Integrand size = 15, antiderivative size = 89 \[ \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx=-\frac {(c+d x)^8}{10 (b c-a d) (a+b x)^{10}}+\frac {d (c+d x)^8}{45 (b c-a d)^2 (a+b x)^9}-\frac {d^2 (c+d x)^8}{360 (b c-a d)^3 (a+b x)^8} \]
-1/10*(d*x+c)^8/(-a*d+b*c)/(b*x+a)^10+1/45*d*(d*x+c)^8/(-a*d+b*c)^2/(b*x+a )^9-1/360*d^2*(d*x+c)^8/(-a*d+b*c)^3/(b*x+a)^8
Leaf count is larger than twice the leaf count of optimal. \(371\) vs. \(2(89)=178\).
Time = 0.08 (sec) , antiderivative size = 371, normalized size of antiderivative = 4.17 \[ \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx=-\frac {a^7 d^7+a^6 b d^6 (3 c+10 d x)+3 a^5 b^2 d^5 \left (2 c^2+10 c d x+15 d^2 x^2\right )+5 a^4 b^3 d^4 \left (2 c^3+12 c^2 d x+27 c d^2 x^2+24 d^3 x^3\right )+5 a^3 b^4 d^3 \left (3 c^4+20 c^3 d x+54 c^2 d^2 x^2+72 c d^3 x^3+42 d^4 x^4\right )+3 a^2 b^5 d^2 \left (7 c^5+50 c^4 d x+150 c^3 d^2 x^2+240 c^2 d^3 x^3+210 c d^4 x^4+84 d^5 x^5\right )+a b^6 d \left (28 c^6+210 c^5 d x+675 c^4 d^2 x^2+1200 c^3 d^3 x^3+1260 c^2 d^4 x^4+756 c d^5 x^5+210 d^6 x^6\right )+b^7 \left (36 c^7+280 c^6 d x+945 c^5 d^2 x^2+1800 c^4 d^3 x^3+2100 c^3 d^4 x^4+1512 c^2 d^5 x^5+630 c d^6 x^6+120 d^7 x^7\right )}{360 b^8 (a+b x)^{10}} \]
-1/360*(a^7*d^7 + a^6*b*d^6*(3*c + 10*d*x) + 3*a^5*b^2*d^5*(2*c^2 + 10*c*d *x + 15*d^2*x^2) + 5*a^4*b^3*d^4*(2*c^3 + 12*c^2*d*x + 27*c*d^2*x^2 + 24*d ^3*x^3) + 5*a^3*b^4*d^3*(3*c^4 + 20*c^3*d*x + 54*c^2*d^2*x^2 + 72*c*d^3*x^ 3 + 42*d^4*x^4) + 3*a^2*b^5*d^2*(7*c^5 + 50*c^4*d*x + 150*c^3*d^2*x^2 + 24 0*c^2*d^3*x^3 + 210*c*d^4*x^4 + 84*d^5*x^5) + a*b^6*d*(28*c^6 + 210*c^5*d* x + 675*c^4*d^2*x^2 + 1200*c^3*d^3*x^3 + 1260*c^2*d^4*x^4 + 756*c*d^5*x^5 + 210*d^6*x^6) + b^7*(36*c^7 + 280*c^6*d*x + 945*c^5*d^2*x^2 + 1800*c^4*d^ 3*x^3 + 2100*c^3*d^4*x^4 + 1512*c^2*d^5*x^5 + 630*c*d^6*x^6 + 120*d^7*x^7) )/(b^8*(a + b*x)^10)
Time = 0.18 (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.
| \(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {d \int \frac {(c+d x)^7}{(a+b x)^{10}}dx}{5 (b c-a d)}-\frac {(c+d x)^8}{10 (a+b x)^{10} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {d \left (-\frac {d \int \frac {(c+d x)^7}{(a+b x)^9}dx}{9 (b c-a d)}-\frac {(c+d x)^8}{9 (a+b x)^9 (b c-a d)}\right )}{5 (b c-a d)}-\frac {(c+d x)^8}{10 (a+b x)^{10} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {(c+d x)^8}{10 (a+b x)^{10} (b c-a d)}-\frac {d \left (\frac {d (c+d x)^8}{72 (a+b x)^8 (b c-a d)^2}-\frac {(c+d x)^8}{9 (a+b x)^9 (b c-a d)}\right )}{5 (b c-a d)}\) |
-1/10*(c + d*x)^8/((b*c - a*d)*(a + b*x)^10) - (d*(-1/9*(c + d*x)^8/((b*c - a*d)*(a + b*x)^9) + (d*(c + d*x)^8)/(72*(b*c - a*d)^2*(a + b*x)^8)))/(5* (b*c - a*d))
3.13.93.3.1 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. \(437\) vs. \(2(83)=166\).
Time = 0.22 (sec) , antiderivative size = 438, normalized size of antiderivative = 4.92
| method | result | size |
| risch | \(\frac {-\frac {d^{7} x^{7}}{3 b}-\frac {7 d^{6} \left (a d +3 b c \right ) x^{6}}{12 b^{2}}-\frac {7 d^{5} \left (a^{2} d^{2}+3 a b c d +6 b^{2} c^{2}\right ) x^{5}}{10 b^{3}}-\frac {7 d^{4} \left (a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d +10 b^{3} c^{3}\right ) x^{4}}{12 b^{4}}-\frac {d^{3} \left (a^{4} d^{4}+3 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+10 a \,b^{3} c^{3} d +15 b^{4} c^{4}\right ) x^{3}}{3 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+3 a^{4} b c \,d^{4}+6 a^{3} b^{2} c^{2} d^{3}+10 a^{2} b^{3} c^{3} d^{2}+15 a \,b^{4} c^{4} d +21 b^{5} c^{5}\right ) x^{2}}{8 b^{6}}-\frac {d \left (a^{6} d^{6}+3 a^{5} b c \,d^{5}+6 a^{4} b^{2} c^{2} d^{4}+10 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}+21 a \,b^{5} c^{5} d +28 b^{6} c^{6}\right ) x}{36 b^{7}}-\frac {a^{7} d^{7}+3 a^{6} b c \,d^{6}+6 a^{5} b^{2} c^{2} d^{5}+10 a^{4} b^{3} c^{3} d^{4}+15 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+28 a \,b^{6} c^{6} d +36 b^{7} c^{7}}{360 b^{8}}}{\left (b x +a \right )^{10}}\) | \(438\) |
| default | \(-\frac {d^{7}}{3 b^{8} \left (b x +a \right )^{3}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{9 b^{8} \left (b x +a \right )^{9}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{6 b^{8} \left (b x +a \right )^{6}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{8 b^{8} \left (b x +a \right )^{8}}+\frac {7 d^{6} \left (a d -b c \right )}{4 b^{8} \left (b x +a \right )^{4}}-\frac {5 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{b^{8} \left (b x +a \right )^{7}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{10 b^{8} \left (b x +a \right )^{10}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{5 b^{8} \left (b x +a \right )^{5}}\) | \(464\) |
| norman | \(\frac {-\frac {d^{7} x^{7}}{3 b}+\frac {7 \left (-a \,b^{2} d^{7}-3 b^{3} c \,d^{6}\right ) x^{6}}{12 b^{4}}+\frac {7 \left (-a^{2} b^{2} d^{7}-3 a \,b^{3} c \,d^{6}-6 b^{4} c^{2} d^{5}\right ) x^{5}}{10 b^{5}}+\frac {7 \left (-a^{3} b^{2} d^{7}-3 a^{2} b^{3} c \,d^{6}-6 a \,b^{4} c^{2} d^{5}-10 b^{5} c^{3} d^{4}\right ) x^{4}}{12 b^{6}}+\frac {\left (-a^{4} b^{2} d^{7}-3 a^{3} b^{3} c \,d^{6}-6 a^{2} b^{4} c^{2} d^{5}-10 a \,b^{5} c^{3} d^{4}-15 b^{6} c^{4} d^{3}\right ) x^{3}}{3 b^{7}}+\frac {\left (-a^{5} b^{2} d^{7}-3 a^{4} b^{3} c \,d^{6}-6 a^{3} b^{4} c^{2} d^{5}-10 a^{2} b^{5} c^{3} d^{4}-15 a \,b^{6} c^{4} d^{3}-21 b^{7} c^{5} d^{2}\right ) x^{2}}{8 b^{8}}+\frac {\left (-a^{6} b^{2} d^{7}-3 a^{5} b^{3} c \,d^{6}-6 a^{4} b^{4} c^{2} d^{5}-10 a^{3} b^{5} c^{3} d^{4}-15 a^{2} b^{6} c^{4} d^{3}-21 a \,b^{7} c^{5} d^{2}-28 b^{8} c^{6} d \right ) x}{36 b^{9}}+\frac {-a^{7} b^{2} d^{7}-3 a^{6} b^{3} c \,d^{6}-6 a^{5} b^{4} c^{2} d^{5}-10 a^{4} b^{5} c^{3} d^{4}-15 a^{3} b^{6} c^{4} d^{3}-21 a^{2} b^{7} c^{5} d^{2}-28 a \,b^{8} c^{6} d -36 b^{9} c^{7}}{360 b^{10}}}{\left (b x +a \right )^{10}}\) | \(492\) |
| gosper | \(-\frac {120 x^{7} d^{7} b^{7}+210 x^{6} a \,b^{6} d^{7}+630 x^{6} b^{7} c \,d^{6}+252 x^{5} a^{2} b^{5} d^{7}+756 x^{5} a \,b^{6} c \,d^{6}+1512 x^{5} b^{7} c^{2} d^{5}+210 x^{4} a^{3} b^{4} d^{7}+630 x^{4} a^{2} b^{5} c \,d^{6}+1260 x^{4} a \,b^{6} c^{2} d^{5}+2100 x^{4} b^{7} c^{3} d^{4}+120 x^{3} a^{4} b^{3} d^{7}+360 x^{3} a^{3} b^{4} c \,d^{6}+720 x^{3} a^{2} b^{5} c^{2} d^{5}+1200 x^{3} a \,b^{6} c^{3} d^{4}+1800 x^{3} b^{7} c^{4} d^{3}+45 x^{2} a^{5} b^{2} d^{7}+135 x^{2} a^{4} b^{3} c \,d^{6}+270 x^{2} a^{3} b^{4} c^{2} d^{5}+450 x^{2} a^{2} b^{5} c^{3} d^{4}+675 x^{2} a \,b^{6} c^{4} d^{3}+945 x^{2} b^{7} c^{5} d^{2}+10 x \,a^{6} b \,d^{7}+30 x \,a^{5} b^{2} c \,d^{6}+60 x \,a^{4} b^{3} c^{2} d^{5}+100 x \,a^{3} b^{4} c^{3} d^{4}+150 x \,a^{2} b^{5} c^{4} d^{3}+210 x a \,b^{6} c^{5} d^{2}+280 x \,b^{7} c^{6} d +a^{7} d^{7}+3 a^{6} b c \,d^{6}+6 a^{5} b^{2} c^{2} d^{5}+10 a^{4} b^{3} c^{3} d^{4}+15 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+28 a \,b^{6} c^{6} d +36 b^{7} c^{7}}{360 b^{8} \left (b x +a \right )^{10}}\) | \(497\) |
| parallelrisch | \(\frac {-120 d^{7} x^{7} b^{9}-210 a \,b^{8} d^{7} x^{6}-630 b^{9} c \,d^{6} x^{6}-252 a^{2} b^{7} d^{7} x^{5}-756 a \,b^{8} c \,d^{6} x^{5}-1512 b^{9} c^{2} d^{5} x^{5}-210 a^{3} b^{6} d^{7} x^{4}-630 a^{2} b^{7} c \,d^{6} x^{4}-1260 a \,b^{8} c^{2} d^{5} x^{4}-2100 b^{9} c^{3} d^{4} x^{4}-120 a^{4} b^{5} d^{7} x^{3}-360 a^{3} b^{6} c \,d^{6} x^{3}-720 a^{2} b^{7} c^{2} d^{5} x^{3}-1200 a \,b^{8} c^{3} d^{4} x^{3}-1800 b^{9} c^{4} d^{3} x^{3}-45 a^{5} b^{4} d^{7} x^{2}-135 a^{4} b^{5} c \,d^{6} x^{2}-270 a^{3} b^{6} c^{2} d^{5} x^{2}-450 a^{2} b^{7} c^{3} d^{4} x^{2}-675 a \,b^{8} c^{4} d^{3} x^{2}-945 b^{9} c^{5} d^{2} x^{2}-10 a^{6} b^{3} d^{7} x -30 a^{5} b^{4} c \,d^{6} x -60 a^{4} b^{5} c^{2} d^{5} x -100 a^{3} b^{6} c^{3} d^{4} x -150 a^{2} b^{7} c^{4} d^{3} x -210 a \,b^{8} c^{5} d^{2} x -280 b^{9} c^{6} d x -a^{7} b^{2} d^{7}-3 a^{6} b^{3} c \,d^{6}-6 a^{5} b^{4} c^{2} d^{5}-10 a^{4} b^{5} c^{3} d^{4}-15 a^{3} b^{6} c^{4} d^{3}-21 a^{2} b^{7} c^{5} d^{2}-28 a \,b^{8} c^{6} d -36 b^{9} c^{7}}{360 b^{10} \left (b x +a \right )^{10}}\) | \(505\) |
(-1/3/b*d^7*x^7-7/12/b^2*d^6*(a*d+3*b*c)*x^6-7/10/b^3*d^5*(a^2*d^2+3*a*b*c *d+6*b^2*c^2)*x^5-7/12/b^4*d^4*(a^3*d^3+3*a^2*b*c*d^2+6*a*b^2*c^2*d+10*b^3 *c^3)*x^4-1/3/b^5*d^3*(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)*x^3-1/8/b^6*d^2*(a^5*d^5+3*a^4*b*c*d^4+6*a^3*b^2*c^2*d^3+1 0*a^2*b^3*c^3*d^2+15*a*b^4*c^4*d+21*b^5*c^5)*x^2-1/36/b^7*d*(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-1/360/b^8*(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))/(b*x+a)^10
Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (83) = 166\).
Time = 0.23 (sec) , antiderivative size = 559, normalized size of antiderivative = 6.28 \[ \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx=-\frac {120 \, b^{7} d^{7} x^{7} + 36 \, b^{7} c^{7} + 28 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} + a^{7} d^{7} + 210 \, {\left (3 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 252 \, {\left (6 \, b^{7} c^{2} d^{5} + 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 210 \, {\left (10 \, b^{7} c^{3} d^{4} + 6 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 120 \, {\left (15 \, b^{7} c^{4} d^{3} + 10 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 3 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 45 \, {\left (21 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 6 \, a^{3} b^{4} c^{2} d^{5} + 3 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 10 \, {\left (28 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{360 \, {\left (b^{18} x^{10} + 10 \, a b^{17} x^{9} + 45 \, a^{2} b^{16} x^{8} + 120 \, a^{3} b^{15} x^{7} + 210 \, a^{4} b^{14} x^{6} + 252 \, a^{5} b^{13} x^{5} + 210 \, a^{6} b^{12} x^{4} + 120 \, a^{7} b^{11} x^{3} + 45 \, a^{8} b^{10} x^{2} + 10 \, a^{9} b^{9} x + a^{10} b^{8}\right )}} \]
-1/360*(120*b^7*d^7*x^7 + 36*b^7*c^7 + 28*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 15*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c^3*d^4 + 6*a^5*b^2*c^2*d^5 + 3*a^6*b*c *d^6 + a^7*d^7 + 210*(3*b^7*c*d^6 + a*b^6*d^7)*x^6 + 252*(6*b^7*c^2*d^5 + 3*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 210*(10*b^7*c^3*d^4 + 6*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 120*(15*b^7*c^4*d^3 + 10*a*b^6*c^3*d ^4 + 6*a^2*b^5*c^2*d^5 + 3*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 45*(21*b^7*c ^5*d^2 + 15*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 + 6*a^3*b^4*c^2*d^5 + 3*a^4 *b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 10*(28*b^7*c^6*d + 21*a*b^6*c^5*d^2 + 15*a ^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4 + 6*a^4*b^3*c^2*d^5 + 3*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^18*x^10 + 10*a*b^17*x^9 + 45*a^2*b^16*x^8 + 120*a^3*b^1 5*x^7 + 210*a^4*b^14*x^6 + 252*a^5*b^13*x^5 + 210*a^6*b^12*x^4 + 120*a^7*b ^11*x^3 + 45*a^8*b^10*x^2 + 10*a^9*b^9*x + a^10*b^8)
Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (83) = 166\).
Time = 0.24 (sec) , antiderivative size = 559, normalized size of antiderivative = 6.28 \[ \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx=-\frac {120 \, b^{7} d^{7} x^{7} + 36 \, b^{7} c^{7} + 28 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} + a^{7} d^{7} + 210 \, {\left (3 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 252 \, {\left (6 \, b^{7} c^{2} d^{5} + 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 210 \, {\left (10 \, b^{7} c^{3} d^{4} + 6 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 120 \, {\left (15 \, b^{7} c^{4} d^{3} + 10 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} + 3 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 45 \, {\left (21 \, b^{7} c^{5} d^{2} + 15 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} + 6 \, a^{3} b^{4} c^{2} d^{5} + 3 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 10 \, {\left (28 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} + 10 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{360 \, {\left (b^{18} x^{10} + 10 \, a b^{17} x^{9} + 45 \, a^{2} b^{16} x^{8} + 120 \, a^{3} b^{15} x^{7} + 210 \, a^{4} b^{14} x^{6} + 252 \, a^{5} b^{13} x^{5} + 210 \, a^{6} b^{12} x^{4} + 120 \, a^{7} b^{11} x^{3} + 45 \, a^{8} b^{10} x^{2} + 10 \, a^{9} b^{9} x + a^{10} b^{8}\right )}} \]
-1/360*(120*b^7*d^7*x^7 + 36*b^7*c^7 + 28*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 15*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c^3*d^4 + 6*a^5*b^2*c^2*d^5 + 3*a^6*b*c *d^6 + a^7*d^7 + 210*(3*b^7*c*d^6 + a*b^6*d^7)*x^6 + 252*(6*b^7*c^2*d^5 + 3*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 210*(10*b^7*c^3*d^4 + 6*a*b^6*c^2*d^5 + 3*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 120*(15*b^7*c^4*d^3 + 10*a*b^6*c^3*d ^4 + 6*a^2*b^5*c^2*d^5 + 3*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 45*(21*b^7*c ^5*d^2 + 15*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 + 6*a^3*b^4*c^2*d^5 + 3*a^4 *b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 10*(28*b^7*c^6*d + 21*a*b^6*c^5*d^2 + 15*a ^2*b^5*c^4*d^3 + 10*a^3*b^4*c^3*d^4 + 6*a^4*b^3*c^2*d^5 + 3*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^18*x^10 + 10*a*b^17*x^9 + 45*a^2*b^16*x^8 + 120*a^3*b^1 5*x^7 + 210*a^4*b^14*x^6 + 252*a^5*b^13*x^5 + 210*a^6*b^12*x^4 + 120*a^7*b ^11*x^3 + 45*a^8*b^10*x^2 + 10*a^9*b^9*x + a^10*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (83) = 166\).
Time = 0.28 (sec) , antiderivative size = 496, normalized size of antiderivative = 5.57 \[ \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx=-\frac {120 \, b^{7} d^{7} x^{7} + 630 \, b^{7} c d^{6} x^{6} + 210 \, a b^{6} d^{7} x^{6} + 1512 \, b^{7} c^{2} d^{5} x^{5} + 756 \, a b^{6} c d^{6} x^{5} + 252 \, a^{2} b^{5} d^{7} x^{5} + 2100 \, b^{7} c^{3} d^{4} x^{4} + 1260 \, a b^{6} c^{2} d^{5} x^{4} + 630 \, a^{2} b^{5} c d^{6} x^{4} + 210 \, a^{3} b^{4} d^{7} x^{4} + 1800 \, b^{7} c^{4} d^{3} x^{3} + 1200 \, a b^{6} c^{3} d^{4} x^{3} + 720 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 360 \, a^{3} b^{4} c d^{6} x^{3} + 120 \, a^{4} b^{3} d^{7} x^{3} + 945 \, b^{7} c^{5} d^{2} x^{2} + 675 \, a b^{6} c^{4} d^{3} x^{2} + 450 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 270 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 135 \, a^{4} b^{3} c d^{6} x^{2} + 45 \, a^{5} b^{2} d^{7} x^{2} + 280 \, b^{7} c^{6} d x + 210 \, a b^{6} c^{5} d^{2} x + 150 \, a^{2} b^{5} c^{4} d^{3} x + 100 \, a^{3} b^{4} c^{3} d^{4} x + 60 \, a^{4} b^{3} c^{2} d^{5} x + 30 \, a^{5} b^{2} c d^{6} x + 10 \, a^{6} b d^{7} x + 36 \, b^{7} c^{7} + 28 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 15 \, a^{3} b^{4} c^{4} d^{3} + 10 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} + a^{7} d^{7}}{360 \, {\left (b x + a\right )}^{10} b^{8}} \]
-1/360*(120*b^7*d^7*x^7 + 630*b^7*c*d^6*x^6 + 210*a*b^6*d^7*x^6 + 1512*b^7 *c^2*d^5*x^5 + 756*a*b^6*c*d^6*x^5 + 252*a^2*b^5*d^7*x^5 + 2100*b^7*c^3*d^ 4*x^4 + 1260*a*b^6*c^2*d^5*x^4 + 630*a^2*b^5*c*d^6*x^4 + 210*a^3*b^4*d^7*x ^4 + 1800*b^7*c^4*d^3*x^3 + 1200*a*b^6*c^3*d^4*x^3 + 720*a^2*b^5*c^2*d^5*x ^3 + 360*a^3*b^4*c*d^6*x^3 + 120*a^4*b^3*d^7*x^3 + 945*b^7*c^5*d^2*x^2 + 6 75*a*b^6*c^4*d^3*x^2 + 450*a^2*b^5*c^3*d^4*x^2 + 270*a^3*b^4*c^2*d^5*x^2 + 135*a^4*b^3*c*d^6*x^2 + 45*a^5*b^2*d^7*x^2 + 280*b^7*c^6*d*x + 210*a*b^6* c^5*d^2*x + 150*a^2*b^5*c^4*d^3*x + 100*a^3*b^4*c^3*d^4*x + 60*a^4*b^3*c^2 *d^5*x + 30*a^5*b^2*c*d^6*x + 10*a^6*b*d^7*x + 36*b^7*c^7 + 28*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 15*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c^3*d^4 + 6*a^5*b^ 2*c^2*d^5 + 3*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^10*b^8)
Time = 0.64 (sec) , antiderivative size = 600, normalized size of antiderivative = 6.74 \[ \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx=-\frac {a^7\,d^7+3\,a^6\,b\,c\,d^6+10\,a^6\,b\,d^7\,x+6\,a^5\,b^2\,c^2\,d^5+30\,a^5\,b^2\,c\,d^6\,x+45\,a^5\,b^2\,d^7\,x^2+10\,a^4\,b^3\,c^3\,d^4+60\,a^4\,b^3\,c^2\,d^5\,x+135\,a^4\,b^3\,c\,d^6\,x^2+120\,a^4\,b^3\,d^7\,x^3+15\,a^3\,b^4\,c^4\,d^3+100\,a^3\,b^4\,c^3\,d^4\,x+270\,a^3\,b^4\,c^2\,d^5\,x^2+360\,a^3\,b^4\,c\,d^6\,x^3+210\,a^3\,b^4\,d^7\,x^4+21\,a^2\,b^5\,c^5\,d^2+150\,a^2\,b^5\,c^4\,d^3\,x+450\,a^2\,b^5\,c^3\,d^4\,x^2+720\,a^2\,b^5\,c^2\,d^5\,x^3+630\,a^2\,b^5\,c\,d^6\,x^4+252\,a^2\,b^5\,d^7\,x^5+28\,a\,b^6\,c^6\,d+210\,a\,b^6\,c^5\,d^2\,x+675\,a\,b^6\,c^4\,d^3\,x^2+1200\,a\,b^6\,c^3\,d^4\,x^3+1260\,a\,b^6\,c^2\,d^5\,x^4+756\,a\,b^6\,c\,d^6\,x^5+210\,a\,b^6\,d^7\,x^6+36\,b^7\,c^7+280\,b^7\,c^6\,d\,x+945\,b^7\,c^5\,d^2\,x^2+1800\,b^7\,c^4\,d^3\,x^3+2100\,b^7\,c^3\,d^4\,x^4+1512\,b^7\,c^2\,d^5\,x^5+630\,b^7\,c\,d^6\,x^6+120\,b^7\,d^7\,x^7}{360\,a^{10}\,b^8+3600\,a^9\,b^9\,x+16200\,a^8\,b^{10}\,x^2+43200\,a^7\,b^{11}\,x^3+75600\,a^6\,b^{12}\,x^4+90720\,a^5\,b^{13}\,x^5+75600\,a^4\,b^{14}\,x^6+43200\,a^3\,b^{15}\,x^7+16200\,a^2\,b^{16}\,x^8+3600\,a\,b^{17}\,x^9+360\,b^{18}\,x^{10}} \]
-(a^7*d^7 + 36*b^7*c^7 + 120*b^7*d^7*x^7 + 210*a*b^6*d^7*x^6 + 630*b^7*c*d ^6*x^6 + 21*a^2*b^5*c^5*d^2 + 15*a^3*b^4*c^4*d^3 + 10*a^4*b^3*c^3*d^4 + 6* a^5*b^2*c^2*d^5 + 45*a^5*b^2*d^7*x^2 + 120*a^4*b^3*d^7*x^3 + 210*a^3*b^4*d ^7*x^4 + 252*a^2*b^5*d^7*x^5 + 945*b^7*c^5*d^2*x^2 + 1800*b^7*c^4*d^3*x^3 + 2100*b^7*c^3*d^4*x^4 + 1512*b^7*c^2*d^5*x^5 + 28*a*b^6*c^6*d + 3*a^6*b*c *d^6 + 10*a^6*b*d^7*x + 280*b^7*c^6*d*x + 450*a^2*b^5*c^3*d^4*x^2 + 270*a^ 3*b^4*c^2*d^5*x^2 + 720*a^2*b^5*c^2*d^5*x^3 + 210*a*b^6*c^5*d^2*x + 30*a^5 *b^2*c*d^6*x + 756*a*b^6*c*d^6*x^5 + 150*a^2*b^5*c^4*d^3*x + 100*a^3*b^4*c ^3*d^4*x + 60*a^4*b^3*c^2*d^5*x + 675*a*b^6*c^4*d^3*x^2 + 135*a^4*b^3*c*d^ 6*x^2 + 1200*a*b^6*c^3*d^4*x^3 + 360*a^3*b^4*c*d^6*x^3 + 1260*a*b^6*c^2*d^ 5*x^4 + 630*a^2*b^5*c*d^6*x^4)/(360*a^10*b^8 + 360*b^18*x^10 + 3600*a^9*b^ 9*x + 3600*a*b^17*x^9 + 16200*a^8*b^10*x^2 + 43200*a^7*b^11*x^3 + 75600*a^ 6*b^12*x^4 + 90720*a^5*b^13*x^5 + 75600*a^4*b^14*x^6 + 43200*a^3*b^15*x^7 + 16200*a^2*b^16*x^8)
Time = 0.00 (sec) , antiderivative size = 596, normalized size of antiderivative = 6.70 \[ \int \frac {(c+d x)^7}{(a+b x)^{11}} \, dx=\frac {-120 b^{7} d^{7} x^{7}-210 a \,b^{6} d^{7} x^{6}-630 b^{7} c \,d^{6} x^{6}-252 a^{2} b^{5} d^{7} x^{5}-756 a \,b^{6} c \,d^{6} x^{5}-1512 b^{7} c^{2} d^{5} x^{5}-210 a^{3} b^{4} d^{7} x^{4}-630 a^{2} b^{5} c \,d^{6} x^{4}-1260 a \,b^{6} c^{2} d^{5} x^{4}-2100 b^{7} c^{3} d^{4} x^{4}-120 a^{4} b^{3} d^{7} x^{3}-360 a^{3} b^{4} c \,d^{6} x^{3}-720 a^{2} b^{5} c^{2} d^{5} x^{3}-1200 a \,b^{6} c^{3} d^{4} x^{3}-1800 b^{7} c^{4} d^{3} x^{3}-45 a^{5} b^{2} d^{7} x^{2}-135 a^{4} b^{3} c \,d^{6} x^{2}-270 a^{3} b^{4} c^{2} d^{5} x^{2}-450 a^{2} b^{5} c^{3} d^{4} x^{2}-675 a \,b^{6} c^{4} d^{3} x^{2}-945 b^{7} c^{5} d^{2} x^{2}-10 a^{6} b \,d^{7} x -30 a^{5} b^{2} c \,d^{6} x -60 a^{4} b^{3} c^{2} d^{5} x -100 a^{3} b^{4} c^{3} d^{4} x -150 a^{2} b^{5} c^{4} d^{3} x -210 a \,b^{6} c^{5} d^{2} x -280 b^{7} c^{6} d x -a^{7} d^{7}-3 a^{6} b c \,d^{6}-6 a^{5} b^{2} c^{2} d^{5}-10 a^{4} b^{3} c^{3} d^{4}-15 a^{3} b^{4} c^{4} d^{3}-21 a^{2} b^{5} c^{5} d^{2}-28 a \,b^{6} c^{6} d -36 b^{7} c^{7}}{360 b^{8} \left (b^{10} x^{10}+10 a \,b^{9} x^{9}+45 a^{2} b^{8} x^{8}+120 a^{3} b^{7} x^{7}+210 a^{4} b^{6} x^{6}+252 a^{5} b^{5} x^{5}+210 a^{6} b^{4} x^{4}+120 a^{7} b^{3} x^{3}+45 a^{8} b^{2} x^{2}+10 a^{9} b x +a^{10}\right )} \]
int((c**7 + 7*c**6*d*x + 21*c**5*d**2*x**2 + 35*c**4*d**3*x**3 + 35*c**3*d **4*x**4 + 21*c**2*d**5*x**5 + 7*c*d**6*x**6 + d**7*x**7)/(a**11 + 11*a**1 0*b*x + 55*a**9*b**2*x**2 + 165*a**8*b**3*x**3 + 330*a**7*b**4*x**4 + 462* a**6*b**5*x**5 + 462*a**5*b**6*x**6 + 330*a**4*b**7*x**7 + 165*a**3*b**8*x **8 + 55*a**2*b**9*x**9 + 11*a*b**10*x**10 + b**11*x**11),x)
( - a**7*d**7 - 3*a**6*b*c*d**6 - 10*a**6*b*d**7*x - 6*a**5*b**2*c**2*d**5 - 30*a**5*b**2*c*d**6*x - 45*a**5*b**2*d**7*x**2 - 10*a**4*b**3*c**3*d**4 - 60*a**4*b**3*c**2*d**5*x - 135*a**4*b**3*c*d**6*x**2 - 120*a**4*b**3*d* *7*x**3 - 15*a**3*b**4*c**4*d**3 - 100*a**3*b**4*c**3*d**4*x - 270*a**3*b* *4*c**2*d**5*x**2 - 360*a**3*b**4*c*d**6*x**3 - 210*a**3*b**4*d**7*x**4 - 21*a**2*b**5*c**5*d**2 - 150*a**2*b**5*c**4*d**3*x - 450*a**2*b**5*c**3*d* *4*x**2 - 720*a**2*b**5*c**2*d**5*x**3 - 630*a**2*b**5*c*d**6*x**4 - 252*a **2*b**5*d**7*x**5 - 28*a*b**6*c**6*d - 210*a*b**6*c**5*d**2*x - 675*a*b** 6*c**4*d**3*x**2 - 1200*a*b**6*c**3*d**4*x**3 - 1260*a*b**6*c**2*d**5*x**4 - 756*a*b**6*c*d**6*x**5 - 210*a*b**6*d**7*x**6 - 36*b**7*c**7 - 280*b**7 *c**6*d*x - 945*b**7*c**5*d**2*x**2 - 1800*b**7*c**4*d**3*x**3 - 2100*b**7 *c**3*d**4*x**4 - 1512*b**7*c**2*d**5*x**5 - 630*b**7*c*d**6*x**6 - 120*b* *7*d**7*x**7)/(360*b**8*(a**10 + 10*a**9*b*x + 45*a**8*b**2*x**2 + 120*a** 7*b**3*x**3 + 210*a**6*b**4*x**4 + 252*a**5*b**5*x**5 + 210*a**4*b**6*x**6 + 120*a**3*b**7*x**7 + 45*a**2*b**8*x**8 + 10*a*b**9*x**9 + b**10*x**10))