,b^6\,c^3\,d^7-4725\,a^2\,b^7\,c^4\,d^6+630\,a\,b^8\,c^5\,d^5+105\,b^9\,c^6\,d^4\right )+\frac {-2131\,a^{10}\,d^{10}+10036\,a^9\,b\,c\,d^9-18414\,a^8\,b^2\,c^2\,d^8+16056\,a^7\,b^3\,c^3\,d^7-6174\,a^6\,b^4\,c^4\,d^6+504\,a^5\,b^5\,c^5\,d^5+84\,a^4\,b^6\,c^6\,d^4+24\,a^3\,b^7\,c^7\,d^3+9\,a^2\,b^8\,c^8\,d^2+4\,a\,b^9\,c^9\,d+2\,b^{10}\,c^{10}}{12\,b}+x\,\left (-\frac {1879\,a^9\,d^{10}}{2}+4458\,a^8\,b\,c\,d^9-8262\,a^7\,b^2\,c^2\,d^8+7308\,a^6\,b^3\,c^3\,d^7-2877\,a^5\,b^4\,c^4\,d^6+252\,a^4\,b^5\,c^5\,d^5+42\,a^3\,b^6\,c^6\,d^4+12\,a^2\,b^7\,c^7\,d^3+\frac {9\,a\,b^8\,c^8\,d^2}{2}+2\,b^9\,c^9\,d\right )+x^3\,\left (-2140\,a^7\,b^2\,d^{10}+10360\,a^6\,b^3\,c\,d^9-19740\,a^5\,b^4\,c^2\,d^8+18200\,a^4\,b^5\,c^3\,d^7-7700\,a^3\,b^6\,c^4\,d^6+840\,a^2\,b^7\,c^5\,d^5+140\,a\,b^8\,c^6\,d^4+40\,b^9\,c^7\,d^3\right )+x^2\,\left (-\frac {7995\,a^8\,b\,d^{10}}{4}+9570\,a^7\,b^2\,c\,d^9-17955\,a^6\,b^3\,c^2\,d^8+16170\,a^5\,b^4\,c^3\,d^7-\frac {13125\,a^4\,b^5\,c^4\,d^6}{2}+630\,a^3\,b^6\,c^5\,d^5+105\,a^2\,b^7\,c^6\,d^4+30\,a\,b^8\,c^7\,d^3+\frac {45\,b^9\,c^8\,d^2}{4}\right )-x^5\,\left (252\,a^5\,b^4\,d^{10}-1260\,a^4\,b^5\,c\,d^9+2520\,a^3\,b^6\,c^2\,d^8-2520\,a^2\,b^7\,c^3\,d^7+1260\,a\,b^8\,c^4\,d^6-252\,b^9\,c^5\,d^5\right )}{a^6\,b^{10}+6\,a^5\,b^{11}\,x+15\,a^4\,b^{12}\,x^2+20\,a^3\,b^{13}\,x^3+15\,a^2\,b^{14}\,x^4+6\,a\,b^{15}\,x^5+b^{16}\,x^6}-x^3\,\left (\frac {7\,a\,d^{10}}{3\,b^8}-\frac {10\,c\,d^9}{3\,b^7}\right )-x\,\left (\frac {7\,a\,\left (\frac {7\,a\,\left (\frac {7\,a\,d^{10}}{b^8}-\frac {10\,c\,d^9}{b^7}\right )}{b}-\frac {21\,a^2\,d^{10}}{b^9}+\frac {45\,c^2\,d^8}{b^7}\right )}{b}+\frac {35\,a^3\,d^{10}}{b^{10}}-\frac {120\,c^3\,d^7}{b^7}-\frac {21\,a^2\,\left (\frac {7\,a\,d^{10}}{b^8}-\frac {10\,c\,d^9}{b^7}\right )}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (210\,a^4\,d^{10}-840\,a^3\,b\,c\,d^9+1260\,a^2\,b^2\,c^2\,d^8-840\,a\,b^3\,c^3\,d^7+210\,b^4\,c^4\,d^6\right )}{b^{11}}+\frac {d^{10}\,x^4}{4\,b^7} \]
x^2*((7*a*((7*a*d^10)/b^8 - (10*c*d^9)/b^7))/(2*b) - (21*a^2*d^10)/(2*b^9) + (45*c^2*d^8)/(2*b^7)) - (x^4*(105*b^9*c^6*d^4 - 1155*a^6*b^3*d^10 + 630 *a*b^8*c^5*d^5 + 5670*a^5*b^4*c*d^9 - 4725*a^2*b^7*c^4*d^6 + 10500*a^3*b^6 *c^3*d^7 - 11025*a^4*b^5*c^2*d^8) + (2*b^10*c^10 - 2131*a^10*d^10 + 9*a^2* b^8*c^8*d^2 + 24*a^3*b^7*c^7*d^3 + 84*a^4*b^6*c^6*d^4 + 504*a^5*b^5*c^5*d^ 5 - 6174*a^6*b^4*c^4*d^6 + 16056*a^7*b^3*c^3*d^7 - 18414*a^8*b^2*c^2*d^8 + 4*a*b^9*c^9*d + 10036*a^9*b*c*d^9)/(12*b) + x*(2*b^9*c^9*d - (1879*a^9*d^ 10)/2 + (9*a*b^8*c^8*d^2)/2 + 12*a^2*b^7*c^7*d^3 + 42*a^3*b^6*c^6*d^4 + 25 2*a^4*b^5*c^5*d^5 - 2877*a^5*b^4*c^4*d^6 + 7308*a^6*b^3*c^3*d^7 - 8262*a^7 *b^2*c^2*d^8 + 4458*a^8*b*c*d^9) + x^3*(40*b^9*c^7*d^3 - 2140*a^7*b^2*d^10 + 140*a*b^8*c^6*d^4 + 10360*a^6*b^3*c*d^9 + 840*a^2*b^7*c^5*d^5 - 7700*a^ 3*b^6*c^4*d^6 + 18200*a^4*b^5*c^3*d^7 - 19740*a^5*b^4*c^2*d^8) + x^2*((45* b^9*c^8*d^2)/4 - (7995*a^8*b*d^10)/4 + 30*a*b^8*c^7*d^3 + 9570*a^7*b^2*c*d ^9 + 105*a^2*b^7*c^6*d^4 + 630*a^3*b^6*c^5*d^5 - (13125*a^4*b^5*c^4*d^6)/2 + 16170*a^5*b^4*c^3*d^7 - 17955*a^6*b^3*c^2*d^8) - x^5*(252*a^5*b^4*d^10 - 252*b^9*c^5*d^5 + 1260*a*b^8*c^4*d^6 - 1260*a^4*b^5*c*d^9 - 2520*a^2*b^7 *c^3*d^7 + 2520*a^3*b^6*c^2*d^8))/(a^6*b^10 + b^16*x^6 + 6*a^5*b^11*x + 6* a*b^15*x^5 + 15*a^4*b^12*x^2 + 20*a^3*b^13*x^3 + 15*a^2*b^14*x^4) - x^3*(( 7*a*d^10)/(3*b^8) - (10*c*d^9)/(3*b^7)) - x*((7*a*((7*a*((7*a*d^10)/b^8 - (10*c*d^9)/b^7))/b - (21*a^2*d^10)/b^9 + (45*c^2*d^8)/b^7))/b + (35*a^3...