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

Optimal. Leaf size=13 \[ d \log \left (a+b x+c x^2\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {628} \begin {gather*} d \log \left (a+b x+c x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

d*Log[a + b*x + c*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 {b d+2 c d x}{a+b x+c x^2} \, dx &=d \log \left (a+b x+c x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.92 \begin {gather*} d \log (a+x (b+c x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

d*Log[a + x*(b + c*x)]

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 13, normalized size = 1.00 \begin {gather*} d \log \left (c x^{2} + b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(c*x^2 + b*x + a)

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giac [A]  time = 0.16, size = 21, normalized size = 1.62 \begin {gather*} d \log \left ({\left | a + \frac {c d x^{2} + b d x}{d} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(abs(a + (c*d*x^2 + b*d*x)/d))

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maple [A]  time = 0.04, size = 14, normalized size = 1.08 \begin {gather*} d \ln \left (c \,x^{2}+b x +a \right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*ln(c*x^2+b*x+a)

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maxima [A]  time = 1.30, size = 13, normalized size = 1.00 \begin {gather*} d \log \left (c x^{2} + b x + a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(c*x^2 + b*x + a)

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mupad [B]  time = 0.04, size = 13, normalized size = 1.00 \begin {gather*} d\,\ln \left (c\,x^2+b\,x+a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(a + b*x + c*x^2)

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sympy [A]  time = 0.20, size = 12, normalized size = 0.92 \begin {gather*} d \log {\left (a + b x + c x^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*log(a + b*x + c*x**2)

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