### 3.1869 $$\int \frac{d+e x}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx$$

Optimal. Leaf size=16 $\frac{\log (a e+c d x)}{c d}$

[Out]

Log[a*e + c*d*x]/(c*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0061629, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.061, Rules used = {626, 31} $\frac{\log (a e+c d x)}{c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[a*e + c*d*x]/(c*d)

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0015131, size = 16, normalized size = 1. $\frac{\log (a e+c d x)}{c d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Log[a*e + c*d*x]/(c*d)

________________________________________________________________________________________

Maple [A]  time = 0.039, size = 17, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( cdx+ae \right ) }{cd}} \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),x)

[Out]

ln(c*d*x+a*e)/c/d

________________________________________________________________________________________

Maxima [A]  time = 1.0236, size = 22, normalized size = 1.38 \begin{align*} \frac{\log \left (c d x + a e\right )}{c d} \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),x, algorithm="maxima")

[Out]

log(c*d*x + a*e)/(c*d)

________________________________________________________________________________________

Fricas [A]  time = 1.62955, size = 32, normalized size = 2. \begin{align*} \frac{\log \left (c d x + a e\right )}{c d} \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),x, algorithm="fricas")

[Out]

log(c*d*x + a*e)/(c*d)

________________________________________________________________________________________

Sympy [A]  time = 0.083825, size = 12, normalized size = 0.75 \begin{align*} \frac{\log{\left (a e + c d x \right )}}{c d} \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),x)

[Out]

log(a*e + c*d*x)/(c*d)

________________________________________________________________________________________

Giac [B]  time = 1.22298, size = 170, normalized size = 10.62 \begin{align*} \frac{{\left (c d^{2} - a e^{2}\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 )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c d} + \frac{\log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c d} \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),x, algorithm="giac")

[Out]

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