Integrand size = 15, antiderivative size = 187 \[ \int \frac {(c+d x)^7}{(a+b x)^4} \, dx=\frac {35 d^4 (b c-a d)^3 x}{b^7}-\frac {(b c-a d)^7}{3 b^8 (a+b x)^3}-\frac {7 d (b c-a d)^6}{2 b^8 (a+b x)^2}-\frac {21 d^2 (b c-a d)^5}{b^8 (a+b x)}+\frac {21 d^5 (b c-a d)^2 (a+b x)^2}{2 b^8}+\frac {7 d^6 (b c-a d) (a+b x)^3}{3 b^8}+\frac {d^7 (a+b x)^4}{4 b^8}+\frac {35 d^3 (b c-a d)^4 \log (a+b x)}{b^8} \] Output:
35*d^4*(-a*d+b*c)^3*x/b^7-1/3*(-a*d+b*c)^7/b^8/(b*x+a)^3-7/2*d*(-a*d+b*c)^ 6/b^8/(b*x+a)^2-21*d^2*(-a*d+b*c)^5/b^8/(b*x+a)+21/2*d^5*(-a*d+b*c)^2*(b*x +a)^2/b^8+7/3*d^6*(-a*d+b*c)*(b*x+a)^3/b^8+1/4*d^7*(b*x+a)^4/b^8+35*d^3*(- a*d+b*c)^4*ln(b*x+a)/b^8
Time = 0.06 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.06 \[ \int \frac {(c+d x)^7}{(a+b x)^4} \, dx=\frac {12 b d^4 \left (35 b^3 c^3-84 a b^2 c^2 d+70 a^2 b c d^2-20 a^3 d^3\right ) x+6 b^2 d^5 \left (21 b^2 c^2-28 a b c d+10 a^2 d^2\right ) x^2+4 b^3 d^6 (7 b c-4 a d) x^3+3 b^4 d^7 x^4-\frac {4 (b c-a d)^7}{(a+b x)^3}-\frac {42 d (b c-a d)^6}{(a+b x)^2}+\frac {252 d^2 (-b c+a d)^5}{a+b x}+420 d^3 (b c-a d)^4 \log (a+b x)}{12 b^8} \] Input:
Integrate[(c + d*x)^7/(a + b*x)^4,x]
Output:
(12*b*d^4*(35*b^3*c^3 - 84*a*b^2*c^2*d + 70*a^2*b*c*d^2 - 20*a^3*d^3)*x + 6*b^2*d^5*(21*b^2*c^2 - 28*a*b*c*d + 10*a^2*d^2)*x^2 + 4*b^3*d^6*(7*b*c - 4*a*d)*x^3 + 3*b^4*d^7*x^4 - (4*(b*c - a*d)^7)/(a + b*x)^3 - (42*d*(b*c - a*d)^6)/(a + b*x)^2 + (252*d^2*(-(b*c) + a*d)^5)/(a + b*x) + 420*d^3*(b*c - a*d)^4*Log[a + b*x])/(12*b^8)
Time = 0.39 (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.
| \(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^4} \, dx\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \int \left (\frac {7 d^6 (a+b x)^2 (b c-a d)}{b^7}+\frac {21 d^5 (a+b x) (b c-a d)^2}{b^7}+\frac {35 d^4 (b c-a d)^3}{b^7}+\frac {35 d^3 (b c-a d)^4}{b^7 (a+b x)}+\frac {21 d^2 (b c-a d)^5}{b^7 (a+b x)^2}+\frac {7 d (b c-a d)^6}{b^7 (a+b x)^3}+\frac {(b c-a d)^7}{b^7 (a+b x)^4}+\frac {d^7 (a+b x)^3}{b^7}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {7 d^6 (a+b x)^3 (b c-a d)}{3 b^8}+\frac {21 d^5 (a+b x)^2 (b c-a d)^2}{2 b^8}+\frac {35 d^3 (b c-a d)^4 \log (a+b x)}{b^8}-\frac {21 d^2 (b c-a d)^5}{b^8 (a+b x)}-\frac {7 d (b c-a d)^6}{2 b^8 (a+b x)^2}-\frac {(b c-a d)^7}{3 b^8 (a+b x)^3}+\frac {d^7 (a+b x)^4}{4 b^8}+\frac {35 d^4 x (b c-a d)^3}{b^7}\) |
Input:
Int[(c + d*x)^7/(a + b*x)^4,x]
Output:
(35*d^4*(b*c - a*d)^3*x)/b^7 - (b*c - a*d)^7/(3*b^8*(a + b*x)^3) - (7*d*(b *c - a*d)^6)/(2*b^8*(a + b*x)^2) - (21*d^2*(b*c - a*d)^5)/(b^8*(a + b*x)) + (21*d^5*(b*c - a*d)^2*(a + b*x)^2)/(2*b^8) + (7*d^6*(b*c - a*d)*(a + b*x )^3)/(3*b^8) + (d^7*(a + b*x)^4)/(4*b^8) + (35*d^3*(b*c - a*d)^4*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. \(447\) vs. \(2(177)=354\).
Time = 0.11 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.40
| method | result | size |
| norman | \(\frac {\frac {385 a^{7} d^{7}-1540 a^{6} b c \,d^{6}+2310 a^{5} b^{2} c^{2} d^{5}-1540 a^{4} b^{3} c^{3} d^{4}+385 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -2 b^{7} c^{7}}{6 b^{8}}+\frac {d^{7} x^{7}}{4 b}+\frac {3 \left (35 a^{5} d^{7}-140 a^{4} b c \,d^{6}+210 a^{3} b^{2} c^{2} d^{5}-140 a^{2} b^{3} c^{3} d^{4}+35 a \,b^{4} c^{4} d^{3}-7 b^{5} c^{5} d^{2}\right ) x^{2}}{b^{6}}+\frac {\left (315 a^{6} d^{7}-1260 a^{5} b c \,d^{6}+1890 a^{4} b^{2} c^{2} d^{5}-1260 a^{3} b^{3} c^{3} d^{4}+315 a^{2} b^{4} c^{4} d^{3}-42 a \,b^{5} c^{5} d^{2}-7 b^{6} c^{6} d \right ) x}{2 b^{7}}-\frac {35 d^{4} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -4 b^{3} c^{3}\right ) x^{4}}{4 b^{4}}+\frac {7 d^{5} \left (a^{2} d^{2}-4 a b c d +6 b^{2} c^{2}\right ) x^{5}}{4 b^{3}}-\frac {7 d^{6} \left (a d -4 b c \right ) x^{6}}{12 b^{2}}}{\left (b x +a \right )^{3}}+\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \ln \left (b x +a \right )}{b^{8}}\) | \(448\) |
| default | \(-\frac {d^{4} \left (-\frac {1}{4} d^{3} x^{4} b^{3}+\frac {4}{3} x^{3} a \,b^{2} d^{3}-\frac {7}{3} x^{3} b^{3} c \,d^{2}-5 x^{2} a^{2} b \,d^{3}+14 x^{2} a \,b^{2} c \,d^{2}-\frac {21}{2} x^{2} b^{3} c^{2} d +20 a^{3} d^{3} x -70 a^{2} b c \,d^{2} x +84 a \,b^{2} c^{2} d x -35 b^{3} c^{3} x \right )}{b^{7}}-\frac {7 d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{2 b^{8} \left (b x +a \right )^{2}}+\frac {21 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{b^{8} \left (b x +a \right )}+\frac {35 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \ln \left (b x +a \right )}{b^{8}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{3 b^{8} \left (b x +a \right )^{3}}\) | \(459\) |
| risch | \(\frac {d^{7} x^{4}}{4 b^{4}}-\frac {4 d^{7} x^{3} a}{3 b^{5}}+\frac {7 d^{6} x^{3} c}{3 b^{4}}+\frac {5 d^{7} x^{2} a^{2}}{b^{6}}-\frac {14 d^{6} x^{2} a c}{b^{5}}+\frac {21 d^{5} x^{2} c^{2}}{2 b^{4}}-\frac {20 d^{7} a^{3} x}{b^{7}}+\frac {70 d^{6} a^{2} c x}{b^{6}}-\frac {84 d^{5} a \,c^{2} x}{b^{5}}+\frac {35 d^{4} c^{3} x}{b^{4}}+\frac {\left (21 a^{5} b \,d^{7}-105 a^{4} b^{2} c \,d^{6}+210 a^{3} b^{3} c^{2} d^{5}-210 a^{2} b^{4} c^{3} d^{4}+105 a \,b^{5} c^{4} d^{3}-21 b^{6} c^{5} d^{2}\right ) x^{2}+\frac {7 d \left (11 a^{6} d^{6}-54 a^{5} b c \,d^{5}+105 a^{4} b^{2} c^{2} d^{4}-100 a^{3} b^{3} c^{3} d^{3}+45 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d -c^{6} b^{6}\right ) x}{2}+\frac {107 a^{7} d^{7}-518 a^{6} b c \,d^{6}+987 a^{5} b^{2} c^{2} d^{5}-910 a^{4} b^{3} c^{3} d^{4}+385 a^{3} b^{4} c^{4} d^{3}-42 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d -2 b^{7} c^{7}}{6 b}}{b^{7} \left (b x +a \right )^{3}}+\frac {35 d^{7} \ln \left (b x +a \right ) a^{4}}{b^{8}}-\frac {140 d^{6} \ln \left (b x +a \right ) a^{3} c}{b^{7}}+\frac {210 d^{5} \ln \left (b x +a \right ) a^{2} c^{2}}{b^{6}}-\frac {140 d^{4} \ln \left (b x +a \right ) a \,c^{3}}{b^{5}}+\frac {35 d^{3} \ln \left (b x +a \right ) c^{4}}{b^{4}}\) | \(488\) |
| parallelrisch | \(\frac {-3080 a^{6} b c \,d^{6}+4620 a^{5} b^{2} c^{2} d^{5}-3080 a^{4} b^{3} c^{3} d^{4}+420 x^{4} a^{2} b^{5} c \,d^{6}-630 x^{4} a \,b^{6} c^{2} d^{5}-84 x^{5} a \,b^{6} c \,d^{6}-1680 \ln \left (b x +a \right ) a^{6} b c \,d^{6}+2520 \ln \left (b x +a \right ) a^{5} b^{2} c^{2} d^{5}-1680 \ln \left (b x +a \right ) a^{4} b^{3} c^{3} d^{4}+420 \ln \left (b x +a \right ) a^{3} b^{4} c^{4} d^{3}+3 x^{7} d^{7} b^{7}+1890 x \,a^{6} b \,d^{7}-42 x \,b^{7} c^{6} d +1260 x^{2} a^{5} b^{2} d^{7}-252 x^{2} b^{7} c^{5} d^{2}-105 x^{4} a^{3} b^{4} d^{7}+420 x^{4} b^{7} c^{3} d^{4}+21 x^{5} a^{2} b^{5} d^{7}+126 x^{5} b^{7} c^{2} d^{5}-7 x^{6} a \,b^{6} d^{7}+28 x^{6} b^{7} c \,d^{6}-4 b^{7} c^{7}+770 a^{7} d^{7}+420 \ln \left (b x +a \right ) x^{3} a^{4} b^{3} d^{7}+420 \ln \left (b x +a \right ) x^{3} b^{7} c^{4} d^{3}+1260 \ln \left (b x +a \right ) x^{2} a^{5} b^{2} d^{7}+770 a^{3} b^{4} c^{4} d^{3}-84 a^{2} b^{5} c^{5} d^{2}-14 a \,b^{6} c^{6} d -7560 x \,a^{5} b^{2} c \,d^{6}+11340 x \,a^{4} b^{3} c^{2} d^{5}-7560 x \,a^{3} b^{4} c^{3} d^{4}+1890 x \,a^{2} b^{5} c^{4} d^{3}-252 x a \,b^{6} c^{5} d^{2}-5040 x^{2} a^{4} b^{3} c \,d^{6}+7560 x^{2} a^{3} b^{4} c^{2} d^{5}-5040 x^{2} a^{2} b^{5} c^{3} d^{4}+1260 x^{2} a \,b^{6} c^{4} d^{3}+7560 \ln \left (b x +a \right ) x \,a^{4} b^{3} c^{2} d^{5}-5040 \ln \left (b x +a \right ) x \,a^{3} b^{4} c^{3} d^{4}+1260 \ln \left (b x +a \right ) x \,a^{2} b^{5} c^{4} d^{3}-5040 \ln \left (b x +a \right ) x^{2} a^{2} b^{5} c^{3} d^{4}+1260 \ln \left (b x +a \right ) x^{2} a \,b^{6} c^{4} d^{3}-5040 \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}+1260 \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}-1680 \ln \left (b x +a \right ) x^{3} a^{3} b^{4} c \,d^{6}+2520 \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 )^{3}}\) | \(824\) |
Input:
int((d*x+c)^7/(b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
(1/6*(385*a^7*d^7-1540*a^6*b*c*d^6+2310*a^5*b^2*c^2*d^5-1540*a^4*b^3*c^3*d ^4+385*a^3*b^4*c^4*d^3-42*a^2*b^5*c^5*d^2-7*a*b^6*c^6*d-2*b^7*c^7)/b^8+1/4 /b*d^7*x^7+3*(35*a^5*d^7-140*a^4*b*c*d^6+210*a^3*b^2*c^2*d^5-140*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+1/2*(315*a^6*d^7-1260*a^5*b *c*d^6+1890*a^4*b^2*c^2*d^5-1260*a^3*b^3*c^3*d^4+315*a^2*b^4*c^4*d^3-42*a* b^5*c^5*d^2-7*b^6*c^6*d)/b^7*x-35/4*d^4*(a^3*d^3-4*a^2*b*c*d^2+6*a*b^2*c^2 *d-4*b^3*c^3)/b^4*x^4+7/4*d^5*(a^2*d^2-4*a*b*c*d+6*b^2*c^2)/b^3*x^5-7/12*d ^6*(a*d-4*b*c)/b^2*x^6)/(b*x+a)^3+35/b^8*d^3*(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)*ln(b*x+a)
Leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (177) = 354\).
Time = 0.08 (sec) , antiderivative size = 739, normalized size of antiderivative = 3.95 \[ \int \frac {(c+d x)^7}{(a+b x)^4} \, dx=\frac {3 \, b^{7} d^{7} x^{7} - 4 \, b^{7} c^{7} - 14 \, a b^{6} c^{6} d - 84 \, a^{2} b^{5} c^{5} d^{2} + 770 \, a^{3} b^{4} c^{4} d^{3} - 1820 \, a^{4} b^{3} c^{3} d^{4} + 1974 \, a^{5} b^{2} c^{2} d^{5} - 1036 \, a^{6} b c d^{6} + 214 \, a^{7} d^{7} + 7 \, {\left (4 \, b^{7} c d^{6} - a b^{6} d^{7}\right )} x^{6} + 21 \, {\left (6 \, b^{7} c^{2} d^{5} - 4 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 105 \, {\left (4 \, b^{7} c^{3} d^{4} - 6 \, a b^{6} c^{2} d^{5} + 4 \, a^{2} b^{5} c d^{6} - a^{3} b^{4} d^{7}\right )} x^{4} + 2 \, {\left (630 \, a b^{6} c^{3} d^{4} - 1323 \, a^{2} b^{5} c^{2} d^{5} + 1022 \, a^{3} b^{4} c d^{6} - 278 \, a^{4} b^{3} d^{7}\right )} x^{3} - 6 \, {\left (42 \, b^{7} c^{5} d^{2} - 210 \, a b^{6} c^{4} d^{3} + 210 \, a^{2} b^{5} c^{3} d^{4} + 63 \, a^{3} b^{4} c^{2} d^{5} - 182 \, a^{4} b^{3} c d^{6} + 68 \, a^{5} b^{2} d^{7}\right )} x^{2} - 6 \, {\left (7 \, b^{7} c^{6} d + 42 \, a b^{6} c^{5} d^{2} - 315 \, a^{2} b^{5} c^{4} d^{3} + 630 \, a^{3} b^{4} c^{3} d^{4} - 567 \, a^{4} b^{3} c^{2} d^{5} + 238 \, a^{5} b^{2} c d^{6} - 37 \, a^{6} b d^{7}\right )} x + 420 \, {\left (a^{3} b^{4} c^{4} d^{3} - 4 \, a^{4} b^{3} c^{3} d^{4} + 6 \, a^{5} b^{2} c^{2} d^{5} - 4 \, a^{6} b c d^{6} + a^{7} d^{7} + {\left (b^{7} c^{4} d^{3} - 4 \, a b^{6} c^{3} d^{4} + 6 \, a^{2} b^{5} c^{2} d^{5} - 4 \, a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 3 \, {\left (a b^{6} c^{4} d^{3} - 4 \, a^{2} b^{5} c^{3} d^{4} + 6 \, a^{3} b^{4} c^{2} d^{5} - 4 \, a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} d^{3} - 4 \, a^{3} b^{4} c^{3} d^{4} + 6 \, a^{4} b^{3} c^{2} d^{5} - 4 \, a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x\right )} \log \left (b x + a\right )}{12 \, {\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} \] Input:
integrate((d*x+c)^7/(b*x+a)^4,x, algorithm="fricas")
Output:
1/12*(3*b^7*d^7*x^7 - 4*b^7*c^7 - 14*a*b^6*c^6*d - 84*a^2*b^5*c^5*d^2 + 77 0*a^3*b^4*c^4*d^3 - 1820*a^4*b^3*c^3*d^4 + 1974*a^5*b^2*c^2*d^5 - 1036*a^6 *b*c*d^6 + 214*a^7*d^7 + 7*(4*b^7*c*d^6 - a*b^6*d^7)*x^6 + 21*(6*b^7*c^2*d ^5 - 4*a*b^6*c*d^6 + a^2*b^5*d^7)*x^5 + 105*(4*b^7*c^3*d^4 - 6*a*b^6*c^2*d ^5 + 4*a^2*b^5*c*d^6 - a^3*b^4*d^7)*x^4 + 2*(630*a*b^6*c^3*d^4 - 1323*a^2* b^5*c^2*d^5 + 1022*a^3*b^4*c*d^6 - 278*a^4*b^3*d^7)*x^3 - 6*(42*b^7*c^5*d^ 2 - 210*a*b^6*c^4*d^3 + 210*a^2*b^5*c^3*d^4 + 63*a^3*b^4*c^2*d^5 - 182*a^4 *b^3*c*d^6 + 68*a^5*b^2*d^7)*x^2 - 6*(7*b^7*c^6*d + 42*a*b^6*c^5*d^2 - 315 *a^2*b^5*c^4*d^3 + 630*a^3*b^4*c^3*d^4 - 567*a^4*b^3*c^2*d^5 + 238*a^5*b^2 *c*d^6 - 37*a^6*b*d^7)*x + 420*(a^3*b^4*c^4*d^3 - 4*a^4*b^3*c^3*d^4 + 6*a^ 5*b^2*c^2*d^5 - 4*a^6*b*c*d^6 + a^7*d^7 + (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 + 3*(a*b^6*c^4*d^3 - 4*a^2*b^5*c^3*d^4 + 6*a^3*b^4*c^2*d^5 - 4*a^4*b^3*c*d^6 + a^5*b^2*d^7)* x^2 + 3*(a^2*b^5*c^4*d^3 - 4*a^3*b^4*c^3*d^4 + 6*a^4*b^3*c^2*d^5 - 4*a^5*b ^2*c*d^6 + a^6*b*d^7)*x)*log(b*x + a))/(b^11*x^3 + 3*a*b^10*x^2 + 3*a^2*b^ 9*x + a^3*b^8)
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (172) = 344\).
Time = 2.91 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.53 \[ \int \frac {(c+d x)^7}{(a+b x)^4} \, dx=x^{3} \left (- \frac {4 a d^{7}}{3 b^{5}} + \frac {7 c d^{6}}{3 b^{4}}\right ) + x^{2} \cdot \left (\frac {5 a^{2} d^{7}}{b^{6}} - \frac {14 a c d^{6}}{b^{5}} + \frac {21 c^{2} d^{5}}{2 b^{4}}\right ) + x \left (- \frac {20 a^{3} d^{7}}{b^{7}} + \frac {70 a^{2} c d^{6}}{b^{6}} - \frac {84 a c^{2} d^{5}}{b^{5}} + \frac {35 c^{3} d^{4}}{b^{4}}\right ) + \frac {107 a^{7} d^{7} - 518 a^{6} b c d^{6} + 987 a^{5} b^{2} c^{2} d^{5} - 910 a^{4} b^{3} c^{3} d^{4} + 385 a^{3} b^{4} c^{4} d^{3} - 42 a^{2} b^{5} c^{5} d^{2} - 7 a b^{6} c^{6} d - 2 b^{7} c^{7} + x^{2} \cdot \left (126 a^{5} b^{2} d^{7} - 630 a^{4} b^{3} c d^{6} + 1260 a^{3} b^{4} c^{2} d^{5} - 1260 a^{2} b^{5} c^{3} d^{4} + 630 a b^{6} c^{4} d^{3} - 126 b^{7} c^{5} d^{2}\right ) + x \left (231 a^{6} b d^{7} - 1134 a^{5} b^{2} c d^{6} + 2205 a^{4} b^{3} c^{2} d^{5} - 2100 a^{3} b^{4} c^{3} d^{4} + 945 a^{2} b^{5} c^{4} d^{3} - 126 a b^{6} c^{5} d^{2} - 21 b^{7} c^{6} d\right )}{6 a^{3} b^{8} + 18 a^{2} b^{9} x + 18 a b^{10} x^{2} + 6 b^{11} x^{3}} + \frac {d^{7} x^{4}}{4 b^{4}} + \frac {35 d^{3} \left (a d - b c\right )^{4} \log {\left (a + b x \right )}}{b^{8}} \] Input:
integrate((d*x+c)**7/(b*x+a)**4,x)
Output:
x**3*(-4*a*d**7/(3*b**5) + 7*c*d**6/(3*b**4)) + x**2*(5*a**2*d**7/b**6 - 1 4*a*c*d**6/b**5 + 21*c**2*d**5/(2*b**4)) + x*(-20*a**3*d**7/b**7 + 70*a**2 *c*d**6/b**6 - 84*a*c**2*d**5/b**5 + 35*c**3*d**4/b**4) + (107*a**7*d**7 - 518*a**6*b*c*d**6 + 987*a**5*b**2*c**2*d**5 - 910*a**4*b**3*c**3*d**4 + 3 85*a**3*b**4*c**4*d**3 - 42*a**2*b**5*c**5*d**2 - 7*a*b**6*c**6*d - 2*b**7 *c**7 + x**2*(126*a**5*b**2*d**7 - 630*a**4*b**3*c*d**6 + 1260*a**3*b**4*c **2*d**5 - 1260*a**2*b**5*c**3*d**4 + 630*a*b**6*c**4*d**3 - 126*b**7*c**5 *d**2) + x*(231*a**6*b*d**7 - 1134*a**5*b**2*c*d**6 + 2205*a**4*b**3*c**2* d**5 - 2100*a**3*b**4*c**3*d**4 + 945*a**2*b**5*c**4*d**3 - 126*a*b**6*c** 5*d**2 - 21*b**7*c**6*d))/(6*a**3*b**8 + 18*a**2*b**9*x + 18*a*b**10*x**2 + 6*b**11*x**3) + d**7*x**4/(4*b**4) + 35*d**3*(a*d - b*c)**4*log(a + b*x) /b**8
Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (177) = 354\).
Time = 0.05 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.59 \[ \int \frac {(c+d x)^7}{(a+b x)^4} \, dx=-\frac {2 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 42 \, a^{2} b^{5} c^{5} d^{2} - 385 \, a^{3} b^{4} c^{4} d^{3} + 910 \, a^{4} b^{3} c^{3} d^{4} - 987 \, a^{5} b^{2} c^{2} d^{5} + 518 \, a^{6} b c d^{6} - 107 \, a^{7} d^{7} + 126 \, {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 21 \, {\left (b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} - 45 \, a^{2} b^{5} c^{4} d^{3} + 100 \, a^{3} b^{4} c^{3} d^{4} - 105 \, a^{4} b^{3} c^{2} d^{5} + 54 \, a^{5} b^{2} c d^{6} - 11 \, a^{6} b d^{7}\right )} x}{6 \, {\left (b^{11} x^{3} + 3 \, a b^{10} x^{2} + 3 \, a^{2} b^{9} x + a^{3} b^{8}\right )}} + \frac {3 \, b^{3} d^{7} x^{4} + 4 \, {\left (7 \, b^{3} c d^{6} - 4 \, a b^{2} d^{7}\right )} x^{3} + 6 \, {\left (21 \, b^{3} c^{2} d^{5} - 28 \, a b^{2} c d^{6} + 10 \, a^{2} b d^{7}\right )} x^{2} + 12 \, {\left (35 \, b^{3} c^{3} d^{4} - 84 \, a b^{2} c^{2} d^{5} + 70 \, a^{2} b c d^{6} - 20 \, a^{3} d^{7}\right )} x}{12 \, b^{7}} + \frac {35 \, {\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} \log \left (b x + a\right )}{b^{8}} \] Input:
integrate((d*x+c)^7/(b*x+a)^4,x, algorithm="maxima")
Output:
-1/6*(2*b^7*c^7 + 7*a*b^6*c^6*d + 42*a^2*b^5*c^5*d^2 - 385*a^3*b^4*c^4*d^3 + 910*a^4*b^3*c^3*d^4 - 987*a^5*b^2*c^2*d^5 + 518*a^6*b*c*d^6 - 107*a^7*d ^7 + 126*(b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^4* c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^2 + 21*(b^7*c^6*d + 6*a*b^6*c^5 *d^2 - 45*a^2*b^5*c^4*d^3 + 100*a^3*b^4*c^3*d^4 - 105*a^4*b^3*c^2*d^5 + 54 *a^5*b^2*c*d^6 - 11*a^6*b*d^7)*x)/(b^11*x^3 + 3*a*b^10*x^2 + 3*a^2*b^9*x + a^3*b^8) + 1/12*(3*b^3*d^7*x^4 + 4*(7*b^3*c*d^6 - 4*a*b^2*d^7)*x^3 + 6*(2 1*b^3*c^2*d^5 - 28*a*b^2*c*d^6 + 10*a^2*b*d^7)*x^2 + 12*(35*b^3*c^3*d^4 - 84*a*b^2*c^2*d^5 + 70*a^2*b*c*d^6 - 20*a^3*d^7)*x)/b^7 + 35*(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^4*d^7)*log(b*x + a)/b^8
Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (177) = 354\).
Time = 0.12 (sec) , antiderivative size = 470, normalized size of antiderivative = 2.51 \[ \int \frac {(c+d x)^7}{(a+b x)^4} \, dx=\frac {35 \, {\left (b^{4} c^{4} d^{3} - 4 \, a b^{3} c^{3} d^{4} + 6 \, a^{2} b^{2} c^{2} d^{5} - 4 \, a^{3} b c d^{6} + a^{4} d^{7}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{8}} - \frac {2 \, b^{7} c^{7} + 7 \, a b^{6} c^{6} d + 42 \, a^{2} b^{5} c^{5} d^{2} - 385 \, a^{3} b^{4} c^{4} d^{3} + 910 \, a^{4} b^{3} c^{3} d^{4} - 987 \, a^{5} b^{2} c^{2} d^{5} + 518 \, a^{6} b c d^{6} - 107 \, a^{7} d^{7} + 126 \, {\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} x^{2} + 21 \, {\left (b^{7} c^{6} d + 6 \, a b^{6} c^{5} d^{2} - 45 \, a^{2} b^{5} c^{4} d^{3} + 100 \, a^{3} b^{4} c^{3} d^{4} - 105 \, a^{4} b^{3} c^{2} d^{5} + 54 \, a^{5} b^{2} c d^{6} - 11 \, a^{6} b d^{7}\right )} x}{6 \, {\left (b x + a\right )}^{3} b^{8}} + \frac {3 \, b^{12} d^{7} x^{4} + 28 \, b^{12} c d^{6} x^{3} - 16 \, a b^{11} d^{7} x^{3} + 126 \, b^{12} c^{2} d^{5} x^{2} - 168 \, a b^{11} c d^{6} x^{2} + 60 \, a^{2} b^{10} d^{7} x^{2} + 420 \, b^{12} c^{3} d^{4} x - 1008 \, a b^{11} c^{2} d^{5} x + 840 \, a^{2} b^{10} c d^{6} x - 240 \, a^{3} b^{9} d^{7} x}{12 \, b^{16}} \] Input:
integrate((d*x+c)^7/(b*x+a)^4,x, algorithm="giac")
Output:
35*(b^4*c^4*d^3 - 4*a*b^3*c^3*d^4 + 6*a^2*b^2*c^2*d^5 - 4*a^3*b*c*d^6 + a^ 4*d^7)*log(abs(b*x + a))/b^8 - 1/6*(2*b^7*c^7 + 7*a*b^6*c^6*d + 42*a^2*b^5 *c^5*d^2 - 385*a^3*b^4*c^4*d^3 + 910*a^4*b^3*c^3*d^4 - 987*a^5*b^2*c^2*d^5 + 518*a^6*b*c*d^6 - 107*a^7*d^7 + 126*(b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10 *a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 - a^5*b^2*d^7)*x^2 + 21*(b^7*c^6*d + 6*a*b^6*c^5*d^2 - 45*a^2*b^5*c^4*d^3 + 100*a^3*b^4*c^3* d^4 - 105*a^4*b^3*c^2*d^5 + 54*a^5*b^2*c*d^6 - 11*a^6*b*d^7)*x)/((b*x + a) ^3*b^8) + 1/12*(3*b^12*d^7*x^4 + 28*b^12*c*d^6*x^3 - 16*a*b^11*d^7*x^3 + 1 26*b^12*c^2*d^5*x^2 - 168*a*b^11*c*d^6*x^2 + 60*a^2*b^10*d^7*x^2 + 420*b^1 2*c^3*d^4*x - 1008*a*b^11*c^2*d^5*x + 840*a^2*b^10*c*d^6*x - 240*a^3*b^9*d ^7*x)/b^16
Time = 0.17 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.99 \[ \int \frac {(c+d x)^7}{(a+b x)^4} \, dx=x^2\,\left (\frac {2\,a\,\left (\frac {4\,a\,d^7}{b^5}-\frac {7\,c\,d^6}{b^4}\right )}{b}-\frac {3\,a^2\,d^7}{b^6}+\frac {21\,c^2\,d^5}{2\,b^4}\right )-x^3\,\left (\frac {4\,a\,d^7}{3\,b^5}-\frac {7\,c\,d^6}{3\,b^4}\right )-\frac {\frac {-107\,a^7\,d^7+518\,a^6\,b\,c\,d^6-987\,a^5\,b^2\,c^2\,d^5+910\,a^4\,b^3\,c^3\,d^4-385\,a^3\,b^4\,c^4\,d^3+42\,a^2\,b^5\,c^5\,d^2+7\,a\,b^6\,c^6\,d+2\,b^7\,c^7}{6\,b}+x\,\left (-\frac {77\,a^6\,d^7}{2}+189\,a^5\,b\,c\,d^6-\frac {735\,a^4\,b^2\,c^2\,d^5}{2}+350\,a^3\,b^3\,c^3\,d^4-\frac {315\,a^2\,b^4\,c^4\,d^3}{2}+21\,a\,b^5\,c^5\,d^2+\frac {7\,b^6\,c^6\,d}{2}\right )-x^2\,\left (21\,a^5\,b\,d^7-105\,a^4\,b^2\,c\,d^6+210\,a^3\,b^3\,c^2\,d^5-210\,a^2\,b^4\,c^3\,d^4+105\,a\,b^5\,c^4\,d^3-21\,b^6\,c^5\,d^2\right )}{a^3\,b^7+3\,a^2\,b^8\,x+3\,a\,b^9\,x^2+b^{10}\,x^3}-x\,\left (\frac {4\,a\,\left (\frac {4\,a\,\left (\frac {4\,a\,d^7}{b^5}-\frac {7\,c\,d^6}{b^4}\right )}{b}-\frac {6\,a^2\,d^7}{b^6}+\frac {21\,c^2\,d^5}{b^4}\right )}{b}+\frac {4\,a^3\,d^7}{b^7}-\frac {35\,c^3\,d^4}{b^4}-\frac {6\,a^2\,\left (\frac {4\,a\,d^7}{b^5}-\frac {7\,c\,d^6}{b^4}\right )}{b^2}\right )+\frac {\ln \left (a+b\,x\right )\,\left (35\,a^4\,d^7-140\,a^3\,b\,c\,d^6+210\,a^2\,b^2\,c^2\,d^5-140\,a\,b^3\,c^3\,d^4+35\,b^4\,c^4\,d^3\right )}{b^8}+\frac {d^7\,x^4}{4\,b^4} \] Input:
int((c + d*x)^7/(a + b*x)^4,x)
Output:
x^2*((2*a*((4*a*d^7)/b^5 - (7*c*d^6)/b^4))/b - (3*a^2*d^7)/b^6 + (21*c^2*d ^5)/(2*b^4)) - x^3*((4*a*d^7)/(3*b^5) - (7*c*d^6)/(3*b^4)) - ((2*b^7*c^7 - 107*a^7*d^7 + 42*a^2*b^5*c^5*d^2 - 385*a^3*b^4*c^4*d^3 + 910*a^4*b^3*c^3* d^4 - 987*a^5*b^2*c^2*d^5 + 7*a*b^6*c^6*d + 518*a^6*b*c*d^6)/(6*b) + x*((7 *b^6*c^6*d)/2 - (77*a^6*d^7)/2 + 21*a*b^5*c^5*d^2 - (315*a^2*b^4*c^4*d^3)/ 2 + 350*a^3*b^3*c^3*d^4 - (735*a^4*b^2*c^2*d^5)/2 + 189*a^5*b*c*d^6) - x^2 *(21*a^5*b*d^7 - 21*b^6*c^5*d^2 + 105*a*b^5*c^4*d^3 - 105*a^4*b^2*c*d^6 - 210*a^2*b^4*c^3*d^4 + 210*a^3*b^3*c^2*d^5))/(a^3*b^7 + b^10*x^3 + 3*a^2*b^ 8*x + 3*a*b^9*x^2) - x*((4*a*((4*a*((4*a*d^7)/b^5 - (7*c*d^6)/b^4))/b - (6 *a^2*d^7)/b^6 + (21*c^2*d^5)/b^4))/b + (4*a^3*d^7)/b^7 - (35*c^3*d^4)/b^4 - (6*a^2*((4*a*d^7)/b^5 - (7*c*d^6)/b^4))/b^2) + (log(a + b*x)*(35*a^4*d^7 + 35*b^4*c^4*d^3 - 140*a*b^3*c^3*d^4 + 210*a^2*b^2*c^2*d^5 - 140*a^3*b*c* d^6))/b^8 + (d^7*x^4)/(4*b^4)
Time = 0.17 (sec) , antiderivative size = 840, normalized size of antiderivative = 4.49 \[ \int \frac {(c+d x)^7}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:
int((d*x+c)^7/(b*x+a)^4,x)
Output:
(420*log(a + b*x)*a**8*d**7 - 1680*log(a + b*x)*a**7*b*c*d**6 + 1260*log(a + b*x)*a**7*b*d**7*x + 2520*log(a + b*x)*a**6*b**2*c**2*d**5 - 5040*log(a + b*x)*a**6*b**2*c*d**6*x + 1260*log(a + b*x)*a**6*b**2*d**7*x**2 - 1680* log(a + b*x)*a**5*b**3*c**3*d**4 + 7560*log(a + b*x)*a**5*b**3*c**2*d**5*x - 5040*log(a + b*x)*a**5*b**3*c*d**6*x**2 + 420*log(a + b*x)*a**5*b**3*d* *7*x**3 + 420*log(a + b*x)*a**4*b**4*c**4*d**3 - 5040*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 - 1680*log(a + b*x)*a**4*b**4*c*d**6*x**3 + 1260*log(a + b*x)*a**3*b**5*c**4*d**3*x - 50 40*log(a + b*x)*a**3*b**5*c**3*d**4*x**2 + 2520*log(a + b*x)*a**3*b**5*c** 2*d**5*x**3 + 1260*log(a + b*x)*a**2*b**6*c**4*d**3*x**2 - 1680*log(a + b* x)*a**2*b**6*c**3*d**4*x**3 + 420*log(a + b*x)*a*b**7*c**4*d**3*x**3 + 350 *a**8*d**7 - 1400*a**7*b*c*d**6 + 630*a**7*b*d**7*x + 2100*a**6*b**2*c**2* d**5 - 2520*a**6*b**2*c*d**6*x - 1400*a**5*b**3*c**3*d**4 + 3780*a**5*b**3 *c**2*d**5*x - 420*a**5*b**3*d**7*x**3 + 350*a**4*b**4*c**4*d**3 - 2520*a* *4*b**4*c**3*d**4*x + 1680*a**4*b**4*c*d**6*x**3 - 105*a**4*b**4*d**7*x**4 + 630*a**3*b**5*c**4*d**3*x - 2520*a**3*b**5*c**2*d**5*x**3 + 420*a**3*b* *5*c*d**6*x**4 + 21*a**3*b**5*d**7*x**5 - 14*a**2*b**6*c**6*d + 1680*a**2* b**6*c**3*d**4*x**3 - 630*a**2*b**6*c**2*d**5*x**4 - 84*a**2*b**6*c*d**6*x **5 - 7*a**2*b**6*d**7*x**6 - 4*a*b**7*c**7 - 42*a*b**7*c**6*d*x - 420*a*b **7*c**4*d**3*x**3 + 420*a*b**7*c**3*d**4*x**4 + 126*a*b**7*c**2*d**5*x...