Integrand size = 15, antiderivative size = 58 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {(c+d x)^8}{9 (b c-a d) (a+b x)^9}+\frac {d (c+d x)^8}{72 (b c-a d)^2 (a+b x)^8} \]
Leaf count is larger than twice the leaf count of optimal. \(367\) vs. \(2(58)=116\).
Time = 0.07 (sec) , antiderivative size = 367, normalized size of antiderivative = 6.33 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {a^7 d^7+a^6 b d^6 (2 c+9 d x)+3 a^5 b^2 d^5 \left (c^2+6 c d x+12 d^2 x^2\right )+a^4 b^3 d^4 \left (4 c^3+27 c^2 d x+72 c d^2 x^2+84 d^3 x^3\right )+a^3 b^4 d^3 \left (5 c^4+36 c^3 d x+108 c^2 d^2 x^2+168 c d^3 x^3+126 d^4 x^4\right )+3 a^2 b^5 d^2 \left (2 c^5+15 c^4 d x+48 c^3 d^2 x^2+84 c^2 d^3 x^3+84 c d^4 x^4+42 d^5 x^5\right )+a b^6 d \left (7 c^6+54 c^5 d x+180 c^4 d^2 x^2+336 c^3 d^3 x^3+378 c^2 d^4 x^4+252 c d^5 x^5+84 d^6 x^6\right )+b^7 \left (8 c^7+63 c^6 d x+216 c^5 d^2 x^2+420 c^4 d^3 x^3+504 c^3 d^4 x^4+378 c^2 d^5 x^5+168 c d^6 x^6+36 d^7 x^7\right )}{72 b^8 (a+b x)^9} \]
-1/72*(a^7*d^7 + a^6*b*d^6*(2*c + 9*d*x) + 3*a^5*b^2*d^5*(c^2 + 6*c*d*x + 12*d^2*x^2) + a^4*b^3*d^4*(4*c^3 + 27*c^2*d*x + 72*c*d^2*x^2 + 84*d^3*x^3) + a^3*b^4*d^3*(5*c^4 + 36*c^3*d*x + 108*c^2*d^2*x^2 + 168*c*d^3*x^3 + 126 *d^4*x^4) + 3*a^2*b^5*d^2*(2*c^5 + 15*c^4*d*x + 48*c^3*d^2*x^2 + 84*c^2*d^ 3*x^3 + 84*c*d^4*x^4 + 42*d^5*x^5) + a*b^6*d*(7*c^6 + 54*c^5*d*x + 180*c^4 *d^2*x^2 + 336*c^3*d^3*x^3 + 378*c^2*d^4*x^4 + 252*c*d^5*x^5 + 84*d^6*x^6) + b^7*(8*c^7 + 63*c^6*d*x + 216*c^5*d^2*x^2 + 420*c^4*d^3*x^3 + 504*c^3*d ^4*x^4 + 378*c^2*d^5*x^5 + 168*c*d^6*x^6 + 36*d^7*x^7))/(b^8*(a + b*x)^9)
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.
| \(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {d \int \frac {(c+d x)^7}{(a+b x)^9}dx}{9 (b c-a d)}-\frac {(c+d x)^8}{9 (a+b x)^9 (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {d (c+d x)^8}{72 (a+b x)^8 (b c-a d)^2}-\frac {(c+d x)^8}{9 (a+b x)^9 (b c-a d)}\) |
3.13.92.3.1 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. \(2(54)=108\).
Time = 0.22 (sec) , antiderivative size = 438, normalized size of antiderivative = 7.55
| method | result | size |
| risch | \(\frac {-\frac {d^{7} x^{7}}{2 b}-\frac {7 d^{6} \left (a d +2 b c \right ) x^{6}}{6 b^{2}}-\frac {7 d^{5} \left (a^{2} d^{2}+2 a b c d +3 b^{2} c^{2}\right ) x^{5}}{4 b^{3}}-\frac {7 d^{4} \left (a^{3} d^{3}+2 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +4 b^{3} c^{3}\right ) x^{4}}{4 b^{4}}-\frac {7 d^{3} \left (a^{4} d^{4}+2 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +5 b^{4} c^{4}\right ) x^{3}}{6 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+2 a^{4} b c \,d^{4}+3 a^{3} b^{2} c^{2} d^{3}+4 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +6 b^{5} c^{5}\right ) x^{2}}{2 b^{6}}-\frac {d \left (a^{6} d^{6}+2 a^{5} b c \,d^{5}+3 a^{4} b^{2} c^{2} d^{4}+4 a^{3} b^{3} c^{3} d^{3}+5 a^{2} b^{4} c^{4} d^{2}+6 a \,b^{5} c^{5} d +7 b^{6} c^{6}\right ) x}{8 b^{7}}-\frac {a^{7} d^{7}+2 a^{6} b c \,d^{6}+3 a^{5} b^{2} c^{2} d^{5}+4 a^{4} b^{3} c^{3} d^{4}+5 a^{3} b^{4} c^{4} d^{3}+6 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +8 b^{7} c^{7}}{72 b^{8}}}{\left (b x +a \right )^{9}}\) | \(438\) |
| default | \(\frac {7 d^{6} \left (a d -b c \right )}{3 b^{8} \left (b x +a \right )^{3}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{9 b^{8} \left (b x +a \right )^{9}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{6 b^{8} \left (b x +a \right )^{6}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}{8 b^{8} \left (b x +a \right )^{8}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{4 b^{8} \left (b x +a \right )^{4}}+\frac {3 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{b^{8} \left (b x +a \right )^{7}}-\frac {d^{7}}{2 b^{8} \left (b x +a \right )^{2}}+\frac {7 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{b^{8} \left (b x +a \right )^{5}}\) | \(464\) |
| norman | \(\frac {-\frac {d^{7} x^{7}}{2 b}+\frac {7 \left (-a b \,d^{7}-2 b^{2} c \,d^{6}\right ) x^{6}}{6 b^{3}}+\frac {7 \left (-a^{2} b \,d^{7}-2 a \,b^{2} c \,d^{6}-3 b^{3} c^{2} d^{5}\right ) x^{5}}{4 b^{4}}+\frac {7 \left (-a^{3} b \,d^{7}-2 a^{2} b^{2} c \,d^{6}-3 a \,b^{3} c^{2} d^{5}-4 b^{4} c^{3} d^{4}\right ) x^{4}}{4 b^{5}}+\frac {7 \left (-a^{4} b \,d^{7}-2 a^{3} b^{2} c \,d^{6}-3 a^{2} b^{3} c^{2} d^{5}-4 a \,b^{4} c^{3} d^{4}-5 b^{5} c^{4} d^{3}\right ) x^{3}}{6 b^{6}}+\frac {\left (-a^{5} b \,d^{7}-2 a^{4} b^{2} c \,d^{6}-3 a^{3} b^{3} c^{2} d^{5}-4 a^{2} b^{4} c^{3} d^{4}-5 a \,b^{5} c^{4} d^{3}-6 b^{6} c^{5} d^{2}\right ) x^{2}}{2 b^{7}}+\frac {\left (-a^{6} b \,d^{7}-2 a^{5} b^{2} c \,d^{6}-3 a^{4} b^{3} c^{2} d^{5}-4 a^{3} b^{4} c^{3} d^{4}-5 a^{2} b^{5} c^{4} d^{3}-6 a \,b^{6} c^{5} d^{2}-7 b^{7} c^{6} d \right ) x}{8 b^{8}}+\frac {-a^{7} b \,d^{7}-2 a^{6} b^{2} c \,d^{6}-3 a^{5} b^{3} c^{2} d^{5}-4 a^{4} b^{4} c^{3} d^{4}-5 a^{3} b^{5} c^{4} d^{3}-6 a^{2} b^{6} c^{5} d^{2}-7 a \,b^{7} c^{6} d -8 b^{8} c^{7}}{72 b^{9}}}{\left (b x +a \right )^{9}}\) | \(478\) |
| gosper | \(-\frac {36 x^{7} d^{7} b^{7}+84 x^{6} a \,b^{6} d^{7}+168 x^{6} b^{7} c \,d^{6}+126 x^{5} a^{2} b^{5} d^{7}+252 x^{5} a \,b^{6} c \,d^{6}+378 x^{5} b^{7} c^{2} d^{5}+126 x^{4} a^{3} b^{4} d^{7}+252 x^{4} a^{2} b^{5} c \,d^{6}+378 x^{4} a \,b^{6} c^{2} d^{5}+504 x^{4} b^{7} c^{3} d^{4}+84 x^{3} a^{4} b^{3} d^{7}+168 x^{3} a^{3} b^{4} c \,d^{6}+252 x^{3} a^{2} b^{5} c^{2} d^{5}+336 x^{3} a \,b^{6} c^{3} d^{4}+420 x^{3} b^{7} c^{4} d^{3}+36 x^{2} a^{5} b^{2} d^{7}+72 x^{2} a^{4} b^{3} c \,d^{6}+108 x^{2} a^{3} b^{4} c^{2} d^{5}+144 x^{2} a^{2} b^{5} c^{3} d^{4}+180 x^{2} a \,b^{6} c^{4} d^{3}+216 x^{2} b^{7} c^{5} d^{2}+9 x \,a^{6} b \,d^{7}+18 x \,a^{5} b^{2} c \,d^{6}+27 x \,a^{4} b^{3} c^{2} d^{5}+36 x \,a^{3} b^{4} c^{3} d^{4}+45 x \,a^{2} b^{5} c^{4} d^{3}+54 x a \,b^{6} c^{5} d^{2}+63 x \,b^{7} c^{6} d +a^{7} d^{7}+2 a^{6} b c \,d^{6}+3 a^{5} b^{2} c^{2} d^{5}+4 a^{4} b^{3} c^{3} d^{4}+5 a^{3} b^{4} c^{4} d^{3}+6 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +8 b^{7} c^{7}}{72 b^{8} \left (b x +a \right )^{9}}\) | \(497\) |
| parallelrisch | \(\frac {-36 d^{7} x^{7} b^{8}-84 a \,b^{7} d^{7} x^{6}-168 b^{8} c \,d^{6} x^{6}-126 a^{2} b^{6} d^{7} x^{5}-252 a \,b^{7} c \,d^{6} x^{5}-378 b^{8} c^{2} d^{5} x^{5}-126 a^{3} b^{5} d^{7} x^{4}-252 a^{2} b^{6} c \,d^{6} x^{4}-378 a \,b^{7} c^{2} d^{5} x^{4}-504 b^{8} c^{3} d^{4} x^{4}-84 a^{4} b^{4} d^{7} x^{3}-168 a^{3} b^{5} c \,d^{6} x^{3}-252 a^{2} b^{6} c^{2} d^{5} x^{3}-336 a \,b^{7} c^{3} d^{4} x^{3}-420 b^{8} c^{4} d^{3} x^{3}-36 a^{5} b^{3} d^{7} x^{2}-72 a^{4} b^{4} c \,d^{6} x^{2}-108 a^{3} b^{5} c^{2} d^{5} x^{2}-144 a^{2} b^{6} c^{3} d^{4} x^{2}-180 a \,b^{7} c^{4} d^{3} x^{2}-216 b^{8} c^{5} d^{2} x^{2}-9 a^{6} b^{2} d^{7} x -18 a^{5} b^{3} c \,d^{6} x -27 a^{4} b^{4} c^{2} d^{5} x -36 a^{3} b^{5} c^{3} d^{4} x -45 a^{2} b^{6} c^{4} d^{3} x -54 a \,b^{7} c^{5} d^{2} x -63 b^{8} c^{6} d x -a^{7} b \,d^{7}-2 a^{6} b^{2} c \,d^{6}-3 a^{5} b^{3} c^{2} d^{5}-4 a^{4} b^{4} c^{3} d^{4}-5 a^{3} b^{5} c^{4} d^{3}-6 a^{2} b^{6} c^{5} d^{2}-7 a \,b^{7} c^{6} d -8 b^{8} c^{7}}{72 b^{9} \left (b x +a \right )^{9}}\) | \(503\) |
(-1/2/b*d^7*x^7-7/6/b^2*d^6*(a*d+2*b*c)*x^6-7/4/b^3*d^5*(a^2*d^2+2*a*b*c*d +3*b^2*c^2)*x^5-7/4/b^4*d^4*(a^3*d^3+2*a^2*b*c*d^2+3*a*b^2*c^2*d+4*b^3*c^3 )*x^4-7/6/b^5*d^3*(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^3-1/2/b^6*d^2*(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^2-1/8/b^7*d*(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-1/72/b^8*(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))/(b*x+a)^9
Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (54) = 108\).
Time = 0.22 (sec) , antiderivative size = 548, normalized size of antiderivative = 9.45 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{7} d^{7} x^{7} + 8 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 4 \, a^{4} b^{3} c^{3} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{5} + 2 \, a^{6} b c d^{6} + a^{7} d^{7} + 84 \, {\left (2 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 126 \, {\left (3 \, b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 126 \, {\left (4 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 84 \, {\left (5 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 3 \, a^{2} b^{5} c^{2} d^{5} + 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 36 \, {\left (6 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 4 \, a^{2} b^{5} c^{3} d^{4} + 3 \, a^{3} b^{4} c^{2} d^{5} + 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 9 \, {\left (7 \, b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 4 \, a^{3} b^{4} c^{3} d^{4} + 3 \, a^{4} b^{3} c^{2} d^{5} + 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{72 \, {\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \]
-1/72*(36*b^7*d^7*x^7 + 8*b^7*c^7 + 7*a*b^6*c^6*d + 6*a^2*b^5*c^5*d^2 + 5* a^3*b^4*c^4*d^3 + 4*a^4*b^3*c^3*d^4 + 3*a^5*b^2*c^2*d^5 + 2*a^6*b*c*d^6 + a^7*d^7 + 84*(2*b^7*c*d^6 + a*b^6*d^7)*x^6 + 126*(3*b^7*c^2*d^5 + 2*a*b^6* c*d^6 + a^2*b^5*d^7)*x^5 + 126*(4*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 2*a^2*b^ 5*c*d^6 + a^3*b^4*d^7)*x^4 + 84*(5*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 3*a^2*b ^5*c^2*d^5 + 2*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 36*(6*b^7*c^5*d^2 + 5*a* b^6*c^4*d^3 + 4*a^2*b^5*c^3*d^4 + 3*a^3*b^4*c^2*d^5 + 2*a^4*b^3*c*d^6 + a^ 5*b^2*d^7)*x^2 + 9*(7*b^7*c^6*d + 6*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 4* a^3*b^4*c^3*d^4 + 3*a^4*b^3*c^2*d^5 + 2*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^1 7*x^9 + 9*a*b^16*x^8 + 36*a^2*b^15*x^7 + 84*a^3*b^14*x^6 + 126*a^4*b^13*x^ 5 + 126*a^5*b^12*x^4 + 84*a^6*b^11*x^3 + 36*a^7*b^10*x^2 + 9*a^8*b^9*x + a ^9*b^8)
Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (54) = 108\).
Time = 0.23 (sec) , antiderivative size = 548, normalized size of antiderivative = 9.45 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{7} d^{7} x^{7} + 8 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 4 \, a^{4} b^{3} c^{3} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{5} + 2 \, a^{6} b c d^{6} + a^{7} d^{7} + 84 \, {\left (2 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 126 \, {\left (3 \, b^{7} c^{2} d^{5} + 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 126 \, {\left (4 \, b^{7} c^{3} d^{4} + 3 \, a b^{6} c^{2} d^{5} + 2 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 84 \, {\left (5 \, b^{7} c^{4} d^{3} + 4 \, a b^{6} c^{3} d^{4} + 3 \, a^{2} b^{5} c^{2} d^{5} + 2 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 36 \, {\left (6 \, b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} + 4 \, a^{2} b^{5} c^{3} d^{4} + 3 \, a^{3} b^{4} c^{2} d^{5} + 2 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 9 \, {\left (7 \, b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} + 4 \, a^{3} b^{4} c^{3} d^{4} + 3 \, a^{4} b^{3} c^{2} d^{5} + 2 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{72 \, {\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \]
-1/72*(36*b^7*d^7*x^7 + 8*b^7*c^7 + 7*a*b^6*c^6*d + 6*a^2*b^5*c^5*d^2 + 5* a^3*b^4*c^4*d^3 + 4*a^4*b^3*c^3*d^4 + 3*a^5*b^2*c^2*d^5 + 2*a^6*b*c*d^6 + a^7*d^7 + 84*(2*b^7*c*d^6 + a*b^6*d^7)*x^6 + 126*(3*b^7*c^2*d^5 + 2*a*b^6* c*d^6 + a^2*b^5*d^7)*x^5 + 126*(4*b^7*c^3*d^4 + 3*a*b^6*c^2*d^5 + 2*a^2*b^ 5*c*d^6 + a^3*b^4*d^7)*x^4 + 84*(5*b^7*c^4*d^3 + 4*a*b^6*c^3*d^4 + 3*a^2*b ^5*c^2*d^5 + 2*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 36*(6*b^7*c^5*d^2 + 5*a* b^6*c^4*d^3 + 4*a^2*b^5*c^3*d^4 + 3*a^3*b^4*c^2*d^5 + 2*a^4*b^3*c*d^6 + a^ 5*b^2*d^7)*x^2 + 9*(7*b^7*c^6*d + 6*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 + 4* a^3*b^4*c^3*d^4 + 3*a^4*b^3*c^2*d^5 + 2*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^1 7*x^9 + 9*a*b^16*x^8 + 36*a^2*b^15*x^7 + 84*a^3*b^14*x^6 + 126*a^4*b^13*x^ 5 + 126*a^5*b^12*x^4 + 84*a^6*b^11*x^3 + 36*a^7*b^10*x^2 + 9*a^8*b^9*x + a ^9*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (54) = 108\).
Time = 0.29 (sec) , antiderivative size = 496, normalized size of antiderivative = 8.55 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{7} d^{7} x^{7} + 168 \, b^{7} c d^{6} x^{6} + 84 \, a b^{6} d^{7} x^{6} + 378 \, b^{7} c^{2} d^{5} x^{5} + 252 \, a b^{6} c d^{6} x^{5} + 126 \, a^{2} b^{5} d^{7} x^{5} + 504 \, b^{7} c^{3} d^{4} x^{4} + 378 \, a b^{6} c^{2} d^{5} x^{4} + 252 \, a^{2} b^{5} c d^{6} x^{4} + 126 \, a^{3} b^{4} d^{7} x^{4} + 420 \, b^{7} c^{4} d^{3} x^{3} + 336 \, a b^{6} c^{3} d^{4} x^{3} + 252 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 168 \, a^{3} b^{4} c d^{6} x^{3} + 84 \, a^{4} b^{3} d^{7} x^{3} + 216 \, b^{7} c^{5} d^{2} x^{2} + 180 \, a b^{6} c^{4} d^{3} x^{2} + 144 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 108 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 72 \, a^{4} b^{3} c d^{6} x^{2} + 36 \, a^{5} b^{2} d^{7} x^{2} + 63 \, b^{7} c^{6} d x + 54 \, a b^{6} c^{5} d^{2} x + 45 \, a^{2} b^{5} c^{4} d^{3} x + 36 \, a^{3} b^{4} c^{3} d^{4} x + 27 \, a^{4} b^{3} c^{2} d^{5} x + 18 \, a^{5} b^{2} c d^{6} x + 9 \, a^{6} b d^{7} x + 8 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} + 5 \, a^{3} b^{4} c^{4} d^{3} + 4 \, a^{4} b^{3} c^{3} d^{4} + 3 \, a^{5} b^{2} c^{2} d^{5} + 2 \, a^{6} b c d^{6} + a^{7} d^{7}}{72 \, {\left (b x + a\right )}^{9} b^{8}} \]
-1/72*(36*b^7*d^7*x^7 + 168*b^7*c*d^6*x^6 + 84*a*b^6*d^7*x^6 + 378*b^7*c^2 *d^5*x^5 + 252*a*b^6*c*d^6*x^5 + 126*a^2*b^5*d^7*x^5 + 504*b^7*c^3*d^4*x^4 + 378*a*b^6*c^2*d^5*x^4 + 252*a^2*b^5*c*d^6*x^4 + 126*a^3*b^4*d^7*x^4 + 4 20*b^7*c^4*d^3*x^3 + 336*a*b^6*c^3*d^4*x^3 + 252*a^2*b^5*c^2*d^5*x^3 + 168 *a^3*b^4*c*d^6*x^3 + 84*a^4*b^3*d^7*x^3 + 216*b^7*c^5*d^2*x^2 + 180*a*b^6* c^4*d^3*x^2 + 144*a^2*b^5*c^3*d^4*x^2 + 108*a^3*b^4*c^2*d^5*x^2 + 72*a^4*b ^3*c*d^6*x^2 + 36*a^5*b^2*d^7*x^2 + 63*b^7*c^6*d*x + 54*a*b^6*c^5*d^2*x + 45*a^2*b^5*c^4*d^3*x + 36*a^3*b^4*c^3*d^4*x + 27*a^4*b^3*c^2*d^5*x + 18*a^ 5*b^2*c*d^6*x + 9*a^6*b*d^7*x + 8*b^7*c^7 + 7*a*b^6*c^6*d + 6*a^2*b^5*c^5* d^2 + 5*a^3*b^4*c^4*d^3 + 4*a^4*b^3*c^3*d^4 + 3*a^5*b^2*c^2*d^5 + 2*a^6*b* c*d^6 + a^7*d^7)/((b*x + a)^9*b^8)
Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.67 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=\frac {{\left (c+d\,x\right )}^8\,\left (9\,a\,d-8\,b\,c+b\,d\,x\right )}{72\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^9} \]
Time = 0.00 (sec) , antiderivative size = 585, normalized size of antiderivative = 10.09 \[ \int \frac {(c+d x)^7}{(a+b x)^{10}} \, dx=\frac {-36 b^{7} d^{7} x^{7}-84 a \,b^{6} d^{7} x^{6}-168 b^{7} c \,d^{6} x^{6}-126 a^{2} b^{5} d^{7} x^{5}-252 a \,b^{6} c \,d^{6} x^{5}-378 b^{7} c^{2} d^{5} x^{5}-126 a^{3} b^{4} d^{7} x^{4}-252 a^{2} b^{5} c \,d^{6} x^{4}-378 a \,b^{6} c^{2} d^{5} x^{4}-504 b^{7} c^{3} d^{4} x^{4}-84 a^{4} b^{3} d^{7} x^{3}-168 a^{3} b^{4} c \,d^{6} x^{3}-252 a^{2} b^{5} c^{2} d^{5} x^{3}-336 a \,b^{6} c^{3} d^{4} x^{3}-420 b^{7} c^{4} d^{3} x^{3}-36 a^{5} b^{2} d^{7} x^{2}-72 a^{4} b^{3} c \,d^{6} x^{2}-108 a^{3} b^{4} c^{2} d^{5} x^{2}-144 a^{2} b^{5} c^{3} d^{4} x^{2}-180 a \,b^{6} c^{4} d^{3} x^{2}-216 b^{7} c^{5} d^{2} x^{2}-9 a^{6} b \,d^{7} x -18 a^{5} b^{2} c \,d^{6} x -27 a^{4} b^{3} c^{2} d^{5} x -36 a^{3} b^{4} c^{3} d^{4} x -45 a^{2} b^{5} c^{4} d^{3} x -54 a \,b^{6} c^{5} d^{2} x -63 b^{7} c^{6} d x -a^{7} d^{7}-2 a^{6} b c \,d^{6}-3 a^{5} b^{2} c^{2} d^{5}-4 a^{4} b^{3} c^{3} d^{4}-5 a^{3} b^{4} c^{4} d^{3}-6 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -8 b^{7} c^{7}}{72 b^{8} \left (b^{9} x^{9}+9 a \,b^{8} x^{8}+36 a^{2} b^{7} x^{7}+84 a^{3} b^{6} x^{6}+126 a^{4} b^{5} x^{5}+126 a^{5} b^{4} x^{4}+84 a^{6} b^{3} x^{3}+36 a^{7} b^{2} x^{2}+9 a^{8} b x +a^{9}\right )} \]
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**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),x)
( - a**7*d**7 - 2*a**6*b*c*d**6 - 9*a**6*b*d**7*x - 3*a**5*b**2*c**2*d**5 - 18*a**5*b**2*c*d**6*x - 36*a**5*b**2*d**7*x**2 - 4*a**4*b**3*c**3*d**4 - 27*a**4*b**3*c**2*d**5*x - 72*a**4*b**3*c*d**6*x**2 - 84*a**4*b**3*d**7*x **3 - 5*a**3*b**4*c**4*d**3 - 36*a**3*b**4*c**3*d**4*x - 108*a**3*b**4*c** 2*d**5*x**2 - 168*a**3*b**4*c*d**6*x**3 - 126*a**3*b**4*d**7*x**4 - 6*a**2 *b**5*c**5*d**2 - 45*a**2*b**5*c**4*d**3*x - 144*a**2*b**5*c**3*d**4*x**2 - 252*a**2*b**5*c**2*d**5*x**3 - 252*a**2*b**5*c*d**6*x**4 - 126*a**2*b**5 *d**7*x**5 - 7*a*b**6*c**6*d - 54*a*b**6*c**5*d**2*x - 180*a*b**6*c**4*d** 3*x**2 - 336*a*b**6*c**3*d**4*x**3 - 378*a*b**6*c**2*d**5*x**4 - 252*a*b** 6*c*d**6*x**5 - 84*a*b**6*d**7*x**6 - 8*b**7*c**7 - 63*b**7*c**6*d*x - 216 *b**7*c**5*d**2*x**2 - 420*b**7*c**4*d**3*x**3 - 504*b**7*c**3*d**4*x**4 - 378*b**7*c**2*d**5*x**5 - 168*b**7*c*d**6*x**6 - 36*b**7*d**7*x**7)/(72*b **8*(a**9 + 9*a**8*b*x + 36*a**7*b**2*x**2 + 84*a**6*b**3*x**3 + 126*a**5* b**4*x**4 + 126*a**4*b**5*x**5 + 84*a**3*b**6*x**6 + 36*a**2*b**7*x**7 + 9 *a*b**8*x**8 + b**9*x**9))