\(\int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx\) [160]

Optimal result

Integrand size = 90, antiderivative size = 111 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx=-\frac {(b c-a d)^4}{4 d^5 (c+d x)^4}+\frac {4 b (b c-a d)^3}{3 d^5 (c+d x)^3}-\frac {3 b^2 (b c-a d)^2}{d^5 (c+d x)^2}+\frac {4 b^3 (b c-a d)}{d^5 (c+d x)}+\frac {b^4 \log (c+d x)}{d^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx=\frac {\frac {(b c-a d) \left (3 a^3 d^3+a^2 b d^2 (7 c+16 d x)+a b^2 d \left (13 c^2+40 c d x+36 d^2 x^2\right )+b^3 \left (25 c^3+88 c^2 d x+108 c d^2 x^2+48 d^3 x^3\right )\right )}{(c+d x)^4}+12 b^4 \log (c+d x)}{12 d^5} \] 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^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),x]
 

Output:

(((b*c - a*d)*(3*a^3*d^3 + a^2*b*d^2*(7*c + 16*d*x) + a*b^2*d*(13*c^2 + 40 
*c*d*x + 36*d^2*x^2) + b^3*(25*c^3 + 88*c^2*d*x + 108*c*d^2*x^2 + 48*d^3*x 
^3)))/(c + d*x)^4 + 12*b^4*Log[c + d*x])/(12*d^5)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {2006, 2007, 49, 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.

\(\displaystyle \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx\)

\(\Big \downarrow \) 2006

\(\displaystyle \int \frac {(a+b x)^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5}dx\)

\(\Big \downarrow \) 2007

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

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^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),x]
 

Output:

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

Defintions of rubi rules used

rule 49

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}, x] 
&& IGtQ[m, 0] && IGtQ[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 = 0.08 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.64

method	result	size
norman	\(\frac {-\frac {3 d^{4} a^{4}+4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+12 a \,b^{3} c^{3} d -25 c^{4} b^{4}}{12 d^{5}}-\frac {4 \left (a \,b^{3} d -b^{4} c \right ) x^{3}}{d^{2}}-\frac {3 \left (a^{2} b^{2} d^{2}+2 a \,b^{3} c d -3 b^{4} c^{2}\right ) x^{2}}{d^{3}}-\frac {2 \left (2 a^{3} b \,d^{3}+3 a^{2} b^{2} c \,d^{2}+6 a \,b^{3} c^{2} d -11 b^{4} c^{3}\right ) x}{3 d^{4}}}{\left (x d +c \right )^{4}}+\frac {b^{4} \ln \left (x d +c \right )}{d^{5}}\)	\(182\)
default	\(-\frac {4 b^{3} \left (a d -b c \right )}{d^{5} \left (x d +c \right )}+\frac {b^{4} \ln \left (x d +c \right )}{d^{5}}-\frac {3 b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{d^{5} \left (x d +c \right )^{2}}-\frac {d^{4} a^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +c^{4} b^{4}}{4 d^{5} \left (x d +c \right )^{4}}-\frac {4 b \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 d^{5} \left (x d +c \right )^{3}}\)	\(184\)
risch	\(\frac {-\frac {4 b^{3} \left (a d -b c \right ) x^{3}}{d^{2}}-\frac {3 b^{2} \left (a^{2} d^{2}+2 a b c d -3 b^{2} c^{2}\right ) x^{2}}{d^{3}}-\frac {2 b \left (2 a^{3} d^{3}+3 a^{2} b c \,d^{2}+6 a \,b^{2} c^{2} d -11 b^{3} c^{3}\right ) x}{3 d^{4}}-\frac {3 d^{4} a^{4}+4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}+12 a \,b^{3} c^{3} d -25 c^{4} b^{4}}{12 d^{5}}}{d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}}+\frac {b^{4} \ln \left (x d +c \right )}{d^{5}}\)	\(209\)
parallelrisch	\(\frac {12 \ln \left (x d +c \right ) x^{4} b^{4} d^{4}+48 \ln \left (x d +c \right ) x^{3} b^{4} c \,d^{3}+72 \ln \left (x d +c \right ) x^{2} b^{4} c^{2} d^{2}-48 a \,b^{3} d^{4} x^{3}+48 b^{4} c \,d^{3} x^{3}+48 \ln \left (x d +c \right ) x \,b^{4} c^{3} d -36 a^{2} b^{2} d^{4} x^{2}-72 a \,b^{3} c \,d^{3} x^{2}+108 b^{4} c^{2} d^{2} x^{2}+12 \ln \left (x d +c \right ) b^{4} c^{4}-16 a^{3} b \,d^{4} x -24 a^{2} b^{2} c \,d^{3} x -48 a \,b^{3} c^{2} d^{2} x +88 b^{4} c^{3} d x -3 d^{4} a^{4}-4 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}-12 a \,b^{3} c^{3} d +25 c^{4} b^{4}}{12 d^{5} \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right )}\)	\(293\)

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^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),x,method=_RETURNVERBOSE)
 

Output:

(-1/12*(3*a^4*d^4+4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2+12*a*b^3*c^3*d-25*b^4*c^ 
4)/d^5-4*(a*b^3*d-b^4*c)/d^2*x^3-3*(a^2*b^2*d^2+2*a*b^3*c*d-3*b^4*c^2)/d^3 
*x^2-2/3*(2*a^3*b*d^3+3*a^2*b^2*c*d^2+6*a*b^3*c^2*d-11*b^4*c^3)/d^4*x)/(d* 
x+c)^4+b^4*ln(d*x+c)/d^5
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (107) = 214\).

Time = 0.10 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.41 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx=\frac {25 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} + 48 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 36 \, {\left (3 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} - a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (11 \, b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} x + 12 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4}\right )} \log \left (d x + c\right )}{12 \, {\left (d^{9} x^{4} + 4 \, c d^{8} x^{3} + 6 \, c^{2} d^{7} x^{2} + 4 \, c^{3} d^{6} x + c^{4} d^{5}\right )}} \] 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^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),x, algorithm="fricas")
 

Output:

1/12*(25*b^4*c^4 - 12*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - 3* 
a^4*d^4 + 48*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 36*(3*b^4*c^2*d^2 - 2*a*b^3*c*d 
^3 - a^2*b^2*d^4)*x^2 + 8*(11*b^4*c^3*d - 6*a*b^3*c^2*d^2 - 3*a^2*b^2*c*d^ 
3 - 2*a^3*b*d^4)*x + 12*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 
 + 4*b^4*c^3*d*x + b^4*c^4)*log(d*x + c))/(d^9*x^4 + 4*c*d^8*x^3 + 6*c^2*d 
^7*x^2 + 4*c^3*d^6*x + c^4*d^5)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (100) = 200\).

Time = 1.22 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.07 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx=\frac {b^{4} \log {\left (c + d x \right )}}{d^{5}} + \frac {- 3 a^{4} d^{4} - 4 a^{3} b c d^{3} - 6 a^{2} b^{2} c^{2} d^{2} - 12 a b^{3} c^{3} d + 25 b^{4} c^{4} + x^{3} \left (- 48 a b^{3} d^{4} + 48 b^{4} c d^{3}\right ) + x^{2} \left (- 36 a^{2} b^{2} d^{4} - 72 a b^{3} c d^{3} + 108 b^{4} c^{2} d^{2}\right ) + x \left (- 16 a^{3} b d^{4} - 24 a^{2} b^{2} c d^{3} - 48 a b^{3} c^{2} d^{2} + 88 b^{4} c^{3} d\right )}{12 c^{4} d^{5} + 48 c^{3} d^{6} x + 72 c^{2} d^{7} x^{2} + 48 c d^{8} x^{3} + 12 d^{9} x^{4}} \] 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**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),x 
)
 

Output:

b**4*log(c + d*x)/d**5 + (-3*a**4*d**4 - 4*a**3*b*c*d**3 - 6*a**2*b**2*c** 
2*d**2 - 12*a*b**3*c**3*d + 25*b**4*c**4 + x**3*(-48*a*b**3*d**4 + 48*b**4 
*c*d**3) + x**2*(-36*a**2*b**2*d**4 - 72*a*b**3*c*d**3 + 108*b**4*c**2*d** 
2) + x*(-16*a**3*b*d**4 - 24*a**2*b**2*c*d**3 - 48*a*b**3*c**2*d**2 + 88*b 
**4*c**3*d))/(12*c**4*d**5 + 48*c**3*d**6*x + 72*c**2*d**7*x**2 + 48*c*d** 
8*x**3 + 12*d**9*x**4)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (107) = 214\).

Time = 0.05 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.98 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx=\frac {25 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4} + 48 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 36 \, {\left (3 \, b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} - a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (11 \, b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} - 2 \, a^{3} b d^{4}\right )} x}{12 \, {\left (d^{9} x^{4} + 4 \, c d^{8} x^{3} + 6 \, c^{2} d^{7} x^{2} + 4 \, c^{3} d^{6} x + c^{4} d^{5}\right )}} + \frac {b^{4} \log \left (d x + c\right )}{d^{5}} \] 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^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),x, algorithm="maxima")
 

Output:

1/12*(25*b^4*c^4 - 12*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - 3* 
a^4*d^4 + 48*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 36*(3*b^4*c^2*d^2 - 2*a*b^3*c*d 
^3 - a^2*b^2*d^4)*x^2 + 8*(11*b^4*c^3*d - 6*a*b^3*c^2*d^2 - 3*a^2*b^2*c*d^ 
3 - 2*a^3*b*d^4)*x)/(d^9*x^4 + 4*c*d^8*x^3 + 6*c^2*d^7*x^2 + 4*c^3*d^6*x + 
 c^4*d^5) + b^4*log(d*x + c)/d^5
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.67 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx=\frac {b^{4} \log \left ({\left | d x + c \right |}\right )}{d^{5}} + \frac {48 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{3} + 36 \, {\left (3 \, b^{4} c^{2} d - 2 \, a b^{3} c d^{2} - a^{2} b^{2} d^{3}\right )} x^{2} + 8 \, {\left (11 \, b^{4} c^{3} - 6 \, a b^{3} c^{2} d - 3 \, a^{2} b^{2} c d^{2} - 2 \, a^{3} b d^{3}\right )} x + \frac {25 \, b^{4} c^{4} - 12 \, a b^{3} c^{3} d - 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} - 3 \, a^{4} d^{4}}{d}}{12 \, {\left (d x + c\right )}^{4} d^{4}} \] 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^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),x, algorithm="giac")
 

Output:

b^4*log(abs(d*x + c))/d^5 + 1/12*(48*(b^4*c*d^2 - a*b^3*d^3)*x^3 + 36*(3*b 
^4*c^2*d - 2*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 8*(11*b^4*c^3 - 6*a*b^3*c^2* 
d - 3*a^2*b^2*c*d^2 - 2*a^3*b*d^3)*x + (25*b^4*c^4 - 12*a*b^3*c^3*d - 6*a^ 
2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 - 3*a^4*d^4)/d)/((d*x + c)^4*d^4)
 

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.92 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx=\frac {b^4\,\ln \left (c+d\,x\right )}{d^5}-\frac {\frac {3\,a^4\,d^4+4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2+12\,a\,b^3\,c^3\,d-25\,b^4\,c^4}{12\,d^5}+\frac {3\,x^2\,\left (a^2\,b^2\,d^2+2\,a\,b^3\,c\,d-3\,b^4\,c^2\right )}{d^3}+\frac {2\,x\,\left (2\,a^3\,b\,d^3+3\,a^2\,b^2\,c\,d^2+6\,a\,b^3\,c^2\,d-11\,b^4\,c^3\right )}{3\,d^4}+\frac {4\,b^3\,x^3\,\left (a\,d-b\,c\right )}{d^2}}{c^4+4\,c^3\,d\,x+6\,c^2\,d^2\,x^2+4\,c\,d^3\,x^3+d^4\,x^4} \] 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^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),x)
 

Output:

(b^4*log(c + d*x))/d^5 - ((3*a^4*d^4 - 25*b^4*c^4 + 6*a^2*b^2*c^2*d^2 + 12 
*a*b^3*c^3*d + 4*a^3*b*c*d^3)/(12*d^5) + (3*x^2*(a^2*b^2*d^2 - 3*b^4*c^2 + 
 2*a*b^3*c*d))/d^3 + (2*x*(2*a^3*b*d^3 - 11*b^4*c^3 + 3*a^2*b^2*c*d^2 + 6* 
a*b^3*c^2*d))/(3*d^4) + (4*b^3*x^3*(a*d - b*c))/d^2)/(c^4 + d^4*x^4 + 4*c* 
d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.42 \[ \int \frac {a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4}{c^5+5 c^4 d x+10 c^3 d^2 x^2+10 c^2 d^3 x^3+5 c d^4 x^4+d^5 x^5} \, dx=\frac {12 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{5}+48 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{4} d x +72 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{3} d^{2} x^{2}+48 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{2} d^{3} x^{3}+12 \,\mathrm {log}\left (d x +c \right ) b^{4} c \,d^{4} x^{4}-3 a^{4} c \,d^{4}-4 a^{3} b \,c^{2} d^{3}-16 a^{3} b c \,d^{4} x -6 a^{2} b^{2} c^{3} d^{2}-24 a^{2} b^{2} c^{2} d^{3} x -36 a^{2} b^{2} c \,d^{4} x^{2}+12 a \,b^{3} d^{5} x^{4}+13 b^{4} c^{5}+40 b^{4} c^{4} d x +36 b^{4} c^{3} d^{2} x^{2}-12 b^{4} c \,d^{4} x^{4}}{12 c \,d^{5} \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right )} \] 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^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),x)
 

Output:

(12*log(c + d*x)*b**4*c**5 + 48*log(c + d*x)*b**4*c**4*d*x + 72*log(c + d* 
x)*b**4*c**3*d**2*x**2 + 48*log(c + d*x)*b**4*c**2*d**3*x**3 + 12*log(c + 
d*x)*b**4*c*d**4*x**4 - 3*a**4*c*d**4 - 4*a**3*b*c**2*d**3 - 16*a**3*b*c*d 
**4*x - 6*a**2*b**2*c**3*d**2 - 24*a**2*b**2*c**2*d**3*x - 36*a**2*b**2*c* 
d**4*x**2 + 12*a*b**3*d**5*x**4 + 13*b**4*c**5 + 40*b**4*c**4*d*x + 36*b** 
4*c**3*d**2*x**2 - 12*b**4*c*d**4*x**4)/(12*c*d**5*(c**4 + 4*c**3*d*x + 6* 
c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4))