Integrand size = 15, antiderivative size = 198 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {d^7}{6 b^8 (a+b x)^6} \] Output:
-1/13*(-a*d+b*c)^7/b^8/(b*x+a)^13-7/12*d*(-a*d+b*c)^6/b^8/(b*x+a)^12-21/11 *d^2*(-a*d+b*c)^5/b^8/(b*x+a)^11-7/2*d^3*(-a*d+b*c)^4/b^8/(b*x+a)^10-35/9* d^4*(-a*d+b*c)^3/b^8/(b*x+a)^9-21/8*d^5*(-a*d+b*c)^2/b^8/(b*x+a)^8-d^6*(-a *d+b*c)/b^8/(b*x+a)^7-1/6*d^7/b^8/(b*x+a)^6
Time = 0.08 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.86 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {a^7 d^7+a^6 b d^6 (6 c+13 d x)+3 a^5 b^2 d^5 \left (7 c^2+26 c d x+26 d^2 x^2\right )+a^4 b^3 d^4 \left (56 c^3+273 c^2 d x+468 c d^2 x^2+286 d^3 x^3\right )+a^3 b^4 d^3 \left (126 c^4+728 c^3 d x+1638 c^2 d^2 x^2+1716 c d^3 x^3+715 d^4 x^4\right )+3 a^2 b^5 d^2 \left (84 c^5+546 c^4 d x+1456 c^3 d^2 x^2+2002 c^2 d^3 x^3+1430 c d^4 x^4+429 d^5 x^5\right )+a b^6 d \left (462 c^6+3276 c^5 d x+9828 c^4 d^2 x^2+16016 c^3 d^3 x^3+15015 c^2 d^4 x^4+7722 c d^5 x^5+1716 d^6 x^6\right )+b^7 \left (792 c^7+6006 c^6 d x+19656 c^5 d^2 x^2+36036 c^4 d^3 x^3+40040 c^3 d^4 x^4+27027 c^2 d^5 x^5+10296 c d^6 x^6+1716 d^7 x^7\right )}{10296 b^8 (a+b x)^{13}} \] Input:
Integrate[(c + d*x)^7/(a + b*x)^14,x]
Output:
-1/10296*(a^7*d^7 + a^6*b*d^6*(6*c + 13*d*x) + 3*a^5*b^2*d^5*(7*c^2 + 26*c *d*x + 26*d^2*x^2) + a^4*b^3*d^4*(56*c^3 + 273*c^2*d*x + 468*c*d^2*x^2 + 2 86*d^3*x^3) + a^3*b^4*d^3*(126*c^4 + 728*c^3*d*x + 1638*c^2*d^2*x^2 + 1716 *c*d^3*x^3 + 715*d^4*x^4) + 3*a^2*b^5*d^2*(84*c^5 + 546*c^4*d*x + 1456*c^3 *d^2*x^2 + 2002*c^2*d^3*x^3 + 1430*c*d^4*x^4 + 429*d^5*x^5) + a*b^6*d*(462 *c^6 + 3276*c^5*d*x + 9828*c^4*d^2*x^2 + 16016*c^3*d^3*x^3 + 15015*c^2*d^4 *x^4 + 7722*c*d^5*x^5 + 1716*d^6*x^6) + b^7*(792*c^7 + 6006*c^6*d*x + 1965 6*c^5*d^2*x^2 + 36036*c^4*d^3*x^3 + 40040*c^3*d^4*x^4 + 27027*c^2*d^5*x^5 + 10296*c*d^6*x^6 + 1716*d^7*x^7))/(b^8*(a + b*x)^13)
Time = 0.36 (sec) , antiderivative size = 198, 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 = {53, 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.
| \(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (\frac {7 d^6 (b c-a d)}{b^7 (a+b x)^8}+\frac {21 d^5 (b c-a d)^2}{b^7 (a+b x)^9}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)^{10}}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^{11}}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^{12}}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^{13}}+\frac {(b c-a d)^7}{b^7 (a+b x)^{14}}+\frac {d^7}{b^7 (a+b x)^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {d^6 (b c-a d)}{b^8 (a+b x)^7}-\frac {21 d^5 (b c-a d)^2}{8 b^8 (a+b x)^8}-\frac {35 d^4 (b c-a d)^3}{9 b^8 (a+b x)^9}-\frac {7 d^3 (b c-a d)^4}{2 b^8 (a+b x)^{10}}-\frac {21 d^2 (b c-a d)^5}{11 b^8 (a+b x)^{11}}-\frac {7 d (b c-a d)^6}{12 b^8 (a+b x)^{12}}-\frac {(b c-a d)^7}{13 b^8 (a+b x)^{13}}-\frac {d^7}{6 b^8 (a+b x)^6}\) |
Input:
Int[(c + d*x)^7/(a + b*x)^14,x]
Output:
-1/13*(b*c - a*d)^7/(b^8*(a + b*x)^13) - (7*d*(b*c - a*d)^6)/(12*b^8*(a + b*x)^12) - (21*d^2*(b*c - a*d)^5)/(11*b^8*(a + b*x)^11) - (7*d^3*(b*c - a* d)^4)/(2*b^8*(a + b*x)^10) - (35*d^4*(b*c - a*d)^3)/(9*b^8*(a + b*x)^9) - (21*d^5*(b*c - a*d)^2)/(8*b^8*(a + b*x)^8) - (d^6*(b*c - a*d))/(b^8*(a + b *x)^7) - d^7/(6*b^8*(a + b*x)^6)
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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(437\) vs. \(2(184)=368\).
Time = 0.18 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.21
| method | result | size |
| risch | \(\frac {-\frac {d^{7} x^{7}}{6 b}-\frac {d^{6} \left (a d +6 b c \right ) x^{6}}{6 b^{2}}-\frac {d^{5} \left (a^{2} d^{2}+6 a b c d +21 b^{2} c^{2}\right ) x^{5}}{8 b^{3}}-\frac {5 d^{4} \left (a^{3} d^{3}+6 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d +56 b^{3} c^{3}\right ) x^{4}}{72 b^{4}}-\frac {d^{3} \left (a^{4} d^{4}+6 a^{3} b c \,d^{3}+21 a^{2} b^{2} c^{2} d^{2}+56 a \,b^{3} c^{3} d +126 c^{4} b^{4}\right ) x^{3}}{36 b^{5}}-\frac {d^{2} \left (a^{5} d^{5}+6 a^{4} b c \,d^{4}+21 a^{3} b^{2} c^{2} d^{3}+56 a^{2} b^{3} c^{3} d^{2}+126 a \,b^{4} c^{4} d +252 c^{5} b^{5}\right ) x^{2}}{132 b^{6}}-\frac {d \left (a^{6} d^{6}+6 a^{5} b c \,d^{5}+21 a^{4} b^{2} c^{2} d^{4}+56 a^{3} b^{3} c^{3} d^{3}+126 a^{2} b^{4} c^{4} d^{2}+252 a \,b^{5} c^{5} d +462 c^{6} b^{6}\right ) x}{792 b^{7}}-\frac {a^{7} d^{7}+6 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+56 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}+252 a^{2} b^{5} c^{5} d^{2}+462 a \,b^{6} c^{6} d +792 b^{7} c^{7}}{10296 b^{8}}}{\left (b x +a \right )^{13}}\) | \(438\) |
| default | \(\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{9 b^{8} \left (b x +a \right )^{9}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{13 b^{8} \left (b x +a \right )^{13}}+\frac {d^{6} \left (a d -b c \right )}{b^{8} \left (b x +a \right )^{7}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{12 b^{8} \left (b x +a \right )^{12}}-\frac {21 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{8 b^{8} \left (b x +a \right )^{8}}-\frac {7 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{2 b^{8} \left (b x +a \right )^{10}}-\frac {d^{7}}{6 b^{8} \left (b x +a \right )^{6}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{11 b^{8} \left (b x +a \right )^{11}}\) | \(463\) |
| norman | \(\frac {-\frac {d^{7} x^{7}}{6 b}+\frac {\left (-a \,b^{5} d^{7}-6 b^{6} c \,d^{6}\right ) x^{6}}{6 b^{7}}+\frac {\left (-a^{2} b^{5} d^{7}-6 a \,b^{6} c \,d^{6}-21 b^{7} c^{2} d^{5}\right ) x^{5}}{8 b^{8}}+\frac {5 \left (-a^{3} b^{5} d^{7}-6 a^{2} b^{6} c \,d^{6}-21 a \,b^{7} c^{2} d^{5}-56 b^{8} c^{3} d^{4}\right ) x^{4}}{72 b^{9}}+\frac {\left (-a^{4} b^{5} d^{7}-6 a^{3} b^{6} c \,d^{6}-21 a^{2} b^{7} c^{2} d^{5}-56 a \,b^{8} c^{3} d^{4}-126 b^{9} c^{4} d^{3}\right ) x^{3}}{36 b^{10}}+\frac {\left (-a^{5} b^{5} d^{7}-6 a^{4} b^{6} c \,d^{6}-21 a^{3} b^{7} c^{2} d^{5}-56 a^{2} b^{8} c^{3} d^{4}-126 a \,b^{9} c^{4} d^{3}-252 b^{10} c^{5} d^{2}\right ) x^{2}}{132 b^{11}}+\frac {\left (-a^{6} b^{5} d^{7}-6 a^{5} b^{6} c \,d^{6}-21 a^{4} b^{7} c^{2} d^{5}-56 a^{3} b^{8} c^{3} d^{4}-126 a^{2} b^{9} c^{4} d^{3}-252 a \,b^{10} c^{5} d^{2}-462 b^{11} c^{6} d \right ) x}{792 b^{12}}+\frac {-a^{7} b^{5} d^{7}-6 a^{6} b^{6} c \,d^{6}-21 a^{5} b^{7} c^{2} d^{5}-56 a^{4} b^{8} c^{3} d^{4}-126 a^{3} b^{9} c^{4} d^{3}-252 a^{2} b^{10} c^{5} d^{2}-462 a \,c^{6} d \,b^{11}-792 b^{12} c^{7}}{10296 b^{13}}}{\left (b x +a \right )^{13}}\) | \(492\) |
| gosper | \(-\frac {1716 x^{7} d^{7} b^{7}+1716 x^{6} a \,b^{6} d^{7}+10296 x^{6} b^{7} c \,d^{6}+1287 x^{5} a^{2} b^{5} d^{7}+7722 x^{5} a \,b^{6} c \,d^{6}+27027 x^{5} b^{7} c^{2} d^{5}+715 x^{4} a^{3} b^{4} d^{7}+4290 x^{4} a^{2} b^{5} c \,d^{6}+15015 x^{4} a \,b^{6} c^{2} d^{5}+40040 x^{4} b^{7} c^{3} d^{4}+286 x^{3} a^{4} b^{3} d^{7}+1716 x^{3} a^{3} b^{4} c \,d^{6}+6006 x^{3} a^{2} b^{5} c^{2} d^{5}+16016 x^{3} a \,b^{6} c^{3} d^{4}+36036 x^{3} b^{7} c^{4} d^{3}+78 x^{2} a^{5} b^{2} d^{7}+468 x^{2} a^{4} b^{3} c \,d^{6}+1638 x^{2} a^{3} b^{4} c^{2} d^{5}+4368 x^{2} a^{2} b^{5} c^{3} d^{4}+9828 x^{2} a \,b^{6} c^{4} d^{3}+19656 x^{2} b^{7} c^{5} d^{2}+13 x \,a^{6} b \,d^{7}+78 x \,a^{5} b^{2} c \,d^{6}+273 x \,a^{4} b^{3} c^{2} d^{5}+728 x \,a^{3} b^{4} c^{3} d^{4}+1638 x \,a^{2} b^{5} c^{4} d^{3}+3276 x a \,b^{6} c^{5} d^{2}+6006 x \,b^{7} c^{6} d +a^{7} d^{7}+6 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+56 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}+252 a^{2} b^{5} c^{5} d^{2}+462 a \,b^{6} c^{6} d +792 b^{7} c^{7}}{10296 b^{8} \left (b x +a \right )^{13}}\) | \(497\) |
| orering | \(-\frac {1716 x^{7} d^{7} b^{7}+1716 x^{6} a \,b^{6} d^{7}+10296 x^{6} b^{7} c \,d^{6}+1287 x^{5} a^{2} b^{5} d^{7}+7722 x^{5} a \,b^{6} c \,d^{6}+27027 x^{5} b^{7} c^{2} d^{5}+715 x^{4} a^{3} b^{4} d^{7}+4290 x^{4} a^{2} b^{5} c \,d^{6}+15015 x^{4} a \,b^{6} c^{2} d^{5}+40040 x^{4} b^{7} c^{3} d^{4}+286 x^{3} a^{4} b^{3} d^{7}+1716 x^{3} a^{3} b^{4} c \,d^{6}+6006 x^{3} a^{2} b^{5} c^{2} d^{5}+16016 x^{3} a \,b^{6} c^{3} d^{4}+36036 x^{3} b^{7} c^{4} d^{3}+78 x^{2} a^{5} b^{2} d^{7}+468 x^{2} a^{4} b^{3} c \,d^{6}+1638 x^{2} a^{3} b^{4} c^{2} d^{5}+4368 x^{2} a^{2} b^{5} c^{3} d^{4}+9828 x^{2} a \,b^{6} c^{4} d^{3}+19656 x^{2} b^{7} c^{5} d^{2}+13 x \,a^{6} b \,d^{7}+78 x \,a^{5} b^{2} c \,d^{6}+273 x \,a^{4} b^{3} c^{2} d^{5}+728 x \,a^{3} b^{4} c^{3} d^{4}+1638 x \,a^{2} b^{5} c^{4} d^{3}+3276 x a \,b^{6} c^{5} d^{2}+6006 x \,b^{7} c^{6} d +a^{7} d^{7}+6 a^{6} b c \,d^{6}+21 a^{5} b^{2} c^{2} d^{5}+56 a^{4} b^{3} c^{3} d^{4}+126 a^{3} b^{4} c^{4} d^{3}+252 a^{2} b^{5} c^{5} d^{2}+462 a \,b^{6} c^{6} d +792 b^{7} c^{7}}{10296 b^{8} \left (b x +a \right )^{13}}\) | \(497\) |
| parallelrisch | \(\frac {-1716 d^{7} x^{7} b^{12}-1716 a \,b^{11} d^{7} x^{6}-10296 b^{12} c \,d^{6} x^{6}-1287 a^{2} b^{10} d^{7} x^{5}-7722 a \,b^{11} c \,d^{6} x^{5}-27027 b^{12} c^{2} d^{5} x^{5}-715 a^{3} b^{9} d^{7} x^{4}-4290 a^{2} b^{10} c \,d^{6} x^{4}-15015 a \,b^{11} c^{2} d^{5} x^{4}-40040 b^{12} c^{3} d^{4} x^{4}-286 a^{4} b^{8} d^{7} x^{3}-1716 a^{3} b^{9} c \,d^{6} x^{3}-6006 a^{2} b^{10} c^{2} d^{5} x^{3}-16016 a \,b^{11} c^{3} d^{4} x^{3}-36036 b^{12} c^{4} d^{3} x^{3}-78 a^{5} b^{7} d^{7} x^{2}-468 a^{4} b^{8} c \,d^{6} x^{2}-1638 a^{3} b^{9} c^{2} d^{5} x^{2}-4368 a^{2} b^{10} c^{3} d^{4} x^{2}-9828 a \,b^{11} c^{4} d^{3} x^{2}-19656 b^{12} c^{5} d^{2} x^{2}-13 a^{6} b^{6} d^{7} x -78 a^{5} b^{7} c \,d^{6} x -273 a^{4} b^{8} c^{2} d^{5} x -728 a^{3} b^{9} c^{3} d^{4} x -1638 a^{2} b^{10} c^{4} d^{3} x -3276 a \,b^{11} c^{5} d^{2} x -6006 b^{12} c^{6} d x -a^{7} b^{5} d^{7}-6 a^{6} b^{6} c \,d^{6}-21 a^{5} b^{7} c^{2} d^{5}-56 a^{4} b^{8} c^{3} d^{4}-126 a^{3} b^{9} c^{4} d^{3}-252 a^{2} b^{10} c^{5} d^{2}-462 a \,c^{6} d \,b^{11}-792 b^{12} c^{7}}{10296 b^{13} \left (b x +a \right )^{13}}\) | \(505\) |
Input:
int((d*x+c)^7/(b*x+a)^14,x,method=_RETURNVERBOSE)
Output:
(-1/6/b*d^7*x^7-1/6/b^2*d^6*(a*d+6*b*c)*x^6-1/8/b^3*d^5*(a^2*d^2+6*a*b*c*d +21*b^2*c^2)*x^5-5/72/b^4*d^4*(a^3*d^3+6*a^2*b*c*d^2+21*a*b^2*c^2*d+56*b^3 *c^3)*x^4-1/36/b^5*d^3*(a^4*d^4+6*a^3*b*c*d^3+21*a^2*b^2*c^2*d^2+56*a*b^3* c^3*d+126*b^4*c^4)*x^3-1/132/b^6*d^2*(a^5*d^5+6*a^4*b*c*d^4+21*a^3*b^2*c^2 *d^3+56*a^2*b^3*c^3*d^2+126*a*b^4*c^4*d+252*b^5*c^5)*x^2-1/792/b^7*d*(a^6* d^6+6*a^5*b*c*d^5+21*a^4*b^2*c^2*d^4+56*a^3*b^3*c^3*d^3+126*a^2*b^4*c^4*d^ 2+252*a*b^5*c^5*d+462*b^6*c^6)*x-1/10296/b^8*(a^7*d^7+6*a^6*b*c*d^6+21*a^5 *b^2*c^2*d^5+56*a^4*b^3*c^3*d^4+126*a^3*b^4*c^4*d^3+252*a^2*b^5*c^5*d^2+46 2*a*b^6*c^6*d+792*b^7*c^7))/(b*x+a)^13
Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (184) = 368\).
Time = 0.11 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.99 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \, {\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \, {\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \, {\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \, {\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \, {\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \, {\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \, {\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \] Input:
integrate((d*x+c)^7/(b*x+a)^14,x, algorithm="fricas")
Output:
-1/10296*(1716*b^7*d^7*x^7 + 792*b^7*c^7 + 462*a*b^6*c^6*d + 252*a^2*b^5*c ^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 6 *a^6*b*c*d^6 + a^7*d^7 + 1716*(6*b^7*c*d^6 + a*b^6*d^7)*x^6 + 1287*(21*b^7 *c^2*d^5 + 6*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 715*(56*b^7*c^3*d^4 + 21*a*b ^6*c^2*d^5 + 6*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 286*(126*b^7*c^4*d^3 + 5 6*a*b^6*c^3*d^4 + 21*a^2*b^5*c^2*d^5 + 6*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 78*(252*b^7*c^5*d^2 + 126*a*b^6*c^4*d^3 + 56*a^2*b^5*c^3*d^4 + 21*a^3*b^ 4*c^2*d^5 + 6*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 13*(462*b^7*c^6*d + 252*a *b^6*c^5*d^2 + 126*a^2*b^5*c^4*d^3 + 56*a^3*b^4*c^3*d^4 + 21*a^4*b^3*c^2*d ^5 + 6*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^21*x^13 + 13*a*b^20*x^12 + 78*a^2* b^19*x^11 + 286*a^3*b^18*x^10 + 715*a^4*b^17*x^9 + 1287*a^5*b^16*x^8 + 171 6*a^6*b^15*x^7 + 1716*a^7*b^14*x^6 + 1287*a^8*b^13*x^5 + 715*a^9*b^12*x^4 + 286*a^10*b^11*x^3 + 78*a^11*b^10*x^2 + 13*a^12*b^9*x + a^13*b^8)
Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=\text {Timed out} \] Input:
integrate((d*x+c)**7/(b*x+a)**14,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (184) = 368\).
Time = 0.07 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.99 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {1716 \, b^{7} d^{7} x^{7} + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7} + 1716 \, {\left (6 \, b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 1287 \, {\left (21 \, b^{7} c^{2} d^{5} + 6 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 715 \, {\left (56 \, b^{7} c^{3} d^{4} + 21 \, a b^{6} c^{2} d^{5} + 6 \, a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 286 \, {\left (126 \, b^{7} c^{4} d^{3} + 56 \, a b^{6} c^{3} d^{4} + 21 \, a^{2} b^{5} c^{2} d^{5} + 6 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 78 \, {\left (252 \, b^{7} c^{5} d^{2} + 126 \, a b^{6} c^{4} d^{3} + 56 \, a^{2} b^{5} c^{3} d^{4} + 21 \, a^{3} b^{4} c^{2} d^{5} + 6 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 13 \, {\left (462 \, b^{7} c^{6} d + 252 \, a b^{6} c^{5} d^{2} + 126 \, a^{2} b^{5} c^{4} d^{3} + 56 \, a^{3} b^{4} c^{3} d^{4} + 21 \, a^{4} b^{3} c^{2} d^{5} + 6 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{10296 \, {\left (b^{21} x^{13} + 13 \, a b^{20} x^{12} + 78 \, a^{2} b^{19} x^{11} + 286 \, a^{3} b^{18} x^{10} + 715 \, a^{4} b^{17} x^{9} + 1287 \, a^{5} b^{16} x^{8} + 1716 \, a^{6} b^{15} x^{7} + 1716 \, a^{7} b^{14} x^{6} + 1287 \, a^{8} b^{13} x^{5} + 715 \, a^{9} b^{12} x^{4} + 286 \, a^{10} b^{11} x^{3} + 78 \, a^{11} b^{10} x^{2} + 13 \, a^{12} b^{9} x + a^{13} b^{8}\right )}} \] Input:
integrate((d*x+c)^7/(b*x+a)^14,x, algorithm="maxima")
Output:
-1/10296*(1716*b^7*d^7*x^7 + 792*b^7*c^7 + 462*a*b^6*c^6*d + 252*a^2*b^5*c ^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 6 *a^6*b*c*d^6 + a^7*d^7 + 1716*(6*b^7*c*d^6 + a*b^6*d^7)*x^6 + 1287*(21*b^7 *c^2*d^5 + 6*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 715*(56*b^7*c^3*d^4 + 21*a*b ^6*c^2*d^5 + 6*a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 286*(126*b^7*c^4*d^3 + 5 6*a*b^6*c^3*d^4 + 21*a^2*b^5*c^2*d^5 + 6*a^3*b^4*c*d^6 + a^4*b^3*d^7)*x^3 + 78*(252*b^7*c^5*d^2 + 126*a*b^6*c^4*d^3 + 56*a^2*b^5*c^3*d^4 + 21*a^3*b^ 4*c^2*d^5 + 6*a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 13*(462*b^7*c^6*d + 252*a *b^6*c^5*d^2 + 126*a^2*b^5*c^4*d^3 + 56*a^3*b^4*c^3*d^4 + 21*a^4*b^3*c^2*d ^5 + 6*a^5*b^2*c*d^6 + a^6*b*d^7)*x)/(b^21*x^13 + 13*a*b^20*x^12 + 78*a^2* b^19*x^11 + 286*a^3*b^18*x^10 + 715*a^4*b^17*x^9 + 1287*a^5*b^16*x^8 + 171 6*a^6*b^15*x^7 + 1716*a^7*b^14*x^6 + 1287*a^8*b^13*x^5 + 715*a^9*b^12*x^4 + 286*a^10*b^11*x^3 + 78*a^11*b^10*x^2 + 13*a^12*b^9*x + a^13*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (184) = 368\).
Time = 0.13 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.51 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {1716 \, b^{7} d^{7} x^{7} + 10296 \, b^{7} c d^{6} x^{6} + 1716 \, a b^{6} d^{7} x^{6} + 27027 \, b^{7} c^{2} d^{5} x^{5} + 7722 \, a b^{6} c d^{6} x^{5} + 1287 \, a^{2} b^{5} d^{7} x^{5} + 40040 \, b^{7} c^{3} d^{4} x^{4} + 15015 \, a b^{6} c^{2} d^{5} x^{4} + 4290 \, a^{2} b^{5} c d^{6} x^{4} + 715 \, a^{3} b^{4} d^{7} x^{4} + 36036 \, b^{7} c^{4} d^{3} x^{3} + 16016 \, a b^{6} c^{3} d^{4} x^{3} + 6006 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 1716 \, a^{3} b^{4} c d^{6} x^{3} + 286 \, a^{4} b^{3} d^{7} x^{3} + 19656 \, b^{7} c^{5} d^{2} x^{2} + 9828 \, a b^{6} c^{4} d^{3} x^{2} + 4368 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 1638 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 468 \, a^{4} b^{3} c d^{6} x^{2} + 78 \, a^{5} b^{2} d^{7} x^{2} + 6006 \, b^{7} c^{6} d x + 3276 \, a b^{6} c^{5} d^{2} x + 1638 \, a^{2} b^{5} c^{4} d^{3} x + 728 \, a^{3} b^{4} c^{3} d^{4} x + 273 \, a^{4} b^{3} c^{2} d^{5} x + 78 \, a^{5} b^{2} c d^{6} x + 13 \, a^{6} b d^{7} x + 792 \, b^{7} c^{7} + 462 \, a b^{6} c^{6} d + 252 \, a^{2} b^{5} c^{5} d^{2} + 126 \, a^{3} b^{4} c^{4} d^{3} + 56 \, a^{4} b^{3} c^{3} d^{4} + 21 \, a^{5} b^{2} c^{2} d^{5} + 6 \, a^{6} b c d^{6} + a^{7} d^{7}}{10296 \, {\left (b x + a\right )}^{13} b^{8}} \] Input:
integrate((d*x+c)^7/(b*x+a)^14,x, algorithm="giac")
Output:
-1/10296*(1716*b^7*d^7*x^7 + 10296*b^7*c*d^6*x^6 + 1716*a*b^6*d^7*x^6 + 27 027*b^7*c^2*d^5*x^5 + 7722*a*b^6*c*d^6*x^5 + 1287*a^2*b^5*d^7*x^5 + 40040* b^7*c^3*d^4*x^4 + 15015*a*b^6*c^2*d^5*x^4 + 4290*a^2*b^5*c*d^6*x^4 + 715*a ^3*b^4*d^7*x^4 + 36036*b^7*c^4*d^3*x^3 + 16016*a*b^6*c^3*d^4*x^3 + 6006*a^ 2*b^5*c^2*d^5*x^3 + 1716*a^3*b^4*c*d^6*x^3 + 286*a^4*b^3*d^7*x^3 + 19656*b ^7*c^5*d^2*x^2 + 9828*a*b^6*c^4*d^3*x^2 + 4368*a^2*b^5*c^3*d^4*x^2 + 1638* a^3*b^4*c^2*d^5*x^2 + 468*a^4*b^3*c*d^6*x^2 + 78*a^5*b^2*d^7*x^2 + 6006*b^ 7*c^6*d*x + 3276*a*b^6*c^5*d^2*x + 1638*a^2*b^5*c^4*d^3*x + 728*a^3*b^4*c^ 3*d^4*x + 273*a^4*b^3*c^2*d^5*x + 78*a^5*b^2*c*d^6*x + 13*a^6*b*d^7*x + 79 2*b^7*c^7 + 462*a*b^6*c^6*d + 252*a^2*b^5*c^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56*a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 6*a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^13*b^8)
Time = 0.15 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.88 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=-\frac {\frac {a^7\,d^7+6\,a^6\,b\,c\,d^6+21\,a^5\,b^2\,c^2\,d^5+56\,a^4\,b^3\,c^3\,d^4+126\,a^3\,b^4\,c^4\,d^3+252\,a^2\,b^5\,c^5\,d^2+462\,a\,b^6\,c^6\,d+792\,b^7\,c^7}{10296\,b^8}+\frac {d^7\,x^7}{6\,b}+\frac {d^2\,x^2\,\left (a^5\,d^5+6\,a^4\,b\,c\,d^4+21\,a^3\,b^2\,c^2\,d^3+56\,a^2\,b^3\,c^3\,d^2+126\,a\,b^4\,c^4\,d+252\,b^5\,c^5\right )}{132\,b^6}+\frac {5\,d^4\,x^4\,\left (a^3\,d^3+6\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d+56\,b^3\,c^3\right )}{72\,b^4}+\frac {d^6\,x^6\,\left (a\,d+6\,b\,c\right )}{6\,b^2}+\frac {d^3\,x^3\,\left (a^4\,d^4+6\,a^3\,b\,c\,d^3+21\,a^2\,b^2\,c^2\,d^2+56\,a\,b^3\,c^3\,d+126\,b^4\,c^4\right )}{36\,b^5}+\frac {d\,x\,\left (a^6\,d^6+6\,a^5\,b\,c\,d^5+21\,a^4\,b^2\,c^2\,d^4+56\,a^3\,b^3\,c^3\,d^3+126\,a^2\,b^4\,c^4\,d^2+252\,a\,b^5\,c^5\,d+462\,b^6\,c^6\right )}{792\,b^7}+\frac {d^5\,x^5\,\left (a^2\,d^2+6\,a\,b\,c\,d+21\,b^2\,c^2\right )}{8\,b^3}}{a^{13}+13\,a^{12}\,b\,x+78\,a^{11}\,b^2\,x^2+286\,a^{10}\,b^3\,x^3+715\,a^9\,b^4\,x^4+1287\,a^8\,b^5\,x^5+1716\,a^7\,b^6\,x^6+1716\,a^6\,b^7\,x^7+1287\,a^5\,b^8\,x^8+715\,a^4\,b^9\,x^9+286\,a^3\,b^{10}\,x^{10}+78\,a^2\,b^{11}\,x^{11}+13\,a\,b^{12}\,x^{12}+b^{13}\,x^{13}} \] Input:
int((c + d*x)^7/(a + b*x)^14,x)
Output:
-((a^7*d^7 + 792*b^7*c^7 + 252*a^2*b^5*c^5*d^2 + 126*a^3*b^4*c^4*d^3 + 56* a^4*b^3*c^3*d^4 + 21*a^5*b^2*c^2*d^5 + 462*a*b^6*c^6*d + 6*a^6*b*c*d^6)/(1 0296*b^8) + (d^7*x^7)/(6*b) + (d^2*x^2*(a^5*d^5 + 252*b^5*c^5 + 56*a^2*b^3 *c^3*d^2 + 21*a^3*b^2*c^2*d^3 + 126*a*b^4*c^4*d + 6*a^4*b*c*d^4))/(132*b^6 ) + (5*d^4*x^4*(a^3*d^3 + 56*b^3*c^3 + 21*a*b^2*c^2*d + 6*a^2*b*c*d^2))/(7 2*b^4) + (d^6*x^6*(a*d + 6*b*c))/(6*b^2) + (d^3*x^3*(a^4*d^4 + 126*b^4*c^4 + 21*a^2*b^2*c^2*d^2 + 56*a*b^3*c^3*d + 6*a^3*b*c*d^3))/(36*b^5) + (d*x*( a^6*d^6 + 462*b^6*c^6 + 126*a^2*b^4*c^4*d^2 + 56*a^3*b^3*c^3*d^3 + 21*a^4* b^2*c^2*d^4 + 252*a*b^5*c^5*d + 6*a^5*b*c*d^5))/(792*b^7) + (d^5*x^5*(a^2* d^2 + 21*b^2*c^2 + 6*a*b*c*d))/(8*b^3))/(a^13 + b^13*x^13 + 13*a*b^12*x^12 + 78*a^11*b^2*x^2 + 286*a^10*b^3*x^3 + 715*a^9*b^4*x^4 + 1287*a^8*b^5*x^5 + 1716*a^7*b^6*x^6 + 1716*a^6*b^7*x^7 + 1287*a^5*b^8*x^8 + 715*a^4*b^9*x^ 9 + 286*a^3*b^10*x^10 + 78*a^2*b^11*x^11 + 13*a^12*b*x)
Time = 0.17 (sec) , antiderivative size = 629, normalized size of antiderivative = 3.18 \[ \int \frac {(c+d x)^7}{(a+b x)^{14}} \, dx=\frac {-1716 b^{7} d^{7} x^{7}-1716 a \,b^{6} d^{7} x^{6}-10296 b^{7} c \,d^{6} x^{6}-1287 a^{2} b^{5} d^{7} x^{5}-7722 a \,b^{6} c \,d^{6} x^{5}-27027 b^{7} c^{2} d^{5} x^{5}-715 a^{3} b^{4} d^{7} x^{4}-4290 a^{2} b^{5} c \,d^{6} x^{4}-15015 a \,b^{6} c^{2} d^{5} x^{4}-40040 b^{7} c^{3} d^{4} x^{4}-286 a^{4} b^{3} d^{7} x^{3}-1716 a^{3} b^{4} c \,d^{6} x^{3}-6006 a^{2} b^{5} c^{2} d^{5} x^{3}-16016 a \,b^{6} c^{3} d^{4} x^{3}-36036 b^{7} c^{4} d^{3} x^{3}-78 a^{5} b^{2} d^{7} x^{2}-468 a^{4} b^{3} c \,d^{6} x^{2}-1638 a^{3} b^{4} c^{2} d^{5} x^{2}-4368 a^{2} b^{5} c^{3} d^{4} x^{2}-9828 a \,b^{6} c^{4} d^{3} x^{2}-19656 b^{7} c^{5} d^{2} x^{2}-13 a^{6} b \,d^{7} x -78 a^{5} b^{2} c \,d^{6} x -273 a^{4} b^{3} c^{2} d^{5} x -728 a^{3} b^{4} c^{3} d^{4} x -1638 a^{2} b^{5} c^{4} d^{3} x -3276 a \,b^{6} c^{5} d^{2} x -6006 b^{7} c^{6} d x -a^{7} d^{7}-6 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}-56 a^{4} b^{3} c^{3} d^{4}-126 a^{3} b^{4} c^{4} d^{3}-252 a^{2} b^{5} c^{5} d^{2}-462 a \,b^{6} c^{6} d -792 b^{7} c^{7}}{10296 b^{8} \left (b^{13} x^{13}+13 a \,b^{12} x^{12}+78 a^{2} b^{11} x^{11}+286 a^{3} b^{10} x^{10}+715 a^{4} b^{9} x^{9}+1287 a^{5} b^{8} x^{8}+1716 a^{6} b^{7} x^{7}+1716 a^{7} b^{6} x^{6}+1287 a^{8} b^{5} x^{5}+715 a^{9} b^{4} x^{4}+286 a^{10} b^{3} x^{3}+78 a^{11} b^{2} x^{2}+13 a^{12} b x +a^{13}\right )} \] Input:
int((d*x+c)^7/(b*x+a)^14,x)
Output:
( - a**7*d**7 - 6*a**6*b*c*d**6 - 13*a**6*b*d**7*x - 21*a**5*b**2*c**2*d** 5 - 78*a**5*b**2*c*d**6*x - 78*a**5*b**2*d**7*x**2 - 56*a**4*b**3*c**3*d** 4 - 273*a**4*b**3*c**2*d**5*x - 468*a**4*b**3*c*d**6*x**2 - 286*a**4*b**3* d**7*x**3 - 126*a**3*b**4*c**4*d**3 - 728*a**3*b**4*c**3*d**4*x - 1638*a** 3*b**4*c**2*d**5*x**2 - 1716*a**3*b**4*c*d**6*x**3 - 715*a**3*b**4*d**7*x* *4 - 252*a**2*b**5*c**5*d**2 - 1638*a**2*b**5*c**4*d**3*x - 4368*a**2*b**5 *c**3*d**4*x**2 - 6006*a**2*b**5*c**2*d**5*x**3 - 4290*a**2*b**5*c*d**6*x* *4 - 1287*a**2*b**5*d**7*x**5 - 462*a*b**6*c**6*d - 3276*a*b**6*c**5*d**2* x - 9828*a*b**6*c**4*d**3*x**2 - 16016*a*b**6*c**3*d**4*x**3 - 15015*a*b** 6*c**2*d**5*x**4 - 7722*a*b**6*c*d**6*x**5 - 1716*a*b**6*d**7*x**6 - 792*b **7*c**7 - 6006*b**7*c**6*d*x - 19656*b**7*c**5*d**2*x**2 - 36036*b**7*c** 4*d**3*x**3 - 40040*b**7*c**3*d**4*x**4 - 27027*b**7*c**2*d**5*x**5 - 1029 6*b**7*c*d**6*x**6 - 1716*b**7*d**7*x**7)/(10296*b**8*(a**13 + 13*a**12*b* x + 78*a**11*b**2*x**2 + 286*a**10*b**3*x**3 + 715*a**9*b**4*x**4 + 1287*a **8*b**5*x**5 + 1716*a**7*b**6*x**6 + 1716*a**6*b**7*x**7 + 1287*a**5*b**8 *x**8 + 715*a**4*b**9*x**9 + 286*a**3*b**10*x**10 + 78*a**2*b**11*x**11 + 13*a*b**12*x**12 + b**13*x**13))