\(\int \frac {(c+d x)^7}{(a+b x)^9} \, dx\) [85]

Optimal result

Integrand size = 15, antiderivative size = 28 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {(c+d x)^8}{8 (b c-a d) (a+b x)^8} \] Output:

-1/8*(d*x+c)^8/(-a*d+b*c)/(b*x+a)^8
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(353\) vs. \(2(28)=56\).

Time = 0.07 (sec) , antiderivative size = 353, normalized size of antiderivative = 12.61 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {a^7 d^7+a^6 b d^6 (c+8 d x)+a^5 b^2 d^5 \left (c^2+8 c d x+28 d^2 x^2\right )+a^4 b^3 d^4 \left (c^3+8 c^2 d x+28 c d^2 x^2+56 d^3 x^3\right )+a^3 b^4 d^3 \left (c^4+8 c^3 d x+28 c^2 d^2 x^2+56 c d^3 x^3+70 d^4 x^4\right )+a^2 b^5 d^2 \left (c^5+8 c^4 d x+28 c^3 d^2 x^2+56 c^2 d^3 x^3+70 c d^4 x^4+56 d^5 x^5\right )+a b^6 d \left (c^6+8 c^5 d x+28 c^4 d^2 x^2+56 c^3 d^3 x^3+70 c^2 d^4 x^4+56 c d^5 x^5+28 d^6 x^6\right )+b^7 \left (c^7+8 c^6 d x+28 c^5 d^2 x^2+56 c^4 d^3 x^3+70 c^3 d^4 x^4+56 c^2 d^5 x^5+28 c d^6 x^6+8 d^7 x^7\right )}{8 b^8 (a+b x)^8} \] Input:

Integrate[(c + d*x)^7/(a + b*x)^9,x]
 

Output:

-1/8*(a^7*d^7 + a^6*b*d^6*(c + 8*d*x) + a^5*b^2*d^5*(c^2 + 8*c*d*x + 28*d^ 
2*x^2) + a^4*b^3*d^4*(c^3 + 8*c^2*d*x + 28*c*d^2*x^2 + 56*d^3*x^3) + a^3*b 
^4*d^3*(c^4 + 8*c^3*d*x + 28*c^2*d^2*x^2 + 56*c*d^3*x^3 + 70*d^4*x^4) + a^ 
2*b^5*d^2*(c^5 + 8*c^4*d*x + 28*c^3*d^2*x^2 + 56*c^2*d^3*x^3 + 70*c*d^4*x^ 
4 + 56*d^5*x^5) + a*b^6*d*(c^6 + 8*c^5*d*x + 28*c^4*d^2*x^2 + 56*c^3*d^3*x 
^3 + 70*c^2*d^4*x^4 + 56*c*d^5*x^5 + 28*d^6*x^6) + b^7*(c^7 + 8*c^6*d*x + 
28*c^5*d^2*x^2 + 56*c^4*d^3*x^3 + 70*c^3*d^4*x^4 + 56*c^2*d^5*x^5 + 28*c*d 
^6*x^6 + 8*d^7*x^7))/(b^8*(a + b*x)^8)
 

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.

\(\displaystyle \int \frac {(c+d x)^7}{(a+b x)^9} \, dx\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {(c+d x)^8}{8 (a+b x)^8 (b c-a d)}\)

Input:

Int[(c + d*x)^7/(a + b*x)^9,x]
 

Output:

-1/8*(c + d*x)^8/((b*c - a*d)*(a + b*x)^8)
 

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. \(409\) vs. \(2(26)=52\).

Time = 0.13 (sec) , antiderivative size = 410, normalized size of antiderivative = 14.64

method	result	size
risch	\(\frac {-\frac {d^{7} x^{7}}{b}-\frac {7 d^{6} \left (a d +b c \right ) x^{6}}{2 b^{2}}-\frac {7 d^{5} \left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{5}}{b^{3}}-\frac {35 d^{4} \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x^{4}}{4 b^{4}}-\frac {7 d^{3} \left (a^{4} d^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +c^{4} b^{4}\right ) x^{3}}{b^{5}}-\frac {7 d^{2} \left (a^{5} d^{5}+a^{4} b c \,d^{4}+a^{3} b^{2} c^{2} d^{3}+a^{2} b^{3} c^{3} d^{2}+a \,b^{4} c^{4} d +c^{5} b^{5}\right ) x^{2}}{2 b^{6}}-\frac {d \left (a^{6} d^{6}+a^{5} b c \,d^{5}+a^{4} b^{2} c^{2} d^{4}+a^{3} b^{3} c^{3} d^{3}+a^{2} b^{4} c^{4} d^{2}+a \,b^{5} c^{5} d +c^{6} b^{6}\right ) x}{b^{7}}-\frac {a^{7} d^{7}+a^{6} b c \,d^{6}+a^{5} b^{2} c^{2} d^{5}+a^{4} b^{3} c^{3} d^{4}+a^{3} b^{4} c^{4} d^{3}+a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +b^{7} c^{7}}{8 b^{8}}}{\left (b x +a \right )^{8}}\)	\(410\)
norman	\(\frac {-\frac {d^{7} x^{7}}{b}+\frac {7 \left (-a \,d^{7}-b c \,d^{6}\right ) x^{6}}{2 b^{2}}+\frac {7 \left (-a^{2} d^{7}-a b c \,d^{6}-b^{2} c^{2} d^{5}\right ) x^{5}}{b^{3}}+\frac {35 \left (-a^{3} d^{7}-a^{2} b c \,d^{6}-a \,b^{2} c^{2} d^{5}-b^{3} c^{3} d^{4}\right ) x^{4}}{4 b^{4}}+\frac {7 \left (-a^{4} d^{7}-a^{3} b c \,d^{6}-a^{2} b^{2} c^{2} d^{5}-a \,b^{3} c^{3} d^{4}-b^{4} c^{4} d^{3}\right ) x^{3}}{b^{5}}+\frac {7 \left (-a^{5} d^{7}-a^{4} b c \,d^{6}-a^{3} b^{2} c^{2} d^{5}-a^{2} b^{3} c^{3} d^{4}-a \,b^{4} c^{4} d^{3}-b^{5} c^{5} d^{2}\right ) x^{2}}{2 b^{6}}+\frac {\left (-a^{6} d^{7}-a^{5} b c \,d^{6}-a^{4} b^{2} c^{2} d^{5}-a^{3} b^{3} c^{3} d^{4}-a^{2} b^{4} c^{4} d^{3}-a \,b^{5} c^{5} d^{2}-b^{6} c^{6} d \right ) x}{b^{7}}+\frac {-a^{7} d^{7}-a^{6} b c \,d^{6}-a^{5} b^{2} c^{2} d^{5}-a^{4} b^{3} c^{3} d^{4}-a^{3} b^{4} c^{4} d^{3}-a^{2} b^{5} c^{5} d^{2}-a \,b^{6} c^{6} d -b^{7} c^{7}}{8 b^{8}}}{\left (b x +a \right )^{8}}\)	\(456\)
default	\(-\frac {7 d^{3} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}\right )}{b^{8} \left (b x +a \right )^{5}}+\frac {35 d^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{4 b^{8} \left (b x +a \right )^{4}}-\frac {d \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +c^{6} b^{6}\right )}{b^{8} \left (b x +a \right )^{7}}-\frac {-a^{7} d^{7}+7 a^{6} b c \,d^{6}-21 a^{5} b^{2} c^{2} d^{5}+35 a^{4} b^{3} c^{3} d^{4}-35 a^{3} b^{4} c^{4} d^{3}+21 a^{2} b^{5} c^{5} d^{2}-7 a \,b^{6} c^{6} d +b^{7} c^{7}}{8 b^{8} \left (b x +a \right )^{8}}+\frac {7 d^{6} \left (a d -b c \right )}{2 b^{8} \left (b x +a \right )^{2}}-\frac {d^{7}}{b^{8} \left (b x +a \right )}-\frac {7 d^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{8} \left (b x +a \right )^{3}}+\frac {7 d^{2} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right )}{2 b^{8} \left (b x +a \right )^{6}}\)	\(464\)
gosper	\(-\frac {8 x^{7} d^{7} b^{7}+28 x^{6} a \,b^{6} d^{7}+28 x^{6} b^{7} c \,d^{6}+56 x^{5} a^{2} b^{5} d^{7}+56 x^{5} a \,b^{6} c \,d^{6}+56 x^{5} b^{7} c^{2} d^{5}+70 x^{4} a^{3} b^{4} d^{7}+70 x^{4} a^{2} b^{5} c \,d^{6}+70 x^{4} a \,b^{6} c^{2} d^{5}+70 x^{4} b^{7} c^{3} d^{4}+56 x^{3} a^{4} b^{3} d^{7}+56 x^{3} a^{3} b^{4} c \,d^{6}+56 x^{3} a^{2} b^{5} c^{2} d^{5}+56 x^{3} a \,b^{6} c^{3} d^{4}+56 x^{3} b^{7} c^{4} d^{3}+28 x^{2} a^{5} b^{2} d^{7}+28 x^{2} a^{4} b^{3} c \,d^{6}+28 x^{2} a^{3} b^{4} c^{2} d^{5}+28 x^{2} a^{2} b^{5} c^{3} d^{4}+28 x^{2} a \,b^{6} c^{4} d^{3}+28 x^{2} b^{7} c^{5} d^{2}+8 x \,a^{6} b \,d^{7}+8 x \,a^{5} b^{2} c \,d^{6}+8 x \,a^{4} b^{3} c^{2} d^{5}+8 x \,a^{3} b^{4} c^{3} d^{4}+8 x \,a^{2} b^{5} c^{4} d^{3}+8 x a \,b^{6} c^{5} d^{2}+8 x \,b^{7} c^{6} d +a^{7} d^{7}+a^{6} b c \,d^{6}+a^{5} b^{2} c^{2} d^{5}+a^{4} b^{3} c^{3} d^{4}+a^{3} b^{4} c^{4} d^{3}+a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +b^{7} c^{7}}{8 \left (b x +a \right )^{8} b^{8}}\)	\(490\)
orering	\(-\frac {8 x^{7} d^{7} b^{7}+28 x^{6} a \,b^{6} d^{7}+28 x^{6} b^{7} c \,d^{6}+56 x^{5} a^{2} b^{5} d^{7}+56 x^{5} a \,b^{6} c \,d^{6}+56 x^{5} b^{7} c^{2} d^{5}+70 x^{4} a^{3} b^{4} d^{7}+70 x^{4} a^{2} b^{5} c \,d^{6}+70 x^{4} a \,b^{6} c^{2} d^{5}+70 x^{4} b^{7} c^{3} d^{4}+56 x^{3} a^{4} b^{3} d^{7}+56 x^{3} a^{3} b^{4} c \,d^{6}+56 x^{3} a^{2} b^{5} c^{2} d^{5}+56 x^{3} a \,b^{6} c^{3} d^{4}+56 x^{3} b^{7} c^{4} d^{3}+28 x^{2} a^{5} b^{2} d^{7}+28 x^{2} a^{4} b^{3} c \,d^{6}+28 x^{2} a^{3} b^{4} c^{2} d^{5}+28 x^{2} a^{2} b^{5} c^{3} d^{4}+28 x^{2} a \,b^{6} c^{4} d^{3}+28 x^{2} b^{7} c^{5} d^{2}+8 x \,a^{6} b \,d^{7}+8 x \,a^{5} b^{2} c \,d^{6}+8 x \,a^{4} b^{3} c^{2} d^{5}+8 x \,a^{3} b^{4} c^{3} d^{4}+8 x \,a^{2} b^{5} c^{4} d^{3}+8 x a \,b^{6} c^{5} d^{2}+8 x \,b^{7} c^{6} d +a^{7} d^{7}+a^{6} b c \,d^{6}+a^{5} b^{2} c^{2} d^{5}+a^{4} b^{3} c^{3} d^{4}+a^{3} b^{4} c^{4} d^{3}+a^{2} b^{5} c^{5} d^{2}+a \,b^{6} c^{6} d +b^{7} c^{7}}{8 \left (b x +a \right )^{8} b^{8}}\)	\(490\)
parallelrisch	\(\frac {-8 x^{7} d^{7} b^{7}-28 x^{6} a \,b^{6} d^{7}-28 x^{6} b^{7} c \,d^{6}-56 x^{5} a^{2} b^{5} d^{7}-56 x^{5} a \,b^{6} c \,d^{6}-56 x^{5} b^{7} c^{2} d^{5}-70 x^{4} a^{3} b^{4} d^{7}-70 x^{4} a^{2} b^{5} c \,d^{6}-70 x^{4} a \,b^{6} c^{2} d^{5}-70 x^{4} b^{7} c^{3} d^{4}-56 x^{3} a^{4} b^{3} d^{7}-56 x^{3} a^{3} b^{4} c \,d^{6}-56 x^{3} a^{2} b^{5} c^{2} d^{5}-56 x^{3} a \,b^{6} c^{3} d^{4}-56 x^{3} b^{7} c^{4} d^{3}-28 x^{2} a^{5} b^{2} d^{7}-28 x^{2} a^{4} b^{3} c \,d^{6}-28 x^{2} a^{3} b^{4} c^{2} d^{5}-28 x^{2} a^{2} b^{5} c^{3} d^{4}-28 x^{2} a \,b^{6} c^{4} d^{3}-28 x^{2} b^{7} c^{5} d^{2}-8 x \,a^{6} b \,d^{7}-8 x \,a^{5} b^{2} c \,d^{6}-8 x \,a^{4} b^{3} c^{2} d^{5}-8 x \,a^{3} b^{4} c^{3} d^{4}-8 x \,a^{2} b^{5} c^{4} d^{3}-8 x a \,b^{6} c^{5} d^{2}-8 x \,b^{7} c^{6} d -a^{7} d^{7}-a^{6} b c \,d^{6}-a^{5} b^{2} c^{2} d^{5}-a^{4} b^{3} c^{3} d^{4}-a^{3} b^{4} c^{4} d^{3}-a^{2} b^{5} c^{5} d^{2}-a \,b^{6} c^{6} d -b^{7} c^{7}}{8 b^{8} \left (b x +a \right )^{8}}\)	\(498\)

Input:

int((d*x+c)^7/(b*x+a)^9,x,method=_RETURNVERBOSE)
 

Output:

(-1/b*d^7*x^7-7/2*d^6*(a*d+b*c)/b^2*x^6-7*d^5*(a^2*d^2+a*b*c*d+b^2*c^2)/b^ 
3*x^5-35/4*d^4*(a^3*d^3+a^2*b*c*d^2+a*b^2*c^2*d+b^3*c^3)/b^4*x^4-7*d^3*(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)/b^5*x^3-7/2*d^2*(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)/b^6 
*x^2-d*(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)/b^7*x-1/8*(a^7*d^7+a^6*b*c*d^6+a^5*b^2*c^2*d^5+a^4* 
b^3*c^3*d^4+a^3*b^4*c^4*d^3+a^2*b^5*c^5*d^2+a*b^6*c^6*d+b^7*c^7)/b^8)/(b*x 
+a)^8
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (26) = 52\).

Time = 0.08 (sec) , antiderivative size = 509, normalized size of antiderivative = 18.18 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {8 \, b^{7} d^{7} x^{7} + b^{7} c^{7} + a b^{6} c^{6} d + a^{2} b^{5} c^{5} d^{2} + a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + a^{7} d^{7} + 28 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 56 \, {\left (b^{7} c^{2} d^{5} + a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \, {\left (b^{7} c^{3} d^{4} + a b^{6} c^{2} d^{5} + a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 56 \, {\left (b^{7} c^{4} d^{3} + a b^{6} c^{3} d^{4} + a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 28 \, {\left (b^{7} c^{5} d^{2} + a b^{6} c^{4} d^{3} + a^{2} b^{5} c^{3} d^{4} + a^{3} b^{4} c^{2} d^{5} + a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 8 \, {\left (b^{7} c^{6} d + a b^{6} c^{5} d^{2} + a^{2} b^{5} c^{4} d^{3} + a^{3} b^{4} c^{3} d^{4} + a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{8 \, {\left (b^{16} x^{8} + 8 \, a b^{15} x^{7} + 28 \, a^{2} b^{14} x^{6} + 56 \, a^{3} b^{13} x^{5} + 70 \, a^{4} b^{12} x^{4} + 56 \, a^{5} b^{11} x^{3} + 28 \, a^{6} b^{10} x^{2} + 8 \, a^{7} b^{9} x + a^{8} b^{8}\right )}} \] Input:

integrate((d*x+c)^7/(b*x+a)^9,x, algorithm="fricas")
 

Output:

-1/8*(8*b^7*d^7*x^7 + b^7*c^7 + a*b^6*c^6*d + a^2*b^5*c^5*d^2 + a^3*b^4*c^ 
4*d^3 + a^4*b^3*c^3*d^4 + a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7 + 28*(b^ 
7*c*d^6 + a*b^6*d^7)*x^6 + 56*(b^7*c^2*d^5 + a*b^6*c*d^6 + a^2*b^5*d^7)*x^ 
5 + 70*(b^7*c^3*d^4 + a*b^6*c^2*d^5 + a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 5 
6*(b^7*c^4*d^3 + a*b^6*c^3*d^4 + a^2*b^5*c^2*d^5 + a^3*b^4*c*d^6 + a^4*b^3 
*d^7)*x^3 + 28*(b^7*c^5*d^2 + a*b^6*c^4*d^3 + a^2*b^5*c^3*d^4 + a^3*b^4*c^ 
2*d^5 + a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 8*(b^7*c^6*d + a*b^6*c^5*d^2 + 
a^2*b^5*c^4*d^3 + a^3*b^4*c^3*d^4 + a^4*b^3*c^2*d^5 + a^5*b^2*c*d^6 + a^6* 
b*d^7)*x)/(b^16*x^8 + 8*a*b^15*x^7 + 28*a^2*b^14*x^6 + 56*a^3*b^13*x^5 + 7 
0*a^4*b^12*x^4 + 56*a^5*b^11*x^3 + 28*a^6*b^10*x^2 + 8*a^7*b^9*x + a^8*b^8 
)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**7/(b*x+a)**9,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (26) = 52\).

Time = 0.06 (sec) , antiderivative size = 509, normalized size of antiderivative = 18.18 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {8 \, b^{7} d^{7} x^{7} + b^{7} c^{7} + a b^{6} c^{6} d + a^{2} b^{5} c^{5} d^{2} + a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + a^{7} d^{7} + 28 \, {\left (b^{7} c d^{6} + a b^{6} d^{7}\right )} x^{6} + 56 \, {\left (b^{7} c^{2} d^{5} + a b^{6} c d^{6} + a^{2} b^{5} d^{7}\right )} x^{5} + 70 \, {\left (b^{7} c^{3} d^{4} + a b^{6} c^{2} d^{5} + a^{2} b^{5} c d^{6} + a^{3} b^{4} d^{7}\right )} x^{4} + 56 \, {\left (b^{7} c^{4} d^{3} + a b^{6} c^{3} d^{4} + a^{2} b^{5} c^{2} d^{5} + a^{3} b^{4} c d^{6} + a^{4} b^{3} d^{7}\right )} x^{3} + 28 \, {\left (b^{7} c^{5} d^{2} + a b^{6} c^{4} d^{3} + a^{2} b^{5} c^{3} d^{4} + a^{3} b^{4} c^{2} d^{5} + a^{4} b^{3} c d^{6} + a^{5} b^{2} d^{7}\right )} x^{2} + 8 \, {\left (b^{7} c^{6} d + a b^{6} c^{5} d^{2} + a^{2} b^{5} c^{4} d^{3} + a^{3} b^{4} c^{3} d^{4} + a^{4} b^{3} c^{2} d^{5} + a^{5} b^{2} c d^{6} + a^{6} b d^{7}\right )} x}{8 \, {\left (b^{16} x^{8} + 8 \, a b^{15} x^{7} + 28 \, a^{2} b^{14} x^{6} + 56 \, a^{3} b^{13} x^{5} + 70 \, a^{4} b^{12} x^{4} + 56 \, a^{5} b^{11} x^{3} + 28 \, a^{6} b^{10} x^{2} + 8 \, a^{7} b^{9} x + a^{8} b^{8}\right )}} \] Input:

integrate((d*x+c)^7/(b*x+a)^9,x, algorithm="maxima")
 

Output:

-1/8*(8*b^7*d^7*x^7 + b^7*c^7 + a*b^6*c^6*d + a^2*b^5*c^5*d^2 + a^3*b^4*c^ 
4*d^3 + a^4*b^3*c^3*d^4 + a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7 + 28*(b^ 
7*c*d^6 + a*b^6*d^7)*x^6 + 56*(b^7*c^2*d^5 + a*b^6*c*d^6 + a^2*b^5*d^7)*x^ 
5 + 70*(b^7*c^3*d^4 + a*b^6*c^2*d^5 + a^2*b^5*c*d^6 + a^3*b^4*d^7)*x^4 + 5 
6*(b^7*c^4*d^3 + a*b^6*c^3*d^4 + a^2*b^5*c^2*d^5 + a^3*b^4*c*d^6 + a^4*b^3 
*d^7)*x^3 + 28*(b^7*c^5*d^2 + a*b^6*c^4*d^3 + a^2*b^5*c^3*d^4 + a^3*b^4*c^ 
2*d^5 + a^4*b^3*c*d^6 + a^5*b^2*d^7)*x^2 + 8*(b^7*c^6*d + a*b^6*c^5*d^2 + 
a^2*b^5*c^4*d^3 + a^3*b^4*c^3*d^4 + a^4*b^3*c^2*d^5 + a^5*b^2*c*d^6 + a^6* 
b*d^7)*x)/(b^16*x^8 + 8*a*b^15*x^7 + 28*a^2*b^14*x^6 + 56*a^3*b^13*x^5 + 7 
0*a^4*b^12*x^4 + 56*a^5*b^11*x^3 + 28*a^6*b^10*x^2 + 8*a^7*b^9*x + a^8*b^8 
)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 489 vs. \(2 (26) = 52\).

Time = 0.13 (sec) , antiderivative size = 489, normalized size of antiderivative = 17.46 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {8 \, b^{7} d^{7} x^{7} + 28 \, b^{7} c d^{6} x^{6} + 28 \, a b^{6} d^{7} x^{6} + 56 \, b^{7} c^{2} d^{5} x^{5} + 56 \, a b^{6} c d^{6} x^{5} + 56 \, a^{2} b^{5} d^{7} x^{5} + 70 \, b^{7} c^{3} d^{4} x^{4} + 70 \, a b^{6} c^{2} d^{5} x^{4} + 70 \, a^{2} b^{5} c d^{6} x^{4} + 70 \, a^{3} b^{4} d^{7} x^{4} + 56 \, b^{7} c^{4} d^{3} x^{3} + 56 \, a b^{6} c^{3} d^{4} x^{3} + 56 \, a^{2} b^{5} c^{2} d^{5} x^{3} + 56 \, a^{3} b^{4} c d^{6} x^{3} + 56 \, a^{4} b^{3} d^{7} x^{3} + 28 \, b^{7} c^{5} d^{2} x^{2} + 28 \, a b^{6} c^{4} d^{3} x^{2} + 28 \, a^{2} b^{5} c^{3} d^{4} x^{2} + 28 \, a^{3} b^{4} c^{2} d^{5} x^{2} + 28 \, a^{4} b^{3} c d^{6} x^{2} + 28 \, a^{5} b^{2} d^{7} x^{2} + 8 \, b^{7} c^{6} d x + 8 \, a b^{6} c^{5} d^{2} x + 8 \, a^{2} b^{5} c^{4} d^{3} x + 8 \, a^{3} b^{4} c^{3} d^{4} x + 8 \, a^{4} b^{3} c^{2} d^{5} x + 8 \, a^{5} b^{2} c d^{6} x + 8 \, a^{6} b d^{7} x + b^{7} c^{7} + a b^{6} c^{6} d + a^{2} b^{5} c^{5} d^{2} + a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4} + a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} + a^{7} d^{7}}{8 \, {\left (b x + a\right )}^{8} b^{8}} \] Input:

integrate((d*x+c)^7/(b*x+a)^9,x, algorithm="giac")
 

Output:

-1/8*(8*b^7*d^7*x^7 + 28*b^7*c*d^6*x^6 + 28*a*b^6*d^7*x^6 + 56*b^7*c^2*d^5 
*x^5 + 56*a*b^6*c*d^6*x^5 + 56*a^2*b^5*d^7*x^5 + 70*b^7*c^3*d^4*x^4 + 70*a 
*b^6*c^2*d^5*x^4 + 70*a^2*b^5*c*d^6*x^4 + 70*a^3*b^4*d^7*x^4 + 56*b^7*c^4* 
d^3*x^3 + 56*a*b^6*c^3*d^4*x^3 + 56*a^2*b^5*c^2*d^5*x^3 + 56*a^3*b^4*c*d^6 
*x^3 + 56*a^4*b^3*d^7*x^3 + 28*b^7*c^5*d^2*x^2 + 28*a*b^6*c^4*d^3*x^2 + 28 
*a^2*b^5*c^3*d^4*x^2 + 28*a^3*b^4*c^2*d^5*x^2 + 28*a^4*b^3*c*d^6*x^2 + 28* 
a^5*b^2*d^7*x^2 + 8*b^7*c^6*d*x + 8*a*b^6*c^5*d^2*x + 8*a^2*b^5*c^4*d^3*x 
+ 8*a^3*b^4*c^3*d^4*x + 8*a^4*b^3*c^2*d^5*x + 8*a^5*b^2*c*d^6*x + 8*a^6*b* 
d^7*x + b^7*c^7 + a*b^6*c^6*d + a^2*b^5*c^5*d^2 + a^3*b^4*c^4*d^3 + a^4*b^ 
3*c^3*d^4 + a^5*b^2*c^2*d^5 + a^6*b*c*d^6 + a^7*d^7)/((b*x + a)^8*b^8)
 

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 571, normalized size of antiderivative = 20.39 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=-\frac {a^7\,d^7+a^6\,b\,c\,d^6+8\,a^6\,b\,d^7\,x+a^5\,b^2\,c^2\,d^5+8\,a^5\,b^2\,c\,d^6\,x+28\,a^5\,b^2\,d^7\,x^2+a^4\,b^3\,c^3\,d^4+8\,a^4\,b^3\,c^2\,d^5\,x+28\,a^4\,b^3\,c\,d^6\,x^2+56\,a^4\,b^3\,d^7\,x^3+a^3\,b^4\,c^4\,d^3+8\,a^3\,b^4\,c^3\,d^4\,x+28\,a^3\,b^4\,c^2\,d^5\,x^2+56\,a^3\,b^4\,c\,d^6\,x^3+70\,a^3\,b^4\,d^7\,x^4+a^2\,b^5\,c^5\,d^2+8\,a^2\,b^5\,c^4\,d^3\,x+28\,a^2\,b^5\,c^3\,d^4\,x^2+56\,a^2\,b^5\,c^2\,d^5\,x^3+70\,a^2\,b^5\,c\,d^6\,x^4+56\,a^2\,b^5\,d^7\,x^5+a\,b^6\,c^6\,d+8\,a\,b^6\,c^5\,d^2\,x+28\,a\,b^6\,c^4\,d^3\,x^2+56\,a\,b^6\,c^3\,d^4\,x^3+70\,a\,b^6\,c^2\,d^5\,x^4+56\,a\,b^6\,c\,d^6\,x^5+28\,a\,b^6\,d^7\,x^6+b^7\,c^7+8\,b^7\,c^6\,d\,x+28\,b^7\,c^5\,d^2\,x^2+56\,b^7\,c^4\,d^3\,x^3+70\,b^7\,c^3\,d^4\,x^4+56\,b^7\,c^2\,d^5\,x^5+28\,b^7\,c\,d^6\,x^6+8\,b^7\,d^7\,x^7}{8\,a^8\,b^8+64\,a^7\,b^9\,x+224\,a^6\,b^{10}\,x^2+448\,a^5\,b^{11}\,x^3+560\,a^4\,b^{12}\,x^4+448\,a^3\,b^{13}\,x^5+224\,a^2\,b^{14}\,x^6+64\,a\,b^{15}\,x^7+8\,b^{16}\,x^8} \] Input:

int((c + d*x)^7/(a + b*x)^9,x)
 

Output:

-(a^7*d^7 + b^7*c^7 + 8*b^7*d^7*x^7 + 28*a*b^6*d^7*x^6 + 28*b^7*c*d^6*x^6 
+ a^2*b^5*c^5*d^2 + a^3*b^4*c^4*d^3 + a^4*b^3*c^3*d^4 + a^5*b^2*c^2*d^5 + 
28*a^5*b^2*d^7*x^2 + 56*a^4*b^3*d^7*x^3 + 70*a^3*b^4*d^7*x^4 + 56*a^2*b^5* 
d^7*x^5 + 28*b^7*c^5*d^2*x^2 + 56*b^7*c^4*d^3*x^3 + 70*b^7*c^3*d^4*x^4 + 5 
6*b^7*c^2*d^5*x^5 + a*b^6*c^6*d + a^6*b*c*d^6 + 8*a^6*b*d^7*x + 8*b^7*c^6* 
d*x + 28*a^2*b^5*c^3*d^4*x^2 + 28*a^3*b^4*c^2*d^5*x^2 + 56*a^2*b^5*c^2*d^5 
*x^3 + 8*a*b^6*c^5*d^2*x + 8*a^5*b^2*c*d^6*x + 56*a*b^6*c*d^6*x^5 + 8*a^2* 
b^5*c^4*d^3*x + 8*a^3*b^4*c^3*d^4*x + 8*a^4*b^3*c^2*d^5*x + 28*a*b^6*c^4*d 
^3*x^2 + 28*a^4*b^3*c*d^6*x^2 + 56*a*b^6*c^3*d^4*x^3 + 56*a^3*b^4*c*d^6*x^ 
3 + 70*a*b^6*c^2*d^5*x^4 + 70*a^2*b^5*c*d^6*x^4)/(8*a^8*b^8 + 8*b^16*x^8 + 
 64*a^7*b^9*x + 64*a*b^15*x^7 + 224*a^6*b^10*x^2 + 448*a^5*b^11*x^3 + 560* 
a^4*b^12*x^4 + 448*a^3*b^13*x^5 + 224*a^2*b^14*x^6)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 504, normalized size of antiderivative = 18.00 \[ \int \frac {(c+d x)^7}{(a+b x)^9} \, dx=\frac {b^{7} d^{7} x^{8}-28 a \,b^{6} c \,d^{6} x^{6}-56 a^{2} b^{5} c \,d^{6} x^{5}-56 a \,b^{6} c^{2} d^{5} x^{5}-70 a^{3} b^{4} c \,d^{6} x^{4}-70 a^{2} b^{5} c^{2} d^{5} x^{4}-70 a \,b^{6} c^{3} d^{4} x^{4}-56 a^{4} b^{3} c \,d^{6} x^{3}-56 a^{3} b^{4} c^{2} d^{5} x^{3}-56 a^{2} b^{5} c^{3} d^{4} x^{3}-56 a \,b^{6} c^{4} d^{3} x^{3}-28 a^{5} b^{2} c \,d^{6} x^{2}-28 a^{4} b^{3} c^{2} d^{5} x^{2}-28 a^{3} b^{4} c^{3} d^{4} x^{2}-28 a^{2} b^{5} c^{4} d^{3} x^{2}-28 a \,b^{6} c^{5} d^{2} x^{2}-8 a^{6} b c \,d^{6} x -8 a^{5} b^{2} c^{2} d^{5} x -8 a^{4} b^{3} c^{3} d^{4} x -8 a^{3} b^{4} c^{4} d^{3} x -8 a^{2} b^{5} c^{5} d^{2} x -8 a \,b^{6} c^{6} d x -a^{7} c \,d^{6}-a^{6} b \,c^{2} d^{5}-a^{5} b^{2} c^{3} d^{4}-a^{4} b^{3} c^{4} d^{3}-a^{3} b^{4} c^{5} d^{2}-a^{2} b^{5} c^{6} d -a \,b^{6} c^{7}}{8 a \,b^{7} \left (b^{8} x^{8}+8 a \,b^{7} x^{7}+28 a^{2} b^{6} x^{6}+56 a^{3} b^{5} x^{5}+70 a^{4} b^{4} x^{4}+56 a^{5} b^{3} x^{3}+28 a^{6} b^{2} x^{2}+8 a^{7} b x +a^{8}\right )} \] Input:

int((d*x+c)^7/(b*x+a)^9,x)
 

Output:

( - a**7*c*d**6 - a**6*b*c**2*d**5 - 8*a**6*b*c*d**6*x - a**5*b**2*c**3*d* 
*4 - 8*a**5*b**2*c**2*d**5*x - 28*a**5*b**2*c*d**6*x**2 - a**4*b**3*c**4*d 
**3 - 8*a**4*b**3*c**3*d**4*x - 28*a**4*b**3*c**2*d**5*x**2 - 56*a**4*b**3 
*c*d**6*x**3 - a**3*b**4*c**5*d**2 - 8*a**3*b**4*c**4*d**3*x - 28*a**3*b** 
4*c**3*d**4*x**2 - 56*a**3*b**4*c**2*d**5*x**3 - 70*a**3*b**4*c*d**6*x**4 
- a**2*b**5*c**6*d - 8*a**2*b**5*c**5*d**2*x - 28*a**2*b**5*c**4*d**3*x**2 
 - 56*a**2*b**5*c**3*d**4*x**3 - 70*a**2*b**5*c**2*d**5*x**4 - 56*a**2*b** 
5*c*d**6*x**5 - a*b**6*c**7 - 8*a*b**6*c**6*d*x - 28*a*b**6*c**5*d**2*x**2 
 - 56*a*b**6*c**4*d**3*x**3 - 70*a*b**6*c**3*d**4*x**4 - 56*a*b**6*c**2*d* 
*5*x**5 - 28*a*b**6*c*d**6*x**6 + b**7*d**7*x**8)/(8*a*b**7*(a**8 + 8*a**7 
*b*x + 28*a**6*b**2*x**2 + 56*a**5*b**3*x**3 + 70*a**4*b**4*x**4 + 56*a**3 
*b**5*x**5 + 28*a**2*b**6*x**6 + 8*a*b**7*x**7 + b**8*x**8))