### 3.1873 $$\int \frac{1}{(d+e x)^3 (a d e+(c d^2+a e^2) x+c d e x^2)} \, dx$$

Optimal. Leaf size=139 $\frac{c^2 d^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac{c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}+\frac{c d}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{1}{3 (d+e x)^3 \left (c d^2-a e^2\right )}$

[Out]

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

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Rubi [A]  time = 0.0989909, antiderivative size = 139, 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} $\frac{c^2 d^2}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{c^3 d^3 \log (a e+c d x)}{\left (c d^2-a e^2\right )^4}-\frac{c^3 d^3 \log (d+e x)}{\left (c d^2-a e^2\right )^4}+\frac{c d}{2 (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{1}{3 (d+e x)^3 \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0684324, size = 135, normalized size = 0.97 $\frac{\left (c d^2-a e^2\right ) \left (2 a^2 e^4-a c d e^2 (7 d+3 e x)+c^2 d^2 \left (11 d^2+15 d e x+6 e^2 x^2\right )\right )+6 c^3 d^3 (d+e x)^3 \log (a e+c d x)-6 c^3 d^3 (d+e x)^3 \log (d+e x)}{6 (d+e x)^3 \left (c d^2-a e^2\right )^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((c*d^2 - a*e^2)*(2*a^2*e^4 - a*c*d*e^2*(7*d + 3*e*x) + c^2*d^2*(11*d^2 + 15*d*e*x + 6*e^2*x^2)) + 6*c^3*d^3*(
d + e*x)^3*Log[a*e + c*d*x] - 6*c^3*d^3*(d + e*x)^3*Log[d + e*x])/(6*(c*d^2 - a*e^2)^4*(d + e*x)^3)

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Maple [A]  time = 0.053, size = 137, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( 3\,a{e}^{2}-3\,c{d}^{2} \right ) \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( ex+d \right ) }}+{\frac{cd}{2\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{3}{d}^{3}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}}+{\frac{{c}^{3}{d}^{3}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{4}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.15512, size = 531, normalized size = 3.82 \begin{align*} \frac{c^{3} d^{3} \log \left (c d x + a e\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} - \frac{c^{3} d^{3} \log \left (e x + d\right )}{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}} + \frac{6 \, c^{2} d^{2} e^{2} x^{2} + 11 \, c^{2} d^{4} - 7 \, a c d^{2} e^{2} + 2 \, a^{2} e^{4} + 3 \,{\left (5 \, c^{2} d^{3} e - a c d e^{3}\right )} x}{6 \,{\left (c^{3} d^{9} - 3 \, a c^{2} d^{7} e^{2} + 3 \, a^{2} c d^{5} e^{4} - a^{3} d^{3} e^{6} +{\left (c^{3} d^{6} e^{3} - 3 \, a c^{2} d^{4} e^{5} + 3 \, a^{2} c d^{2} e^{7} - a^{3} e^{9}\right )} x^{3} + 3 \,{\left (c^{3} d^{7} e^{2} - 3 \, a c^{2} d^{5} e^{4} + 3 \, a^{2} c d^{3} e^{6} - a^{3} d e^{8}\right )} x^{2} + 3 \,{\left (c^{3} d^{8} e - 3 \, a c^{2} d^{6} e^{3} + 3 \, a^{2} c d^{4} e^{5} - a^{3} d^{2} e^{7}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*d^3*log(c*d*x + a*e)/(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8) - c^3*d^3
*log(e*x + d)/(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8) + 1/6*(6*c^2*d^2*e^2
*x^2 + 11*c^2*d^4 - 7*a*c*d^2*e^2 + 2*a^2*e^4 + 3*(5*c^2*d^3*e - a*c*d*e^3)*x)/(c^3*d^9 - 3*a*c^2*d^7*e^2 + 3*
a^2*c*d^5*e^4 - a^3*d^3*e^6 + (c^3*d^6*e^3 - 3*a*c^2*d^4*e^5 + 3*a^2*c*d^2*e^7 - a^3*e^9)*x^3 + 3*(c^3*d^7*e^2
- 3*a*c^2*d^5*e^4 + 3*a^2*c*d^3*e^6 - a^3*d*e^8)*x^2 + 3*(c^3*d^8*e - 3*a*c^2*d^6*e^3 + 3*a^2*c*d^4*e^5 - a^3
*d^2*e^7)*x)

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Fricas [B]  time = 1.86856, size = 906, normalized size = 6.52 \begin{align*} \frac{11 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} - 2 \, a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 3 \,{\left (5 \, c^{3} d^{5} e - 6 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x + 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (c d x + a e\right ) - 6 \,{\left (c^{3} d^{3} e^{3} x^{3} + 3 \, c^{3} d^{4} e^{2} x^{2} + 3 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{6 \,{\left (c^{4} d^{11} - 4 \, a c^{3} d^{9} e^{2} + 6 \, a^{2} c^{2} d^{7} e^{4} - 4 \, a^{3} c d^{5} e^{6} + a^{4} d^{3} e^{8} +{\left (c^{4} d^{8} e^{3} - 4 \, a c^{3} d^{6} e^{5} + 6 \, a^{2} c^{2} d^{4} e^{7} - 4 \, a^{3} c d^{2} e^{9} + a^{4} e^{11}\right )} x^{3} + 3 \,{\left (c^{4} d^{9} e^{2} - 4 \, a c^{3} d^{7} e^{4} + 6 \, a^{2} c^{2} d^{5} e^{6} - 4 \, a^{3} c d^{3} e^{8} + a^{4} d e^{10}\right )} x^{2} + 3 \,{\left (c^{4} d^{10} e - 4 \, a c^{3} d^{8} e^{3} + 6 \, a^{2} c^{2} d^{6} e^{5} - 4 \, a^{3} c d^{4} e^{7} + a^{4} d^{2} e^{9}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(11*c^3*d^6 - 18*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 2*a^3*e^6 + 6*(c^3*d^4*e^2 - a*c^2*d^2*e^4)*x^2 + 3*(5*
c^3*d^5*e - 6*a*c^2*d^3*e^3 + a^2*c*d*e^5)*x + 6*(c^3*d^3*e^3*x^3 + 3*c^3*d^4*e^2*x^2 + 3*c^3*d^5*e*x + c^3*d^
6)*log(c*d*x + a*e) - 6*(c^3*d^3*e^3*x^3 + 3*c^3*d^4*e^2*x^2 + 3*c^3*d^5*e*x + c^3*d^6)*log(e*x + d))/(c^4*d^1
1 - 4*a*c^3*d^9*e^2 + 6*a^2*c^2*d^7*e^4 - 4*a^3*c*d^5*e^6 + a^4*d^3*e^8 + (c^4*d^8*e^3 - 4*a*c^3*d^6*e^5 + 6*a
^2*c^2*d^4*e^7 - 4*a^3*c*d^2*e^9 + a^4*e^11)*x^3 + 3*(c^4*d^9*e^2 - 4*a*c^3*d^7*e^4 + 6*a^2*c^2*d^5*e^6 - 4*a^
3*c*d^3*e^8 + a^4*d*e^10)*x^2 + 3*(c^4*d^10*e - 4*a*c^3*d^8*e^3 + 6*a^2*c^2*d^6*e^5 - 4*a^3*c*d^4*e^7 + a^4*d^
2*e^9)*x)

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Sympy [B]  time = 2.40193, size = 672, normalized size = 4.83 \begin{align*} - \frac{c^{3} d^{3} \log{\left (x + \frac{- \frac{a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} + \frac{c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{c^{3} d^{3} \log{\left (x + \frac{\frac{a^{5} c^{3} d^{3} e^{10}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{5 a^{4} c^{4} d^{5} e^{8}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{10 a^{3} c^{5} d^{7} e^{6}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{10 a^{2} c^{6} d^{9} e^{4}}{\left (a e^{2} - c d^{2}\right )^{4}} + \frac{5 a c^{7} d^{11} e^{2}}{\left (a e^{2} - c d^{2}\right )^{4}} + a c^{3} d^{3} e^{2} - \frac{c^{8} d^{13}}{\left (a e^{2} - c d^{2}\right )^{4}} + c^{4} d^{5}}{2 c^{4} d^{4} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{4}} - \frac{2 a^{2} e^{4} - 7 a c d^{2} e^{2} + 11 c^{2} d^{4} + 6 c^{2} d^{2} e^{2} x^{2} + x \left (- 3 a c d e^{3} + 15 c^{2} d^{3} e\right )}{6 a^{3} d^{3} e^{6} - 18 a^{2} c d^{5} e^{4} + 18 a c^{2} d^{7} e^{2} - 6 c^{3} d^{9} + x^{3} \left (6 a^{3} e^{9} - 18 a^{2} c d^{2} e^{7} + 18 a c^{2} d^{4} e^{5} - 6 c^{3} d^{6} e^{3}\right ) + x^{2} \left (18 a^{3} d e^{8} - 54 a^{2} c d^{3} e^{6} + 54 a c^{2} d^{5} e^{4} - 18 c^{3} d^{7} e^{2}\right ) + x \left (18 a^{3} d^{2} e^{7} - 54 a^{2} c d^{4} e^{5} + 54 a c^{2} d^{6} e^{3} - 18 c^{3} d^{8} e\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c**3*d**3*log(x + (-a**5*c**3*d**3*e**10/(a*e**2 - c*d**2)**4 + 5*a**4*c**4*d**5*e**8/(a*e**2 - c*d**2)**4 -
10*a**3*c**5*d**7*e**6/(a*e**2 - c*d**2)**4 + 10*a**2*c**6*d**9*e**4/(a*e**2 - c*d**2)**4 - 5*a*c**7*d**11*e**
2/(a*e**2 - c*d**2)**4 + a*c**3*d**3*e**2 + c**8*d**13/(a*e**2 - c*d**2)**4 + c**4*d**5)/(2*c**4*d**4*e))/(a*e
**2 - c*d**2)**4 + c**3*d**3*log(x + (a**5*c**3*d**3*e**10/(a*e**2 - c*d**2)**4 - 5*a**4*c**4*d**5*e**8/(a*e**
2 - c*d**2)**4 + 10*a**3*c**5*d**7*e**6/(a*e**2 - c*d**2)**4 - 10*a**2*c**6*d**9*e**4/(a*e**2 - c*d**2)**4 + 5
*a*c**7*d**11*e**2/(a*e**2 - c*d**2)**4 + a*c**3*d**3*e**2 - c**8*d**13/(a*e**2 - c*d**2)**4 + c**4*d**5)/(2*c
**4*d**4*e))/(a*e**2 - c*d**2)**4 - (2*a**2*e**4 - 7*a*c*d**2*e**2 + 11*c**2*d**4 + 6*c**2*d**2*e**2*x**2 + x*
(-3*a*c*d*e**3 + 15*c**2*d**3*e))/(6*a**3*d**3*e**6 - 18*a**2*c*d**5*e**4 + 18*a*c**2*d**7*e**2 - 6*c**3*d**9
+ x**3*(6*a**3*e**9 - 18*a**2*c*d**2*e**7 + 18*a*c**2*d**4*e**5 - 6*c**3*d**6*e**3) + x**2*(18*a**3*d*e**8 - 5
4*a**2*c*d**3*e**6 + 54*a*c**2*d**5*e**4 - 18*c**3*d**7*e**2) + x*(18*a**3*d**2*e**7 - 54*a**2*c*d**4*e**5 + 5
4*a*c**2*d**6*e**3 - 18*c**3*d**8*e))

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

[Out]

Exception raised: NotImplementedError