Integrand size = 15, antiderivative size = 151 \[ \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx=-\frac {(c+d x)^8}{12 (b c-a d) (a+b x)^{12}}+\frac {d (c+d x)^8}{33 (b c-a d)^2 (a+b x)^{11}}-\frac {d^2 (c+d x)^8}{110 (b c-a d)^3 (a+b x)^{10}}+\frac {d^3 (c+d x)^8}{495 (b c-a d)^4 (a+b x)^9}-\frac {d^4 (c+d x)^8}{3960 (b c-a d)^5 (a+b x)^8} \] Output:
-1/12*(d*x+c)^8/(-a*d+b*c)/(b*x+a)^12+1/33*d*(d*x+c)^8/(-a*d+b*c)^2/(b*x+a )^11-1/110*d^2*(d*x+c)^8/(-a*d+b*c)^3/(b*x+a)^10+1/495*d^3*(d*x+c)^8/(-a*d +b*c)^4/(b*x+a)^9-1/3960*d^4*(d*x+c)^8/(-a*d+b*c)^5/(b*x+a)^8
Leaf count is larger than twice the leaf count of optimal. \(371\) vs. \(2(151)=302\).
Time = 0.08 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.46 \[ \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx=-\frac {a^7 d^7+a^6 b d^6 (5 c+12 d x)+3 a^5 b^2 d^5 \left (5 c^2+20 c d x+22 d^2 x^2\right )+5 a^4 b^3 d^4 \left (7 c^3+36 c^2 d x+66 c d^2 x^2+44 d^3 x^3\right )+5 a^3 b^4 d^3 \left (14 c^4+84 c^3 d x+198 c^2 d^2 x^2+220 c d^3 x^3+99 d^4 x^4\right )+3 a^2 b^5 d^2 \left (42 c^5+280 c^4 d x+770 c^3 d^2 x^2+1100 c^2 d^3 x^3+825 c d^4 x^4+264 d^5 x^5\right )+a b^6 d \left (210 c^6+1512 c^5 d x+4620 c^4 d^2 x^2+7700 c^3 d^3 x^3+7425 c^2 d^4 x^4+3960 c d^5 x^5+924 d^6 x^6\right )+b^7 \left (330 c^7+2520 c^6 d x+8316 c^5 d^2 x^2+15400 c^4 d^3 x^3+17325 c^3 d^4 x^4+11880 c^2 d^5 x^5+4620 c d^6 x^6+792 d^7 x^7\right )}{3960 b^8 (a+b x)^{12}} \] Input:
Integrate[(c + d*x)^7/(a + b*x)^13,x]
Output:
-1/3960*(a^7*d^7 + a^6*b*d^6*(5*c + 12*d*x) + 3*a^5*b^2*d^5*(5*c^2 + 20*c* d*x + 22*d^2*x^2) + 5*a^4*b^3*d^4*(7*c^3 + 36*c^2*d*x + 66*c*d^2*x^2 + 44* d^3*x^3) + 5*a^3*b^4*d^3*(14*c^4 + 84*c^3*d*x + 198*c^2*d^2*x^2 + 220*c*d^ 3*x^3 + 99*d^4*x^4) + 3*a^2*b^5*d^2*(42*c^5 + 280*c^4*d*x + 770*c^3*d^2*x^ 2 + 1100*c^2*d^3*x^3 + 825*c*d^4*x^4 + 264*d^5*x^5) + a*b^6*d*(210*c^6 + 1 512*c^5*d*x + 4620*c^4*d^2*x^2 + 7700*c^3*d^3*x^3 + 7425*c^2*d^4*x^4 + 396 0*c*d^5*x^5 + 924*d^6*x^6) + b^7*(330*c^7 + 2520*c^6*d*x + 8316*c^5*d^2*x^ 2 + 15400*c^4*d^3*x^3 + 17325*c^3*d^4*x^4 + 11880*c^2*d^5*x^5 + 4620*c*d^6 *x^6 + 792*d^7*x^7))/(b^8*(a + b*x)^12)
Time = 0.23 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {55, 55, 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.
| \(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {d \int \frac {(c+d x)^7}{(a+b x)^{12}}dx}{3 (b c-a d)}-\frac {(c+d x)^8}{12 (a+b x)^{12} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {d \left (-\frac {3 d \int \frac {(c+d x)^7}{(a+b x)^{11}}dx}{11 (b c-a d)}-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)}\right )}{3 (b c-a d)}-\frac {(c+d x)^8}{12 (a+b x)^{12} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {d \left (-\frac {3 d \left (-\frac {d \int \frac {(c+d x)^7}{(a+b x)^{10}}dx}{5 (b c-a d)}-\frac {(c+d x)^8}{10 (a+b x)^{10} (b c-a d)}\right )}{11 (b c-a d)}-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)}\right )}{3 (b c-a d)}-\frac {(c+d x)^8}{12 (a+b x)^{12} (b c-a d)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle -\frac {d \left (-\frac {3 d \left (-\frac {d \left (-\frac {d \int \frac {(c+d x)^7}{(a+b x)^9}dx}{9 (b c-a d)}-\frac {(c+d x)^8}{9 (a+b x)^9 (b c-a d)}\right )}{5 (b c-a d)}-\frac {(c+d x)^8}{10 (a+b x)^{10} (b c-a d)}\right )}{11 (b c-a d)}-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)}\right )}{3 (b c-a d)}-\frac {(c+d x)^8}{12 (a+b x)^{12} (b c-a d)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle -\frac {(c+d x)^8}{12 (a+b x)^{12} (b c-a d)}-\frac {d \left (-\frac {(c+d x)^8}{11 (a+b x)^{11} (b c-a d)}-\frac {3 d \left (-\frac {(c+d x)^8}{10 (a+b x)^{10} (b c-a d)}-\frac {d \left (\frac {d (c+d x)^8}{72 (a+b x)^8 (b c-a d)^2}-\frac {(c+d x)^8}{9 (a+b x)^9 (b c-a d)}\right )}{5 (b c-a d)}\right )}{11 (b c-a d)}\right )}{3 (b c-a d)}\) |
Input:
Int[(c + d*x)^7/(a + b*x)^13,x]
Output:
-1/12*(c + d*x)^8/((b*c - a*d)*(a + b*x)^12) - (d*(-1/11*(c + d*x)^8/((b*c - a*d)*(a + b*x)^11) - (3*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))))/(11*(b*c - a*d))))/(3*(b*c - a*d))
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. \(2(141)=282\).
Time = 0.17 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.90
| method | result | size |
| risch | \(\frac {-\frac {d^{7} x^{7}}{5 b}-\frac {7 d^{6} \left (a d +5 b c \right ) x^{6}}{30 b^{2}}-\frac {d^{5} \left (a^{2} d^{2}+5 a b c d +15 b^{2} c^{2}\right ) x^{5}}{5 b^{3}}-\frac {d^{4} \left (a^{3} d^{3}+5 a^{2} b c \,d^{2}+15 a \,b^{2} c^{2} d +35 b^{3} c^{3}\right ) x^{4}}{8 b^{4}}-\frac {d^{3} \left (a^{4} d^{4}+5 a^{3} b c \,d^{3}+15 a^{2} b^{2} c^{2} d^{2}+35 a \,b^{3} c^{3} d +70 c^{4} b^{4}\right ) x^{3}}{18 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+5 a^{4} b c \,d^{4}+15 a^{3} b^{2} c^{2} d^{3}+35 a^{2} b^{3} c^{3} d^{2}+70 a \,b^{4} c^{4} d +126 c^{5} b^{5}\right ) x^{2}}{60 b^{6}}-\frac {d \left (a^{6} d^{6}+5 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}+35 a^{3} b^{3} c^{3} d^{3}+70 a^{2} b^{4} c^{4} d^{2}+126 a \,b^{5} c^{5} d +210 c^{6} b^{6}\right ) x}{330 b^{7}}-\frac {a^{7} d^{7}+5 a^{6} b c \,d^{6}+15 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+126 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d +330 b^{7} c^{7}}{3960 b^{8}}}{\left (b x +a \right )^{12}}\) | \(438\) |
| default | \(-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{9 b^{8} \left (b x +a \right )^{9}}-\frac {d^{7}}{5 b^{8} \left (b x +a \right )^{5}}-\frac {3 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{8} \left (b x +a \right )^{7}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{12 b^{8} \left (b x +a \right )^{12}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{8 b^{8} \left (b x +a \right )^{8}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{10 b^{8} \left (b x +a \right )^{10}}+\frac {7 d^{6} \left (a d -b c \right )}{6 b^{8} \left (b x +a \right )^{6}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{11 b^{8} \left (b x +a \right )^{11}}\) | \(464\) |
| norman | \(\frac {-\frac {d^{7} x^{7}}{5 b}+\frac {7 \left (-a \,b^{4} d^{7}-5 b^{5} c \,d^{6}\right ) x^{6}}{30 b^{6}}+\frac {\left (-a^{2} b^{4} d^{7}-5 a \,b^{5} c \,d^{6}-15 b^{6} c^{2} d^{5}\right ) x^{5}}{5 b^{7}}+\frac {\left (-a^{3} b^{4} d^{7}-5 a^{2} b^{5} c \,d^{6}-15 a \,b^{6} c^{2} d^{5}-35 b^{7} c^{3} d^{4}\right ) x^{4}}{8 b^{8}}+\frac {\left (-a^{4} b^{4} d^{7}-5 a^{3} b^{5} c \,d^{6}-15 a^{2} b^{6} c^{2} d^{5}-35 a \,b^{7} c^{3} d^{4}-70 b^{8} c^{4} d^{3}\right ) x^{3}}{18 b^{9}}+\frac {\left (-a^{5} b^{4} d^{7}-5 a^{4} b^{5} c \,d^{6}-15 a^{3} b^{6} c^{2} d^{5}-35 a^{2} b^{7} c^{3} d^{4}-70 a \,b^{8} c^{4} d^{3}-126 b^{9} c^{5} d^{2}\right ) x^{2}}{60 b^{10}}+\frac {\left (-a^{6} b^{4} d^{7}-5 a^{5} b^{5} c \,d^{6}-15 a^{4} b^{6} c^{2} d^{5}-35 a^{3} b^{7} c^{3} d^{4}-70 a^{2} b^{8} c^{4} d^{3}-126 a \,b^{9} c^{5} d^{2}-210 b^{10} c^{6} d \right ) x}{330 b^{11}}+\frac {-a^{7} b^{4} d^{7}-5 a^{6} b^{5} c \,d^{6}-15 a^{5} b^{6} c^{2} d^{5}-35 a^{4} b^{7} c^{3} d^{4}-70 a^{3} b^{8} c^{4} d^{3}-126 a^{2} b^{9} c^{5} d^{2}-210 a \,c^{6} d \,b^{10}-330 b^{11} c^{7}}{3960 b^{12}}}{\left (b x +a \right )^{12}}\) | \(492\) |
| gosper | \(-\frac {792 x^{7} d^{7} b^{7}+924 x^{6} a \,b^{6} d^{7}+4620 x^{6} b^{7} c \,d^{6}+792 x^{5} a^{2} b^{5} d^{7}+3960 x^{5} a \,b^{6} c \,d^{6}+11880 x^{5} b^{7} c^{2} d^{5}+495 x^{4} a^{3} b^{4} d^{7}+2475 x^{4} a^{2} b^{5} c \,d^{6}+7425 x^{4} a \,b^{6} c^{2} d^{5}+17325 x^{4} b^{7} c^{3} d^{4}+220 x^{3} a^{4} b^{3} d^{7}+1100 x^{3} a^{3} b^{4} c \,d^{6}+3300 x^{3} a^{2} b^{5} c^{2} d^{5}+7700 x^{3} a \,b^{6} c^{3} d^{4}+15400 x^{3} b^{7} c^{4} d^{3}+66 x^{2} a^{5} b^{2} d^{7}+330 x^{2} a^{4} b^{3} c \,d^{6}+990 x^{2} a^{3} b^{4} c^{2} d^{5}+2310 x^{2} a^{2} b^{5} c^{3} d^{4}+4620 x^{2} a \,b^{6} c^{4} d^{3}+8316 x^{2} b^{7} c^{5} d^{2}+12 x \,a^{6} b \,d^{7}+60 x \,a^{5} b^{2} c \,d^{6}+180 x \,a^{4} b^{3} c^{2} d^{5}+420 x \,a^{3} b^{4} c^{3} d^{4}+840 x \,a^{2} b^{5} c^{4} d^{3}+1512 x a \,b^{6} c^{5} d^{2}+2520 x \,b^{7} c^{6} d +a^{7} d^{7}+5 a^{6} b c \,d^{6}+15 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+126 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d +330 b^{7} c^{7}}{3960 b^{8} \left (b x +a \right )^{12}}\) | \(497\) |
| orering | \(-\frac {792 x^{7} d^{7} b^{7}+924 x^{6} a \,b^{6} d^{7}+4620 x^{6} b^{7} c \,d^{6}+792 x^{5} a^{2} b^{5} d^{7}+3960 x^{5} a \,b^{6} c \,d^{6}+11880 x^{5} b^{7} c^{2} d^{5}+495 x^{4} a^{3} b^{4} d^{7}+2475 x^{4} a^{2} b^{5} c \,d^{6}+7425 x^{4} a \,b^{6} c^{2} d^{5}+17325 x^{4} b^{7} c^{3} d^{4}+220 x^{3} a^{4} b^{3} d^{7}+1100 x^{3} a^{3} b^{4} c \,d^{6}+3300 x^{3} a^{2} b^{5} c^{2} d^{5}+7700 x^{3} a \,b^{6} c^{3} d^{4}+15400 x^{3} b^{7} c^{4} d^{3}+66 x^{2} a^{5} b^{2} d^{7}+330 x^{2} a^{4} b^{3} c \,d^{6}+990 x^{2} a^{3} b^{4} c^{2} d^{5}+2310 x^{2} a^{2} b^{5} c^{3} d^{4}+4620 x^{2} a \,b^{6} c^{4} d^{3}+8316 x^{2} b^{7} c^{5} d^{2}+12 x \,a^{6} b \,d^{7}+60 x \,a^{5} b^{2} c \,d^{6}+180 x \,a^{4} b^{3} c^{2} d^{5}+420 x \,a^{3} b^{4} c^{3} d^{4}+840 x \,a^{2} b^{5} c^{4} d^{3}+1512 x a \,b^{6} c^{5} d^{2}+2520 x \,b^{7} c^{6} d +a^{7} d^{7}+5 a^{6} b c \,d^{6}+15 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}+70 a^{3} b^{4} c^{4} d^{3}+126 a^{2} b^{5} c^{5} d^{2}+210 a \,b^{6} c^{6} d +330 b^{7} c^{7}}{3960 b^{8} \left (b x +a \right )^{12}}\) | \(497\) |
| parallelrisch | \(\frac {-792 d^{7} x^{7} b^{11}-924 a \,b^{10} d^{7} x^{6}-4620 b^{11} c \,d^{6} x^{6}-792 a^{2} b^{9} d^{7} x^{5}-3960 a \,b^{10} c \,d^{6} x^{5}-11880 b^{11} c^{2} d^{5} x^{5}-495 a^{3} b^{8} d^{7} x^{4}-2475 a^{2} b^{9} c \,d^{6} x^{4}-7425 a \,b^{10} c^{2} d^{5} x^{4}-17325 b^{11} c^{3} d^{4} x^{4}-220 a^{4} b^{7} d^{7} x^{3}-1100 a^{3} b^{8} c \,d^{6} x^{3}-3300 a^{2} b^{9} c^{2} d^{5} x^{3}-7700 a \,b^{10} c^{3} d^{4} x^{3}-15400 b^{11} c^{4} d^{3} x^{3}-66 a^{5} b^{6} d^{7} x^{2}-330 a^{4} b^{7} c \,d^{6} x^{2}-990 a^{3} b^{8} c^{2} d^{5} x^{2}-2310 a^{2} b^{9} c^{3} d^{4} x^{2}-4620 a \,b^{10} c^{4} d^{3} x^{2}-8316 b^{11} c^{5} d^{2} x^{2}-12 a^{6} b^{5} d^{7} x -60 a^{5} b^{6} c \,d^{6} x -180 a^{4} b^{7} c^{2} d^{5} x -420 a^{3} b^{8} c^{3} d^{4} x -840 a^{2} b^{9} c^{4} d^{3} x -1512 a \,b^{10} c^{5} d^{2} x -2520 b^{11} c^{6} d x -a^{7} b^{4} d^{7}-5 a^{6} b^{5} c \,d^{6}-15 a^{5} b^{6} c^{2} d^{5}-35 a^{4} b^{7} c^{3} d^{4}-70 a^{3} b^{8} c^{4} d^{3}-126 a^{2} b^{9} c^{5} d^{2}-210 a \,c^{6} d \,b^{10}-330 b^{11} c^{7}}{3960 b^{12} \left (b x +a \right )^{12}}\) | \(505\) |
Input:
int((d*x+c)^7/(b*x+a)^13,x,method=_RETURNVERBOSE)
Output:
(-1/5/b*d^7*x^7-7/30/b^2*d^6*(a*d+5*b*c)*x^6-1/5/b^3*d^5*(a^2*d^2+5*a*b*c* d+15*b^2*c^2)*x^5-1/8/b^4*d^4*(a^3*d^3+5*a^2*b*c*d^2+15*a*b^2*c^2*d+35*b^3 *c^3)*x^4-1/18/b^5*d^3*(a^4*d^4+5*a^3*b*c*d^3+15*a^2*b^2*c^2*d^2+35*a*b^3* c^3*d+70*b^4*c^4)*x^3-1/60/b^6*d^2*(a^5*d^5+5*a^4*b*c*d^4+15*a^3*b^2*c^2*d ^3+35*a^2*b^3*c^3*d^2+70*a*b^4*c^4*d+126*b^5*c^5)*x^2-1/330/b^7*d*(a^6*d^6 +5*a^5*b*c*d^5+15*a^4*b^2*c^2*d^4+35*a^3*b^3*c^3*d^3+70*a^2*b^4*c^4*d^2+12 6*a*b^5*c^5*d+210*b^6*c^6)*x-1/3960/b^8*(a^7*d^7+5*a^6*b*c*d^6+15*a^5*b^2* c^2*d^5+35*a^4*b^3*c^3*d^4+70*a^3*b^4*c^4*d^3+126*a^2*b^5*c^5*d^2+210*a*b^ 6*c^6*d+330*b^7*c^7))/(b*x+a)^12
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (141) = 282\).
Time = 0.08 (sec) , antiderivative size = 581, normalized size of antiderivative = 3.85 \[ \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx=-\frac {792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \, {\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \, {\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \, {\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \, {\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \, {\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \, {\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \, {\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \] Input:
integrate((d*x+c)^7/(b*x+a)^13,x, algorithm="fricas")
Output:
-1/3960*(792*b^7*d^7*x^7 + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5 *d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^ 6*b*c*d^6 + a^7*d^7 + 924*(5*b^7*c*d^6 + a*b^6*d^7)*x^6 + 792*(15*b^7*c^2* d^5 + 5*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 495*(35*b^7*c^3*d^4 + 15*a*b^6*c^ 2*d^5 + 5*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 220*(70*b^7*c^4*d^3 + 35*a*b^ 6*c^3*d^4 + 15*a^2*b^5*c^2*d^5 + 5*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 66*( 126*b^7*c^5*d^2 + 70*a*b^6*c^4*d^3 + 35*a^2*b^5*c^3*d^4 + 15*a^3*b^4*c^2*d ^5 + 5*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 12*(210*b^7*c^6*d + 126*a*b^6*c^ 5*d^2 + 70*a^2*b^5*c^4*d^3 + 35*a^3*b^4*c^3*d^4 + 15*a^4*b^3*c^2*d^5 + 5*a ^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^20*x^12 + 12*a*b^19*x^11 + 66*a^2*b^18*x^1 0 + 220*a^3*b^17*x^9 + 495*a^4*b^16*x^8 + 792*a^5*b^15*x^7 + 924*a^6*b^14* x^6 + 792*a^7*b^13*x^5 + 495*a^8*b^12*x^4 + 220*a^9*b^11*x^3 + 66*a^10*b^1 0*x^2 + 12*a^11*b^9*x + a^12*b^8)
Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**7/(b*x+a)**13,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (141) = 282\).
Time = 0.06 (sec) , antiderivative size = 581, normalized size of antiderivative = 3.85 \[ \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx=-\frac {792 \, b^{7} d^{7} x^{7} + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7} + 924 \, {\left (5 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 792 \, {\left (15 \, b^{7} c^{2} d^{5} + 5 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 495 \, {\left (35 \, b^{7} c^{3} d^{4} + 15 \, a b^{6} c^{2} d^{5} + 5 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 220 \, {\left (70 \, b^{7} c^{4} d^{3} + 35 \, a b^{6} c^{3} d^{4} + 15 \, a^{2} b^{5} c^{2} d^{5} + 5 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 66 \, {\left (126 \, b^{7} c^{5} d^{2} + 70 \, a b^{6} c^{4} d^{3} + 35 \, a^{2} b^{5} c^{3} d^{4} + 15 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 12 \, {\left (210 \, b^{7} c^{6} d + 126 \, a b^{6} c^{5} d^{2} + 70 \, a^{2} b^{5} c^{4} d^{3} + 35 \, a^{3} b^{4} c^{3} d^{4} + 15 \, a^{4} b^{3} c^{2} d^{5} + 5 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{3960 \, {\left (b^{20} x^{12} + 12 \, a b^{19} x^{11} + 66 \, a^{2} b^{18} x^{10} + 220 \, a^{3} b^{17} x^{9} + 495 \, a^{4} b^{16} x^{8} + 792 \, a^{5} b^{15} x^{7} + 924 \, a^{6} b^{14} x^{6} + 792 \, a^{7} b^{13} x^{5} + 495 \, a^{8} b^{12} x^{4} + 220 \, a^{9} b^{11} x^{3} + 66 \, a^{10} b^{10} x^{2} + 12 \, a^{11} b^{9} x + a^{12} b^{8}\right )}} \] Input:
integrate((d*x+c)^7/(b*x+a)^13,x, algorithm="maxima")
Output:
-1/3960*(792*b^7*d^7*x^7 + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5 *d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^ 6*b*c*d^6 + a^7*d^7 + 924*(5*b^7*c*d^6 + a*b^6*d^7)*x^6 + 792*(15*b^7*c^2* d^5 + 5*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 495*(35*b^7*c^3*d^4 + 15*a*b^6*c^ 2*d^5 + 5*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 220*(70*b^7*c^4*d^3 + 35*a*b^ 6*c^3*d^4 + 15*a^2*b^5*c^2*d^5 + 5*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 66*( 126*b^7*c^5*d^2 + 70*a*b^6*c^4*d^3 + 35*a^2*b^5*c^3*d^4 + 15*a^3*b^4*c^2*d ^5 + 5*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 12*(210*b^7*c^6*d + 126*a*b^6*c^ 5*d^2 + 70*a^2*b^5*c^4*d^3 + 35*a^3*b^4*c^3*d^4 + 15*a^4*b^3*c^2*d^5 + 5*a ^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^20*x^12 + 12*a*b^19*x^11 + 66*a^2*b^18*x^1 0 + 220*a^3*b^17*x^9 + 495*a^4*b^16*x^8 + 792*a^5*b^15*x^7 + 924*a^6*b^14* x^6 + 792*a^7*b^13*x^5 + 495*a^8*b^12*x^4 + 220*a^9*b^11*x^3 + 66*a^10*b^1 0*x^2 + 12*a^11*b^9*x + a^12*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (141) = 282\).
Time = 0.13 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.28 \[ \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx=-\frac {792 \, b^{7} d^{7} x^{7} + 4620 \, b^{7} c d^{6} x^{6} + 924 \, a b^{6} d^{7} x^{6} + 11880 \, b^{7} c^{2} d^{5} x^{5} + 3960 \, a b^{6} c d^{6} x^{5} + 792 \, a^{2} b^{5} d^{7} x^{5} + 17325 \, b^{7} c^{3} d^{4} x^{4} + 7425 \, a b^{6} c^{2} d^{5} x^{4} + 2475 \, a^{2} b^{5} c d^{6} x^{4} + 495 \, a^{3} b^{4} d^{7} x^{4} + 15400 \, b^{7} c^{4} d^{3} x^{3} + 7700 \, a b^{6} c^{3} d^{4} x^{3} + 3300 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1100 \, a^{3} b^{4} c d^{6} x^{3} + 220 \, a^{4} b^{3} d^{7} x^{3} + 8316 \, b^{7} c^{5} d^{2} x^{2} + 4620 \, a b^{6} c^{4} d^{3} x^{2} + 2310 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 990 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 330 \, a^{4} b^{3} c d^{6} x^{2} + 66 \, a^{5} b^{2} d^{7} x^{2} + 2520 \, b^{7} c^{6} d x + 1512 \, a b^{6} c^{5} d^{2} x + 840 \, a^{2} b^{5} c^{4} d^{3} x + 420 \, a^{3} b^{4} c^{3} d^{4} x + 180 \, a^{4} b^{3} c^{2} d^{5} x + 60 \, a^{5} b^{2} c d^{6} x + 12 \, a^{6} b d^{7} x + 330 \, b^{7} c^{7} + 210 \, a b^{6} c^{6} d + 126 \, a^{2} b^{5} c^{5} d^{2} + 70 \, a^{3} b^{4} c^{4} d^{3} + 35 \, a^{4} b^{3} c^{3} d^{4} + 15 \, a^{5} b^{2} c^{2} d^{5} + 5 \, a^{6} b c d^{6} + a^{7} d^{7}}{3960 \, {\left (b x + a\right )}^{12} b^{8}} \] Input:
integrate((d*x+c)^7/(b*x+a)^13,x, algorithm="giac")
Output:
-1/3960*(792*b^7*d^7*x^7 + 4620*b^7*c*d^6*x^6 + 924*a*b^6*d^7*x^6 + 11880* b^7*c^2*d^5*x^5 + 3960*a*b^6*c*d^6*x^5 + 792*a^2*b^5*d^7*x^5 + 17325*b^7*c ^3*d^4*x^4 + 7425*a*b^6*c^2*d^5*x^4 + 2475*a^2*b^5*c*d^6*x^4 + 495*a^3*b^4 *d^7*x^4 + 15400*b^7*c^4*d^3*x^3 + 7700*a*b^6*c^3*d^4*x^3 + 3300*a^2*b^5*c ^2*d^5*x^3 + 1100*a^3*b^4*c*d^6*x^3 + 220*a^4*b^3*d^7*x^3 + 8316*b^7*c^5*d ^2*x^2 + 4620*a*b^6*c^4*d^3*x^2 + 2310*a^2*b^5*c^3*d^4*x^2 + 990*a^3*b^4*c ^2*d^5*x^2 + 330*a^4*b^3*c*d^6*x^2 + 66*a^5*b^2*d^7*x^2 + 2520*b^7*c^6*d*x + 1512*a*b^6*c^5*d^2*x + 840*a^2*b^5*c^4*d^3*x + 420*a^3*b^4*c^3*d^4*x + 180*a^4*b^3*c^2*d^5*x + 60*a^5*b^2*c*d^6*x + 12*a^6*b*d^7*x + 330*b^7*c^7 + 210*a*b^6*c^6*d + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c^4*d^3 + 35*a^4*b^3* c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^12*b^8)
Time = 0.22 (sec) , antiderivative size = 559, normalized size of antiderivative = 3.70 \[ \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx=-\frac {\frac {a^7\,d^7+5\,a^6\,b\,c\,d^6+15\,a^5\,b^2\,c^2\,d^5+35\,a^4\,b^3\,c^3\,d^4+70\,a^3\,b^4\,c^4\,d^3+126\,a^2\,b^5\,c^5\,d^2+210\,a\,b^6\,c^6\,d+330\,b^7\,c^7}{3960\,b^8}+\frac {d^7\,x^7}{5\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+5\,a^4\,b\,c\,d^4+15\,a^3\,b^2\,c^2\,d^3+35\,a^2\,b^3\,c^3\,d^2+70\,a\,b^4\,c^4\,d+126\,b^5\,c^5\right )}{60\,b^6}+\frac {d^4\,x^4\,\left (a^3\,d^3+5\,a^2\,b\,c\,d^2+15\,a\,b^2\,c^2\,d+35\,b^3\,c^3\right )}{8\,b^4}+\frac {7\,d^6\,x^6\,\left (a\,d+5\,b\,c\right )}{30\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+5\,a^3\,b\,c\,d^3+15\,a^2\,b^2\,c^2\,d^2+35\,a\,b^3\,c^3\,d+70\,b^4\,c^4\right )}{18\,b^5}+\frac {d\,x\,\left (a^6\,d^6+5\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4+35\,a^3\,b^3\,c^3\,d^3+70\,a^2\,b^4\,c^4\,d^2+126\,a\,b^5\,c^5\,d+210\,b^6\,c^6\right )}{330\,b^7}+\frac {d^5\,x^5\,\left (a^2\,d^2+5\,a\,b\,c\,d+15\,b^2\,c^2\right )}{5\,b^3}}{a^{12}+12\,a^{11}\,b\,x+66\,a^{10}\,b^2\,x^2+220\,a^9\,b^3\,x^3+495\,a^8\,b^4\,x^4+792\,a^7\,b^5\,x^5+924\,a^6\,b^6\,x^6+792\,a^5\,b^7\,x^7+495\,a^4\,b^8\,x^8+220\,a^3\,b^9\,x^9+66\,a^2\,b^{10}\,x^{10}+12\,a\,b^{11}\,x^{11}+b^{12}\,x^{12}} \] Input:
int((c + d*x)^7/(a + b*x)^13,x)
Output:
-((a^7*d^7 + 330*b^7*c^7 + 126*a^2*b^5*c^5*d^2 + 70*a^3*b^4*c^4*d^3 + 35*a ^4*b^3*c^3*d^4 + 15*a^5*b^2*c^2*d^5 + 210*a*b^6*c^6*d + 5*a^6*b*c*d^6)/(39 60*b^8) + (d^7*x^7)/(5*b) + (d^2*x^2*(a^5*d^5 + 126*b^5*c^5 + 35*a^2*b^3*c ^3*d^2 + 15*a^3*b^2*c^2*d^3 + 70*a*b^4*c^4*d + 5*a^4*b*c*d^4))/(60*b^6) + (d^4*x^4*(a^3*d^3 + 35*b^3*c^3 + 15*a*b^2*c^2*d + 5*a^2*b*c*d^2))/(8*b^4) + (7*d^6*x^6*(a*d + 5*b*c))/(30*b^2) + (d^3*x^3*(a^4*d^4 + 70*b^4*c^4 + 15 *a^2*b^2*c^2*d^2 + 35*a*b^3*c^3*d + 5*a^3*b*c*d^3))/(18*b^5) + (d*x*(a^6*d ^6 + 210*b^6*c^6 + 70*a^2*b^4*c^4*d^2 + 35*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^ 2*d^4 + 126*a*b^5*c^5*d + 5*a^5*b*c*d^5))/(330*b^7) + (d^5*x^5*(a^2*d^2 + 15*b^2*c^2 + 5*a*b*c*d))/(5*b^3))/(a^12 + b^12*x^12 + 12*a*b^11*x^11 + 66* a^10*b^2*x^2 + 220*a^9*b^3*x^3 + 495*a^8*b^4*x^4 + 792*a^7*b^5*x^5 + 924*a ^6*b^6*x^6 + 792*a^5*b^7*x^7 + 495*a^4*b^8*x^8 + 220*a^3*b^9*x^9 + 66*a^2* b^10*x^10 + 12*a^11*b*x)
Time = 0.18 (sec) , antiderivative size = 618, normalized size of antiderivative = 4.09 \[ \int \frac {(c+d x)^7}{(a+b x)^{13}} \, dx=\frac {-792 b^{7} d^{7} x^{7}-924 a \,b^{6} d^{7} x^{6}-4620 b^{7} c \,d^{6} x^{6}-792 a^{2} b^{5} d^{7} x^{5}-3960 a \,b^{6} c \,d^{6} x^{5}-11880 b^{7} c^{2} d^{5} x^{5}-495 a^{3} b^{4} d^{7} x^{4}-2475 a^{2} b^{5} c \,d^{6} x^{4}-7425 a \,b^{6} c^{2} d^{5} x^{4}-17325 b^{7} c^{3} d^{4} x^{4}-220 a^{4} b^{3} d^{7} x^{3}-1100 a^{3} b^{4} c \,d^{6} x^{3}-3300 a^{2} b^{5} c^{2} d^{5} x^{3}-7700 a \,b^{6} c^{3} d^{4} x^{3}-15400 b^{7} c^{4} d^{3} x^{3}-66 a^{5} b^{2} d^{7} x^{2}-330 a^{4} b^{3} c \,d^{6} x^{2}-990 a^{3} b^{4} c^{2} d^{5} x^{2}-2310 a^{2} b^{5} c^{3} d^{4} x^{2}-4620 a \,b^{6} c^{4} d^{3} x^{2}-8316 b^{7} c^{5} d^{2} x^{2}-12 a^{6} b \,d^{7} x -60 a^{5} b^{2} c \,d^{6} x -180 a^{4} b^{3} c^{2} d^{5} x -420 a^{3} b^{4} c^{3} d^{4} x -840 a^{2} b^{5} c^{4} d^{3} x -1512 a \,b^{6} c^{5} d^{2} x -2520 b^{7} c^{6} d x -a^{7} d^{7}-5 a^{6} b c \,d^{6}-15 a^{5} b^{2} c^{2} d^{5}-35 a^{4} b^{3} c^{3} d^{4}-70 a^{3} b^{4} c^{4} d^{3}-126 a^{2} b^{5} c^{5} d^{2}-210 a \,b^{6} c^{6} d -330 b^{7} c^{7}}{3960 b^{8} \left (b^{12} x^{12}+12 a \,b^{11} x^{11}+66 a^{2} b^{10} x^{10}+220 a^{3} b^{9} x^{9}+495 a^{4} b^{8} x^{8}+792 a^{5} b^{7} x^{7}+924 a^{6} b^{6} x^{6}+792 a^{7} b^{5} x^{5}+495 a^{8} b^{4} x^{4}+220 a^{9} b^{3} x^{3}+66 a^{10} b^{2} x^{2}+12 a^{11} b x +a^{12}\right )} \] Input:
int((d*x+c)^7/(b*x+a)^13,x)
Output:
( - a**7*d**7 - 5*a**6*b*c*d**6 - 12*a**6*b*d**7*x - 15*a**5*b**2*c**2*d** 5 - 60*a**5*b**2*c*d**6*x - 66*a**5*b**2*d**7*x**2 - 35*a**4*b**3*c**3*d** 4 - 180*a**4*b**3*c**2*d**5*x - 330*a**4*b**3*c*d**6*x**2 - 220*a**4*b**3* d**7*x**3 - 70*a**3*b**4*c**4*d**3 - 420*a**3*b**4*c**3*d**4*x - 990*a**3* b**4*c**2*d**5*x**2 - 1100*a**3*b**4*c*d**6*x**3 - 495*a**3*b**4*d**7*x**4 - 126*a**2*b**5*c**5*d**2 - 840*a**2*b**5*c**4*d**3*x - 2310*a**2*b**5*c* *3*d**4*x**2 - 3300*a**2*b**5*c**2*d**5*x**3 - 2475*a**2*b**5*c*d**6*x**4 - 792*a**2*b**5*d**7*x**5 - 210*a*b**6*c**6*d - 1512*a*b**6*c**5*d**2*x - 4620*a*b**6*c**4*d**3*x**2 - 7700*a*b**6*c**3*d**4*x**3 - 7425*a*b**6*c**2 *d**5*x**4 - 3960*a*b**6*c*d**6*x**5 - 924*a*b**6*d**7*x**6 - 330*b**7*c** 7 - 2520*b**7*c**6*d*x - 8316*b**7*c**5*d**2*x**2 - 15400*b**7*c**4*d**3*x **3 - 17325*b**7*c**3*d**4*x**4 - 11880*b**7*c**2*d**5*x**5 - 4620*b**7*c* d**6*x**6 - 792*b**7*d**7*x**7)/(3960*b**8*(a**12 + 12*a**11*b*x + 66*a**1 0*b**2*x**2 + 220*a**9*b**3*x**3 + 495*a**8*b**4*x**4 + 792*a**7*b**5*x**5 + 924*a**6*b**6*x**6 + 792*a**5*b**7*x**7 + 495*a**4*b**8*x**8 + 220*a**3 *b**9*x**9 + 66*a**2*b**10*x**10 + 12*a*b**11*x**11 + b**12*x**12))