3.1141 \(\int \frac{A+B x}{b x+c x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac{(b B-A c) \log (b+c x)}{b c}+\frac{A \log (x)}{b} \]

[Out]

(A*Log[x])/b + ((b*B - A*c)*Log[b + c*x])/(b*c)

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Rubi [A]  time = 0.0204292, antiderivative size = 29, 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} \[ \frac{(b B-A c) \log (b+c x)}{b c}+\frac{A \log (x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(b*x + c*x^2),x]

[Out]

(A*Log[x])/b + ((b*B - A*c)*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{A+B x}{b x+c x^2} \, dx &=\int \left (\frac{A}{b x}+\frac{b B-A c}{b (b+c x)}\right ) \, dx\\ &=\frac{A \log (x)}{b}+\frac{(b B-A c) \log (b+c x)}{b c}\\ \end{align*}

Mathematica [A]  time = 0.0091392, size = 29, normalized size = 1. \[ \frac{(b B-A c) \log (b+c x)}{b c}+\frac{A \log (x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(b*x + c*x^2),x]

[Out]

(A*Log[x])/b + ((b*B - A*c)*Log[b + c*x])/(b*c)

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Maple [A]  time = 0.005, size = 32, normalized size = 1.1 \begin{align*}{\frac{A\ln \left ( x \right ) }{b}}-{\frac{\ln \left ( cx+b \right ) A}{b}}+{\frac{\ln \left ( cx+b \right ) B}{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(c*x^2+b*x),x)

[Out]

A*ln(x)/b-1/b*ln(c*x+b)*A+1/c*ln(c*x+b)*B

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Maxima [A]  time = 1.1099, size = 39, normalized size = 1.34 \begin{align*} \frac{A \log \left (x\right )}{b} + \frac{{\left (B b - A c\right )} \log \left (c x + b\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*log(x)/b + (B*b - A*c)*log(c*x + b)/(b*c)

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Fricas [A]  time = 1.4723, size = 63, normalized size = 2.17 \begin{align*} \frac{A c \log \left (x\right ) +{\left (B b - A c\right )} \log \left (c x + b\right )}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*c*log(x) + (B*b - A*c)*log(c*x + b))/(b*c)

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Sympy [A]  time = 0.430276, size = 41, normalized size = 1.41 \begin{align*} \frac{A \log{\left (x \right )}}{b} + \frac{\left (- A c + B b\right ) \log{\left (x + \frac{- A b + \frac{b \left (- A c + B b\right )}{c}}{- 2 A c + B b} \right )}}{b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(c*x**2+b*x),x)

[Out]

A*log(x)/b + (-A*c + B*b)*log(x + (-A*b + b*(-A*c + B*b)/c)/(-2*A*c + B*b))/(b*c)

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Giac [A]  time = 1.24458, size = 42, normalized size = 1.45 \begin{align*} \frac{A \log \left ({\left | x \right |}\right )}{b} + \frac{{\left (B b - A c\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((B*x+A)/(c*x^2+b*x),x, algorithm="giac")

[Out]

A*log(abs(x))/b + (B*b - A*c)*log(abs(c*x + b))/(b*c)