### 3.262 $$\int \frac{d+e x}{b x+c x^2} \, dx$$

Optimal. Leaf size=30 $\frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c}$

[Out]

(d*Log[x])/b - ((c*d - b*e)*Log[b + c*x])/(b*c)

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Rubi [A]  time = 0.0207153, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {631} $\frac{d \log (x)}{b}-\frac{(c d-b e) \log (b+c x)}{b c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)/(b*x + c*x^2),x]

[Out]

(d*Log[x])/b - ((c*d - b*e)*Log[b + c*x])/(b*c)

Rule 631

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)
*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0]
|| EqQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0097819, size = 29, normalized size = 0.97 $\frac{(b e-c d) \log (b+c x)}{b c}+\frac{d \log (x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)/(b*x + c*x^2),x]

[Out]

(d*Log[x])/b + ((-(c*d) + b*e)*Log[b + c*x])/(b*c)

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Maple [A]  time = 0.052, size = 32, normalized size = 1.1 \begin{align*}{\frac{d\ln \left ( x \right ) }{b}}+{\frac{\ln \left ( cx+b \right ) e}{c}}-{\frac{\ln \left ( cx+b \right ) d}{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*d*ln(x)+1/c*ln(c*x+b)*e-1/b*ln(c*x+b)*d

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Maxima [A]  time = 1.13507, size = 41, normalized size = 1.37 \begin{align*} \frac{d \log \left (x\right )}{b} - \frac{{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(x)/b - (c*d - b*e)*log(c*x + b)/(b*c)

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Fricas [A]  time = 1.67953, size = 63, normalized size = 2.1 \begin{align*} \frac{c d \log \left (x\right ) -{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*d*log(x) - (c*d - b*e)*log(c*x + b))/(b*c)

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Sympy [A]  time = 2.02744, size = 41, normalized size = 1.37 \begin{align*} \frac{d \log{\left (x \right )}}{b} + \frac{\left (b e - c d\right ) \log{\left (x + \frac{- b d + \frac{b \left (b e - c d\right )}{c}}{b e - 2 c d} \right )}}{b c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(x)/b + (b*e - c*d)*log(x + (-b*d + b*(b*e - c*d)/c)/(b*e - 2*c*d))/(b*c)

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Giac [A]  time = 1.32039, size = 45, normalized size = 1.5 \begin{align*} \frac{d \log \left ({\left | x \right |}\right )}{b} - \frac{{\left (c d - b e\right )} \log \left ({\left | c x + b \right |}\right )}{b c} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(abs(x))/b - (c*d - b*e)*log(abs(c*x + b))/(b*c)