### 3.1871 $$\int \frac{1}{(d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)} \, dx$$

Optimal. Leaf size=73 $\frac{1}{(d+e x) \left (c d^2-a e^2\right )}+\frac{c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac{c d \log (d+e x)}{\left (c d^2-a e^2\right )^2}$

[Out]

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

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Rubi [A]  time = 0.0505175, antiderivative size = 73, 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} $\frac{1}{(d+e x) \left (c d^2-a e^2\right )}+\frac{c d \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}-\frac{c d \log (d+e x)}{\left (c d^2-a e^2\right )^2}$

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)),x]

[Out]

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

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

Mathematica [A]  time = 0.0292618, size = 66, normalized size = 0.9 $\frac{c d (d+e x) \log (a e+c d x)-a e^2+c d^2-c d (d+e x) \log (d+e x)}{(d+e x) \left (c d^2-a e^2\right )^2}$

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)),x]

[Out]

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

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Maple [A]  time = 0.076, size = 75, normalized size = 1. \begin{align*} -{\frac{1}{ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( ex+d \right ) }}-{\frac{cd\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}}+{\frac{cd\ln \left ( cdx+ae \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}} \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),x)

[Out]

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

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Maxima [A]  time = 0.998684, size = 144, normalized size = 1.97 \begin{align*} \frac{c d \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac{c d \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac{1}{c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x} \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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.91676, size = 225, normalized size = 3.08 \begin{align*} \frac{c d^{2} - a e^{2} +{\left (c d e x + c d^{2}\right )} \log \left (c d x + a e\right ) -{\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x} \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),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.00351, size = 301, normalized size = 4.12 \begin{align*} - \frac{c d \log{\left (x + \frac{- \frac{a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} + \frac{c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{c d \log{\left (x + \frac{\frac{a^{3} c d e^{6}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a^{2} c^{2} d^{3} e^{4}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a c^{3} d^{5} e^{2}}{\left (a e^{2} - c d^{2}\right )^{2}} + a c d e^{2} - \frac{c^{4} d^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + c^{2} d^{3}}{2 c^{2} d^{2} e} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{1}{a d e^{2} - c d^{3} + x \left (a e^{3} - c d^{2} e\right )} \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),x)

[Out]

-c*d*log(x + (-a**3*c*d*e**6/(a*e**2 - c*d**2)**2 + 3*a**2*c**2*d**3*e**4/(a*e**2 - c*d**2)**2 - 3*a*c**3*d**5
*e**2/(a*e**2 - c*d**2)**2 + a*c*d*e**2 + c**4*d**7/(a*e**2 - c*d**2)**2 + c**2*d**3)/(2*c**2*d**2*e))/(a*e**2
- c*d**2)**2 + c*d*log(x + (a**3*c*d*e**6/(a*e**2 - c*d**2)**2 - 3*a**2*c**2*d**3*e**4/(a*e**2 - c*d**2)**2 +
3*a*c**3*d**5*e**2/(a*e**2 - c*d**2)**2 + a*c*d*e**2 - c**4*d**7/(a*e**2 - c*d**2)**2 + c**2*d**3)/(2*c**2*d*
*2*e))/(a*e**2 - c*d**2)**2 - 1/(a*d*e**2 - c*d**3 + x*(a*e**3 - c*d**2*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)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError