Integrand size = 15, antiderivative size = 187 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=\frac {21 d^5 (b c-a d)^2 x}{b^7}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}+\frac {7 d^6 (b c-a d) (a+b x)^2}{2 b^8}+\frac {d^7 (a+b x)^3}{3 b^8}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8} \] Output:
21*d^5*(-a*d+b*c)^2*x/b^7-1/4*(-a*d+b*c)^7/b^8/(b*x+a)^4-7/3*d*(-a*d+b*c)^ 6/b^8/(b*x+a)^3-21/2*d^2*(-a*d+b*c)^5/b^8/(b*x+a)^2-35*d^3*(-a*d+b*c)^4/b^ 8/(b*x+a)+7/2*d^6*(-a*d+b*c)*(b*x+a)^2/b^8+1/3*d^7*(b*x+a)^3/b^8+35*d^4*(- a*d+b*c)^3*ln(b*x+a)/b^8
Time = 0.06 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.93 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=\frac {12 b d^5 \left (21 b^2 c^2-35 a b c d+15 a^2 d^2\right ) x+6 b^2 d^6 (7 b c-5 a d) x^2+4 b^3 d^7 x^3-\frac {3 (b c-a d)^7}{(a+b x)^4}-\frac {28 d (b c-a d)^6}{(a+b x)^3}+\frac {126 d^2 (-b c+a d)^5}{(a+b x)^2}-\frac {420 d^3 (b c-a d)^4}{a+b x}+420 d^4 (b c-a d)^3 \log (a+b x)}{12 b^8} \] Input:
Integrate[(c + d*x)^7/(a + b*x)^5,x]
Output:
(12*b*d^5*(21*b^2*c^2 - 35*a*b*c*d + 15*a^2*d^2)*x + 6*b^2*d^6*(7*b*c - 5* a*d)*x^2 + 4*b^3*d^7*x^3 - (3*(b*c - a*d)^7)/(a + b*x)^4 - (28*d*(b*c - a* d)^6)/(a + b*x)^3 + (126*d^2*(-(b*c) + a*d)^5)/(a + b*x)^2 - (420*d^3*(b*c - a*d)^4)/(a + b*x) + 420*d^4*(b*c - a*d)^3*Log[a + b*x])/(12*b^8)
Time = 0.37 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {49, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^5} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {7 d^6 (a+b x) (b c-a d)}{b^7}+\frac {21 d^5 (b c-a d)^2}{b^7}+\frac {35 d^4 (b c-a d)^3}{b^7 (a+b x)}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)^2}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^3}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^4}+\frac {(b c-a d)^7}{b^7 (a+b x)^5}+\frac {d^7 (a+b x)^2}{b^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 d^6 (a+b x)^2 (b c-a d)}{2 b^8}+\frac {35 d^4 (b c-a d)^3 \log (a+b x)}{b^8}-\frac {35 d^3 (b c-a d)^4}{b^8 (a+b x)}-\frac {21 d^2 (b c-a d)^5}{2 b^8 (a+b x)^2}-\frac {7 d (b c-a d)^6}{3 b^8 (a+b x)^3}-\frac {(b c-a d)^7}{4 b^8 (a+b x)^4}+\frac {d^7 (a+b x)^3}{3 b^8}+\frac {21 d^5 x (b c-a d)^2}{b^7}\) |
Input:
Int[(c + d*x)^7/(a + b*x)^5,x]
Output:
(21*d^5*(b*c - a*d)^2*x)/b^7 - (b*c - a*d)^7/(4*b^8*(a + b*x)^4) - (7*d*(b *c - a*d)^6)/(3*b^8*(a + b*x)^3) - (21*d^2*(b*c - a*d)^5)/(2*b^8*(a + b*x) ^2) - (35*d^3*(b*c - a*d)^4)/(b^8*(a + b*x)) + (7*d^6*(b*c - a*d)*(a + b*x )^2)/(2*b^8) + (d^7*(a + b*x)^3)/(3*b^8) + (35*d^4*(b*c - a*d)^3*Log[a + b *x])/b^8
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(451\) vs. \(2(177)=354\).
Time = 0.12 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.42
| method | result | size |
| norman | \(\frac {-\frac {875 a^{7} d^{7}-2625 a^{6} b c \,d^{6}+2625 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +3 b^{7} c^{7}}{12 b^{8}}+\frac {d^{7} x^{7}}{3 b}-\frac {\left (140 a^{4} d^{7}-420 a^{3} b c \,d^{6}+420 a^{2} b^{2} c^{2} d^{5}-140 a \,b^{3} c^{3} d^{4}+35 b^{4} c^{4} d^{3}\right ) x^{3}}{b^{5}}-\frac {3 \left (210 a^{5} d^{7}-630 a^{4} b c \,d^{6}+630 a^{3} b^{2} c^{2} d^{5}-210 a^{2} b^{3} c^{3} d^{4}+35 a \,b^{4} c^{4} d^{3}+7 b^{5} c^{5} d^{2}\right ) x^{2}}{2 b^{6}}-\frac {\left (770 a^{6} d^{7}-2310 a^{5} b c \,d^{6}+2310 a^{4} b^{2} c^{2} d^{5}-770 a^{3} b^{3} c^{3} d^{4}+105 a^{2} b^{4} c^{4} d^{3}+21 a \,b^{5} c^{5} d^{2}+7 b^{6} c^{6} d \right ) x}{3 b^{7}}+\frac {7 d^{5} \left (a^{2} d^{2}-3 a b c d +3 b^{2} c^{2}\right ) x^{5}}{b^{3}}-\frac {7 d^{6} \left (a d -3 b c \right ) x^{6}}{6 b^{2}}}{\left (b x +a \right )^{4}}-\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{8}}\) | \(452\) |
| default | \(\frac {d^{5} \left (\frac {1}{3} d^{2} x^{3} b^{2}-\frac {5}{2} x^{2} a b \,d^{2}+\frac {7}{2} x^{2} b^{2} c d +15 a^{2} d^{2} x -35 a b c d x +21 b^{2} c^{2} x \right )}{b^{7}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{4 b^{8} \left (b x +a \right )^{4}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{b^{8} \left (b x +a \right )}-\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (b x +a \right )}{b^{8}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{3 b^{8} \left (b x +a \right )^{3}}\) | \(453\) |
| risch | \(\frac {d^{7} x^{3}}{3 b^{5}}-\frac {5 d^{7} x^{2} a}{2 b^{6}}+\frac {7 d^{6} x^{2} c}{2 b^{5}}+\frac {15 d^{7} a^{2} x}{b^{7}}-\frac {35 d^{6} a c x}{b^{6}}+\frac {21 d^{5} c^{2} x}{b^{5}}+\frac {\left (-35 a^{4} b^{2} d^{7}+140 a^{3} b^{3} c \,d^{6}-210 a^{2} b^{4} c^{2} d^{5}+140 a \,b^{5} c^{3} d^{4}-35 b^{6} c^{4} d^{3}\right ) x^{3}-\frac {21 b \,d^{2} \left (9 a^{5} d^{5}-35 a^{4} b c \,d^{4}+50 a^{3} b^{2} c^{2} d^{3}-30 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +c^{5} b^{5}\right ) x^{2}}{2}-\frac {7 d \left (37 a^{6} d^{6}-141 a^{5} b c \,d^{5}+195 a^{4} b^{2} c^{2} d^{4}-110 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}+3 a \,b^{5} c^{5} d +c^{6} b^{6}\right ) x}{3}-\frac {319 a^{7} d^{7}-1197 a^{6} b c \,d^{6}+1617 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d +3 b^{7} c^{7}}{12 b}}{b^{7} \left (b x +a \right )^{4}}-\frac {35 d^{7} \ln \left (b x +a \right ) a^{3}}{b^{8}}+\frac {105 d^{6} \ln \left (b x +a \right ) a^{2} c}{b^{7}}-\frac {105 d^{5} \ln \left (b x +a \right ) a \,c^{2}}{b^{6}}+\frac {35 d^{4} \ln \left (b x +a \right ) c^{3}}{b^{5}}\) | \(472\) |
| parallelrisch | \(-\frac {-2625 a^{6} b c \,d^{6}+2625 a^{5} b^{2} c^{2} d^{5}-875 a^{4} b^{3} c^{3} d^{4}+252 x^{5} a \,b^{6} c \,d^{6}-1260 \ln \left (b x +a \right ) a^{6} b c \,d^{6}+1260 \ln \left (b x +a \right ) a^{5} b^{2} c^{2} d^{5}-420 \ln \left (b x +a \right ) a^{4} b^{3} c^{3} d^{4}-4 x^{7} d^{7} b^{7}+3080 x \,a^{6} b \,d^{7}+28 x \,b^{7} c^{6} d +3780 x^{2} a^{5} b^{2} d^{7}+126 x^{2} b^{7} c^{5} d^{2}+1680 x^{3} a^{4} b^{3} d^{7}+420 x^{3} b^{7} c^{4} d^{3}-84 x^{5} a^{2} b^{5} d^{7}-252 x^{5} b^{7} c^{2} d^{5}+14 x^{6} a \,b^{6} d^{7}-42 x^{6} b^{7} c \,d^{6}+3 b^{7} c^{7}+875 a^{7} d^{7}+1680 \ln \left (b x +a \right ) x^{3} a^{4} b^{3} d^{7}+2520 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} d^{7}+105 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}+7 a \,b^{6} c^{6} d -9240 x \,a^{5} b^{2} c \,d^{6}+9240 x \,a^{4} b^{3} c^{2} d^{5}-3080 x \,a^{3} b^{4} c^{3} d^{4}+420 x \,a^{2} b^{5} c^{4} d^{3}+84 x a \,b^{6} c^{5} d^{2}-11340 x^{2} a^{4} b^{3} c \,d^{6}+11340 x^{2} a^{3} b^{4} c^{2} d^{5}-3780 x^{2} a^{2} b^{5} c^{3} d^{4}+630 x^{2} a \,b^{6} c^{4} d^{3}-5040 x^{3} a^{3} b^{4} c \,d^{6}+5040 x^{3} a^{2} b^{5} c^{2} d^{5}-1680 x^{3} a \,b^{6} c^{3} d^{4}+5040 \ln \left (b x +a \right ) x \,a^{4} b^{3} c^{2} d^{5}-1680 \ln \left (b x +a \right ) x \,a^{3} b^{4} c^{3} d^{4}-2520 \ln \left (b x +a \right ) x^{2} a^{2} b^{5} c^{3} d^{4}-7560 \ln \left (b x +a \right ) x^{2} a^{4} b^{3} c \,d^{6}+7560 \ln \left (b x +a \right ) x^{2} a^{3} b^{4} c^{2} d^{5}+1680 \ln \left (b x +a \right ) x \,a^{6} b \,d^{7}-5040 \ln \left (b x +a \right ) x \,a^{5} b^{2} c \,d^{6}+420 \ln \left (b x +a \right ) a^{7} d^{7}+420 \ln \left (b x +a \right ) x^{4} a^{3} b^{4} d^{7}-420 \ln \left (b x +a \right ) x^{4} b^{7} c^{3} d^{4}-1260 \ln \left (b x +a \right ) x^{4} a^{2} b^{5} c \,d^{6}+1260 \ln \left (b x +a \right ) x^{4} a \,b^{6} c^{2} d^{5}-5040 \ln \left (b x +a \right ) x^{3} a^{3} b^{4} c \,d^{6}+5040 \ln \left (b x +a \right ) x^{3} a^{2} b^{5} c^{2} d^{5}-1680 \ln \left (b x +a \right ) x^{3} a \,b^{6} c^{3} d^{4}}{12 b^{8} \left (b x +a \right )^{4}}\) | \(841\) |
Input:
int((d*x+c)^7/(b*x+a)^5,x,method=_RETURNVERBOSE)
Output:
(-1/12*(875*a^7*d^7-2625*a^6*b*c*d^6+2625*a^5*b^2*c^2*d^5-875*a^4*b^3*c^3* d^4+105*a^3*b^4*c^4*d^3+21*a^2*b^5*c^5*d^2+7*a*b^6*c^6*d+3*b^7*c^7)/b^8+1/ 3/b*d^7*x^7-(140*a^4*d^7-420*a^3*b*c*d^6+420*a^2*b^2*c^2*d^5-140*a*b^3*c^3 *d^4+35*b^4*c^4*d^3)/b^5*x^3-3/2*(210*a^5*d^7-630*a^4*b*c*d^6+630*a^3*b^2* c^2*d^5-210*a^2*b^3*c^3*d^4+35*a*b^4*c^4*d^3+7*b^5*c^5*d^2)/b^6*x^2-1/3*(7 70*a^6*d^7-2310*a^5*b*c*d^6+2310*a^4*b^2*c^2*d^5-770*a^3*b^3*c^3*d^4+105*a ^2*b^4*c^4*d^3+21*a*b^5*c^5*d^2+7*b^6*c^6*d)/b^7*x+7*d^5*(a^2*d^2-3*a*b*c* d+3*b^2*c^2)/b^3*x^5-7/6*d^6*(a*d-3*b*c)/b^2*x^6)/(b*x+a)^4-35/b^8*d^4*(a^ 3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 754 vs. \(2 (177) = 354\).
Time = 0.08 (sec) , antiderivative size = 754, normalized size of antiderivative = 4.03 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=\frac {4 \, b^{7} d^{7} x^{7} - 3 \, b^{7} c^{7} - 7 \, a b^{6} c^{6} d - 21 \, a^{2} b^{5} c^{5} d^{2} - 105 \, a^{3} b^{4} c^{4} d^{3} + 875 \, a^{4} b^{3} c^{3} d^{4} - 1617 \, a^{5} b^{2} c^{2} d^{5} + 1197 \, a^{6} b c d^{6} - 319 \, a^{7} d^{7} + 14 \, {\left (3 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 84 \, {\left (3 \, b^{7} c^{2} d^{5} - 3 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 4 \, {\left (252 \, a b^{6} c^{2} d^{5} - 357 \, a^{2} b^{5} c d^{6} + 139 \, a^{3} b^{4} d^{7}\right )} x^{4} - 4 \, {\left (105 \, b^{7} c^{4} d^{3} - 420 \, a b^{6} c^{3} d^{4} + 252 \, a^{2} b^{5} c^{2} d^{5} + 168 \, a^{3} b^{4} c d^{6} - 136 \, a^{4} b^{3} d^{7}\right )} x^{3} - 6 \, {\left (21 \, b^{7} c^{5} d^{2} + 105 \, a b^{6} c^{4} d^{3} - 630 \, a^{2} b^{5} c^{3} d^{4} + 882 \, a^{3} b^{4} c^{2} d^{5} - 462 \, a^{4} b^{3} c d^{6} + 74 \, a^{5} b^{2} d^{7}\right )} x^{2} - 4 \, {\left (7 \, b^{7} c^{6} d + 21 \, a b^{6} c^{5} d^{2} + 105 \, a^{2} b^{5} c^{4} d^{3} - 770 \, a^{3} b^{4} c^{3} d^{4} + 1302 \, a^{4} b^{3} c^{2} d^{5} - 882 \, a^{5} b^{2} c d^{6} + 214 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{4} b^{3} c^{3} d^{4} - 3 \, a^{5} b^{2} c^{2} d^{5} + 3 \, a^{6} b c d^{6} - a^{7} d^{7} + {\left (b^{7} c^{3} d^{4} - 3 \, a b^{6} c^{2} d^{5} + 3 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 4 \, {\left (a b^{6} c^{3} d^{4} - 3 \, a^{2} b^{5} c^{2} d^{5} + 3 \, a^{3} b^{4} c d^{6} - a^{4} b^{3} d^{7}\right )} x^{3} + 6 \, {\left (a^{2} b^{5} c^{3} d^{4} - 3 \, a^{3} b^{4} c^{2} d^{5} + 3 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 4 \, {\left (a^{3} b^{4} c^{3} d^{4} - 3 \, a^{4} b^{3} c^{2} d^{5} + 3 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} \] Input:
integrate((d*x+c)^7/(b*x+a)^5,x, algorithm="fricas")
Output:
1/12*(4*b^7*d^7*x^7 - 3*b^7*c^7 - 7*a*b^6*c^6*d - 21*a^2*b^5*c^5*d^2 - 105 *a^3*b^4*c^4*d^3 + 875*a^4*b^3*c^3*d^4 - 1617*a^5*b^2*c^2*d^5 + 1197*a^6*b *c*d^6 - 319*a^7*d^7 + 14*(3*b^7*c*d^6 - a*b^6*d^7)*x^6 + 84*(3*b^7*c^2*d^ 5 - 3*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 4*(252*a*b^6*c^2*d^5 - 357*a^2*b^5* c*d^6 + 139*a^3*b^4*d^7)*x^4 - 4*(105*b^7*c^4*d^3 - 420*a*b^6*c^3*d^4 + 25 2*a^2*b^5*c^2*d^5 + 168*a^3*b^4*c*d^6 - 136*a^4*b^3*d^7)*x^3 - 6*(21*b^7*c ^5*d^2 + 105*a*b^6*c^4*d^3 - 630*a^2*b^5*c^3*d^4 + 882*a^3*b^4*c^2*d^5 - 4 62*a^4*b^3*c*d^6 + 74*a^5*b^2*d^7)*x^2 - 4*(7*b^7*c^6*d + 21*a*b^6*c^5*d^2 + 105*a^2*b^5*c^4*d^3 - 770*a^3*b^4*c^3*d^4 + 1302*a^4*b^3*c^2*d^5 - 882* a^5*b^2*c*d^6 + 214*a^6*b*d^7)*x + 420*(a^4*b^3*c^3*d^4 - 3*a^5*b^2*c^2*d^ 5 + 3*a^6*b*c*d^6 - a^7*d^7 + (b^7*c^3*d^4 - 3*a*b^6*c^2*d^5 + 3*a^2*b^5*c *d^6 - a^3*b^4*d^7)*x^4 + 4*(a*b^6*c^3*d^4 - 3*a^2*b^5*c^2*d^5 + 3*a^3*b^4 *c*d^6 - a^4*b^3*d^7)*x^3 + 6*(a^2*b^5*c^3*d^4 - 3*a^3*b^4*c^2*d^5 + 3*a^4 *b^3*c*d^6 - a^5*b^2*d^7)*x^2 + 4*(a^3*b^4*c^3*d^4 - 3*a^4*b^3*c^2*d^5 + 3 *a^5*b^2*c*d^6 - a^6*b*d^7)*x)*log(b*x + a))/(b^12*x^4 + 4*a*b^11*x^3 + 6* a^2*b^10*x^2 + 4*a^3*b^9*x + a^4*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 500 vs. \(2 (172) = 344\).
Time = 13.07 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.67 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=x^{2} \left (- \frac {5 a d^{7}}{2 b^{6}} + \frac {7 c d^{6}}{2 b^{5}}\right ) + x \left (\frac {15 a^{2} d^{7}}{b^{7}} - \frac {35 a c d^{6}}{b^{6}} + \frac {21 c^{2} d^{5}}{b^{5}}\right ) + \frac {- 319 a^{7} d^{7} + 1197 a^{6} b c d^{6} - 1617 a^{5} b^{2} c^{2} d^{5} + 875 a^{4} b^{3} c^{3} d^{4} - 105 a^{3} b^{4} c^{4} d^{3} - 21 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - 3 b^{7} c^{7} + x^{3} \left (- 420 a^{4} b^{3} d^{7} + 1680 a^{3} b^{4} c d^{6} - 2520 a^{2} b^{5} c^{2} d^{5} + 1680 a b^{6} c^{3} d^{4} - 420 b^{7} c^{4} d^{3}\right ) + x^{2} \left (- 1134 a^{5} b^{2} d^{7} + 4410 a^{4} b^{3} c d^{6} - 6300 a^{3} b^{4} c^{2} d^{5} + 3780 a^{2} b^{5} c^{3} d^{4} - 630 a b^{6} c^{4} d^{3} - 126 b^{7} c^{5} d^{2}\right ) + x \left (- 1036 a^{6} b d^{7} + 3948 a^{5} b^{2} c d^{6} - 5460 a^{4} b^{3} c^{2} d^{5} + 3080 a^{3} b^{4} c^{3} d^{4} - 420 a^{2} b^{5} c^{4} d^{3} - 84 a b^{6} c^{5} d^{2} - 28 b^{7} c^{6} d\right )}{12 a^{4} b^{8} + 48 a^{3} b^{9} x + 72 a^{2} b^{10} x^{2} + 48 a b^{11} x^{3} + 12 b^{12} x^{4}} + \frac {d^{7} x^{3}}{3 b^{5}} - \frac {35 d^{4} \left (a d - b c\right )^{3} \log {\left (a + b x \right )}}{b^{8}} \] Input:
integrate((d*x+c)**7/(b*x+a)**5,x)
Output:
x**2*(-5*a*d**7/(2*b**6) + 7*c*d**6/(2*b**5)) + x*(15*a**2*d**7/b**7 - 35* a*c*d**6/b**6 + 21*c**2*d**5/b**5) + (-319*a**7*d**7 + 1197*a**6*b*c*d**6 - 1617*a**5*b**2*c**2*d**5 + 875*a**4*b**3*c**3*d**4 - 105*a**3*b**4*c**4* d**3 - 21*a**2*b**5*c**5*d**2 - 7*a*b**6*c**6*d - 3*b**7*c**7 + x**3*(-420 *a**4*b**3*d**7 + 1680*a**3*b**4*c*d**6 - 2520*a**2*b**5*c**2*d**5 + 1680* a*b**6*c**3*d**4 - 420*b**7*c**4*d**3) + x**2*(-1134*a**5*b**2*d**7 + 4410 *a**4*b**3*c*d**6 - 6300*a**3*b**4*c**2*d**5 + 3780*a**2*b**5*c**3*d**4 - 630*a*b**6*c**4*d**3 - 126*b**7*c**5*d**2) + x*(-1036*a**6*b*d**7 + 3948*a **5*b**2*c*d**6 - 5460*a**4*b**3*c**2*d**5 + 3080*a**3*b**4*c**3*d**4 - 42 0*a**2*b**5*c**4*d**3 - 84*a*b**6*c**5*d**2 - 28*b**7*c**6*d))/(12*a**4*b* *8 + 48*a**3*b**9*x + 72*a**2*b**10*x**2 + 48*a*b**11*x**3 + 12*b**12*x**4 ) + d**7*x**3/(3*b**5) - 35*d**4*(a*d - b*c)**3*log(a + b*x)/b**8
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (177) = 354\).
Time = 0.05 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.64 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=-\frac {3 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 21 \, a^{2} b^{5} c^{5} d^{2} + 105 \, a^{3} b^{4} c^{4} d^{3} - 875 \, a^{4} b^{3} c^{3} d^{4} + 1617 \, a^{5} b^{2} c^{2} d^{5} - 1197 \, a^{6} b c d^{6} + 319 \, a^{7} d^{7} + 420 \, {\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 126 \, {\left (b^{7} c^{5} d^{2} + 5 \, a b^{6} c^{4} d^{3} - 30 \, a^{2} b^{5} c^{3} d^{4} + 50 \, a^{3} b^{4} c^{2} d^{5} - 35 \, a^{4} b^{3} c d^{6} + 9 \, a^{5} b^{2} d^{7}\right )} x^{2} + 28 \, {\left (b^{7} c^{6} d + 3 \, a b^{6} c^{5} d^{2} + 15 \, a^{2} b^{5} c^{4} d^{3} - 110 \, a^{3} b^{4} c^{3} d^{4} + 195 \, a^{4} b^{3} c^{2} d^{5} - 141 \, a^{5} b^{2} c d^{6} + 37 \, a^{6} b d^{7}\right )} x}{12 \, {\left (b^{12} x^{4} + 4 \, a b^{11} x^{3} + 6 \, a^{2} b^{10} x^{2} + 4 \, a^{3} b^{9} x + a^{4} b^{8}\right )}} + \frac {2 \, b^{2} d^{7} x^{3} + 3 \, {\left (7 \, b^{2} c d^{6} - 5 \, a b d^{7}\right )} x^{2} + 6 \, {\left (21 \, b^{2} c^{2} d^{5} - 35 \, a b c d^{6} + 15 \, a^{2} d^{7}\right )} x}{6 \, b^{7}} + \frac {35 \, {\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \] Input:
integrate((d*x+c)^7/(b*x+a)^5,x, algorithm="maxima")
Output:
-1/12*(3*b^7*c^7 + 7*a*b^6*c^6*d + 21*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^ 3 - 875*a^4*b^3*c^3*d^4 + 1617*a^5*b^2*c^2*d^5 - 1197*a^6*b*c*d^6 + 319*a^ 7*d^7 + 420*(b^7*c^4*d^3 - 4*a*b^6*c^3*d^4 + 6*a^2*b^5*c^2*d^5 - 4*a^3*b^4 *c*d^6 + a^4*b^3*d^7)*x^3 + 126*(b^7*c^5*d^2 + 5*a*b^6*c^4*d^3 - 30*a^2*b^ 5*c^3*d^4 + 50*a^3*b^4*c^2*d^5 - 35*a^4*b^3*c*d^6 + 9*a^5*b^2*d^7)*x^2 + 2 8*(b^7*c^6*d + 3*a*b^6*c^5*d^2 + 15*a^2*b^5*c^4*d^3 - 110*a^3*b^4*c^3*d^4 + 195*a^4*b^3*c^2*d^5 - 141*a^5*b^2*c*d^6 + 37*a^6*b*d^7)*x)/(b^12*x^4 + 4 *a*b^11*x^3 + 6*a^2*b^10*x^2 + 4*a^3*b^9*x + a^4*b^8) + 1/6*(2*b^2*d^7*x^3 + 3*(7*b^2*c*d^6 - 5*a*b*d^7)*x^2 + 6*(21*b^2*c^2*d^5 - 35*a*b*c*d^6 + 15 *a^2*d^7)*x)/b^7 + 35*(b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3 *d^7)*log(b*x + a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (177) = 354\).
Time = 0.12 (sec) , antiderivative size = 660, normalized size of antiderivative = 3.53 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^7/(b*x+a)^5,x, algorithm="giac")
Output:
1/6*(2*d^7 + 21*(b^2*c*d^6 - a*b*d^7)/((b*x + a)*b) + 126*(b^4*c^2*d^5 - 2 *a*b^3*c*d^6 + a^2*b^2*d^7)/((b*x + a)^2*b^2))*(b*x + a)^3/b^8 - 35*(b^3*c ^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*log(abs(b*x + a)/((b*x + a)^2*abs(b)))/b^8 - 1/12*(3*b^43*c^7/(b*x + a)^4 + 28*b^42*c^6*d/(b*x + a)^3 - 21*a*b^42*c^6*d/(b*x + a)^4 + 126*b^41*c^5*d^2/(b*x + a)^2 - 168*a *b^41*c^5*d^2/(b*x + a)^3 + 63*a^2*b^41*c^5*d^2/(b*x + a)^4 + 420*b^40*c^4 *d^3/(b*x + a) - 630*a*b^40*c^4*d^3/(b*x + a)^2 + 420*a^2*b^40*c^4*d^3/(b* x + a)^3 - 105*a^3*b^40*c^4*d^3/(b*x + a)^4 - 1680*a*b^39*c^3*d^4/(b*x + a ) + 1260*a^2*b^39*c^3*d^4/(b*x + a)^2 - 560*a^3*b^39*c^3*d^4/(b*x + a)^3 + 105*a^4*b^39*c^3*d^4/(b*x + a)^4 + 2520*a^2*b^38*c^2*d^5/(b*x + a) - 1260 *a^3*b^38*c^2*d^5/(b*x + a)^2 + 420*a^4*b^38*c^2*d^5/(b*x + a)^3 - 63*a^5* b^38*c^2*d^5/(b*x + a)^4 - 1680*a^3*b^37*c*d^6/(b*x + a) + 630*a^4*b^37*c* d^6/(b*x + a)^2 - 168*a^5*b^37*c*d^6/(b*x + a)^3 + 21*a^6*b^37*c*d^6/(b*x + a)^4 + 420*a^4*b^36*d^7/(b*x + a) - 126*a^5*b^36*d^7/(b*x + a)^2 + 28*a^ 6*b^36*d^7/(b*x + a)^3 - 3*a^7*b^36*d^7/(b*x + a)^4)/b^44
Time = 0.18 (sec) , antiderivative size = 512, normalized size of antiderivative = 2.74 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx=x\,\left (\frac {5\,a\,\left (\frac {5\,a\,d^7}{b^6}-\frac {7\,c\,d^6}{b^5}\right )}{b}-\frac {10\,a^2\,d^7}{b^7}+\frac {21\,c^2\,d^5}{b^5}\right )-x^2\,\left (\frac {5\,a\,d^7}{2\,b^6}-\frac {7\,c\,d^6}{2\,b^5}\right )-\frac {\frac {319\,a^7\,d^7-1197\,a^6\,b\,c\,d^6+1617\,a^5\,b^2\,c^2\,d^5-875\,a^4\,b^3\,c^3\,d^4+105\,a^3\,b^4\,c^4\,d^3+21\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+3\,b^7\,c^7}{12\,b}+x\,\left (\frac {259\,a^6\,d^7}{3}-329\,a^5\,b\,c\,d^6+455\,a^4\,b^2\,c^2\,d^5-\frac {770\,a^3\,b^3\,c^3\,d^4}{3}+35\,a^2\,b^4\,c^4\,d^3+7\,a\,b^5\,c^5\,d^2+\frac {7\,b^6\,c^6\,d}{3}\right )+x^3\,\left (35\,a^4\,b^2\,d^7-140\,a^3\,b^3\,c\,d^6+210\,a^2\,b^4\,c^2\,d^5-140\,a\,b^5\,c^3\,d^4+35\,b^6\,c^4\,d^3\right )+x^2\,\left (\frac {189\,a^5\,b\,d^7}{2}-\frac {735\,a^4\,b^2\,c\,d^6}{2}+525\,a^3\,b^3\,c^2\,d^5-315\,a^2\,b^4\,c^3\,d^4+\frac {105\,a\,b^5\,c^4\,d^3}{2}+\frac {21\,b^6\,c^5\,d^2}{2}\right )}{a^4\,b^7+4\,a^3\,b^8\,x+6\,a^2\,b^9\,x^2+4\,a\,b^{10}\,x^3+b^{11}\,x^4}-\frac {\ln \left (a+b\,x\right )\,\left (35\,a^3\,d^7-105\,a^2\,b\,c\,d^6+105\,a\,b^2\,c^2\,d^5-35\,b^3\,c^3\,d^4\right )}{b^8}+\frac {d^7\,x^3}{3\,b^5} \] Input:
int((c + d*x)^7/(a + b*x)^5,x)
Output:
x*((5*a*((5*a*d^7)/b^6 - (7*c*d^6)/b^5))/b - (10*a^2*d^7)/b^7 + (21*c^2*d^ 5)/b^5) - x^2*((5*a*d^7)/(2*b^6) - (7*c*d^6)/(2*b^5)) - ((319*a^7*d^7 + 3* b^7*c^7 + 21*a^2*b^5*c^5*d^2 + 105*a^3*b^4*c^4*d^3 - 875*a^4*b^3*c^3*d^4 + 1617*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d - 1197*a^6*b*c*d^6)/(12*b) + x*((259 *a^6*d^7)/3 + (7*b^6*c^6*d)/3 + 7*a*b^5*c^5*d^2 + 35*a^2*b^4*c^4*d^3 - (77 0*a^3*b^3*c^3*d^4)/3 + 455*a^4*b^2*c^2*d^5 - 329*a^5*b*c*d^6) + x^3*(35*a^ 4*b^2*d^7 + 35*b^6*c^4*d^3 - 140*a*b^5*c^3*d^4 - 140*a^3*b^3*c*d^6 + 210*a ^2*b^4*c^2*d^5) + x^2*((189*a^5*b*d^7)/2 + (21*b^6*c^5*d^2)/2 + (105*a*b^5 *c^4*d^3)/2 - (735*a^4*b^2*c*d^6)/2 - 315*a^2*b^4*c^3*d^4 + 525*a^3*b^3*c^ 2*d^5))/(a^4*b^7 + b^11*x^4 + 4*a^3*b^8*x + 4*a*b^10*x^3 + 6*a^2*b^9*x^2) - (log(a + b*x)*(35*a^3*d^7 - 35*b^3*c^3*d^4 + 105*a*b^2*c^2*d^5 - 105*a^2 *b*c*d^6))/b^8 + (d^7*x^3)/(3*b^5)
Time = 0.17 (sec) , antiderivative size = 851, normalized size of antiderivative = 4.55 \[ \int \frac {(c+d x)^7}{(a+b x)^5} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^7/(b*x+a)^5,x)
Output:
( - 420*log(a + b*x)*a**8*d**7 + 1260*log(a + b*x)*a**7*b*c*d**6 - 1680*lo g(a + b*x)*a**7*b*d**7*x - 1260*log(a + b*x)*a**6*b**2*c**2*d**5 + 5040*lo g(a + b*x)*a**6*b**2*c*d**6*x - 2520*log(a + b*x)*a**6*b**2*d**7*x**2 + 42 0*log(a + b*x)*a**5*b**3*c**3*d**4 - 5040*log(a + b*x)*a**5*b**3*c**2*d**5 *x + 7560*log(a + b*x)*a**5*b**3*c*d**6*x**2 - 1680*log(a + b*x)*a**5*b**3 *d**7*x**3 + 1680*log(a + b*x)*a**4*b**4*c**3*d**4*x - 7560*log(a + b*x)*a **4*b**4*c**2*d**5*x**2 + 5040*log(a + b*x)*a**4*b**4*c*d**6*x**3 - 420*lo g(a + b*x)*a**4*b**4*d**7*x**4 + 2520*log(a + b*x)*a**3*b**5*c**3*d**4*x** 2 - 5040*log(a + b*x)*a**3*b**5*c**2*d**5*x**3 + 1260*log(a + b*x)*a**3*b* *5*c*d**6*x**4 + 1680*log(a + b*x)*a**2*b**6*c**3*d**4*x**3 - 1260*log(a + b*x)*a**2*b**6*c**2*d**5*x**4 + 420*log(a + b*x)*a*b**7*c**3*d**4*x**4 - 455*a**8*d**7 + 1365*a**7*b*c*d**6 - 1400*a**7*b*d**7*x - 1365*a**6*b**2*c **2*d**5 + 4200*a**6*b**2*c*d**6*x - 1260*a**6*b**2*d**7*x**2 + 455*a**5*b **3*c**3*d**4 - 4200*a**5*b**3*c**2*d**5*x + 3780*a**5*b**3*c*d**6*x**2 + 1400*a**4*b**4*c**3*d**4*x - 3780*a**4*b**4*c**2*d**5*x**2 + 420*a**4*b**4 *d**7*x**4 - 21*a**3*b**5*c**5*d**2 + 1260*a**3*b**5*c**3*d**4*x**2 - 1260 *a**3*b**5*c*d**6*x**4 + 84*a**3*b**5*d**7*x**5 - 7*a**2*b**6*c**6*d - 84* a**2*b**6*c**5*d**2*x + 1260*a**2*b**6*c**2*d**5*x**4 - 252*a**2*b**6*c*d* *6*x**5 - 14*a**2*b**6*d**7*x**6 - 3*a*b**7*c**7 - 28*a*b**7*c**6*d*x - 12 6*a*b**7*c**5*d**2*x**2 - 420*a*b**7*c**3*d**4*x**4 + 252*a*b**7*c**2*d...