\(\int \frac {(d+e x)^4}{(b x+c x^2)^3} \, dx\) [74]

Optimal result

Integrand size = 19, antiderivative size = 136 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=-\frac {d^4}{2 b^3 x^2}+\frac {d^3 (3 c d-4 b e)}{b^4 x}+\frac {(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}+\frac {(c d-b e)^3 (3 c d+b e)}{b^4 c^2 (b+c x)}+\frac {6 d^2 (c d-b e)^2 \log (x)}{b^5}-\frac {6 d^2 (c d-b e)^2 \log (b+c x)}{b^5} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {b^2 d^4}{x^2}+\frac {2 b d^3 (-3 c d+4 b e)}{x}-\frac {b^2 (c d-b e)^4}{c^2 (b+c x)^2}+\frac {2 b (-c d+b e)^3 (3 c d+b e)}{c^2 (b+c x)}-12 d^2 (c d-b e)^2 \log (x)+12 d^2 (c d-b e)^2 \log (b+c x)}{2 b^5} \] Input:

Integrate[(d + e*x)^4/(b*x + c*x^2)^3,x]
 

Output:

-1/2*((b^2*d^4)/x^2 + (2*b*d^3*(-3*c*d + 4*b*e))/x - (b^2*(c*d - b*e)^4)/( 
c^2*(b + c*x)^2) + (2*b*(-(c*d) + b*e)^3*(3*c*d + b*e))/(c^2*(b + c*x)) - 
12*d^2*(c*d - b*e)^2*Log[x] + 12*d^2*(c*d - b*e)^2*Log[b + c*x])/b^5
 

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1141, 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.

\(\displaystyle \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1141

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

\(\Big \downarrow \) 2009

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

Input:

Int[(d + e*x)^4/(b*x + c*x^2)^3,x]
 

Output:

c^3*(-1/2*d^4/(b^3*c^3*x^2) + (d^3*(3*c*d - 4*b*e))/(b^4*c^3*x) + (c*d - b 
*e)^4/(2*b^3*c^5*(b + c*x)^2) + ((c*d - b*e)^3*(3*c*d + b*e))/(b^4*c^5*(b 
+ c*x)) + (6*d^2*(c*d - b*e)^2*Log[x])/(b^5*c^3) - (6*d^2*(c*d - b*e)^2*Lo 
g[b + c*x])/(b^5*c^3))
 

Defintions of rubi rules used

rule 1141

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ 
Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p   Int[ExpandIntegrand[ 
(d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 
1] ||  !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 
0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
 

rule 2009

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

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.56

method	result	size
norman	\(\frac {-\frac {d^{4}}{2 b}+\frac {2 \left (2 d \,e^{3} b^{3}-6 d^{2} e^{2} b^{2} c +12 d^{3} e b \,c^{2}-6 d^{4} c^{3}\right ) x^{3}}{b^{4}}+\frac {\left (e^{4} b^{4}+4 d \,e^{3} b^{3} c -18 d^{2} e^{2} b^{2} c^{2}+36 d^{3} e b \,c^{3}-18 d^{4} c^{4}\right ) x^{4}}{2 b^{5}}-\frac {2 d^{3} \left (2 b e -c d \right ) x}{b^{2}}}{x^{2} \left (c x +b \right )^{2}}+\frac {6 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}-\frac {6 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \ln \left (c x +b \right )}{b^{5}}\)	\(212\)
default	\(-\frac {e^{4} b^{4}-6 d^{2} e^{2} b^{2} c^{2}+8 d^{3} e b \,c^{3}-3 d^{4} c^{4}}{b^{4} c^{2} \left (c x +b \right )}-\frac {-e^{4} b^{4}+4 d \,e^{3} b^{3} c -6 d^{2} e^{2} b^{2} c^{2}+4 d^{3} e b \,c^{3}-d^{4} c^{4}}{2 b^{3} c^{2} \left (c x +b \right )^{2}}-\frac {6 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \ln \left (c x +b \right )}{b^{5}}-\frac {d^{4}}{2 b^{3} x^{2}}-\frac {d^{3} \left (4 b e -3 c d \right )}{b^{4} x}+\frac {6 d^{2} \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \ln \left (x \right )}{b^{5}}\)	\(220\)
risch	\(\frac {-\frac {\left (e^{4} b^{4}-6 d^{2} e^{2} b^{2} c^{2}+12 d^{3} e b \,c^{3}-6 d^{4} c^{4}\right ) x^{3}}{b^{4} c}-\frac {\left (e^{4} b^{4}+4 d \,e^{3} b^{3} c -18 d^{2} e^{2} b^{2} c^{2}+36 d^{3} e b \,c^{3}-18 d^{4} c^{4}\right ) x^{2}}{2 b^{3} c^{2}}-\frac {2 d^{3} \left (2 b e -c d \right ) x}{b^{2}}-\frac {d^{4}}{2 b}}{x^{2} \left (c x +b \right )^{2}}+\frac {6 d^{2} \ln \left (-x \right ) e^{2}}{b^{3}}-\frac {12 d^{3} \ln \left (-x \right ) c e}{b^{4}}+\frac {6 d^{4} \ln \left (-x \right ) c^{2}}{b^{5}}-\frac {6 d^{2} \ln \left (c x +b \right ) e^{2}}{b^{3}}+\frac {12 d^{3} \ln \left (c x +b \right ) c e}{b^{4}}-\frac {6 d^{4} \ln \left (c x +b \right ) c^{2}}{b^{5}}\)	\(246\)
parallelrisch	\(\frac {-24 \ln \left (x \right ) x^{2} b^{3} c \,d^{3} e +24 \ln \left (c x +b \right ) x^{2} b^{3} c \,d^{3} e +48 \ln \left (c x +b \right ) x^{3} b^{2} c^{2} d^{3} e +12 \ln \left (x \right ) x^{4} b^{2} c^{2} d^{2} e^{2}-24 \ln \left (x \right ) x^{4} b \,c^{3} d^{3} e -12 \ln \left (c x +b \right ) x^{4} b^{2} c^{2} d^{2} e^{2}+24 \ln \left (c x +b \right ) x^{4} b \,c^{3} d^{3} e +24 \ln \left (x \right ) x^{3} b^{3} c \,d^{2} e^{2}-48 \ln \left (x \right ) x^{3} b^{2} c^{2} d^{3} e -24 \ln \left (c x +b \right ) x^{3} b^{3} c \,d^{2} e^{2}-18 c^{4} x^{4} d^{4}-b^{4} d^{4}+12 \ln \left (x \right ) x^{4} c^{4} d^{4}-12 \ln \left (c x +b \right ) x^{4} c^{4} d^{4}+8 x^{3} b^{4} d \,e^{3}-24 x^{3} b \,c^{3} d^{4}-8 x \,b^{4} d^{3} e +4 x \,b^{3} c \,d^{4}-12 \ln \left (c x +b \right ) x^{2} b^{2} c^{2} d^{4}+4 x^{4} b^{3} c d \,e^{3}-18 x^{4} b^{2} c^{2} d^{2} e^{2}+36 x^{4} b \,c^{3} d^{3} e -24 x^{3} b^{3} c \,d^{2} e^{2}+48 x^{3} b^{2} c^{2} d^{3} e +24 \ln \left (x \right ) x^{3} b \,c^{3} d^{4}-24 \ln \left (c x +b \right ) x^{3} b \,c^{3} d^{4}+12 \ln \left (x \right ) x^{2} b^{4} d^{2} e^{2}+12 \ln \left (x \right ) x^{2} b^{2} c^{2} d^{4}-12 \ln \left (c x +b \right ) x^{2} b^{4} d^{2} e^{2}+x^{4} b^{4} e^{4}}{2 b^{5} x^{2} \left (c x +b \right )^{2}}\)	\(483\)

Input:

int((e*x+d)^4/(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)
 

Output:

(-1/2*d^4/b+2*(2*b^3*d*e^3-6*b^2*c*d^2*e^2+12*b*c^2*d^3*e-6*c^3*d^4)/b^4*x 
^3+1/2*(b^4*e^4+4*b^3*c*d*e^3-18*b^2*c^2*d^2*e^2+36*b*c^3*d^3*e-18*c^4*d^4 
)/b^5*x^4-2*d^3*(2*b*e-c*d)/b^2*x)/x^2/(c*x+b)^2+6*d^2*(b^2*e^2-2*b*c*d*e+ 
c^2*d^2)/b^5*ln(x)-6*d^2*(b^2*e^2-2*b*c*d*e+c^2*d^2)/b^5*ln(c*x+b)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (132) = 264\).

Time = 0.09 (sec) , antiderivative size = 426, normalized size of antiderivative = 3.13 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{4} c^{2} d^{4} - 2 \, {\left (6 \, b c^{5} d^{4} - 12 \, b^{2} c^{4} d^{3} e + 6 \, b^{3} c^{3} d^{2} e^{2} - b^{5} c e^{4}\right )} x^{3} - {\left (18 \, b^{2} c^{4} d^{4} - 36 \, b^{3} c^{3} d^{3} e + 18 \, b^{4} c^{2} d^{2} e^{2} - 4 \, b^{5} c d e^{3} - b^{6} e^{4}\right )} x^{2} - 4 \, {\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e\right )} x + 12 \, {\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} + {\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 12 \, {\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} + {\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \] Input:

integrate((e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="fricas")
 

Output:

-1/2*(b^4*c^2*d^4 - 2*(6*b*c^5*d^4 - 12*b^2*c^4*d^3*e + 6*b^3*c^3*d^2*e^2 
- b^5*c*e^4)*x^3 - (18*b^2*c^4*d^4 - 36*b^3*c^3*d^3*e + 18*b^4*c^2*d^2*e^2 
 - 4*b^5*c*d*e^3 - b^6*e^4)*x^2 - 4*(b^3*c^3*d^4 - 2*b^4*c^2*d^3*e)*x + 12 
*((c^6*d^4 - 2*b*c^5*d^3*e + b^2*c^4*d^2*e^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c 
^4*d^3*e + b^3*c^3*d^2*e^2)*x^3 + (b^2*c^4*d^4 - 2*b^3*c^3*d^3*e + b^4*c^2 
*d^2*e^2)*x^2)*log(c*x + b) - 12*((c^6*d^4 - 2*b*c^5*d^3*e + b^2*c^4*d^2*e 
^2)*x^4 + 2*(b*c^5*d^4 - 2*b^2*c^4*d^3*e + b^3*c^3*d^2*e^2)*x^3 + (b^2*c^4 
*d^4 - 2*b^3*c^3*d^3*e + b^4*c^2*d^2*e^2)*x^2)*log(x))/(b^5*c^4*x^4 + 2*b^ 
6*c^3*x^3 + b^7*c^2*x^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (126) = 252\).

Time = 1.31 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.86 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=\frac {- b^{3} c^{2} d^{4} + x^{3} \left (- 2 b^{4} c e^{4} + 12 b^{2} c^{3} d^{2} e^{2} - 24 b c^{4} d^{3} e + 12 c^{5} d^{4}\right ) + x^{2} \left (- b^{5} e^{4} - 4 b^{4} c d e^{3} + 18 b^{3} c^{2} d^{2} e^{2} - 36 b^{2} c^{3} d^{3} e + 18 b c^{4} d^{4}\right ) + x \left (- 8 b^{3} c^{2} d^{3} e + 4 b^{2} c^{3} d^{4}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac {6 d^{2} \left (b e - c d\right )^{2} \log {\left (x + \frac {6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} - 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} - \frac {6 d^{2} \left (b e - c d\right )^{2} \log {\left (x + \frac {6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} + 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} \] Input:

integrate((e*x+d)**4/(c*x**2+b*x)**3,x)
 

Output:

(-b**3*c**2*d**4 + x**3*(-2*b**4*c*e**4 + 12*b**2*c**3*d**2*e**2 - 24*b*c* 
*4*d**3*e + 12*c**5*d**4) + x**2*(-b**5*e**4 - 4*b**4*c*d*e**3 + 18*b**3*c 
**2*d**2*e**2 - 36*b**2*c**3*d**3*e + 18*b*c**4*d**4) + x*(-8*b**3*c**2*d* 
*3*e + 4*b**2*c**3*d**4))/(2*b**6*c**2*x**2 + 4*b**5*c**3*x**3 + 2*b**4*c* 
*4*x**4) + 6*d**2*(b*e - c*d)**2*log(x + (6*b**3*d**2*e**2 - 12*b**2*c*d** 
3*e + 6*b*c**2*d**4 - 6*b*d**2*(b*e - c*d)**2)/(12*b**2*c*d**2*e**2 - 24*b 
*c**2*d**3*e + 12*c**3*d**4))/b**5 - 6*d**2*(b*e - c*d)**2*log(x + (6*b**3 
*d**2*e**2 - 12*b**2*c*d**3*e + 6*b*c**2*d**4 + 6*b*d**2*(b*e - c*d)**2)/( 
12*b**2*c*d**2*e**2 - 24*b*c**2*d**3*e + 12*c**3*d**4))/b**5
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.84 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{3} c^{2} d^{4} - 2 \, {\left (6 \, c^{5} d^{4} - 12 \, b c^{4} d^{3} e + 6 \, b^{2} c^{3} d^{2} e^{2} - b^{4} c e^{4}\right )} x^{3} - {\left (18 \, b c^{4} d^{4} - 36 \, b^{2} c^{3} d^{3} e + 18 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x^{2} - 4 \, {\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )} x}{2 \, {\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (x\right )}{b^{5}} \] Input:

integrate((e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="maxima")
 

Output:

-1/2*(b^3*c^2*d^4 - 2*(6*c^5*d^4 - 12*b*c^4*d^3*e + 6*b^2*c^3*d^2*e^2 - b^ 
4*c*e^4)*x^3 - (18*b*c^4*d^4 - 36*b^2*c^3*d^3*e + 18*b^3*c^2*d^2*e^2 - 4*b 
^4*c*d*e^3 - b^5*e^4)*x^2 - 4*(b^2*c^3*d^4 - 2*b^3*c^2*d^3*e)*x)/(b^4*c^4* 
x^4 + 2*b^5*c^3*x^3 + b^6*c^2*x^2) - 6*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^ 
2)*log(c*x + b)/b^5 + 6*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*log(x)/b^5
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=\frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {6 \, {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac {12 \, c^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} e x^{3} + 12 \, b^{2} c^{3} d^{2} e^{2} x^{3} - 2 \, b^{4} c e^{4} x^{3} + 18 \, b c^{4} d^{4} x^{2} - 36 \, b^{2} c^{3} d^{3} e x^{2} + 18 \, b^{3} c^{2} d^{2} e^{2} x^{2} - 4 \, b^{4} c d e^{3} x^{2} - b^{5} e^{4} x^{2} + 4 \, b^{2} c^{3} d^{4} x - 8 \, b^{3} c^{2} d^{3} e x - b^{3} c^{2} d^{4}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \] Input:

integrate((e*x+d)^4/(c*x^2+b*x)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

6*(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*log(abs(x))/b^5 - 6*(c^3*d^4 - 2*b 
*c^2*d^3*e + b^2*c*d^2*e^2)*log(abs(c*x + b))/(b^5*c) + 1/2*(12*c^5*d^4*x^ 
3 - 24*b*c^4*d^3*e*x^3 + 12*b^2*c^3*d^2*e^2*x^3 - 2*b^4*c*e^4*x^3 + 18*b*c 
^4*d^4*x^2 - 36*b^2*c^3*d^3*e*x^2 + 18*b^3*c^2*d^2*e^2*x^2 - 4*b^4*c*d*e^3 
*x^2 - b^5*e^4*x^2 + 4*b^2*c^3*d^4*x - 8*b^3*c^2*d^3*e*x - b^3*c^2*d^4)/(( 
c*x^2 + b*x)^2*b^4*c^2)
 

Mupad [B] (verification not implemented)

Time = 9.14 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.75 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {d^4}{2\,b}+\frac {2\,d^3\,x\,\left (2\,b\,e-c\,d\right )}{b^2}+\frac {x^2\,\left (b^4\,e^4+4\,b^3\,c\,d\,e^3-18\,b^2\,c^2\,d^2\,e^2+36\,b\,c^3\,d^3\,e-18\,c^4\,d^4\right )}{2\,b^3\,c^2}+\frac {x^3\,\left (b^4\,e^4-6\,b^2\,c^2\,d^2\,e^2+12\,b\,c^3\,d^3\,e-6\,c^4\,d^4\right )}{b^4\,c}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {12\,d^2\,\mathrm {atanh}\left (\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b+2\,c\,x\right )}{b\,\left (6\,b^2\,d^2\,e^2-12\,b\,c\,d^3\,e+6\,c^2\,d^4\right )}\right )\,{\left (b\,e-c\,d\right )}^2}{b^5} \] Input:

int((d + e*x)^4/(b*x + c*x^2)^3,x)
 

Output:

- (d^4/(2*b) + (2*d^3*x*(2*b*e - c*d))/b^2 + (x^2*(b^4*e^4 - 18*c^4*d^4 - 
18*b^2*c^2*d^2*e^2 + 36*b*c^3*d^3*e + 4*b^3*c*d*e^3))/(2*b^3*c^2) + (x^3*( 
b^4*e^4 - 6*c^4*d^4 - 6*b^2*c^2*d^2*e^2 + 12*b*c^3*d^3*e))/(b^4*c))/(b^2*x 
^2 + c^2*x^4 + 2*b*c*x^3) - (12*d^2*atanh((6*d^2*(b*e - c*d)^2*(b + 2*c*x) 
)/(b*(6*c^2*d^4 + 6*b^2*d^2*e^2 - 12*b*c*d^3*e)))*(b*e - c*d)^2)/b^5
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 500, normalized size of antiderivative = 3.68 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=\frac {-12 \,\mathrm {log}\left (c x +b \right ) b^{4} c \,d^{2} e^{2} x^{2}+24 \,\mathrm {log}\left (c x +b \right ) b^{3} c^{2} d^{3} e \,x^{2}-24 \,\mathrm {log}\left (c x +b \right ) b^{3} c^{2} d^{2} e^{2} x^{3}+48 \,\mathrm {log}\left (c x +b \right ) b^{2} c^{3} d^{3} e \,x^{3}-12 \,\mathrm {log}\left (c x +b \right ) b^{2} c^{3} d^{2} e^{2} x^{4}+24 \,\mathrm {log}\left (c x +b \right ) b \,c^{4} d^{3} e \,x^{4}+12 \,\mathrm {log}\left (x \right ) b^{4} c \,d^{2} e^{2} x^{2}-24 \,\mathrm {log}\left (x \right ) b^{3} c^{2} d^{3} e \,x^{2}+24 \,\mathrm {log}\left (x \right ) b^{3} c^{2} d^{2} e^{2} x^{3}-48 \,\mathrm {log}\left (x \right ) b^{2} c^{3} d^{3} e \,x^{3}+12 \,\mathrm {log}\left (x \right ) b^{2} c^{3} d^{2} e^{2} x^{4}-24 \,\mathrm {log}\left (x \right ) b \,c^{4} d^{3} e \,x^{4}-6 c^{5} d^{4} x^{4}-b^{4} c \,d^{4}+b^{4} c \,e^{4} x^{4}-12 \,\mathrm {log}\left (c x +b \right ) c^{5} d^{4} x^{4}+12 \,\mathrm {log}\left (x \right ) c^{5} d^{4} x^{4}-4 b^{5} d \,e^{3} x^{2}+4 b^{3} c^{2} d^{4} x +12 b^{2} c^{3} d^{4} x^{2}-12 \,\mathrm {log}\left (c x +b \right ) b^{2} c^{3} d^{4} x^{2}-24 \,\mathrm {log}\left (c x +b \right ) b \,c^{4} d^{4} x^{3}+12 \,\mathrm {log}\left (x \right ) b^{2} c^{3} d^{4} x^{2}+24 \,\mathrm {log}\left (x \right ) b \,c^{4} d^{4} x^{3}-8 b^{4} c \,d^{3} e x +12 b^{4} c \,d^{2} e^{2} x^{2}-24 b^{3} c^{2} d^{3} e \,x^{2}-6 b^{2} c^{3} d^{2} e^{2} x^{4}+12 b \,c^{4} d^{3} e \,x^{4}}{2 b^{5} c \,x^{2} \left (c^{2} x^{2}+2 b c x +b^{2}\right )} \] Input:

int((e*x+d)^4/(c*x^2+b*x)^3,x)
 

Output:

( - 12*log(b + c*x)*b**4*c*d**2*e**2*x**2 + 24*log(b + c*x)*b**3*c**2*d**3 
*e*x**2 - 24*log(b + c*x)*b**3*c**2*d**2*e**2*x**3 - 12*log(b + c*x)*b**2* 
c**3*d**4*x**2 + 48*log(b + c*x)*b**2*c**3*d**3*e*x**3 - 12*log(b + c*x)*b 
**2*c**3*d**2*e**2*x**4 - 24*log(b + c*x)*b*c**4*d**4*x**3 + 24*log(b + c* 
x)*b*c**4*d**3*e*x**4 - 12*log(b + c*x)*c**5*d**4*x**4 + 12*log(x)*b**4*c* 
d**2*e**2*x**2 - 24*log(x)*b**3*c**2*d**3*e*x**2 + 24*log(x)*b**3*c**2*d** 
2*e**2*x**3 + 12*log(x)*b**2*c**3*d**4*x**2 - 48*log(x)*b**2*c**3*d**3*e*x 
**3 + 12*log(x)*b**2*c**3*d**2*e**2*x**4 + 24*log(x)*b*c**4*d**4*x**3 - 24 
*log(x)*b*c**4*d**3*e*x**4 + 12*log(x)*c**5*d**4*x**4 - 4*b**5*d*e**3*x**2 
 - b**4*c*d**4 - 8*b**4*c*d**3*e*x + 12*b**4*c*d**2*e**2*x**2 + b**4*c*e** 
4*x**4 + 4*b**3*c**2*d**4*x - 24*b**3*c**2*d**3*e*x**2 + 12*b**2*c**3*d**4 
*x**2 - 6*b**2*c**3*d**2*e**2*x**4 + 12*b*c**4*d**3*e*x**4 - 6*c**5*d**4*x 
**4)/(2*b**5*c*x**2*(b**2 + 2*b*c*x + c**2*x**2))