\(\int \frac {(a+b x)^6}{(c+d x)^8} \, dx\) [197]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=\frac {(a+b x)^7}{7 (b c-a d) (c+d x)^7} \] Output:

1/7*(b*x+a)^7/(-a*d+b*c)/(d*x+c)^7
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(271\) vs. \(2(28)=56\).

Time = 0.05 (sec) , antiderivative size = 271, normalized size of antiderivative = 9.68 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {a^6 d^6+a^5 b d^5 (c+7 d x)+a^4 b^2 d^4 \left (c^2+7 c d x+21 d^2 x^2\right )+a^3 b^3 d^3 \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+a^2 b^4 d^2 \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )+a b^5 d \left (c^5+7 c^4 d x+21 c^3 d^2 x^2+35 c^2 d^3 x^3+35 c d^4 x^4+21 d^5 x^5\right )+b^6 \left (c^6+7 c^5 d x+21 c^4 d^2 x^2+35 c^3 d^3 x^3+35 c^2 d^4 x^4+21 c d^5 x^5+7 d^6 x^6\right )}{7 d^7 (c+d x)^7} \] Input:

Integrate[(a + b*x)^6/(c + d*x)^8,x]
 

Output:

-1/7*(a^6*d^6 + a^5*b*d^5*(c + 7*d*x) + a^4*b^2*d^4*(c^2 + 7*c*d*x + 21*d^ 
2*x^2) + a^3*b^3*d^3*(c^3 + 7*c^2*d*x + 21*c*d^2*x^2 + 35*d^3*x^3) + a^2*b 
^4*d^2*(c^4 + 7*c^3*d*x + 21*c^2*d^2*x^2 + 35*c*d^3*x^3 + 35*d^4*x^4) + a* 
b^5*d*(c^5 + 7*c^4*d*x + 21*c^3*d^2*x^2 + 35*c^2*d^3*x^3 + 35*c*d^4*x^4 + 
21*d^5*x^5) + b^6*(c^6 + 7*c^5*d*x + 21*c^4*d^2*x^2 + 35*c^3*d^3*x^3 + 35* 
c^2*d^4*x^4 + 21*c*d^5*x^5 + 7*d^6*x^6))/(d^7*(c + d*x)^7)
 

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {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.

\(\displaystyle \int \frac {(a+b x)^6}{(c+d x)^8} \, dx\)

\(\Big \downarrow \) 48

\(\displaystyle \frac {(a+b x)^7}{7 (c+d x)^7 (b c-a d)}\)

Input:

Int[(a + b*x)^6/(c + d*x)^8,x]
 

Output:

(a + b*x)^7/(7*(b*c - a*d)*(c + d*x)^7)
 

Defintions of rubi rules used

rule 48

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(26)=52\).

Time = 0.12 (sec) , antiderivative size = 314, normalized size of antiderivative = 11.21

method	result	size
risch	\(\frac {-\frac {b^{6} x^{6}}{d}-\frac {3 b^{5} \left (a d +b c \right ) x^{5}}{d^{2}}-\frac {5 b^{4} \left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{4}}{d^{3}}-\frac {5 b^{3} \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{3}}{d^{4}}-\frac {3 b^{2} \left (d^{4} a^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +c^{4} b^{4}\right ) x^{2}}{d^{5}}-\frac {b \left (a^{5} d^{5}+a^{4} b c \,d^{4}+a^{3} b^{2} c^{2} d^{3}+a^{2} b^{3} c^{3} d^{2}+a \,b^{4} c^{4} d +c^{5} b^{5}\right ) x}{d^{6}}-\frac {a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +c^{6} b^{6}}{7 d^{7}}}{\left (x d +c \right )^{7}}\)	\(314\)
norman	\(\frac {-\frac {b^{6} x^{6}}{d}-\frac {3 \left (a \,b^{5} d +b^{6} c \right ) x^{5}}{d^{2}}-\frac {5 \left (a^{2} b^{4} d^{2}+a c \,b^{5} d +b^{6} c^{2}\right ) x^{4}}{d^{3}}-\frac {5 \left (a^{3} b^{3} d^{3}+a^{2} c \,b^{4} d^{2}+a \,c^{2} b^{5} d +b^{6} c^{3}\right ) x^{3}}{d^{4}}-\frac {3 \left (a^{4} b^{2} d^{4}+a^{3} b^{3} d^{3} c +a^{2} b^{4} c^{2} d^{2}+a \,b^{5} c^{3} d +b^{6} c^{4}\right ) x^{2}}{d^{5}}-\frac {\left (a^{5} b \,d^{5}+a^{4} c \,b^{2} d^{4}+a^{3} c^{2} b^{3} d^{3}+a^{2} c^{3} b^{4} d^{2}+a \,c^{4} b^{5} d +b^{6} c^{5}\right ) x}{d^{6}}-\frac {a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +c^{6} b^{6}}{7 d^{7}}}{\left (x d +c \right )^{7}}\)	\(324\)
default	\(-\frac {b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{d^{7} \left (x d +c \right )^{6}}-\frac {3 b^{2} \left (d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{d^{7} \left (x d +c \right )^{5}}-\frac {b^{6}}{d^{7} \left (x d +c \right )}-\frac {a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}}{7 d^{7} \left (x d +c \right )^{7}}-\frac {3 b^{5} \left (a d -b c \right )}{d^{7} \left (x d +c \right )^{2}}-\frac {5 b^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{7} \left (x d +c \right )^{4}}-\frac {5 b^{4} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{7} \left (x d +c \right )^{3}}\)	\(357\)
gosper	\(-\frac {7 x^{6} b^{6} d^{6}+21 x^{5} a \,b^{5} d^{6}+21 x^{5} b^{6} c \,d^{5}+35 x^{4} a^{2} b^{4} d^{6}+35 x^{4} a \,b^{5} c \,d^{5}+35 x^{4} b^{6} c^{2} d^{4}+35 x^{3} a^{3} b^{3} d^{6}+35 x^{3} a^{2} b^{4} c \,d^{5}+35 x^{3} a \,b^{5} c^{2} d^{4}+35 x^{3} b^{6} c^{3} d^{3}+21 a^{4} b^{2} d^{6} x^{2}+21 a^{3} b^{3} c \,d^{5} x^{2}+21 a^{2} b^{4} c^{2} d^{4} x^{2}+21 a \,b^{5} c^{3} d^{3} x^{2}+21 b^{6} c^{4} d^{2} x^{2}+7 a^{5} b \,d^{6} x +7 x \,a^{4} b^{2} c \,d^{5}+7 a^{3} b^{3} c^{2} d^{4} x +7 a^{2} b^{4} c^{3} d^{3} x +7 a \,b^{5} c^{4} d^{2} x +7 b^{6} c^{5} d x +a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +c^{6} b^{6}}{7 \left (x d +c \right )^{7} d^{7}}\)	\(370\)
orering	\(-\frac {7 x^{6} b^{6} d^{6}+21 x^{5} a \,b^{5} d^{6}+21 x^{5} b^{6} c \,d^{5}+35 x^{4} a^{2} b^{4} d^{6}+35 x^{4} a \,b^{5} c \,d^{5}+35 x^{4} b^{6} c^{2} d^{4}+35 x^{3} a^{3} b^{3} d^{6}+35 x^{3} a^{2} b^{4} c \,d^{5}+35 x^{3} a \,b^{5} c^{2} d^{4}+35 x^{3} b^{6} c^{3} d^{3}+21 a^{4} b^{2} d^{6} x^{2}+21 a^{3} b^{3} c \,d^{5} x^{2}+21 a^{2} b^{4} c^{2} d^{4} x^{2}+21 a \,b^{5} c^{3} d^{3} x^{2}+21 b^{6} c^{4} d^{2} x^{2}+7 a^{5} b \,d^{6} x +7 x \,a^{4} b^{2} c \,d^{5}+7 a^{3} b^{3} c^{2} d^{4} x +7 a^{2} b^{4} c^{3} d^{3} x +7 a \,b^{5} c^{4} d^{2} x +7 b^{6} c^{5} d x +a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +c^{6} b^{6}}{7 \left (x d +c \right )^{7} d^{7}}\)	\(370\)
parallelrisch	\(\frac {-7 x^{6} b^{6} d^{6}-21 x^{5} a \,b^{5} d^{6}-21 x^{5} b^{6} c \,d^{5}-35 x^{4} a^{2} b^{4} d^{6}-35 x^{4} a \,b^{5} c \,d^{5}-35 x^{4} b^{6} c^{2} d^{4}-35 x^{3} a^{3} b^{3} d^{6}-35 x^{3} a^{2} b^{4} c \,d^{5}-35 x^{3} a \,b^{5} c^{2} d^{4}-35 x^{3} b^{6} c^{3} d^{3}-21 a^{4} b^{2} d^{6} x^{2}-21 a^{3} b^{3} c \,d^{5} x^{2}-21 a^{2} b^{4} c^{2} d^{4} x^{2}-21 a \,b^{5} c^{3} d^{3} x^{2}-21 b^{6} c^{4} d^{2} x^{2}-7 a^{5} b \,d^{6} x -7 x \,a^{4} b^{2} c \,d^{5}-7 a^{3} b^{3} c^{2} d^{4} x -7 a^{2} b^{4} c^{3} d^{3} x -7 a \,b^{5} c^{4} d^{2} x -7 b^{6} c^{5} d x -a^{6} d^{6}-a^{5} b c \,d^{5}-a^{4} b^{2} c^{2} d^{4}-a^{3} b^{3} c^{3} d^{3}-a^{2} b^{4} c^{4} d^{2}-a \,b^{5} c^{5} d -c^{6} b^{6}}{7 d^{7} \left (x d +c \right )^{7}}\)	\(377\)

Input:

int((b*x+a)^6/(d*x+c)^8,x,method=_RETURNVERBOSE)
 

Output:

(-b^6/d*x^6-3*b^5*(a*d+b*c)/d^2*x^5-5*b^4*(a^2*d^2+a*b*c*d+b^2*c^2)/d^3*x^ 
4-5*b^3*(a^3*d^3+a^2*b*c*d^2+a*b^2*c^2*d+b^3*c^3)/d^4*x^3-3*b^2*(a^4*d^4+a 
^3*b*c*d^3+a^2*b^2*c^2*d^2+a*b^3*c^3*d+b^4*c^4)/d^5*x^2-b*(a^5*d^5+a^4*b*c 
*d^4+a^3*b^2*c^2*d^3+a^2*b^3*c^3*d^2+a*b^4*c^4*d+b^5*c^5)/d^6*x-1/7*(a^6*d 
^6+a^5*b*c*d^5+a^4*b^2*c^2*d^4+a^3*b^3*c^3*d^3+a^2*b^4*c^4*d^2+a*b^5*c^5*d 
+b^6*c^6)/d^7)/(d*x+c)^7
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (26) = 52\).

Time = 0.08 (sec) , antiderivative size = 398, normalized size of antiderivative = 14.21 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \, {\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \, {\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \, {\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \, {\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \, {\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\right )}} \] Input:

integrate((b*x+a)^6/(d*x+c)^8,x, algorithm="fricas")
 

Output:

-1/7*(7*b^6*d^6*x^6 + b^6*c^6 + a*b^5*c^5*d + a^2*b^4*c^4*d^2 + a^3*b^3*c^ 
3*d^3 + a^4*b^2*c^2*d^4 + a^5*b*c*d^5 + a^6*d^6 + 21*(b^6*c*d^5 + a*b^5*d^ 
6)*x^5 + 35*(b^6*c^2*d^4 + a*b^5*c*d^5 + a^2*b^4*d^6)*x^4 + 35*(b^6*c^3*d^ 
3 + a*b^5*c^2*d^4 + a^2*b^4*c*d^5 + a^3*b^3*d^6)*x^3 + 21*(b^6*c^4*d^2 + a 
*b^5*c^3*d^3 + a^2*b^4*c^2*d^4 + a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^2 + 7*(b^6 
*c^5*d + a*b^5*c^4*d^2 + a^2*b^4*c^3*d^3 + a^3*b^3*c^2*d^4 + a^4*b^2*c*d^5 
 + a^5*b*d^6)*x)/(d^14*x^7 + 7*c*d^13*x^6 + 21*c^2*d^12*x^5 + 35*c^3*d^11* 
x^4 + 35*c^4*d^10*x^3 + 21*c^5*d^9*x^2 + 7*c^6*d^8*x + c^7*d^7)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=\text {Timed out} \] Input:

integrate((b*x+a)**6/(d*x+c)**8,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (26) = 52\).

Time = 0.05 (sec) , antiderivative size = 398, normalized size of antiderivative = 14.21 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {7 \, b^{6} d^{6} x^{6} + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6} + 21 \, {\left (b^{6} c d^{5} + a b^{5} d^{6}\right )} x^{5} + 35 \, {\left (b^{6} c^{2} d^{4} + a b^{5} c d^{5} + a^{2} b^{4} d^{6}\right )} x^{4} + 35 \, {\left (b^{6} c^{3} d^{3} + a b^{5} c^{2} d^{4} + a^{2} b^{4} c d^{5} + a^{3} b^{3} d^{6}\right )} x^{3} + 21 \, {\left (b^{6} c^{4} d^{2} + a b^{5} c^{3} d^{3} + a^{2} b^{4} c^{2} d^{4} + a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{2} + 7 \, {\left (b^{6} c^{5} d + a b^{5} c^{4} d^{2} + a^{2} b^{4} c^{3} d^{3} + a^{3} b^{3} c^{2} d^{4} + a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x}{7 \, {\left (d^{14} x^{7} + 7 \, c d^{13} x^{6} + 21 \, c^{2} d^{12} x^{5} + 35 \, c^{3} d^{11} x^{4} + 35 \, c^{4} d^{10} x^{3} + 21 \, c^{5} d^{9} x^{2} + 7 \, c^{6} d^{8} x + c^{7} d^{7}\right )}} \] Input:

integrate((b*x+a)^6/(d*x+c)^8,x, algorithm="maxima")
 

Output:

-1/7*(7*b^6*d^6*x^6 + b^6*c^6 + a*b^5*c^5*d + a^2*b^4*c^4*d^2 + a^3*b^3*c^ 
3*d^3 + a^4*b^2*c^2*d^4 + a^5*b*c*d^5 + a^6*d^6 + 21*(b^6*c*d^5 + a*b^5*d^ 
6)*x^5 + 35*(b^6*c^2*d^4 + a*b^5*c*d^5 + a^2*b^4*d^6)*x^4 + 35*(b^6*c^3*d^ 
3 + a*b^5*c^2*d^4 + a^2*b^4*c*d^5 + a^3*b^3*d^6)*x^3 + 21*(b^6*c^4*d^2 + a 
*b^5*c^3*d^3 + a^2*b^4*c^2*d^4 + a^3*b^3*c*d^5 + a^4*b^2*d^6)*x^2 + 7*(b^6 
*c^5*d + a*b^5*c^4*d^2 + a^2*b^4*c^3*d^3 + a^3*b^3*c^2*d^4 + a^4*b^2*c*d^5 
 + a^5*b*d^6)*x)/(d^14*x^7 + 7*c*d^13*x^6 + 21*c^2*d^12*x^5 + 35*c^3*d^11* 
x^4 + 35*c^4*d^10*x^3 + 21*c^5*d^9*x^2 + 7*c^6*d^8*x + c^7*d^7)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (26) = 52\).

Time = 0.12 (sec) , antiderivative size = 369, normalized size of antiderivative = 13.18 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {7 \, b^{6} d^{6} x^{6} + 21 \, b^{6} c d^{5} x^{5} + 21 \, a b^{5} d^{6} x^{5} + 35 \, b^{6} c^{2} d^{4} x^{4} + 35 \, a b^{5} c d^{5} x^{4} + 35 \, a^{2} b^{4} d^{6} x^{4} + 35 \, b^{6} c^{3} d^{3} x^{3} + 35 \, a b^{5} c^{2} d^{4} x^{3} + 35 \, a^{2} b^{4} c d^{5} x^{3} + 35 \, a^{3} b^{3} d^{6} x^{3} + 21 \, b^{6} c^{4} d^{2} x^{2} + 21 \, a b^{5} c^{3} d^{3} x^{2} + 21 \, a^{2} b^{4} c^{2} d^{4} x^{2} + 21 \, a^{3} b^{3} c d^{5} x^{2} + 21 \, a^{4} b^{2} d^{6} x^{2} + 7 \, b^{6} c^{5} d x + 7 \, a b^{5} c^{4} d^{2} x + 7 \, a^{2} b^{4} c^{3} d^{3} x + 7 \, a^{3} b^{3} c^{2} d^{4} x + 7 \, a^{4} b^{2} c d^{5} x + 7 \, a^{5} b d^{6} x + b^{6} c^{6} + a b^{5} c^{5} d + a^{2} b^{4} c^{4} d^{2} + a^{3} b^{3} c^{3} d^{3} + a^{4} b^{2} c^{2} d^{4} + a^{5} b c d^{5} + a^{6} d^{6}}{7 \, {\left (d x + c\right )}^{7} d^{7}} \] Input:

integrate((b*x+a)^6/(d*x+c)^8,x, algorithm="giac")
 

Output:

-1/7*(7*b^6*d^6*x^6 + 21*b^6*c*d^5*x^5 + 21*a*b^5*d^6*x^5 + 35*b^6*c^2*d^4 
*x^4 + 35*a*b^5*c*d^5*x^4 + 35*a^2*b^4*d^6*x^4 + 35*b^6*c^3*d^3*x^3 + 35*a 
*b^5*c^2*d^4*x^3 + 35*a^2*b^4*c*d^5*x^3 + 35*a^3*b^3*d^6*x^3 + 21*b^6*c^4* 
d^2*x^2 + 21*a*b^5*c^3*d^3*x^2 + 21*a^2*b^4*c^2*d^4*x^2 + 21*a^3*b^3*c*d^5 
*x^2 + 21*a^4*b^2*d^6*x^2 + 7*b^6*c^5*d*x + 7*a*b^5*c^4*d^2*x + 7*a^2*b^4* 
c^3*d^3*x + 7*a^3*b^3*c^2*d^4*x + 7*a^4*b^2*c*d^5*x + 7*a^5*b*d^6*x + b^6* 
c^6 + a*b^5*c^5*d + a^2*b^4*c^4*d^2 + a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4 + 
a^5*b*c*d^5 + a^6*d^6)/((d*x + c)^7*d^7)
 

Mupad [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 378, normalized size of antiderivative = 13.50 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=-\frac {\frac {a^6\,d^6+a^5\,b\,c\,d^5+a^4\,b^2\,c^2\,d^4+a^3\,b^3\,c^3\,d^3+a^2\,b^4\,c^4\,d^2+a\,b^5\,c^5\,d+b^6\,c^6}{7\,d^7}+\frac {b^6\,x^6}{d}+\frac {5\,b^3\,x^3\,\left (a^3\,d^3+a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3\right )}{d^4}+\frac {b\,x\,\left (a^5\,d^5+a^4\,b\,c\,d^4+a^3\,b^2\,c^2\,d^3+a^2\,b^3\,c^3\,d^2+a\,b^4\,c^4\,d+b^5\,c^5\right )}{d^6}+\frac {3\,b^5\,x^5\,\left (a\,d+b\,c\right )}{d^2}+\frac {3\,b^2\,x^2\,\left (a^4\,d^4+a^3\,b\,c\,d^3+a^2\,b^2\,c^2\,d^2+a\,b^3\,c^3\,d+b^4\,c^4\right )}{d^5}+\frac {5\,b^4\,x^4\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{d^3}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \] Input:

int((a + b*x)^6/(c + d*x)^8,x)
 

Output:

-((a^6*d^6 + b^6*c^6 + a^2*b^4*c^4*d^2 + a^3*b^3*c^3*d^3 + a^4*b^2*c^2*d^4 
 + a*b^5*c^5*d + a^5*b*c*d^5)/(7*d^7) + (b^6*x^6)/d + (5*b^3*x^3*(a^3*d^3 
+ b^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2))/d^4 + (b*x*(a^5*d^5 + b^5*c^5 + a^ 
2*b^3*c^3*d^2 + a^3*b^2*c^2*d^3 + a*b^4*c^4*d + a^4*b*c*d^4))/d^6 + (3*b^5 
*x^5*(a*d + b*c))/d^2 + (3*b^2*x^2*(a^4*d^4 + b^4*c^4 + a^2*b^2*c^2*d^2 + 
a*b^3*c^3*d + a^3*b*c*d^3))/d^5 + (5*b^4*x^4*(a^2*d^2 + b^2*c^2 + a*b*c*d) 
)/d^3)/(c^7 + d^7*x^7 + 7*c*d^6*x^6 + 21*c^5*d^2*x^2 + 35*c^4*d^3*x^3 + 35 
*c^3*d^4*x^4 + 21*c^2*d^5*x^5 + 7*c^6*d*x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 383, normalized size of antiderivative = 13.68 \[ \int \frac {(a+b x)^6}{(c+d x)^8} \, dx=\frac {b^{6} d^{6} x^{7}-21 a \,b^{5} c \,d^{5} x^{5}-35 a^{2} b^{4} c \,d^{5} x^{4}-35 a \,b^{5} c^{2} d^{4} x^{4}-35 a^{3} b^{3} c \,d^{5} x^{3}-35 a^{2} b^{4} c^{2} d^{4} x^{3}-35 a \,b^{5} c^{3} d^{3} x^{3}-21 a^{4} b^{2} c \,d^{5} x^{2}-21 a^{3} b^{3} c^{2} d^{4} x^{2}-21 a^{2} b^{4} c^{3} d^{3} x^{2}-21 a \,b^{5} c^{4} d^{2} x^{2}-7 a^{5} b c \,d^{5} x -7 a^{4} b^{2} c^{2} d^{4} x -7 a^{3} b^{3} c^{3} d^{3} x -7 a^{2} b^{4} c^{4} d^{2} x -7 a \,b^{5} c^{5} d x -a^{6} c \,d^{5}-a^{5} b \,c^{2} d^{4}-a^{4} b^{2} c^{3} d^{3}-a^{3} b^{3} c^{4} d^{2}-a^{2} b^{4} c^{5} d -a \,b^{5} c^{6}}{7 c \,d^{6} \left (d^{7} x^{7}+7 c \,d^{6} x^{6}+21 c^{2} d^{5} x^{5}+35 c^{3} d^{4} x^{4}+35 c^{4} d^{3} x^{3}+21 c^{5} d^{2} x^{2}+7 c^{6} d x +c^{7}\right )} \] Input:

int((b*x+a)^6/(d*x+c)^8,x)
 

Output:

( - a**6*c*d**5 - a**5*b*c**2*d**4 - 7*a**5*b*c*d**5*x - a**4*b**2*c**3*d* 
*3 - 7*a**4*b**2*c**2*d**4*x - 21*a**4*b**2*c*d**5*x**2 - a**3*b**3*c**4*d 
**2 - 7*a**3*b**3*c**3*d**3*x - 21*a**3*b**3*c**2*d**4*x**2 - 35*a**3*b**3 
*c*d**5*x**3 - a**2*b**4*c**5*d - 7*a**2*b**4*c**4*d**2*x - 21*a**2*b**4*c 
**3*d**3*x**2 - 35*a**2*b**4*c**2*d**4*x**3 - 35*a**2*b**4*c*d**5*x**4 - a 
*b**5*c**6 - 7*a*b**5*c**5*d*x - 21*a*b**5*c**4*d**2*x**2 - 35*a*b**5*c**3 
*d**3*x**3 - 35*a*b**5*c**2*d**4*x**4 - 21*a*b**5*c*d**5*x**5 + b**6*d**6* 
x**7)/(7*c*d**6*(c**7 + 7*c**6*d*x + 21*c**5*d**2*x**2 + 35*c**4*d**3*x**3 
 + 35*c**3*d**4*x**4 + 21*c**2*d**5*x**5 + 7*c*d**6*x**6 + d**7*x**7))