### 3.1895 $$\int \frac{1}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx$$

Optimal. Leaf size=223 $\frac{4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac{c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac{6 c^2 d^2 e^2}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac{10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac{10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6}+\frac{3 c d e^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}+\frac{e^2}{3 (d+e x)^3 \left (c d^2-a e^2\right )^3}$

[Out]

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

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Rubi [A]  time = 0.223859, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 44} $\frac{4 c^3 d^3 e}{\left (c d^2-a e^2\right )^5 (a e+c d x)}-\frac{c^3 d^3}{2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2}+\frac{6 c^2 d^2 e^2}{(d+e x) \left (c d^2-a e^2\right )^5}+\frac{10 c^3 d^3 e^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^6}-\frac{10 c^3 d^3 e^2 \log (d+e x)}{\left (c d^2-a e^2\right )^6}+\frac{3 c d e^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )^4}+\frac{e^2}{3 (d+e x)^3 \left (c d^2-a e^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3),x]

[Out]

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

Mathematica [A]  time = 0.194587, size = 201, normalized size = 0.9 $\frac{\frac{24 c^3 d^3 e \left (c d^2-a e^2\right )}{a e+c d x}-\frac{3 c^3 d^3 \left (c d^2-a e^2\right )^2}{(a e+c d x)^2}+\frac{36 c^2 d^2 e^2 \left (c d^2-a e^2\right )}{d+e x}+60 c^3 d^3 e^2 \log (a e+c d x)+\frac{9 c d \left (c d^2 e-a e^3\right )^2}{(d+e x)^2}-\frac{2 e^2 \left (a e^2-c d^2\right )^3}{(d+e x)^3}-60 c^3 d^3 e^2 \log (d+e x)}{6 \left (c d^2-a e^2\right )^6}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[1/((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3),x]

[Out]

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

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Maple [A]  time = 0.053, size = 218, normalized size = 1. \begin{align*} -{\frac{{e}^{2}}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( ex+d \right ) ^{3}}}-10\,{\frac{{c}^{3}{d}^{3}{e}^{2}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6}}}-6\,{\frac{{c}^{2}{e}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( ex+d \right ) }}+{\frac{3\,cd{e}^{2}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{3}}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{4} \left ( cdx+ae \right ) ^{2}}}+10\,{\frac{{c}^{3}{d}^{3}{e}^{2}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{6}}}-4\,{\frac{{c}^{3}{d}^{3}e}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{5} \left ( cdx+ae \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

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

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Maxima [B]  time = 1.27441, size = 1278, normalized size = 5.73 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.41755, size = 2485, 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(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 6.329, size = 1353, normalized size = 6.07 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError