Integrand size = 101, antiderivative size = 28 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6} \, dx=\frac {(a+b x)^5}{5 (b c-a d) (c+d x)^5} \] Output:
1/5*(b*x+a)^5/(-a*d+b*c)/(d*x+c)^5
Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(28)=56\).
Time = 0.01 (sec) , antiderivative size = 140, normalized size of antiderivative = 5.00 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6} \, dx=-\frac {a^4 d^4+a^3 b d^3 (c+5 d x)+a^2 b^2 d^2 \left (c^2+5 c d x+10 d^2 x^2\right )+a b^3 d \left (c^3+5 c^2 d x+10 c d^2 x^2+10 d^3 x^3\right )+b^4 \left (c^4+5 c^3 d x+10 c^2 d^2 x^2+10 c d^3 x^3+5 d^4 x^4\right )}{5 d^5 (c+d x)^5} \] 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^6 + 6*c^5*d*x + 15*c^4*d^2*x^2 + 20*c^3*d^3*x^3 + 15*c^2*d^4*x^4 + 6*c*d^5*x^ 5 + d^6*x^6),x]
Output:
-1/5*(a^4*d^4 + a^3*b*d^3*(c + 5*d*x) + a^2*b^2*d^2*(c^2 + 5*c*d*x + 10*d^ 2*x^2) + a*b^3*d*(c^3 + 5*c^2*d*x + 10*c*d^2*x^2 + 10*d^3*x^3) + b^4*(c^4 + 5*c^3*d*x + 10*c^2*d^2*x^2 + 10*c*d^3*x^3 + 5*d^4*x^4))/(d^5*(c + d*x)^5 )
Time = 0.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2006, 2007, 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.
\(\displaystyle \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6} \, dx\) |
\(\Big \downarrow \) 2006 |
\(\displaystyle \int \frac {(a+b x)^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {(a+b x)^4}{(c+d x)^6}dx\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {(a+b x)^5}{5 (c+d x)^5 (b c-a d)}\) |
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^6 + 6*c^5 *d*x + 15*c^4*d^2*x^2 + 20*c^3*d^3*x^3 + 15*c^2*d^4*x^4 + 6*c*d^5*x^5 + d^ 6*x^6),x]
Output:
(a + b*x)^5/(5*(b*c - a*d)*(c + d*x)^5)
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]
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]]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(166\) vs. \(2(26)=52\).
Time = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 5.96
method | result | size |
norman | \(\frac {-\frac {b^{4} x^{4}}{d}-\frac {2 \left (a \,b^{3} d +b^{4} c \right ) x^{3}}{d^{2}}-\frac {2 \left (a^{2} b^{2} d^{2}+a \,b^{3} c d +b^{4} c^{2}\right ) x^{2}}{d^{3}}-\frac {\left (a^{3} b \,d^{3}+a^{2} b^{2} c \,d^{2}+a \,b^{3} c^{2} d +b^{4} c^{3}\right ) x}{d^{4}}-\frac {d^{4} a^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +c^{4} b^{4}}{5 d^{5}}}{\left (x d +c \right )^{5}}\) | \(167\) |
default | \(-\frac {d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}{5 d^{5} \left (x d +c \right )^{5}}-\frac {b^{4}}{d^{5} \left (x d +c \right )}-\frac {2 b^{3} \left (a d -b c \right )}{d^{5} \left (x d +c \right )^{2}}-\frac {b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{d^{5} \left (x d +c \right )^{4}}-\frac {2 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{5} \left (x d +c \right )^{3}}\) | \(186\) |
risch | \(\frac {-\frac {b^{4} x^{4}}{d}-\frac {2 b^{3} \left (a d +b c \right ) x^{3}}{d^{2}}-\frac {2 b^{2} \left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{2}}{d^{3}}-\frac {b \left (a^{3} d^{3}+a^{2} b c \,d^{2}+a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{d^{4}}-\frac {d^{4} a^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +c^{4} b^{4}}{5 d^{5}}}{d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}}\) | \(205\) |
gosper | \(-\frac {5 d^{4} x^{4} b^{4}+10 a \,b^{3} d^{4} x^{3}+10 b^{4} c \,d^{3} x^{3}+10 a^{2} b^{2} d^{4} x^{2}+10 a \,b^{3} c \,d^{3} x^{2}+10 b^{4} c^{2} d^{2} x^{2}+5 a^{3} b \,d^{4} x +5 a^{2} b^{2} c \,d^{3} x +5 a \,b^{3} c^{2} d^{2} x +5 b^{4} c^{3} d x +d^{4} a^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +c^{4} b^{4}}{5 \left (d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}\right ) d^{5}}\) | \(225\) |
parallelrisch | \(\frac {-5 d^{4} x^{4} b^{4}-10 a \,b^{3} d^{4} x^{3}-10 b^{4} c \,d^{3} x^{3}-10 a^{2} b^{2} d^{4} x^{2}-10 a \,b^{3} c \,d^{3} x^{2}-10 b^{4} c^{2} d^{2} x^{2}-5 a^{3} b \,d^{4} x -5 a^{2} b^{2} c \,d^{3} x -5 a \,b^{3} c^{2} d^{2} x -5 b^{4} c^{3} d x -d^{4} a^{4}-a^{3} b c \,d^{3}-a^{2} b^{2} c^{2} d^{2}-a \,b^{3} c^{3} d -c^{4} b^{4}}{5 d^{5} \left (d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}\right )}\) | \(230\) |
orering | \(-\frac {\left (5 d^{4} x^{4} b^{4}+10 a \,b^{3} d^{4} x^{3}+10 b^{4} c \,d^{3} x^{3}+10 a^{2} b^{2} d^{4} x^{2}+10 a \,b^{3} c \,d^{3} x^{2}+10 b^{4} c^{2} d^{2} x^{2}+5 a^{3} b \,d^{4} x +5 a^{2} b^{2} c \,d^{3} x +5 a \,b^{3} c^{2} d^{2} x +5 b^{4} c^{3} d x +d^{4} a^{4}+a^{3} b c \,d^{3}+a^{2} b^{2} c^{2} d^{2}+a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \left (x d +c \right ) \left (b^{4} x^{4}+4 a \,x^{3} b^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )}{5 d^{5} \left (b x +a \right )^{4} \left (d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}\right )}\) | \(286\) |
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^6*x^6+6*c*d^5*x^5 +15*c^2*d^4*x^4+20*c^3*d^3*x^3+15*c^4*d^2*x^2+6*c^5*d*x+c^6),x,method=_RET URNVERBOSE)
Output:
(-b^4/d*x^4-2*(a*b^3*d+b^4*c)/d^2*x^3-2*(a^2*b^2*d^2+a*b^3*c*d+b^4*c^2)/d^ 3*x^2-(a^3*b*d^3+a^2*b^2*c*d^2+a*b^3*c^2*d+b^4*c^3)/d^4*x-1/5*(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)/(d*x+c)^5
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (26) = 52\).
Time = 0.08 (sec) , antiderivative size = 215, normalized size of antiderivative = 7.68 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6} \, dx=-\frac {5 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} + a^{4} d^{4} + 10 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 10 \, {\left (b^{4} c^{2} d^{2} + a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 5 \, {\left (b^{4} c^{3} d + a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x}{5 \, {\left (d^{10} x^{5} + 5 \, c d^{9} x^{4} + 10 \, c^{2} d^{8} x^{3} + 10 \, c^{3} d^{7} x^{2} + 5 \, c^{4} d^{6} x + c^{5} d^{5}\right )}} \] 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^6*x^6+6*c*d ^5*x^5+15*c^2*d^4*x^4+20*c^3*d^3*x^3+15*c^4*d^2*x^2+6*c^5*d*x+c^6),x, algo rithm="fricas")
Output:
-1/5*(5*b^4*d^4*x^4 + b^4*c^4 + a*b^3*c^3*d + a^2*b^2*c^2*d^2 + a^3*b*c*d^ 3 + a^4*d^4 + 10*(b^4*c*d^3 + a*b^3*d^4)*x^3 + 10*(b^4*c^2*d^2 + a*b^3*c*d ^3 + a^2*b^2*d^4)*x^2 + 5*(b^4*c^3*d + a*b^3*c^2*d^2 + a^2*b^2*c*d^3 + a^3 *b*d^4)*x)/(d^10*x^5 + 5*c*d^9*x^4 + 10*c^2*d^8*x^3 + 10*c^3*d^7*x^2 + 5*c ^4*d^6*x + c^5*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (20) = 40\).
Time = 1.78 (sec) , antiderivative size = 236, normalized size of antiderivative = 8.43 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6} \, dx=\frac {- a^{4} d^{4} - a^{3} b c d^{3} - a^{2} b^{2} c^{2} d^{2} - a b^{3} c^{3} d - b^{4} c^{4} - 5 b^{4} d^{4} x^{4} + x^{3} \left (- 10 a b^{3} d^{4} - 10 b^{4} c d^{3}\right ) + x^{2} \left (- 10 a^{2} b^{2} d^{4} - 10 a b^{3} c d^{3} - 10 b^{4} c^{2} d^{2}\right ) + x \left (- 5 a^{3} b d^{4} - 5 a^{2} b^{2} c d^{3} - 5 a b^{3} c^{2} d^{2} - 5 b^{4} c^{3} d\right )}{5 c^{5} d^{5} + 25 c^{4} d^{6} x + 50 c^{3} d^{7} x^{2} + 50 c^{2} d^{8} x^{3} + 25 c d^{9} x^{4} + 5 d^{10} x^{5}} \] 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**6 *x**6+6*c*d**5*x**5+15*c**2*d**4*x**4+20*c**3*d**3*x**3+15*c**4*d**2*x**2+ 6*c**5*d*x+c**6),x)
Output:
(-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 - 5*b**4*d**4*x**4 + x**3*(-10*a*b**3*d**4 - 10*b**4*c*d**3) + x**2*(- 10*a**2*b**2*d**4 - 10*a*b**3*c*d**3 - 10*b**4*c**2*d**2) + x*(-5*a**3*b*d **4 - 5*a**2*b**2*c*d**3 - 5*a*b**3*c**2*d**2 - 5*b**4*c**3*d))/(5*c**5*d* *5 + 25*c**4*d**6*x + 50*c**3*d**7*x**2 + 50*c**2*d**8*x**3 + 25*c*d**9*x* *4 + 5*d**10*x**5)
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (26) = 52\).
Time = 0.04 (sec) , antiderivative size = 215, normalized size of antiderivative = 7.68 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6} \, dx=-\frac {5 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} + a^{4} d^{4} + 10 \, {\left (b^{4} c d^{3} + a b^{3} d^{4}\right )} x^{3} + 10 \, {\left (b^{4} c^{2} d^{2} + a b^{3} c d^{3} + a^{2} b^{2} d^{4}\right )} x^{2} + 5 \, {\left (b^{4} c^{3} d + a b^{3} c^{2} d^{2} + a^{2} b^{2} c d^{3} + a^{3} b d^{4}\right )} x}{5 \, {\left (d^{10} x^{5} + 5 \, c d^{9} x^{4} + 10 \, c^{2} d^{8} x^{3} + 10 \, c^{3} d^{7} x^{2} + 5 \, c^{4} d^{6} x + c^{5} d^{5}\right )}} \] 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^6*x^6+6*c*d ^5*x^5+15*c^2*d^4*x^4+20*c^3*d^3*x^3+15*c^4*d^2*x^2+6*c^5*d*x+c^6),x, algo rithm="maxima")
Output:
-1/5*(5*b^4*d^4*x^4 + b^4*c^4 + a*b^3*c^3*d + a^2*b^2*c^2*d^2 + a^3*b*c*d^ 3 + a^4*d^4 + 10*(b^4*c*d^3 + a*b^3*d^4)*x^3 + 10*(b^4*c^2*d^2 + a*b^3*c*d ^3 + a^2*b^2*d^4)*x^2 + 5*(b^4*c^3*d + a*b^3*c^2*d^2 + a^2*b^2*c*d^3 + a^3 *b*d^4)*x)/(d^10*x^5 + 5*c*d^9*x^4 + 10*c^2*d^8*x^3 + 10*c^3*d^7*x^2 + 5*c ^4*d^6*x + c^5*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (26) = 52\).
Time = 0.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 6.43 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6} \, dx=-\frac {5 \, b^{4} d^{4} x^{4} + 10 \, b^{4} c d^{3} x^{3} + 10 \, a b^{3} d^{4} x^{3} + 10 \, b^{4} c^{2} d^{2} x^{2} + 10 \, a b^{3} c d^{3} x^{2} + 10 \, a^{2} b^{2} d^{4} x^{2} + 5 \, b^{4} c^{3} d x + 5 \, a b^{3} c^{2} d^{2} x + 5 \, a^{2} b^{2} c d^{3} x + 5 \, a^{3} b d^{4} x + b^{4} c^{4} + a b^{3} c^{3} d + a^{2} b^{2} c^{2} d^{2} + a^{3} b c d^{3} + a^{4} d^{4}}{5 \, {\left (d x + c\right )}^{5} d^{5}} \] 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^6*x^6+6*c*d ^5*x^5+15*c^2*d^4*x^4+20*c^3*d^3*x^3+15*c^4*d^2*x^2+6*c^5*d*x+c^6),x, algo rithm="giac")
Output:
-1/5*(5*b^4*d^4*x^4 + 10*b^4*c*d^3*x^3 + 10*a*b^3*d^4*x^3 + 10*b^4*c^2*d^2 *x^2 + 10*a*b^3*c*d^3*x^2 + 10*a^2*b^2*d^4*x^2 + 5*b^4*c^3*d*x + 5*a*b^3*c ^2*d^2*x + 5*a^2*b^2*c*d^3*x + 5*a^3*b*d^4*x + b^4*c^4 + a*b^3*c^3*d + a^2 *b^2*c^2*d^2 + a^3*b*c*d^3 + a^4*d^4)/((d*x + c)^5*d^5)
Time = 0.16 (sec) , antiderivative size = 203, normalized size of antiderivative = 7.25 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6} \, dx=-\frac {\frac {a^4\,d^4+a^3\,b\,c\,d^3+a^2\,b^2\,c^2\,d^2+a\,b^3\,c^3\,d+b^4\,c^4}{5\,d^5}+\frac {b^4\,x^4}{d}+\frac {2\,b^3\,x^3\,\left (a\,d+b\,c\right )}{d^2}+\frac {b\,x\,\left (a^3\,d^3+a^2\,b\,c\,d^2+a\,b^2\,c^2\,d+b^3\,c^3\right )}{d^4}+\frac {2\,b^2\,x^2\,\left (a^2\,d^2+a\,b\,c\,d+b^2\,c^2\right )}{d^3}}{c^5+5\,c^4\,d\,x+10\,c^3\,d^2\,x^2+10\,c^2\,d^3\,x^3+5\,c\,d^4\,x^4+d^5\,x^5} \] 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^6 + d^6*x ^6 + 6*c*d^5*x^5 + 15*c^4*d^2*x^2 + 20*c^3*d^3*x^3 + 15*c^2*d^4*x^4 + 6*c^ 5*d*x),x)
Output:
-((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)/(5*d^5 ) + (b^4*x^4)/d + (2*b^3*x^3*(a*d + b*c))/d^2 + (b*x*(a^3*d^3 + b^3*c^3 + a*b^2*c^2*d + a^2*b*c*d^2))/d^4 + (2*b^2*x^2*(a^2*d^2 + b^2*c^2 + a*b*c*d) )/d^3)/(c^5 + d^5*x^5 + 5*c*d^4*x^4 + 10*c^3*d^2*x^2 + 10*c^2*d^3*x^3 + 5* c^4*d*x)
Time = 0.16 (sec) , antiderivative size = 192, normalized size of antiderivative = 6.86 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^6+6 c^5 d x+15 c^4 d^2 x^2+20 c^3 d^3 x^3+15 c^2 d^4 x^4+6 c d^5 x^5+d^6 x^6} \, dx=\frac {b^{4} d^{4} x^{5}-10 a \,b^{3} c \,d^{3} x^{3}-10 a^{2} b^{2} c \,d^{3} x^{2}-10 a \,b^{3} c^{2} d^{2} x^{2}-5 a^{3} b c \,d^{3} x -5 a^{2} b^{2} c^{2} d^{2} x -5 a \,b^{3} c^{3} d x -a^{4} c \,d^{3}-a^{3} b \,c^{2} d^{2}-a^{2} b^{2} c^{3} d -a \,b^{3} c^{4}}{5 c \,d^{4} \left (d^{5} x^{5}+5 c \,d^{4} x^{4}+10 c^{2} d^{3} x^{3}+10 c^{3} d^{2} x^{2}+5 c^{4} d x +c^{5}\right )} \] 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^6*x^6+6*c*d^5*x^5 +15*c^2*d^4*x^4+20*c^3*d^3*x^3+15*c^4*d^2*x^2+6*c^5*d*x+c^6),x)
Output:
( - a**4*c*d**3 - a**3*b*c**2*d**2 - 5*a**3*b*c*d**3*x - a**2*b**2*c**3*d - 5*a**2*b**2*c**2*d**2*x - 10*a**2*b**2*c*d**3*x**2 - a*b**3*c**4 - 5*a*b **3*c**3*d*x - 10*a*b**3*c**2*d**2*x**2 - 10*a*b**3*c*d**3*x**3 + b**4*d** 4*x**5)/(5*c*d**4*(c**5 + 5*c**4*d*x + 10*c**3*d**2*x**2 + 10*c**2*d**3*x* *3 + 5*c*d**4*x**4 + d**5*x**5))