\(\int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx\) [162]

Optimal result

Integrand size = 112, antiderivative size = 58 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx=\frac {(a+b x)^5}{6 (b c-a d) (c+d x)^6}+\frac {b (a+b x)^5}{30 (b c-a d)^2 (c+d x)^5} \] Output:

1/6*(b*x+a)^5/(-a*d+b*c)/(d*x+c)^6+1/30*b*(b*x+a)^5/(-a*d+b*c)^2/(d*x+c)^5
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(58)=116\).

Time = 0.01 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.48 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx=-\frac {5 a^4 d^4+4 a^3 b d^3 (c+6 d x)+3 a^2 b^2 d^2 \left (c^2+6 c d x+15 d^2 x^2\right )+2 a b^3 d \left (c^3+6 c^2 d x+15 c d^2 x^2+20 d^3 x^3\right )+b^4 \left (c^4+6 c^3 d x+15 c^2 d^2 x^2+20 c d^3 x^3+15 d^4 x^4\right )}{30 d^5 (c+d x)^6} \] 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^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),x]
 

Output:

-1/30*(5*a^4*d^4 + 4*a^3*b*d^3*(c + 6*d*x) + 3*a^2*b^2*d^2*(c^2 + 6*c*d*x 
+ 15*d^2*x^2) + 2*a*b^3*d*(c^3 + 6*c^2*d*x + 15*c*d^2*x^2 + 20*d^3*x^3) + 
b^4*(c^4 + 6*c^3*d*x + 15*c^2*d^2*x^2 + 20*c*d^3*x^3 + 15*d^4*x^4))/(d^5*( 
c + d*x)^6)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2006, 2007, 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.

\(\displaystyle \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx\)

\(\Big \downarrow \) 2006

\(\displaystyle \int \frac {(a+b x)^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7}dx\)

\(\Big \downarrow \) 2007

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

\(\Big \downarrow \) 55

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

\(\Big \downarrow \) 48

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

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^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),x]
 

Output:

(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)
 

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. \(2(54)=108\).

Time = 0.10 (sec) , antiderivative size = 186, normalized size of antiderivative = 3.21

method	result	size
default	\(-\frac {d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}{6 d^{5} \left (x d +c \right )^{6}}-\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{5 d^{5} \left (x d +c \right )^{5}}-\frac {b^{4}}{2 d^{5} \left (x d +c \right )^{2}}-\frac {3 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{2 d^{5} \left (x d +c \right )^{4}}-\frac {4 b^{3} \left (a d -b c \right )}{3 d^{5} \left (x d +c \right )^{3}}\)	\(186\)
norman	\(\frac {-\frac {b^{4} x^{4}}{2 d}-\frac {2 \left (2 a \,b^{3} d^{2}+b^{4} c d \right ) x^{3}}{3 d^{3}}-\frac {\left (3 a^{2} b^{2} d^{3}+2 a \,b^{3} c \,d^{2}+b^{4} c^{2} d \right ) x^{2}}{2 d^{4}}-\frac {\left (4 d^{4} a^{3} b +3 a^{2} b^{2} c \,d^{3}+2 a \,b^{3} c^{2} d^{2}+b^{4} c^{3} d \right ) x}{5 d^{5}}-\frac {5 a^{4} d^{5}+4 a^{3} b c \,d^{4}+3 a^{2} b^{2} c^{2} d^{3}+2 a \,b^{3} c^{3} d^{2}+b^{4} c^{4} d}{30 d^{6}}}{\left (x d +c \right )^{6}}\)	\(189\)
risch	\(\frac {-\frac {b^{4} x^{4}}{2 d}-\frac {2 b^{3} \left (2 a d +b c \right ) x^{3}}{3 d^{2}}-\frac {b^{2} \left (3 a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) x^{2}}{2 d^{3}}-\frac {b \left (4 a^{3} d^{3}+3 a^{2} b c \,d^{2}+2 a \,b^{2} c^{2} d +b^{3} c^{3}\right ) x}{5 d^{4}}-\frac {5 d^{4} a^{4}+4 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +c^{4} b^{4}}{30 d^{5}}}{d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}}\)	\(226\)
gosper	\(-\frac {15 d^{4} x^{4} b^{4}+40 a \,b^{3} d^{4} x^{3}+20 b^{4} c \,d^{3} x^{3}+45 a^{2} b^{2} d^{4} x^{2}+30 a \,b^{3} c \,d^{3} x^{2}+15 b^{4} c^{2} d^{2} x^{2}+24 a^{3} b \,d^{4} x +18 a^{2} b^{2} c \,d^{3} x +12 a \,b^{3} c^{2} d^{2} x +6 b^{4} c^{3} d x +5 d^{4} a^{4}+4 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +c^{4} b^{4}}{30 d^{5} \left (d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}\right )}\)	\(240\)
parallelrisch	\(\frac {-15 b^{4} x^{4} d^{5}-40 a \,b^{3} d^{5} x^{3}-20 b^{4} c \,d^{4} x^{3}-45 a^{2} b^{2} d^{5} x^{2}-30 a \,b^{3} c \,d^{4} x^{2}-15 b^{4} c^{2} d^{3} x^{2}-24 a^{3} b \,d^{5} x -18 a^{2} b^{2} c \,d^{4} x -12 a \,b^{3} c^{2} d^{3} x -6 b^{4} c^{3} d^{2} x -5 a^{4} d^{5}-4 a^{3} b c \,d^{4}-3 a^{2} b^{2} c^{2} d^{3}-2 a \,b^{3} c^{3} d^{2}-b^{4} c^{4} d}{30 d^{6} \left (d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}\right )}\)	\(246\)
orering	\(-\frac {\left (15 d^{4} x^{4} b^{4}+40 a \,b^{3} d^{4} x^{3}+20 b^{4} c \,d^{3} x^{3}+45 a^{2} b^{2} d^{4} x^{2}+30 a \,b^{3} c \,d^{3} x^{2}+15 b^{4} c^{2} d^{2} x^{2}+24 a^{3} b \,d^{4} x +18 a^{2} b^{2} c \,d^{3} x +12 a \,b^{3} c^{2} d^{2} x +6 b^{4} c^{3} d x +5 d^{4} a^{4}+4 a^{3} b c \,d^{3}+3 a^{2} b^{2} c^{2} d^{2}+2 a \,b^{3} c^{3} d +c^{4} b^{4}\right ) \left (x d +c \right ) \left (b^{4} x^{4}+4 a \,x^{3} b^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )}{30 d^{5} \left (b x +a \right )^{4} \left (d^{7} x^{7}+7 c \,d^{6} x^{6}+21 c^{2} d^{5} x^{5}+35 c^{3} d^{4} x^{4}+35 c^{4} d^{3} x^{3}+21 c^{5} d^{2} x^{2}+7 c^{6} d x +c^{7}\right )}\)	\(301\)

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^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 
),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (54) = 108\).

Time = 0.08 (sec) , antiderivative size = 236, normalized size of antiderivative = 4.07 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx=-\frac {15 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 2 \, a b^{3} c^{3} d + 3 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4} + 20 \, {\left (b^{4} c d^{3} + 2 \, a b^{3} d^{4}\right )} x^{3} + 15 \, {\left (b^{4} c^{2} d^{2} + 2 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{2} + 6 \, {\left (b^{4} c^{3} d + 2 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} + 4 \, a^{3} b d^{4}\right )} x}{30 \, {\left (d^{11} x^{6} + 6 \, c d^{10} x^{5} + 15 \, c^{2} d^{9} x^{4} + 20 \, c^{3} d^{8} x^{3} + 15 \, c^{4} d^{7} x^{2} + 6 \, c^{5} d^{6} x + c^{6} d^{5}\right )}} \] 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^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),x, algorithm="fricas")
 

Output:

-1/30*(15*b^4*d^4*x^4 + b^4*c^4 + 2*a*b^3*c^3*d + 3*a^2*b^2*c^2*d^2 + 4*a^ 
3*b*c*d^3 + 5*a^4*d^4 + 20*(b^4*c*d^3 + 2*a*b^3*d^4)*x^3 + 15*(b^4*c^2*d^2 
 + 2*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*x^2 + 6*(b^4*c^3*d + 2*a*b^3*c^2*d^2 + 3 
*a^2*b^2*c*d^3 + 4*a^3*b*d^4)*x)/(d^11*x^6 + 6*c*d^10*x^5 + 15*c^2*d^9*x^4 
 + 20*c^3*d^8*x^3 + 15*c^4*d^7*x^2 + 6*c^5*d^6*x + c^6*d^5)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (46) = 92\).

Time = 2.68 (sec) , antiderivative size = 255, normalized size of antiderivative = 4.40 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx=\frac {- 5 a^{4} d^{4} - 4 a^{3} b c d^{3} - 3 a^{2} b^{2} c^{2} d^{2} - 2 a b^{3} c^{3} d - b^{4} c^{4} - 15 b^{4} d^{4} x^{4} + x^{3} \left (- 40 a b^{3} d^{4} - 20 b^{4} c d^{3}\right ) + x^{2} \left (- 45 a^{2} b^{2} d^{4} - 30 a b^{3} c d^{3} - 15 b^{4} c^{2} d^{2}\right ) + x \left (- 24 a^{3} b d^{4} - 18 a^{2} b^{2} c d^{3} - 12 a b^{3} c^{2} d^{2} - 6 b^{4} c^{3} d\right )}{30 c^{6} d^{5} + 180 c^{5} d^{6} x + 450 c^{4} d^{7} x^{2} + 600 c^{3} d^{8} x^{3} + 450 c^{2} d^{9} x^{4} + 180 c d^{10} x^{5} + 30 d^{11} x^{6}} \] 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**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),x)
 

Output:

(-5*a**4*d**4 - 4*a**3*b*c*d**3 - 3*a**2*b**2*c**2*d**2 - 2*a*b**3*c**3*d 
- b**4*c**4 - 15*b**4*d**4*x**4 + x**3*(-40*a*b**3*d**4 - 20*b**4*c*d**3) 
+ x**2*(-45*a**2*b**2*d**4 - 30*a*b**3*c*d**3 - 15*b**4*c**2*d**2) + x*(-2 
4*a**3*b*d**4 - 18*a**2*b**2*c*d**3 - 12*a*b**3*c**2*d**2 - 6*b**4*c**3*d) 
)/(30*c**6*d**5 + 180*c**5*d**6*x + 450*c**4*d**7*x**2 + 600*c**3*d**8*x** 
3 + 450*c**2*d**9*x**4 + 180*c*d**10*x**5 + 30*d**11*x**6)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (54) = 108\).

Time = 0.04 (sec) , antiderivative size = 236, normalized size of antiderivative = 4.07 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx=-\frac {15 \, b^{4} d^{4} x^{4} + b^{4} c^{4} + 2 \, a b^{3} c^{3} d + 3 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4} + 20 \, {\left (b^{4} c d^{3} + 2 \, a b^{3} d^{4}\right )} x^{3} + 15 \, {\left (b^{4} c^{2} d^{2} + 2 \, a b^{3} c d^{3} + 3 \, a^{2} b^{2} d^{4}\right )} x^{2} + 6 \, {\left (b^{4} c^{3} d + 2 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} + 4 \, a^{3} b d^{4}\right )} x}{30 \, {\left (d^{11} x^{6} + 6 \, c d^{10} x^{5} + 15 \, c^{2} d^{9} x^{4} + 20 \, c^{3} d^{8} x^{3} + 15 \, c^{4} d^{7} x^{2} + 6 \, c^{5} d^{6} x + c^{6} d^{5}\right )}} \] 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^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),x, algorithm="maxima")
 

Output:

-1/30*(15*b^4*d^4*x^4 + b^4*c^4 + 2*a*b^3*c^3*d + 3*a^2*b^2*c^2*d^2 + 4*a^ 
3*b*c*d^3 + 5*a^4*d^4 + 20*(b^4*c*d^3 + 2*a*b^3*d^4)*x^3 + 15*(b^4*c^2*d^2 
 + 2*a*b^3*c*d^3 + 3*a^2*b^2*d^4)*x^2 + 6*(b^4*c^3*d + 2*a*b^3*c^2*d^2 + 3 
*a^2*b^2*c*d^3 + 4*a^3*b*d^4)*x)/(d^11*x^6 + 6*c*d^10*x^5 + 15*c^2*d^9*x^4 
 + 20*c^3*d^8*x^3 + 15*c^4*d^7*x^2 + 6*c^5*d^6*x + c^6*d^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (54) = 108\).

Time = 0.12 (sec) , antiderivative size = 184, normalized size of antiderivative = 3.17 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx=-\frac {15 \, b^{4} d^{4} x^{4} + 20 \, b^{4} c d^{3} x^{3} + 40 \, a b^{3} d^{4} x^{3} + 15 \, b^{4} c^{2} d^{2} x^{2} + 30 \, a b^{3} c d^{3} x^{2} + 45 \, a^{2} b^{2} d^{4} x^{2} + 6 \, b^{4} c^{3} d x + 12 \, a b^{3} c^{2} d^{2} x + 18 \, a^{2} b^{2} c d^{3} x + 24 \, a^{3} b d^{4} x + b^{4} c^{4} + 2 \, a b^{3} c^{3} d + 3 \, a^{2} b^{2} c^{2} d^{2} + 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}}{30 \, {\left (d x + c\right )}^{6} d^{5}} \] 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^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),x, algorithm="giac")
 

Output:

-1/30*(15*b^4*d^4*x^4 + 20*b^4*c*d^3*x^3 + 40*a*b^3*d^4*x^3 + 15*b^4*c^2*d 
^2*x^2 + 30*a*b^3*c*d^3*x^2 + 45*a^2*b^2*d^4*x^2 + 6*b^4*c^3*d*x + 12*a*b^ 
3*c^2*d^2*x + 18*a^2*b^2*c*d^3*x + 24*a^3*b*d^4*x + b^4*c^4 + 2*a*b^3*c^3* 
d + 3*a^2*b^2*c^2*d^2 + 4*a^3*b*c*d^3 + 5*a^4*d^4)/((d*x + c)^6*d^5)
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.90 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx=-\frac {\frac {5\,a^4\,d^4+4\,a^3\,b\,c\,d^3+3\,a^2\,b^2\,c^2\,d^2+2\,a\,b^3\,c^3\,d+b^4\,c^4}{30\,d^5}+\frac {b^4\,x^4}{2\,d}+\frac {2\,b^3\,x^3\,\left (2\,a\,d+b\,c\right )}{3\,d^2}+\frac {b\,x\,\left (4\,a^3\,d^3+3\,a^2\,b\,c\,d^2+2\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{5\,d^4}+\frac {b^2\,x^2\,\left (3\,a^2\,d^2+2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,d^3}}{c^6+6\,c^5\,d\,x+15\,c^4\,d^2\,x^2+20\,c^3\,d^3\,x^3+15\,c^2\,d^4\,x^4+6\,c\,d^5\,x^5+d^6\,x^6} \] 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^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),x)
 

Output:

-((5*a^4*d^4 + b^4*c^4 + 3*a^2*b^2*c^2*d^2 + 2*a*b^3*c^3*d + 4*a^3*b*c*d^3 
)/(30*d^5) + (b^4*x^4)/(2*d) + (2*b^3*x^3*(2*a*d + b*c))/(3*d^2) + (b*x*(4 
*a^3*d^3 + b^3*c^3 + 2*a*b^2*c^2*d + 3*a^2*b*c*d^2))/(5*d^4) + (b^2*x^2*(3 
*a^2*d^2 + b^2*c^2 + 2*a*b*c*d))/(2*d^3))/(c^6 + d^6*x^6 + 6*c*d^5*x^5 + 1 
5*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)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 240, normalized size of antiderivative = 4.14 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^7+7 c^6 d x+21 c^5 d^2 x^2+35 c^4 d^3 x^3+35 c^3 d^4 x^4+21 c^2 d^5 x^5+7 c d^6 x^6+d^7 x^7} \, dx=\frac {-15 b^{4} d^{4} x^{4}-40 a \,b^{3} d^{4} x^{3}-20 b^{4} c \,d^{3} x^{3}-45 a^{2} b^{2} d^{4} x^{2}-30 a \,b^{3} c \,d^{3} x^{2}-15 b^{4} c^{2} d^{2} x^{2}-24 a^{3} b \,d^{4} x -18 a^{2} b^{2} c \,d^{3} x -12 a \,b^{3} c^{2} d^{2} x -6 b^{4} c^{3} d x -5 a^{4} d^{4}-4 a^{3} b c \,d^{3}-3 a^{2} b^{2} c^{2} d^{2}-2 a \,b^{3} c^{3} d -b^{4} c^{4}}{30 d^{5} \left (d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}\right )} \] 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^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 
),x)
 

Output:

( - 5*a**4*d**4 - 4*a**3*b*c*d**3 - 24*a**3*b*d**4*x - 3*a**2*b**2*c**2*d* 
*2 - 18*a**2*b**2*c*d**3*x - 45*a**2*b**2*d**4*x**2 - 2*a*b**3*c**3*d - 12 
*a*b**3*c**2*d**2*x - 30*a*b**3*c*d**3*x**2 - 40*a*b**3*d**4*x**3 - b**4*c 
**4 - 6*b**4*c**3*d*x - 15*b**4*c**2*d**2*x**2 - 20*b**4*c*d**3*x**3 - 15* 
b**4*d**4*x**4)/(30*d**5*(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))