\(\int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx\) [164]

Optimal result

Integrand size = 134, antiderivative size = 117 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx=-\frac {(b c-a d)^4}{8 d^5 (c+d x)^8}+\frac {4 b (b c-a d)^3}{7 d^5 (c+d x)^7}-\frac {b^2 (b c-a d)^2}{d^5 (c+d x)^6}+\frac {4 b^3 (b c-a d)}{5 d^5 (c+d x)^5}-\frac {b^4}{4 d^5 (c+d x)^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.23 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx=-\frac {35 a^4 d^4+20 a^3 b d^3 (c+8 d x)+10 a^2 b^2 d^2 \left (c^2+8 c d x+28 d^2 x^2\right )+4 a b^3 d \left (c^3+8 c^2 d x+28 c d^2 x^2+56 d^3 x^3\right )+b^4 \left (c^4+8 c^3 d x+28 c^2 d^2 x^2+56 c d^3 x^3+70 d^4 x^4\right )}{280 d^5 (c+d x)^8} \] Input:

Integrate[(a^4 + 4*a^3*b*x + 6*a^2*b^2*x^2 + 4*a*b^3*x^3 + b^4*x^4)/(c^9 + 
 9*c^8*d*x + 36*c^7*d^2*x^2 + 84*c^6*d^3*x^3 + 126*c^5*d^4*x^4 + 126*c^4*d 
^5*x^5 + 84*c^3*d^6*x^6 + 36*c^2*d^7*x^7 + 9*c*d^8*x^8 + d^9*x^9),x]
 

Output:

-1/280*(35*a^4*d^4 + 20*a^3*b*d^3*(c + 8*d*x) + 10*a^2*b^2*d^2*(c^2 + 8*c* 
d*x + 28*d^2*x^2) + 4*a*b^3*d*(c^3 + 8*c^2*d*x + 28*c*d^2*x^2 + 56*d^3*x^3 
) + b^4*(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))/(d 
^5*(c + d*x)^8)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2006, 2007, 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.

\(\displaystyle \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx\)

\(\Big \downarrow \) 2006

\(\displaystyle \int \frac {(a+b x)^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {(a+b x)^4}{(c+d x)^9}dx\)

\(\Big \downarrow \) 53

\(\displaystyle \int \left (-\frac {4 b^3 (b c-a d)}{d^4 (c+d x)^6}+\frac {6 b^2 (b c-a d)^2}{d^4 (c+d x)^7}-\frac {4 b (b c-a d)^3}{d^4 (c+d x)^8}+\frac {(a d-b c)^4}{d^4 (c+d x)^9}+\frac {b^4}{d^4 (c+d x)^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 b^3 (b c-a d)}{5 d^5 (c+d x)^5}-\frac {b^2 (b c-a d)^2}{d^5 (c+d x)^6}+\frac {4 b (b c-a d)^3}{7 d^5 (c+d x)^7}-\frac {(b c-a d)^4}{8 d^5 (c+d x)^8}-\frac {b^4}{4 d^5 (c+d x)^4}\)

Input:

Int[(a^4 + 4*a^3*b*x + 6*a^2*b^2*x^2 + 4*a*b^3*x^3 + b^4*x^4)/(c^9 + 9*c^8 
*d*x + 36*c^7*d^2*x^2 + 84*c^6*d^3*x^3 + 126*c^5*d^4*x^4 + 126*c^4*d^5*x^5 
 + 84*c^3*d^6*x^6 + 36*c^2*d^7*x^7 + 9*c*d^8*x^8 + d^9*x^9),x]
 

Output:

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

Defintions of rubi rules used

rule 53

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])
 

rule 2006

Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], 
b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px 
, x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P 
x, x], 1] && NeQ[Coeff[Px, x, 0], 0] &&  !MatchQ[Px, (a_.)*(v_)^Expon[Px, x 
] /; FreeQ[a, x] && LinearQ[v, x]]
 

rule 2007

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, 
x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex 
pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol 
yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.59

method	result	size
default	\(-\frac {b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{5} \left (x d +c \right )^{6}}-\frac {4 b^{3} \left (a d -b c \right )}{5 d^{5} \left (x d +c \right )^{5}}-\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{7 d^{5} \left (x d +c \right )^{7}}-\frac {b^{4}}{4 d^{5} \left (x d +c \right )^{4}}-\frac {d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}{8 d^{5} \left (x d +c \right )^{8}}\)	\(186\)
norman	\(\frac {-\frac {b^{4} x^{4}}{4 d}-\frac {\left (4 a \,b^{3} d^{4}+b^{4} c \,d^{3}\right ) x^{3}}{5 d^{5}}-\frac {\left (10 a^{2} b^{2} d^{5}+4 a \,b^{3} c \,d^{4}+b^{4} c^{2} d^{3}\right ) x^{2}}{10 d^{6}}-\frac {\left (20 a^{3} b \,d^{6}+10 a^{2} b^{2} c \,d^{5}+4 a \,b^{3} c^{2} d^{4}+b^{4} c^{3} d^{3}\right ) x}{35 d^{7}}-\frac {35 a^{4} d^{7}+20 a^{3} b c \,d^{6}+10 a^{2} b^{2} c^{2} d^{5}+4 a \,b^{3} c^{3} d^{4}+b^{4} c^{4} d^{3}}{280 d^{8}}}{\left (x d +c \right )^{8}}\)	\(197\)
risch	\(\frac {-\frac {b^{4} x^{4}}{4 d}-\frac {b^{3} \left (4 a d +b c \right ) x^{3}}{5 d^{2}}-\frac {b^{2} \left (10 a^{2} d^{2}+4 a b c d +b^{2} c^{2}\right ) x^{2}}{10 d^{3}}-\frac {b \left (20 a^{3} d^{3}+10 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{35 d^{4}}-\frac {35 d^{4} a^{4}+20 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +c^{4} b^{4}}{280 d^{5}}}{d^{8} x^{8}+8 c \,d^{7} x^{7}+28 c^{2} d^{6} x^{6}+56 c^{3} d^{5} x^{5}+70 c^{4} d^{4} x^{4}+56 c^{5} d^{3} x^{3}+28 c^{6} d^{2} x^{2}+8 c^{7} d x +c^{8}}\)	\(248\)
gosper	\(-\frac {70 d^{4} x^{4} b^{4}+224 a \,b^{3} d^{4} x^{3}+56 b^{4} c \,d^{3} x^{3}+280 a^{2} b^{2} d^{4} x^{2}+112 a \,b^{3} c \,d^{3} x^{2}+28 b^{4} c^{2} d^{2} x^{2}+160 a^{3} b \,d^{4} x +80 a^{2} b^{2} c \,d^{3} x +32 a \,b^{3} c^{2} d^{2} x +8 b^{4} c^{3} d x +35 d^{4} a^{4}+20 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +c^{4} b^{4}}{280 d^{5} \left (d^{8} x^{8}+8 c \,d^{7} x^{7}+28 c^{2} d^{6} x^{6}+56 c^{3} d^{5} x^{5}+70 c^{4} d^{4} x^{4}+56 c^{5} d^{3} x^{3}+28 c^{6} d^{2} x^{2}+8 c^{7} d x +c^{8}\right )}\)	\(262\)
parallelrisch	\(\frac {-70 b^{4} x^{4} d^{7}-224 a \,b^{3} d^{7} x^{3}-56 b^{4} c \,d^{6} x^{3}-280 a^{2} b^{2} d^{7} x^{2}-112 a \,b^{3} c \,d^{6} x^{2}-28 b^{4} c^{2} d^{5} x^{2}-160 a^{3} b \,d^{7} x -80 a^{2} b^{2} c \,d^{6} x -32 a \,b^{3} c^{2} d^{5} x -8 b^{4} c^{3} d^{4} x -35 a^{4} d^{7}-20 a^{3} b c \,d^{6}-10 a^{2} b^{2} c^{2} d^{5}-4 a \,b^{3} c^{3} d^{4}-b^{4} c^{4} d^{3}}{280 d^{8} \left (d^{8} x^{8}+8 c \,d^{7} x^{7}+28 c^{2} d^{6} x^{6}+56 c^{3} d^{5} x^{5}+70 c^{4} d^{4} x^{4}+56 c^{5} d^{3} x^{3}+28 c^{6} d^{2} x^{2}+8 c^{7} d x +c^{8}\right )}\)	\(270\)
orering	\(-\frac {\left (70 d^{4} x^{4} b^{4}+224 a \,b^{3} d^{4} x^{3}+56 b^{4} c \,d^{3} x^{3}+280 a^{2} b^{2} d^{4} x^{2}+112 a \,b^{3} c \,d^{3} x^{2}+28 b^{4} c^{2} d^{2} x^{2}+160 a^{3} b \,d^{4} x +80 a^{2} b^{2} c \,d^{3} x +32 a \,b^{3} c^{2} d^{2} x +8 b^{4} c^{3} d x +35 d^{4} a^{4}+20 a^{3} b c \,d^{3}+10 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \left (x d +c \right ) \left (b^{4} x^{4}+4 a \,x^{3} b^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )}{280 d^{5} \left (b x +a \right )^{4} \left (d^{9} x^{9}+9 c \,d^{8} x^{8}+36 c^{2} d^{7} x^{7}+84 c^{3} d^{6} x^{6}+126 c^{4} d^{5} x^{5}+126 c^{5} d^{4} x^{4}+84 c^{6} d^{3} x^{3}+36 c^{7} d^{2} x^{2}+9 c^{8} d x +c^{9}\right )}\)	\(323\)

Input:

int((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d^9*x^9+9*c*d^8*x^8 
+36*c^2*d^7*x^7+84*c^3*d^6*x^6+126*c^4*d^5*x^5+126*c^5*d^4*x^4+84*c^6*d^3* 
x^3+36*c^7*d^2*x^2+9*c^8*d*x+c^9),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (109) = 218\).

Time = 0.09 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.21 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx=-\frac {70 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 10 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4} + 56 \, {\left (b^{4} c d^{3} + 4 \, a b^{3} d^{4}\right )} x^{3} + 28 \, {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + 10 \, a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (b^{4} c^{3} d + 4 \, a b^{3} c^{2} d^{2} + 10 \, a^{2} b^{2} c d^{3} + 20 \, a^{3} b d^{4}\right )} x}{280 \, {\left (d^{13} x^{8} + 8 \, c d^{12} x^{7} + 28 \, c^{2} d^{11} x^{6} + 56 \, c^{3} d^{10} x^{5} + 70 \, c^{4} d^{9} x^{4} + 56 \, c^{5} d^{8} x^{3} + 28 \, c^{6} d^{7} x^{2} + 8 \, c^{7} d^{6} x + c^{8} d^{5}\right )}} \] Input:

integrate((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d^9*x^9+9*c*d 
^8*x^8+36*c^2*d^7*x^7+84*c^3*d^6*x^6+126*c^4*d^5*x^5+126*c^5*d^4*x^4+84*c^ 
6*d^3*x^3+36*c^7*d^2*x^2+9*c^8*d*x+c^9),x, algorithm="fricas")
 

Output:

-1/280*(70*b^4*d^4*x^4 + b^4*c^4 + 4*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 + 20 
*a^3*b*c*d^3 + 35*a^4*d^4 + 56*(b^4*c*d^3 + 4*a*b^3*d^4)*x^3 + 28*(b^4*c^2 
*d^2 + 4*a*b^3*c*d^3 + 10*a^2*b^2*d^4)*x^2 + 8*(b^4*c^3*d + 4*a*b^3*c^2*d^ 
2 + 10*a^2*b^2*c*d^3 + 20*a^3*b*d^4)*x)/(d^13*x^8 + 8*c*d^12*x^7 + 28*c^2* 
d^11*x^6 + 56*c^3*d^10*x^5 + 70*c^4*d^9*x^4 + 56*c^5*d^8*x^3 + 28*c^6*d^7* 
x^2 + 8*c^7*d^6*x + c^8*d^5)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (104) = 208\).

Time = 9.16 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.38 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx=\frac {- 35 a^{4} d^{4} - 20 a^{3} b c d^{3} - 10 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4} - 70 b^{4} d^{4} x^{4} + x^{3} \left (- 224 a b^{3} d^{4} - 56 b^{4} c d^{3}\right ) + x^{2} \left (- 280 a^{2} b^{2} d^{4} - 112 a b^{3} c d^{3} - 28 b^{4} c^{2} d^{2}\right ) + x \left (- 160 a^{3} b d^{4} - 80 a^{2} b^{2} c d^{3} - 32 a b^{3} c^{2} d^{2} - 8 b^{4} c^{3} d\right )}{280 c^{8} d^{5} + 2240 c^{7} d^{6} x + 7840 c^{6} d^{7} x^{2} + 15680 c^{5} d^{8} x^{3} + 19600 c^{4} d^{9} x^{4} + 15680 c^{3} d^{10} x^{5} + 7840 c^{2} d^{11} x^{6} + 2240 c d^{12} x^{7} + 280 d^{13} x^{8}} \] Input:

integrate((b**4*x**4+4*a*b**3*x**3+6*a**2*b**2*x**2+4*a**3*b*x+a**4)/(d**9 
*x**9+9*c*d**8*x**8+36*c**2*d**7*x**7+84*c**3*d**6*x**6+126*c**4*d**5*x**5 
+126*c**5*d**4*x**4+84*c**6*d**3*x**3+36*c**7*d**2*x**2+9*c**8*d*x+c**9),x 
)
 

Output:

(-35*a**4*d**4 - 20*a**3*b*c*d**3 - 10*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3 
*d - b**4*c**4 - 70*b**4*d**4*x**4 + x**3*(-224*a*b**3*d**4 - 56*b**4*c*d* 
*3) + x**2*(-280*a**2*b**2*d**4 - 112*a*b**3*c*d**3 - 28*b**4*c**2*d**2) + 
 x*(-160*a**3*b*d**4 - 80*a**2*b**2*c*d**3 - 32*a*b**3*c**2*d**2 - 8*b**4* 
c**3*d))/(280*c**8*d**5 + 2240*c**7*d**6*x + 7840*c**6*d**7*x**2 + 15680*c 
**5*d**8*x**3 + 19600*c**4*d**9*x**4 + 15680*c**3*d**10*x**5 + 7840*c**2*d 
**11*x**6 + 2240*c*d**12*x**7 + 280*d**13*x**8)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (109) = 218\).

Time = 0.05 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.21 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx=-\frac {70 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 10 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4} + 56 \, {\left (b^{4} c d^{3} + 4 \, a b^{3} d^{4}\right )} x^{3} + 28 \, {\left (b^{4} c^{2} d^{2} + 4 \, a b^{3} c d^{3} + 10 \, a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (b^{4} c^{3} d + 4 \, a b^{3} c^{2} d^{2} + 10 \, a^{2} b^{2} c d^{3} + 20 \, a^{3} b d^{4}\right )} x}{280 \, {\left (d^{13} x^{8} + 8 \, c d^{12} x^{7} + 28 \, c^{2} d^{11} x^{6} + 56 \, c^{3} d^{10} x^{5} + 70 \, c^{4} d^{9} x^{4} + 56 \, c^{5} d^{8} x^{3} + 28 \, c^{6} d^{7} x^{2} + 8 \, c^{7} d^{6} x + c^{8} d^{5}\right )}} \] Input:

integrate((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d^9*x^9+9*c*d 
^8*x^8+36*c^2*d^7*x^7+84*c^3*d^6*x^6+126*c^4*d^5*x^5+126*c^5*d^4*x^4+84*c^ 
6*d^3*x^3+36*c^7*d^2*x^2+9*c^8*d*x+c^9),x, algorithm="maxima")
 

Output:

-1/280*(70*b^4*d^4*x^4 + b^4*c^4 + 4*a*b^3*c^3*d + 10*a^2*b^2*c^2*d^2 + 20 
*a^3*b*c*d^3 + 35*a^4*d^4 + 56*(b^4*c*d^3 + 4*a*b^3*d^4)*x^3 + 28*(b^4*c^2 
*d^2 + 4*a*b^3*c*d^3 + 10*a^2*b^2*d^4)*x^2 + 8*(b^4*c^3*d + 4*a*b^3*c^2*d^ 
2 + 10*a^2*b^2*c*d^3 + 20*a^3*b*d^4)*x)/(d^13*x^8 + 8*c*d^12*x^7 + 28*c^2* 
d^11*x^6 + 56*c^3*d^10*x^5 + 70*c^4*d^9*x^4 + 56*c^5*d^8*x^3 + 28*c^6*d^7* 
x^2 + 8*c^7*d^6*x + c^8*d^5)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.57 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx=-\frac {70 \, b^{4} d^{4} x^{4} + 56 \, b^{4} c d^{3} x^{3} + 224 \, a b^{3} d^{4} x^{3} + 28 \, b^{4} c^{2} d^{2} x^{2} + 112 \, a b^{3} c d^{3} x^{2} + 280 \, a^{2} b^{2} d^{4} x^{2} + 8 \, b^{4} c^{3} d x + 32 \, a b^{3} c^{2} d^{2} x + 80 \, a^{2} b^{2} c d^{3} x + 160 \, a^{3} b d^{4} x + b^{4} c^{4} + 4 \, a b^{3} c^{3} d + 10 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4}}{280 \, {\left (d x + c\right )}^{8} d^{5}} \] Input:

integrate((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d^9*x^9+9*c*d 
^8*x^8+36*c^2*d^7*x^7+84*c^3*d^6*x^6+126*c^4*d^5*x^5+126*c^5*d^4*x^4+84*c^ 
6*d^3*x^3+36*c^7*d^2*x^2+9*c^8*d*x+c^9),x, algorithm="giac")
 

Output:

-1/280*(70*b^4*d^4*x^4 + 56*b^4*c*d^3*x^3 + 224*a*b^3*d^4*x^3 + 28*b^4*c^2 
*d^2*x^2 + 112*a*b^3*c*d^3*x^2 + 280*a^2*b^2*d^4*x^2 + 8*b^4*c^3*d*x + 32* 
a*b^3*c^2*d^2*x + 80*a^2*b^2*c*d^3*x + 160*a^3*b*d^4*x + b^4*c^4 + 4*a*b^3 
*c^3*d + 10*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 + 35*a^4*d^4)/((d*x + c)^8*d^ 
5)
 

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.12 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx=-\frac {\frac {35\,a^4\,d^4+20\,a^3\,b\,c\,d^3+10\,a^2\,b^2\,c^2\,d^2+4\,a\,b^3\,c^3\,d+b^4\,c^4}{280\,d^5}+\frac {b^4\,x^4}{4\,d}+\frac {b^3\,x^3\,\left (4\,a\,d+b\,c\right )}{5\,d^2}+\frac {b\,x\,\left (20\,a^3\,d^3+10\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{35\,d^4}+\frac {b^2\,x^2\,\left (10\,a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{10\,d^3}}{c^8+8\,c^7\,d\,x+28\,c^6\,d^2\,x^2+56\,c^5\,d^3\,x^3+70\,c^4\,d^4\,x^4+56\,c^3\,d^5\,x^5+28\,c^2\,d^6\,x^6+8\,c\,d^7\,x^7+d^8\,x^8} \] Input:

int((a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x)/(c^9 + d^9*x 
^9 + 9*c*d^8*x^8 + 36*c^7*d^2*x^2 + 84*c^6*d^3*x^3 + 126*c^5*d^4*x^4 + 126 
*c^4*d^5*x^5 + 84*c^3*d^6*x^6 + 36*c^2*d^7*x^7 + 9*c^8*d*x),x)
 

Output:

-((35*a^4*d^4 + b^4*c^4 + 10*a^2*b^2*c^2*d^2 + 4*a*b^3*c^3*d + 20*a^3*b*c* 
d^3)/(280*d^5) + (b^4*x^4)/(4*d) + (b^3*x^3*(4*a*d + b*c))/(5*d^2) + (b*x* 
(20*a^3*d^3 + b^3*c^3 + 4*a*b^2*c^2*d + 10*a^2*b*c*d^2))/(35*d^4) + (b^2*x 
^2*(10*a^2*d^2 + b^2*c^2 + 4*a*b*c*d))/(10*d^3))/(c^8 + d^8*x^8 + 8*c*d^7* 
x^7 + 28*c^6*d^2*x^2 + 56*c^5*d^3*x^3 + 70*c^4*d^4*x^4 + 56*c^3*d^5*x^5 + 
28*c^2*d^6*x^6 + 8*c^7*d*x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.24 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^9+9 c^8 d x+36 c^7 d^2 x^2+84 c^6 d^3 x^3+126 c^5 d^4 x^4+126 c^4 d^5 x^5+84 c^3 d^6 x^6+36 c^2 d^7 x^7+9 c d^8 x^8+d^9 x^9} \, dx=\frac {-70 b^{4} d^{4} x^{4}-224 a \,b^{3} d^{4} x^{3}-56 b^{4} c \,d^{3} x^{3}-280 a^{2} b^{2} d^{4} x^{2}-112 a \,b^{3} c \,d^{3} x^{2}-28 b^{4} c^{2} d^{2} x^{2}-160 a^{3} b \,d^{4} x -80 a^{2} b^{2} c \,d^{3} x -32 a \,b^{3} c^{2} d^{2} x -8 b^{4} c^{3} d x -35 a^{4} d^{4}-20 a^{3} b c \,d^{3}-10 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d -b^{4} c^{4}}{280 d^{5} \left (d^{8} x^{8}+8 c \,d^{7} x^{7}+28 c^{2} d^{6} x^{6}+56 c^{3} d^{5} x^{5}+70 c^{4} d^{4} x^{4}+56 c^{5} d^{3} x^{3}+28 c^{6} d^{2} x^{2}+8 c^{7} d x +c^{8}\right )} \] Input:

int((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(d^9*x^9+9*c*d^8*x^8 
+36*c^2*d^7*x^7+84*c^3*d^6*x^6+126*c^4*d^5*x^5+126*c^5*d^4*x^4+84*c^6*d^3* 
x^3+36*c^7*d^2*x^2+9*c^8*d*x+c^9),x)
 

Output:

( - 35*a**4*d**4 - 20*a**3*b*c*d**3 - 160*a**3*b*d**4*x - 10*a**2*b**2*c** 
2*d**2 - 80*a**2*b**2*c*d**3*x - 280*a**2*b**2*d**4*x**2 - 4*a*b**3*c**3*d 
 - 32*a*b**3*c**2*d**2*x - 112*a*b**3*c*d**3*x**2 - 224*a*b**3*d**4*x**3 - 
 b**4*c**4 - 8*b**4*c**3*d*x - 28*b**4*c**2*d**2*x**2 - 56*b**4*c*d**3*x** 
3 - 70*b**4*d**4*x**4)/(280*d**5*(c**8 + 8*c**7*d*x + 28*c**6*d**2*x**2 + 
56*c**5*d**3*x**3 + 70*c**4*d**4*x**4 + 56*c**3*d**5*x**5 + 28*c**2*d**6*x 
**6 + 8*c*d**7*x**7 + d**8*x**8))