3.2.98 \(\int \frac {a+2 b x}{a x+b x^2} \, dx\)

Optimal. Leaf size=10 \[ \log \left (a x+b x^2\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {628} \begin {gather*} \log \left (a x+b x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + 2*b*x)/(a*x + b*x^2),x]

[Out]

Log[a*x + b*x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.90 \begin {gather*} \log (a+b x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + 2*b*x)/(a*x + b*x^2),x]

[Out]

Log[x] + Log[a + b*x]

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

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + 2*b*x)/(a*x + b*x^2),x]

[Out]

IntegrateAlgebraic[(a + 2*b*x)/(a*x + b*x^2), x]

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fricas [A]  time = 0.40, size = 10, normalized size = 1.00 \begin {gather*} \log \left (b x^{2} + a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*b*x+a)/(b*x^2+a*x),x, algorithm="fricas")

[Out]

log(b*x^2 + a*x)

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giac [A]  time = 0.16, size = 11, normalized size = 1.10 \begin {gather*} \log \left ({\left | b x^{2} + a x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*b*x+a)/(b*x^2+a*x),x, algorithm="giac")

[Out]

log(abs(b*x^2 + a*x))

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maple [A]  time = 0.05, size = 9, normalized size = 0.90 \begin {gather*} \ln \left (\left (b x +a \right ) x \right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*b*x+a)/(b*x^2+a*x),x)

[Out]

ln((b*x+a)*x)

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maxima [A]  time = 1.43, size = 10, normalized size = 1.00 \begin {gather*} \log \left (b x^{2} + a x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*b*x+a)/(b*x^2+a*x),x, algorithm="maxima")

[Out]

log(b*x^2 + a*x)

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mupad [B]  time = 0.06, size = 8, normalized size = 0.80 \begin {gather*} \ln \left (x\,\left (a+b\,x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + 2*b*x)/(a*x + b*x^2),x)

[Out]

log(x*(a + b*x))

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sympy [A]  time = 0.12, size = 8, normalized size = 0.80 \begin {gather*} \log {\left (a x + b x^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*b*x+a)/(b*x**2+a*x),x)

[Out]

log(a*x + b*x**2)

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