\(\int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx\) [163]

Optimal result

Integrand size = 123, antiderivative size = 89 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx=\frac {(a+b x)^5}{7 (b c-a d) (c+d x)^7}+\frac {b (a+b x)^5}{21 (b c-a d)^2 (c+d x)^6}+\frac {b^2 (a+b x)^5}{105 (b c-a d)^3 (c+d x)^5} \] Output:

1/7*(b*x+a)^5/(-a*d+b*c)/(d*x+c)^7+1/21*b*(b*x+a)^5/(-a*d+b*c)^2/(d*x+c)^6 
+1/105*b^2*(b*x+a)^5/(-a*d+b*c)^3/(d*x+c)^5
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.62 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx=-\frac {15 a^4 d^4+10 a^3 b d^3 (c+7 d x)+6 a^2 b^2 d^2 \left (c^2+7 c d x+21 d^2 x^2\right )+3 a b^3 d \left (c^3+7 c^2 d x+21 c d^2 x^2+35 d^3 x^3\right )+b^4 \left (c^4+7 c^3 d x+21 c^2 d^2 x^2+35 c d^3 x^3+35 d^4 x^4\right )}{105 d^5 (c+d x)^7} \] 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^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),x]
 

Output:

-1/105*(15*a^4*d^4 + 10*a^3*b*d^3*(c + 7*d*x) + 6*a^2*b^2*d^2*(c^2 + 7*c*d 
*x + 21*d^2*x^2) + 3*a*b^3*d*(c^3 + 7*c^2*d*x + 21*c*d^2*x^2 + 35*d^3*x^3) 
 + b^4*(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))/(d^ 
5*(c + d*x)^7)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {2006, 2007, 55, 55, 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.

\(\displaystyle \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx\)

\(\Big \downarrow \) 2006

\(\displaystyle \int \frac {(a+b x)^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {(a+b x)^4}{(c+d x)^8}dx\)

\(\Big \downarrow \) 55

\(\displaystyle \frac {2 b \int \frac {(a+b x)^4}{(c+d x)^7}dx}{7 (b c-a d)}+\frac {(a+b x)^5}{7 (c+d x)^7 (b c-a d)}\)

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

\(\displaystyle \frac {(a+b x)^5}{7 (c+d x)^7 (b c-a d)}+\frac {2 b \left (\frac {b (a+b x)^5}{30 (c+d x)^5 (b c-a d)^2}+\frac {(a+b x)^5}{6 (c+d x)^6 (b c-a d)}\right )}{7 (b c-a d)}\)

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^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),x]
 

Output:

(a + b*x)^5/(7*(b*c - a*d)*(c + d*x)^7) + (2*b*((a + b*x)^5/(6*(b*c - a*d) 
*(c + d*x)^6) + (b*(a + b*x)^5)/(30*(b*c - a*d)^2*(c + d*x)^5)))/(7*(b*c - 
 a*d))
 

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]
 

rule 55

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] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs. \(2(83)=166\).

Time = 0.11 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.09

method	result	size
default	\(-\frac {2 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 d^{5} \left (x d +c \right )^{6}}-\frac {6 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{5 d^{5} \left (x d +c \right )^{5}}-\frac {d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}{7 d^{5} \left (x d +c \right )^{7}}-\frac {b^{3} \left (a d -b c \right )}{d^{5} \left (x d +c \right )^{4}}-\frac {b^{4}}{3 d^{5} \left (x d +c \right )^{3}}\)	\(186\)
norman	\(\frac {-\frac {b^{4} x^{4}}{3 d}-\frac {\left (3 a \,b^{3} d^{3}+b^{4} c \,d^{2}\right ) x^{3}}{3 d^{4}}-\frac {\left (6 a^{2} b^{2} d^{4}+3 a \,b^{3} c \,d^{3}+b^{4} c^{2} d^{2}\right ) x^{2}}{5 d^{5}}-\frac {\left (10 a^{3} b \,d^{5}+6 a^{2} b^{2} c \,d^{4}+3 a \,b^{3} c^{2} d^{3}+b^{4} c^{3} d^{2}\right ) x}{15 d^{6}}-\frac {15 a^{4} d^{6}+10 a^{3} b c \,d^{5}+6 a^{2} b^{2} c^{2} d^{4}+3 a \,b^{3} c^{3} d^{3}+b^{4} c^{4} d^{2}}{105 d^{7}}}{\left (x d +c \right )^{7}}\)	\(197\)
risch	\(\frac {-\frac {b^{4} x^{4}}{3 d}-\frac {b^{3} \left (3 a d +b c \right ) x^{3}}{3 d^{2}}-\frac {b^{2} \left (6 a^{2} d^{2}+3 a b c d +b^{2} c^{2}\right ) x^{2}}{5 d^{3}}-\frac {b \left (10 a^{3} d^{3}+6 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{15 d^{4}}-\frac {15 d^{4} a^{4}+10 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+3 a \,b^{3} c^{3} d +c^{4} b^{4}}{105 d^{5}}}{d^{7} x^{7}+7 c \,d^{6} x^{6}+21 c^{2} d^{5} x^{5}+35 c^{3} d^{4} x^{4}+35 c^{4} d^{3} x^{3}+21 c^{5} d^{2} x^{2}+7 c^{6} d x +c^{7}}\)	\(237\)
gosper	\(-\frac {35 d^{4} x^{4} b^{4}+105 a \,b^{3} d^{4} x^{3}+35 b^{4} c \,d^{3} x^{3}+126 a^{2} b^{2} d^{4} x^{2}+63 a \,b^{3} c \,d^{3} x^{2}+21 b^{4} c^{2} d^{2} x^{2}+70 a^{3} b \,d^{4} x +42 a^{2} b^{2} c \,d^{3} x +21 a \,b^{3} c^{2} d^{2} x +7 b^{4} c^{3} d x +15 d^{4} a^{4}+10 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+3 a \,b^{3} c^{3} d +c^{4} b^{4}}{105 d^{5} \left (d^{7} x^{7}+7 c \,d^{6} x^{6}+21 c^{2} d^{5} x^{5}+35 c^{3} d^{4} x^{4}+35 c^{4} d^{3} x^{3}+21 c^{5} d^{2} x^{2}+7 c^{6} d x +c^{7}\right )}\)	\(251\)
parallelrisch	\(\frac {-35 b^{4} x^{4} d^{6}-105 a \,b^{3} d^{6} x^{3}-35 b^{4} c \,d^{5} x^{3}-126 a^{2} b^{2} d^{6} x^{2}-63 a \,b^{3} c \,d^{5} x^{2}-21 b^{4} c^{2} d^{4} x^{2}-70 a^{3} b \,d^{6} x -42 a^{2} b^{2} c \,d^{5} x -21 a \,b^{3} c^{2} d^{4} x -7 b^{4} c^{3} d^{3} x -15 a^{4} d^{6}-10 a^{3} b c \,d^{5}-6 a^{2} b^{2} c^{2} d^{4}-3 a \,b^{3} c^{3} d^{3}-b^{4} c^{4} d^{2}}{105 d^{7} \left (d^{7} x^{7}+7 c \,d^{6} x^{6}+21 c^{2} d^{5} x^{5}+35 c^{3} d^{4} x^{4}+35 c^{4} d^{3} x^{3}+21 c^{5} d^{2} x^{2}+7 c^{6} d x +c^{7}\right )}\)	\(259\)
orering	\(-\frac {\left (35 d^{4} x^{4} b^{4}+105 a \,b^{3} d^{4} x^{3}+35 b^{4} c \,d^{3} x^{3}+126 a^{2} b^{2} d^{4} x^{2}+63 a \,b^{3} c \,d^{3} x^{2}+21 b^{4} c^{2} d^{2} x^{2}+70 a^{3} b \,d^{4} x +42 a^{2} b^{2} c \,d^{3} x +21 a \,b^{3} c^{2} d^{2} x +7 b^{4} c^{3} d x +15 d^{4} a^{4}+10 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+3 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \left (x d +c \right ) \left (b^{4} x^{4}+4 a \,x^{3} b^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )}{105 d^{5} \left (b x +a \right )^{4} \left (d^{8} x^{8}+8 c \,d^{7} x^{7}+28 c^{2} d^{6} x^{6}+56 c^{3} d^{5} x^{5}+70 c^{4} d^{4} x^{4}+56 c^{5} d^{3} x^{3}+28 c^{6} d^{2} x^{2}+8 c^{7} d x +c^{8}\right )}\)	\(312\)

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^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),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (83) = 166\).

Time = 0.07 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.78 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx=-\frac {35 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 3 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4} + 35 \, {\left (b^{4} c d^{3} + 3 \, a b^{3} d^{4}\right )} x^{3} + 21 \, {\left (b^{4} c^{2} d^{2} + 3 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} + 7 \, {\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} + 10 \, a^{3} b d^{4}\right )} x}{105 \, {\left (d^{12} x^{7} + 7 \, c d^{11} x^{6} + 21 \, c^{2} d^{10} x^{5} + 35 \, c^{3} d^{9} x^{4} + 35 \, c^{4} d^{8} x^{3} + 21 \, c^{5} d^{7} x^{2} + 7 \, c^{6} d^{6} x + c^{7} d^{5}\right )}} \] 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^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),x, algorithm="fricas")
 

Output:

-1/105*(35*b^4*d^4*x^4 + b^4*c^4 + 3*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 10* 
a^3*b*c*d^3 + 15*a^4*d^4 + 35*(b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 21*(b^4*c^2* 
d^2 + 3*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*x^2 + 7*(b^4*c^3*d + 3*a*b^3*c^2*d^2 
+ 6*a^2*b^2*c*d^3 + 10*a^3*b*d^4)*x)/(d^12*x^7 + 7*c*d^11*x^6 + 21*c^2*d^1 
0*x^5 + 35*c^3*d^9*x^4 + 35*c^4*d^8*x^3 + 21*c^5*d^7*x^2 + 7*c^6*d^6*x + c 
^7*d^5)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (73) = 146\).

Time = 4.96 (sec) , antiderivative size = 267, normalized size of antiderivative = 3.00 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx=\frac {- 15 a^{4} d^{4} - 10 a^{3} b c d^{3} - 6 a^{2} b^{2} c^{2} d^{2} - 3 a b^{3} c^{3} d - b^{4} c^{4} - 35 b^{4} d^{4} x^{4} + x^{3} \left (- 105 a b^{3} d^{4} - 35 b^{4} c d^{3}\right ) + x^{2} \left (- 126 a^{2} b^{2} d^{4} - 63 a b^{3} c d^{3} - 21 b^{4} c^{2} d^{2}\right ) + x \left (- 70 a^{3} b d^{4} - 42 a^{2} b^{2} c d^{3} - 21 a b^{3} c^{2} d^{2} - 7 b^{4} c^{3} d\right )}{105 c^{7} d^{5} + 735 c^{6} d^{6} x + 2205 c^{5} d^{7} x^{2} + 3675 c^{4} d^{8} x^{3} + 3675 c^{3} d^{9} x^{4} + 2205 c^{2} d^{10} x^{5} + 735 c d^{11} x^{6} + 105 d^{12} x^{7}} \] 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**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),x)
 

Output:

(-15*a**4*d**4 - 10*a**3*b*c*d**3 - 6*a**2*b**2*c**2*d**2 - 3*a*b**3*c**3* 
d - b**4*c**4 - 35*b**4*d**4*x**4 + x**3*(-105*a*b**3*d**4 - 35*b**4*c*d** 
3) + x**2*(-126*a**2*b**2*d**4 - 63*a*b**3*c*d**3 - 21*b**4*c**2*d**2) + x 
*(-70*a**3*b*d**4 - 42*a**2*b**2*c*d**3 - 21*a*b**3*c**2*d**2 - 7*b**4*c** 
3*d))/(105*c**7*d**5 + 735*c**6*d**6*x + 2205*c**5*d**7*x**2 + 3675*c**4*d 
**8*x**3 + 3675*c**3*d**9*x**4 + 2205*c**2*d**10*x**5 + 735*c*d**11*x**6 + 
 105*d**12*x**7)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (83) = 166\).

Time = 0.08 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.78 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx=-\frac {35 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 3 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4} + 35 \, {\left (b^{4} c d^{3} + 3 \, a b^{3} d^{4}\right )} x^{3} + 21 \, {\left (b^{4} c^{2} d^{2} + 3 \, a b^{3} c d^{3} + 6 \, a^{2} b^{2} d^{4}\right )} x^{2} + 7 \, {\left (b^{4} c^{3} d + 3 \, a b^{3} c^{2} d^{2} + 6 \, a^{2} b^{2} c d^{3} + 10 \, a^{3} b d^{4}\right )} x}{105 \, {\left (d^{12} x^{7} + 7 \, c d^{11} x^{6} + 21 \, c^{2} d^{10} x^{5} + 35 \, c^{3} d^{9} x^{4} + 35 \, c^{4} d^{8} x^{3} + 21 \, c^{5} d^{7} x^{2} + 7 \, c^{6} d^{6} x + c^{7} d^{5}\right )}} \] 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^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),x, algorithm="maxima")
 

Output:

-1/105*(35*b^4*d^4*x^4 + b^4*c^4 + 3*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 + 10* 
a^3*b*c*d^3 + 15*a^4*d^4 + 35*(b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 21*(b^4*c^2* 
d^2 + 3*a*b^3*c*d^3 + 6*a^2*b^2*d^4)*x^2 + 7*(b^4*c^3*d + 3*a*b^3*c^2*d^2 
+ 6*a^2*b^2*c*d^3 + 10*a^3*b*d^4)*x)/(d^12*x^7 + 7*c*d^11*x^6 + 21*c^2*d^1 
0*x^5 + 35*c^3*d^9*x^4 + 35*c^4*d^8*x^3 + 21*c^5*d^7*x^2 + 7*c^6*d^6*x + c 
^7*d^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (83) = 166\).

Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.07 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx=-\frac {35 \, b^{4} d^{4} x^{4} + 35 \, b^{4} c d^{3} x^{3} + 105 \, a b^{3} d^{4} x^{3} + 21 \, b^{4} c^{2} d^{2} x^{2} + 63 \, a b^{3} c d^{3} x^{2} + 126 \, a^{2} b^{2} d^{4} x^{2} + 7 \, b^{4} c^{3} d x + 21 \, a b^{3} c^{2} d^{2} x + 42 \, a^{2} b^{2} c d^{3} x + 70 \, a^{3} b d^{4} x + b^{4} c^{4} + 3 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} + 10 \, a^{3} b c d^{3} + 15 \, a^{4} d^{4}}{105 \, {\left (d x + c\right )}^{7} d^{5}} \] 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^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),x, algorithm="giac")
 

Output:

-1/105*(35*b^4*d^4*x^4 + 35*b^4*c*d^3*x^3 + 105*a*b^3*d^4*x^3 + 21*b^4*c^2 
*d^2*x^2 + 63*a*b^3*c*d^3*x^2 + 126*a^2*b^2*d^4*x^2 + 7*b^4*c^3*d*x + 21*a 
*b^3*c^2*d^2*x + 42*a^2*b^2*c*d^3*x + 70*a^3*b*d^4*x + b^4*c^4 + 3*a*b^3*c 
^3*d + 6*a^2*b^2*c^2*d^2 + 10*a^3*b*c*d^3 + 15*a^4*d^4)/((d*x + c)^7*d^5)
 

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.66 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx=-\frac {\frac {15\,a^4\,d^4+10\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+3\,a\,b^3\,c^3\,d+b^4\,c^4}{105\,d^5}+\frac {b^4\,x^4}{3\,d}+\frac {b^3\,x^3\,\left (3\,a\,d+b\,c\right )}{3\,d^2}+\frac {b\,x\,\left (10\,a^3\,d^3+6\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{15\,d^4}+\frac {b^2\,x^2\,\left (6\,a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right )}{5\,d^3}}{c^7+7\,c^6\,d\,x+21\,c^5\,d^2\,x^2+35\,c^4\,d^3\,x^3+35\,c^3\,d^4\,x^4+21\,c^2\,d^5\,x^5+7\,c\,d^6\,x^6+d^7\,x^7} \] Input:

int((a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x)/(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),x)
 

Output:

-((15*a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 + 3*a*b^3*c^3*d + 10*a^3*b*c*d 
^3)/(105*d^5) + (b^4*x^4)/(3*d) + (b^3*x^3*(3*a*d + b*c))/(3*d^2) + (b*x*( 
10*a^3*d^3 + b^3*c^3 + 3*a*b^2*c^2*d + 6*a^2*b*c*d^2))/(15*d^4) + (b^2*x^2 
*(6*a^2*d^2 + b^2*c^2 + 3*a*b*c*d))/(5*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 = 0.16 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.82 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^8+8 c^7 d x+28 c^6 d^2 x^2+56 c^5 d^3 x^3+70 c^4 d^4 x^4+56 c^3 d^5 x^5+28 c^2 d^6 x^6+8 c d^7 x^7+d^8 x^8} \, dx=\frac {-35 b^{4} d^{4} x^{4}-105 a \,b^{3} d^{4} x^{3}-35 b^{4} c \,d^{3} x^{3}-126 a^{2} b^{2} d^{4} x^{2}-63 a \,b^{3} c \,d^{3} x^{2}-21 b^{4} c^{2} d^{2} x^{2}-70 a^{3} b \,d^{4} x -42 a^{2} b^{2} c \,d^{3} x -21 a \,b^{3} c^{2} d^{2} x -7 b^{4} c^{3} d x -15 a^{4} d^{4}-10 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-3 a \,b^{3} c^{3} d -b^{4} c^{4}}{105 d^{5} \left (d^{7} x^{7}+7 c \,d^{6} x^{6}+21 c^{2} d^{5} x^{5}+35 c^{3} d^{4} x^{4}+35 c^{4} d^{3} x^{3}+21 c^{5} d^{2} x^{2}+7 c^{6} d x +c^{7}\right )} \] Input:

int((b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4)/(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),x)
 

Output:

( - 15*a**4*d**4 - 10*a**3*b*c*d**3 - 70*a**3*b*d**4*x - 6*a**2*b**2*c**2* 
d**2 - 42*a**2*b**2*c*d**3*x - 126*a**2*b**2*d**4*x**2 - 3*a*b**3*c**3*d - 
 21*a*b**3*c**2*d**2*x - 63*a*b**3*c*d**3*x**2 - 105*a*b**3*d**4*x**3 - b* 
*4*c**4 - 7*b**4*c**3*d*x - 21*b**4*c**2*d**2*x**2 - 35*b**4*c*d**3*x**3 - 
 35*b**4*d**4*x**4)/(105*d**5*(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))