\(\int \frac {(d+e x)^4}{(b x+c x^2)^3} \, dx\) [278]

Optimal result

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

[Out]

-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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=\frac {6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac {6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}+\frac {(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac {d^3 (3 c d-4 b e)}{b^4 x}+\frac {(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}-\frac {d^4}{2 b^3 x^2} \]

[In]

Int[(d + e*x)^4/(b*x + c*x^2)^3,x]

[Out]

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

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^4}{b^3 x^3}+\frac {d^3 (-3 c d+4 b e)}{b^4 x^2}+\frac {6 d^2 (-c d+b e)^2}{b^5 x}-\frac {(-c d+b e)^4}{b^3 c (b+c x)^3}+\frac {(-c d+b e)^3 (3 c d+b e)}{b^4 c (b+c x)^2}-\frac {6 c d^2 (-c d+b e)^2}{b^5 (b+c x)}\right ) \, 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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.96 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {b^2 d^4}{x^2}+\frac {2 b d^3 (-3 c d+4 b e)}{x}-\frac {b^2 (c d-b e)^4}{c^2 (b+c x)^2}+\frac {2 b (-c d+b e)^3 (3 c d+b e)}{c^2 (b+c x)}-12 d^2 (c d-b e)^2 \log (x)+12 d^2 (c d-b e)^2 \log (b+c x)}{2 b^5} \]

[In]

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

[Out]

-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

Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.56

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

[In]

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

[Out]

(-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 = 0.27 (sec) , antiderivative size = 426, normalized size of antiderivative = 3.13 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{4} c^{2} d^{4} - 2 \, {\left (6 \, b c^{5} d^{4} - 12 \, b^{2} c^{4} d^{3} e + 6 \, b^{3} c^{3} d^{2} e^{2} - b^{5} c e^{4}\right )} x^{3} - {\left (18 \, b^{2} c^{4} d^{4} - 36 \, b^{3} c^{3} d^{3} e + 18 \, b^{4} c^{2} d^{2} e^{2} - 4 \, b^{5} c d e^{3} - b^{6} e^{4}\right )} x^{2} - 4 \, {\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e\right )} x + 12 \, {\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} + {\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 12 \, {\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \, {\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} + {\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \, {\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \]

[In]

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

[Out]

-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 = 1.68 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.86 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=\frac {- b^{3} c^{2} d^{4} + x^{3} \left (- 2 b^{4} c e^{4} + 12 b^{2} c^{3} d^{2} e^{2} - 24 b c^{4} d^{3} e + 12 c^{5} d^{4}\right ) + x^{2} \left (- b^{5} e^{4} - 4 b^{4} c d e^{3} + 18 b^{3} c^{2} d^{2} e^{2} - 36 b^{2} c^{3} d^{3} e + 18 b c^{4} d^{4}\right ) + x \left (- 8 b^{3} c^{2} d^{3} e + 4 b^{2} c^{3} d^{4}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac {6 d^{2} \left (b e - c d\right )^{2} \log {\left (x + \frac {6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} - 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} - \frac {6 d^{2} \left (b e - c d\right )^{2} \log {\left (x + \frac {6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} + 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} \]

[In]

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

[Out]

(-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*l
og(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)

none

Time = 0.19 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.84 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=-\frac {b^{3} c^{2} d^{4} - 2 \, {\left (6 \, c^{5} d^{4} - 12 \, b c^{4} d^{3} e + 6 \, b^{2} c^{3} d^{2} e^{2} - b^{4} c e^{4}\right )} x^{3} - {\left (18 \, b c^{4} d^{4} - 36 \, b^{2} c^{3} d^{3} e + 18 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x^{2} - 4 \, {\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )} x}{2 \, {\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (x\right )}{b^{5}} \]

[In]

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

[Out]

-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)

none

Time = 0.27 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=\frac {6 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left ({\left | x \right |}\right )}{b^{5}} - \frac {6 \, {\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac {12 \, c^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} e x^{3} + 12 \, b^{2} c^{3} d^{2} e^{2} x^{3} - 2 \, b^{4} c e^{4} x^{3} + 18 \, b c^{4} d^{4} x^{2} - 36 \, b^{2} c^{3} d^{3} e x^{2} + 18 \, b^{3} c^{2} d^{2} e^{2} x^{2} - 4 \, b^{4} c d e^{3} x^{2} - b^{5} e^{4} x^{2} + 4 \, b^{2} c^{3} d^{4} x - 8 \, b^{3} c^{2} d^{3} e x - b^{3} c^{2} d^{4}}{2 \, {\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \]

[In]

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

[Out]

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 = 0.15 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.75 \[ \int \frac {(d+e x)^4}{\left (b x+c x^2\right )^3} \, dx=-\frac {\frac {d^4}{2\,b}+\frac {2\,d^3\,x\,\left (2\,b\,e-c\,d\right )}{b^2}+\frac {x^2\,\left (b^4\,e^4+4\,b^3\,c\,d\,e^3-18\,b^2\,c^2\,d^2\,e^2+36\,b\,c^3\,d^3\,e-18\,c^4\,d^4\right )}{2\,b^3\,c^2}+\frac {x^3\,\left (b^4\,e^4-6\,b^2\,c^2\,d^2\,e^2+12\,b\,c^3\,d^3\,e-6\,c^4\,d^4\right )}{b^4\,c}}{b^2\,x^2+2\,b\,c\,x^3+c^2\,x^4}-\frac {12\,d^2\,\mathrm {atanh}\left (\frac {6\,d^2\,{\left (b\,e-c\,d\right )}^2\,\left (b+2\,c\,x\right )}{b\,\left (6\,b^2\,d^2\,e^2-12\,b\,c\,d^3\,e+6\,c^2\,d^4\right )}\right )\,{\left (b\,e-c\,d\right )}^2}{b^5} \]

[In]

int((d + e*x)^4/(b*x + c*x^2)^3,x)

[Out]

- (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