### 3.1872 $$\int \frac{1}{(d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)} \, dx$$

Optimal. Leaf size=108 $\frac{c^2 d^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac{c^2 d^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3}+\frac{c d}{(d+e x) \left (c d^2-a e^2\right )^2}+\frac{1}{2 (d+e x)^2 \left (c d^2-a e^2\right )}$

[Out]

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

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Rubi [A]  time = 0.0720214, antiderivative size = 108, 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} $\frac{c^2 d^2 \log (a e+c d x)}{\left (c d^2-a e^2\right )^3}-\frac{c^2 d^2 \log (d+e x)}{\left (c d^2-a e^2\right )^3}+\frac{c d}{(d+e x) \left (c d^2-a e^2\right )^2}+\frac{1}{2 (d+e x)^2 \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0476664, size = 102, normalized size = 0.94 $\frac{2 c^2 d^2 (d+e x)^2 \log (a e+c d x)+\left (c d^2-a e^2\right ) \left (c d (3 d+2 e x)-a e^2\right )-2 c^2 d^2 (d+e x)^2 \log (d+e x)}{2 (d+e x)^2 \left (c d^2-a e^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.05, size = 107, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ) \left ( ex+d \right ) ^{2}}}+{\frac{{c}^{2}{d}^{2}\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}}+{\frac{cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{2}\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.14348, size = 308, normalized size = 2.85 \begin{align*} \frac{c^{2} d^{2} \log \left (c d x + a e\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} - \frac{c^{2} d^{2} \log \left (e x + d\right )}{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}} + \frac{2 \, c d e x + 3 \, c d^{2} - a e^{2}}{2 \,{\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} +{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.94723, size = 528, normalized size = 4.89 \begin{align*} \frac{3 \, c^{2} d^{4} - 4 \, a c d^{2} e^{2} + a^{2} e^{4} + 2 \,{\left (c^{2} d^{3} e - a c d e^{3}\right )} x + 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (c d x + a e\right ) - 2 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{2 \,{\left (c^{3} d^{8} - 3 \, a c^{2} d^{6} e^{2} + 3 \, a^{2} c d^{4} e^{4} - a^{3} d^{2} e^{6} +{\left (c^{3} d^{6} e^{2} - 3 \, a c^{2} d^{4} e^{4} + 3 \, a^{2} c d^{2} e^{6} - a^{3} e^{8}\right )} x^{2} + 2 \,{\left (c^{3} d^{7} e - 3 \, a c^{2} d^{5} e^{3} + 3 \, a^{2} c d^{3} e^{5} - a^{3} d e^{7}\right )} x\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.52632, size = 471, normalized size = 4.36 \begin{align*} \frac{c^{2} d^{2} \log{\left (x + \frac{- \frac{a^{4} c^{2} d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{4 a^{3} c^{3} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{6 a^{2} c^{4} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{4 a c^{5} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + a c^{2} d^{2} e^{2} - \frac{c^{6} d^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} + c^{3} d^{4}}{2 c^{3} d^{3} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{c^{2} d^{2} \log{\left (x + \frac{\frac{a^{4} c^{2} d^{2} e^{8}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{4 a^{3} c^{3} d^{4} e^{6}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{6 a^{2} c^{4} d^{6} e^{4}}{\left (a e^{2} - c d^{2}\right )^{3}} - \frac{4 a c^{5} d^{8} e^{2}}{\left (a e^{2} - c d^{2}\right )^{3}} + a c^{2} d^{2} e^{2} + \frac{c^{6} d^{10}}{\left (a e^{2} - c d^{2}\right )^{3}} + c^{3} d^{4}}{2 c^{3} d^{3} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{3}} + \frac{- a e^{2} + 3 c d^{2} + 2 c d e x}{2 a^{2} d^{2} e^{4} - 4 a c d^{4} e^{2} + 2 c^{2} d^{6} + x^{2} \left (2 a^{2} e^{6} - 4 a c d^{2} e^{4} + 2 c^{2} d^{4} e^{2}\right ) + x \left (4 a^{2} d e^{5} - 8 a c d^{3} e^{3} + 4 c^{2} d^{5} e\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*d**2*log(x + (-a**4*c**2*d**2*e**8/(a*e**2 - c*d**2)**3 + 4*a**3*c**3*d**4*e**6/(a*e**2 - c*d**2)**3 - 6*
a**2*c**4*d**6*e**4/(a*e**2 - c*d**2)**3 + 4*a*c**5*d**8*e**2/(a*e**2 - c*d**2)**3 + a*c**2*d**2*e**2 - c**6*d
**10/(a*e**2 - c*d**2)**3 + c**3*d**4)/(2*c**3*d**3*e))/(a*e**2 - c*d**2)**3 - c**2*d**2*log(x + (a**4*c**2*d*
*2*e**8/(a*e**2 - c*d**2)**3 - 4*a**3*c**3*d**4*e**6/(a*e**2 - c*d**2)**3 + 6*a**2*c**4*d**6*e**4/(a*e**2 - c*
d**2)**3 - 4*a*c**5*d**8*e**2/(a*e**2 - c*d**2)**3 + a*c**2*d**2*e**2 + c**6*d**10/(a*e**2 - c*d**2)**3 + c**3
*d**4)/(2*c**3*d**3*e))/(a*e**2 - c*d**2)**3 + (-a*e**2 + 3*c*d**2 + 2*c*d*e*x)/(2*a**2*d**2*e**4 - 4*a*c*d**4
*e**2 + 2*c**2*d**6 + x**2*(2*a**2*e**6 - 4*a*c*d**2*e**4 + 2*c**2*d**4*e**2) + x*(4*a**2*d*e**5 - 8*a*c*d**3*
e**3 + 4*c**2*d**5*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)^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError