### 3.1905 $$\int \frac{d+e x}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx$$

Optimal. Leaf size=226 $-\frac{6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac{3 c^2 d^2 e}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac{c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}-\frac{10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac{10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6}-\frac{4 c d e^3}{(d+e x) \left (c d^2-a e^2\right )^5}-\frac{e^3}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}$

[Out]

-(c^2*d^2)/(3*(c*d^2 - a*e^2)^3*(a*e + c*d*x)^3) + (3*c^2*d^2*e)/(2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2) - (6*c^
2*d^2*e^2)/((c*d^2 - a*e^2)^5*(a*e + c*d*x)) - e^3/(2*(c*d^2 - a*e^2)^4*(d + e*x)^2) - (4*c*d*e^3)/((c*d^2 - a
*e^2)^5*(d + e*x)) - (10*c^2*d^2*e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2)^6 + (10*c^2*d^2*e^3*Log[d + e*x])/(c*d^
2 - a*e^2)^6

________________________________________________________________________________________

Rubi [A]  time = 0.207764, antiderivative size = 226, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.061, Rules used = {626, 44} $-\frac{6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}+\frac{3 c^2 d^2 e}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac{c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}-\frac{10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac{10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6}-\frac{4 c d e^3}{(d+e x) \left (c d^2-a e^2\right )^5}-\frac{e^3}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(c^2*d^2)/(3*(c*d^2 - a*e^2)^3*(a*e + c*d*x)^3) + (3*c^2*d^2*e)/(2*(c*d^2 - a*e^2)^4*(a*e + c*d*x)^2) - (6*c^
2*d^2*e^2)/((c*d^2 - a*e^2)^5*(a*e + c*d*x)) - e^3/(2*(c*d^2 - a*e^2)^4*(d + e*x)^2) - (4*c*d*e^3)/((c*d^2 - a
*e^2)^5*(d + e*x)) - (10*c^2*d^2*e^3*Log[a*e + c*d*x])/(c*d^2 - a*e^2)^6 + (10*c^2*d^2*e^3*Log[d + e*x])/(c*d^
2 - a*e^2)^6

Rule 626

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

Rule 44

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] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{d+e x}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac{1}{(a e+c d x)^4 (d+e x)^3} \, dx\\ &=\int \left (\frac{c^3 d^3}{\left (c d^2-a e^2\right )^3 (a e+c d x)^4}-\frac{3 c^3 d^3 e}{\left (c d^2-a e^2\right )^4 (a e+c d x)^3}+\frac{6 c^3 d^3 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)^2}-\frac{10 c^3 d^3 e^3}{\left (c d^2-a e^2\right )^6 (a e+c d x)}+\frac{e^4}{\left (c d^2-a e^2\right )^4 (d+e x)^3}+\frac{4 c d e^4}{\left (c d^2-a e^2\right )^5 (d+e x)^2}+\frac{10 c^2 d^2 e^4}{\left (c d^2-a e^2\right )^6 (d+e x)}\right ) \, dx\\ &=-\frac{c^2 d^2}{3 \left (c d^2-a e^2\right )^3 (a e+c d x)^3}+\frac{3 c^2 d^2 e}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}-\frac{6 c^2 d^2 e^2}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac{e^3}{2 \left (c d^2-a e^2\right )^4 (d+e x)^2}-\frac{4 c d e^3}{\left (c d^2-a e^2\right )^5 (d+e x)}-\frac{10 c^2 d^2 e^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}+\frac{10 c^2 d^2 e^3 \log (d+e x)}{\left (c d^2-a e^2\right )^6}\\ \end{align*}

Mathematica [A]  time = 0.193569, size = 206, normalized size = 0.91 $\frac{\frac{36 c^2 d^2 e^2 \left (a e^2-c d^2\right )}{a e+c d x}+\frac{9 c^2 d^2 e \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac{2 c^2 d^2 \left (a e^2-c d^2\right )^3}{(a e+c d x)^3}-60 c^2 d^2 e^3 \log (a e+c d x)+\frac{24 c d e^3 \left (a e^2-c d^2\right )}{d+e x}-\frac{3 e^3 \left (c d^2-a e^2\right )^2}{(d+e x)^2}+60 c^2 d^2 e^3 \log (d+e x)}{6 \left (c d^2-a e^2\right )^6}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((2*c^2*d^2*(-(c*d^2) + a*e^2)^3)/(a*e + c*d*x)^3 + (9*c^2*d^2*e*(c*d^2 - a*e^2)^2)/(a*e + c*d*x)^2 + (36*c^2*
d^2*e^2*(-(c*d^2) + a*e^2))/(a*e + c*d*x) - (3*e^3*(c*d^2 - a*e^2)^2)/(d + e*x)^2 + (24*c*d*e^3*(-(c*d^2) + a*
e^2))/(d + e*x) - 60*c^2*d^2*e^3*Log[a*e + c*d*x] + 60*c^2*d^2*e^3*Log[d + e*x])/(6*(c*d^2 - a*e^2)^6)

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 221, normalized size = 1. \begin{align*} -{\frac{{e}^{3}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{e}^{3}{c}^{2}{d}^{2}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6}}}+4\,{\frac{{e}^{3}cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( ex+d \right ) }}+{\frac{{c}^{2}{d}^{2}}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdx+ae \right ) ^{3}}}-10\,{\frac{{e}^{3}{c}^{2}{d}^{2}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6}}}+6\,{\frac{{c}^{2}{d}^{2}{e}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( cdx+ae \right ) }}+{\frac{3\,e{c}^{2}{d}^{2}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdx+ae \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^3/(a*e^2-c*d^2)^4/(e*x+d)^2+10*e^3/(a*e^2-c*d^2)^6*c^2*d^2*ln(e*x+d)+4*e^3/(a*e^2-c*d^2)^5*c*d/(e*x+d)+
1/3*c^2*d^2/(a*e^2-c*d^2)^3/(c*d*x+a*e)^3-10*e^3/(a*e^2-c*d^2)^6*c^2*d^2*ln(c*d*x+a*e)+6*c^2*d^2/(a*e^2-c*d^2)
^5*e^2/(c*d*x+a*e)+3/2*c^2*d^2/(a*e^2-c*d^2)^4*e/(c*d*x+a*e)^2

________________________________________________________________________________________

Maxima [B]  time = 1.28939, size = 1291, normalized size = 5.71 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*c^2*d^2*e^3*log(c*d*x + a*e)/(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a
^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12) + 10*c^2*d^2*e^3*log(e*x + d)/(c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*
a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12) - 1/6*(60*c^4*d^4*e^4
*x^4 + 2*c^4*d^8 - 13*a*c^3*d^6*e^2 + 47*a^2*c^2*d^4*e^4 + 27*a^3*c*d^2*e^6 - 3*a^4*e^8 + 30*(3*c^4*d^5*e^3 +
5*a*c^3*d^3*e^5)*x^3 + 10*(2*c^4*d^6*e^2 + 23*a*c^3*d^4*e^4 + 11*a^2*c^2*d^2*e^6)*x^2 - 5*(c^4*d^7*e - 11*a*c^
3*d^5*e^3 - 35*a^2*c^2*d^3*e^5 - 3*a^3*c*d*e^7)*x)/(a^3*c^5*d^12*e^3 - 5*a^4*c^4*d^10*e^5 + 10*a^5*c^3*d^8*e^7
- 10*a^6*c^2*d^6*e^9 + 5*a^7*c*d^4*e^11 - a^8*d^2*e^13 + (c^8*d^13*e^2 - 5*a*c^7*d^11*e^4 + 10*a^2*c^6*d^9*e^
6 - 10*a^3*c^5*d^7*e^8 + 5*a^4*c^4*d^5*e^10 - a^5*c^3*d^3*e^12)*x^5 + (2*c^8*d^14*e - 7*a*c^7*d^12*e^3 + 5*a^2
*c^6*d^10*e^5 + 10*a^3*c^5*d^8*e^7 - 20*a^4*c^4*d^6*e^9 + 13*a^5*c^3*d^4*e^11 - 3*a^6*c^2*d^2*e^13)*x^4 + (c^8
*d^15 + a*c^7*d^13*e^2 - 17*a^2*c^6*d^11*e^4 + 35*a^3*c^5*d^9*e^6 - 25*a^4*c^4*d^7*e^8 - a^5*c^3*d^5*e^10 + 9*
a^6*c^2*d^3*e^12 - 3*a^7*c*d*e^14)*x^3 + (3*a*c^7*d^14*e - 9*a^2*c^6*d^12*e^3 + a^3*c^5*d^10*e^5 + 25*a^4*c^4*
d^8*e^7 - 35*a^5*c^3*d^6*e^9 + 17*a^6*c^2*d^4*e^11 - a^7*c*d^2*e^13 - a^8*e^15)*x^2 + (3*a^2*c^6*d^13*e^2 - 13
*a^3*c^5*d^11*e^4 + 20*a^4*c^4*d^9*e^6 - 10*a^5*c^3*d^7*e^8 - 5*a^6*c^2*d^5*e^10 + 7*a^7*c*d^3*e^12 - 2*a^8*d*
e^14)*x)

________________________________________________________________________________________

Fricas [B]  time = 2.37401, size = 2518, normalized size = 11.14 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(2*c^5*d^10 - 15*a*c^4*d^8*e^2 + 60*a^2*c^3*d^6*e^4 - 20*a^3*c^2*d^4*e^6 - 30*a^4*c*d^2*e^8 + 3*a^5*e^10
+ 60*(c^5*d^6*e^4 - a*c^4*d^4*e^6)*x^4 + 30*(3*c^5*d^7*e^3 + 2*a*c^4*d^5*e^5 - 5*a^2*c^3*d^3*e^7)*x^3 + 10*(2*
c^5*d^8*e^2 + 21*a*c^4*d^6*e^4 - 12*a^2*c^3*d^4*e^6 - 11*a^3*c^2*d^2*e^8)*x^2 - 5*(c^5*d^9*e - 12*a*c^4*d^7*e^
3 - 24*a^2*c^3*d^5*e^5 + 32*a^3*c^2*d^3*e^7 + 3*a^4*c*d*e^9)*x + 60*(c^5*d^5*e^5*x^5 + a^3*c^2*d^4*e^6 + (2*c^
5*d^6*e^4 + 3*a*c^4*d^4*e^6)*x^4 + (c^5*d^7*e^3 + 6*a*c^4*d^5*e^5 + 3*a^2*c^3*d^3*e^7)*x^3 + (3*a*c^4*d^6*e^4
+ 6*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e^8)*x^2 + (3*a^2*c^3*d^5*e^5 + 2*a^3*c^2*d^3*e^7)*x)*log(c*d*x + a*e) - 60*
(c^5*d^5*e^5*x^5 + a^3*c^2*d^4*e^6 + (2*c^5*d^6*e^4 + 3*a*c^4*d^4*e^6)*x^4 + (c^5*d^7*e^3 + 6*a*c^4*d^5*e^5 +
3*a^2*c^3*d^3*e^7)*x^3 + (3*a*c^4*d^6*e^4 + 6*a^2*c^3*d^4*e^6 + a^3*c^2*d^2*e^8)*x^2 + (3*a^2*c^3*d^5*e^5 + 2*
a^3*c^2*d^3*e^7)*x)*log(e*x + d))/(a^3*c^6*d^14*e^3 - 6*a^4*c^5*d^12*e^5 + 15*a^5*c^4*d^10*e^7 - 20*a^6*c^3*d^
8*e^9 + 15*a^7*c^2*d^6*e^11 - 6*a^8*c*d^4*e^13 + a^9*d^2*e^15 + (c^9*d^15*e^2 - 6*a*c^8*d^13*e^4 + 15*a^2*c^7*
d^11*e^6 - 20*a^3*c^6*d^9*e^8 + 15*a^4*c^5*d^7*e^10 - 6*a^5*c^4*d^5*e^12 + a^6*c^3*d^3*e^14)*x^5 + (2*c^9*d^16
*e - 9*a*c^8*d^14*e^3 + 12*a^2*c^7*d^12*e^5 + 5*a^3*c^6*d^10*e^7 - 30*a^4*c^5*d^8*e^9 + 33*a^5*c^4*d^6*e^11 -
16*a^6*c^3*d^4*e^13 + 3*a^7*c^2*d^2*e^15)*x^4 + (c^9*d^17 - 18*a^2*c^7*d^13*e^4 + 52*a^3*c^6*d^11*e^6 - 60*a^4
*c^5*d^9*e^8 + 24*a^5*c^4*d^7*e^10 + 10*a^6*c^3*d^5*e^12 - 12*a^7*c^2*d^3*e^14 + 3*a^8*c*d*e^16)*x^3 + (3*a*c^
8*d^16*e - 12*a^2*c^7*d^14*e^3 + 10*a^3*c^6*d^12*e^5 + 24*a^4*c^5*d^10*e^7 - 60*a^5*c^4*d^8*e^9 + 52*a^6*c^3*d
^6*e^11 - 18*a^7*c^2*d^4*e^13 + a^9*e^17)*x^2 + (3*a^2*c^7*d^15*e^2 - 16*a^3*c^6*d^13*e^4 + 33*a^4*c^5*d^11*e^
6 - 30*a^5*c^4*d^9*e^8 + 5*a^6*c^3*d^7*e^10 + 12*a^7*c^2*d^5*e^12 - 9*a^8*c*d^3*e^14 + 2*a^9*d*e^16)*x)

________________________________________________________________________________________

Sympy [B]  time = 7.05294, size = 1363, normalized size = 6.03 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10*c**2*d**2*e**3*log(x + (-10*a**7*c**2*d**2*e**17/(a*e**2 - c*d**2)**6 + 70*a**6*c**3*d**4*e**15/(a*e**2 - c
*d**2)**6 - 210*a**5*c**4*d**6*e**13/(a*e**2 - c*d**2)**6 + 350*a**4*c**5*d**8*e**11/(a*e**2 - c*d**2)**6 - 35
0*a**3*c**6*d**10*e**9/(a*e**2 - c*d**2)**6 + 210*a**2*c**7*d**12*e**7/(a*e**2 - c*d**2)**6 - 70*a*c**8*d**14*
e**5/(a*e**2 - c*d**2)**6 + 10*a*c**2*d**2*e**5 + 10*c**9*d**16*e**3/(a*e**2 - c*d**2)**6 + 10*c**3*d**4*e**3)
/(20*c**3*d**3*e**4))/(a*e**2 - c*d**2)**6 - 10*c**2*d**2*e**3*log(x + (10*a**7*c**2*d**2*e**17/(a*e**2 - c*d*
*2)**6 - 70*a**6*c**3*d**4*e**15/(a*e**2 - c*d**2)**6 + 210*a**5*c**4*d**6*e**13/(a*e**2 - c*d**2)**6 - 350*a*
*4*c**5*d**8*e**11/(a*e**2 - c*d**2)**6 + 350*a**3*c**6*d**10*e**9/(a*e**2 - c*d**2)**6 - 210*a**2*c**7*d**12*
e**7/(a*e**2 - c*d**2)**6 + 70*a*c**8*d**14*e**5/(a*e**2 - c*d**2)**6 + 10*a*c**2*d**2*e**5 - 10*c**9*d**16*e*
*3/(a*e**2 - c*d**2)**6 + 10*c**3*d**4*e**3)/(20*c**3*d**3*e**4))/(a*e**2 - c*d**2)**6 + (-3*a**4*e**8 + 27*a*
*3*c*d**2*e**6 + 47*a**2*c**2*d**4*e**4 - 13*a*c**3*d**6*e**2 + 2*c**4*d**8 + 60*c**4*d**4*e**4*x**4 + x**3*(1
50*a*c**3*d**3*e**5 + 90*c**4*d**5*e**3) + x**2*(110*a**2*c**2*d**2*e**6 + 230*a*c**3*d**4*e**4 + 20*c**4*d**6
*e**2) + x*(15*a**3*c*d*e**7 + 175*a**2*c**2*d**3*e**5 + 55*a*c**3*d**5*e**3 - 5*c**4*d**7*e))/(6*a**8*d**2*e*
*13 - 30*a**7*c*d**4*e**11 + 60*a**6*c**2*d**6*e**9 - 60*a**5*c**3*d**8*e**7 + 30*a**4*c**4*d**10*e**5 - 6*a**
3*c**5*d**12*e**3 + x**5*(6*a**5*c**3*d**3*e**12 - 30*a**4*c**4*d**5*e**10 + 60*a**3*c**5*d**7*e**8 - 60*a**2*
c**6*d**9*e**6 + 30*a*c**7*d**11*e**4 - 6*c**8*d**13*e**2) + x**4*(18*a**6*c**2*d**2*e**13 - 78*a**5*c**3*d**4
*e**11 + 120*a**4*c**4*d**6*e**9 - 60*a**3*c**5*d**8*e**7 - 30*a**2*c**6*d**10*e**5 + 42*a*c**7*d**12*e**3 - 1
2*c**8*d**14*e) + x**3*(18*a**7*c*d*e**14 - 54*a**6*c**2*d**3*e**12 + 6*a**5*c**3*d**5*e**10 + 150*a**4*c**4*d
**7*e**8 - 210*a**3*c**5*d**9*e**6 + 102*a**2*c**6*d**11*e**4 - 6*a*c**7*d**13*e**2 - 6*c**8*d**15) + x**2*(6*
a**8*e**15 + 6*a**7*c*d**2*e**13 - 102*a**6*c**2*d**4*e**11 + 210*a**5*c**3*d**6*e**9 - 150*a**4*c**4*d**8*e**
7 - 6*a**3*c**5*d**10*e**5 + 54*a**2*c**6*d**12*e**3 - 18*a*c**7*d**14*e) + x*(12*a**8*d*e**14 - 42*a**7*c*d**
3*e**12 + 30*a**6*c**2*d**5*e**10 + 60*a**5*c**3*d**7*e**8 - 120*a**4*c**4*d**9*e**6 + 78*a**3*c**5*d**11*e**4
- 18*a**2*c**6*d**13*e**2))

________________________________________________________________________________________

Giac [B]  time = 1.23536, size = 791, normalized size = 3.5 \begin{align*} -\frac{20 \,{\left (c^{3} d^{4} e^{3} - a c^{2} d^{2} e^{5}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac{60 \, c^{5} d^{6} x^{5} e^{5} + 150 \, c^{5} d^{7} x^{4} e^{4} + 110 \, c^{5} d^{8} x^{3} e^{3} + 15 \, c^{5} d^{9} x^{2} e^{2} - 3 \, c^{5} d^{10} x e + 2 \, c^{5} d^{11} - 60 \, a c^{4} d^{4} x^{5} e^{7} + 270 \, a c^{4} d^{6} x^{3} e^{5} + 270 \, a c^{4} d^{7} x^{2} e^{4} + 45 \, a c^{4} d^{8} x e^{3} - 15 \, a c^{4} d^{9} e^{2} - 150 \, a^{2} c^{3} d^{3} x^{4} e^{8} - 270 \, a^{2} c^{3} d^{4} x^{3} e^{7} + 180 \, a^{2} c^{3} d^{6} x e^{5} + 60 \, a^{2} c^{3} d^{7} e^{4} - 110 \, a^{3} c^{2} d^{2} x^{3} e^{9} - 270 \, a^{3} c^{2} d^{3} x^{2} e^{8} - 180 \, a^{3} c^{2} d^{4} x e^{7} - 20 \, a^{3} c^{2} d^{5} e^{6} - 15 \, a^{4} c d x^{2} e^{10} - 45 \, a^{4} c d^{2} x e^{9} - 30 \, a^{4} c d^{3} e^{8} + 3 \, a^{5} x e^{11} + 3 \, a^{5} d e^{10}}{6 \,{\left (c^{6} d^{12} - 6 \, a c^{5} d^{10} e^{2} + 15 \, a^{2} c^{4} d^{8} e^{4} - 20 \, a^{3} c^{3} d^{6} e^{6} + 15 \, a^{4} c^{2} d^{4} e^{8} - 6 \, a^{5} c d^{2} e^{10} + a^{6} e^{12}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-20*(c^3*d^4*e^3 - a*c^2*d^2*e^5)*arctan((2*c*d*x*e + c*d^2 + a*e^2)/sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4))
/((c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2*d^4*e^8 - 6*a^5*c*d^2*e^
10 + a^6*e^12)*sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)) - 1/6*(60*c^5*d^6*x^5*e^5 + 150*c^5*d^7*x^4*e^4 + 110
*c^5*d^8*x^3*e^3 + 15*c^5*d^9*x^2*e^2 - 3*c^5*d^10*x*e + 2*c^5*d^11 - 60*a*c^4*d^4*x^5*e^7 + 270*a*c^4*d^6*x^3
*e^5 + 270*a*c^4*d^7*x^2*e^4 + 45*a*c^4*d^8*x*e^3 - 15*a*c^4*d^9*e^2 - 150*a^2*c^3*d^3*x^4*e^8 - 270*a^2*c^3*d
^4*x^3*e^7 + 180*a^2*c^3*d^6*x*e^5 + 60*a^2*c^3*d^7*e^4 - 110*a^3*c^2*d^2*x^3*e^9 - 270*a^3*c^2*d^3*x^2*e^8 -
180*a^3*c^2*d^4*x*e^7 - 20*a^3*c^2*d^5*e^6 - 15*a^4*c*d*x^2*e^10 - 45*a^4*c*d^2*x*e^9 - 30*a^4*c*d^3*e^8 + 3*a
^5*x*e^11 + 3*a^5*d*e^10)/((c^6*d^12 - 6*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 - 20*a^3*c^3*d^6*e^6 + 15*a^4*c^2
*d^4*e^8 - 6*a^5*c*d^2*e^10 + a^6*e^12)*(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)^3)