3.1.47 \(\int \frac {d+e x}{b x+c x^2} \, dx\)

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

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {d \log (x)}{b}-\frac {(c d-b e) \log (b+c x)}{b c} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 0.97 \begin {gather*} \frac {(b e-c d) \log (b+c x)}{b c}+\frac {d \log (x)}{b} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{b x+c x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)/(b*x + c*x^2),x]

[Out]

IntegrateAlgebraic[(d + e*x)/(b*x + c*x^2), x]

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fricas [A]  time = 0.40, size = 29, normalized size = 0.97 \begin {gather*} \frac {c d \log \relax (x) - {\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \end {gather*}

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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giac [A]  time = 0.15, size = 33, normalized size = 1.10 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 32, normalized size = 1.07 \begin {gather*} \frac {d \ln \relax (x )}{b}-\frac {d \ln \left (c x +b \right )}{b}+\frac {e \ln \left (c x +b \right )}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.85, size = 30, normalized size = 1.00 \begin {gather*} \frac {d \log \relax (x)}{b} - \frac {{\left (c d - b e\right )} \log \left (c x + b\right )}{b c} \end {gather*}

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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mupad [B]  time = 0.10, size = 28, normalized size = 0.93 \begin {gather*} \frac {d\,\ln \relax (x)}{b}-\ln \left (b+c\,x\right )\,\left (\frac {d}{b}-\frac {e}{c}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(b*x + c*x^2),x)

[Out]

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

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sympy [A]  time = 0.42, size = 41, normalized size = 1.37 \begin {gather*} \frac {d \log {\relax (x )}}{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 {gather*}

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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