### 3.1881 $$\int \frac{d+e x}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx$$

Optimal. Leaf size=75 $-\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac{e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac{e \log (d+e x)}{\left (c d^2-a e^2\right )^2}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0446562, antiderivative size = 75, 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} $-\frac{1}{\left (c d^2-a e^2\right ) (a e+c d x)}-\frac{e \log (a e+c d x)}{\left (c d^2-a e^2\right )^2}+\frac{e \log (d+e x)}{\left (c d^2-a e^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0297919, size = 74, normalized size = 0.99 $\frac{1}{\left (a e^2-c d^2\right ) (a e+c d x)}-\frac{e \log (a e+c d x)}{\left (a e^2-c d^2\right )^2}+\frac{e \log (d+e x)}{\left (a e^2-c d^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.074, size = 75, normalized size = 1. \begin{align*}{\frac{e\ln \left ( ex+d \right ) }{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}}+{\frac{1}{ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( cdx+ae \right ) }}-{\frac{e\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((e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.02519, size = 153, normalized size = 2.04 \begin{align*} -\frac{e \log \left (c d x + a e\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} + \frac{e \log \left (e x + d\right )}{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}} - \frac{1}{a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x} \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)^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.90353, size = 238, normalized size = 3.17 \begin{align*} -\frac{c d^{2} - a e^{2} +{\left (c d e x + a e^{2}\right )} \log \left (c d x + a e\right ) -{\left (c d e x + a e^{2}\right )} \log \left (e x + d\right )}{a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} +{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x} \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)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 1.05649, size = 287, normalized size = 3.83 \begin{align*} \frac{e \log{\left (x + \frac{- \frac{a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} + \frac{c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{e \log{\left (x + \frac{\frac{a^{3} e^{7}}{\left (a e^{2} - c d^{2}\right )^{2}} - \frac{3 a^{2} c d^{2} e^{5}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{3 a c^{2} d^{4} e^{3}}{\left (a e^{2} - c d^{2}\right )^{2}} + a e^{3} - \frac{c^{3} d^{6} e}{\left (a e^{2} - c d^{2}\right )^{2}} + c d^{2} e}{2 c d e^{2}} \right )}}{\left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{a^{2} e^{3} - a c d^{2} e + x \left (a c d e^{2} - c^{2} d^{3}\right )} \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)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.17913, size = 262, normalized size = 3.49 \begin{align*} -\frac{2 \,{\left (c d^{2} e - a e^{3}\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^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}} - \frac{c d^{2} x e + c d^{3} - a x e^{3} - a d e^{2}}{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}} \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)^2,x, algorithm="giac")

[Out]

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