3.13.42 \(\int \frac {d+e x}{9+12 x+4 x^2} \, dx\)

Optimal. Leaf size=30 \[ \frac {1}{4} e \log (2 x+3)-\frac {2 d-3 e}{4 (2 x+3)} \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 43} \begin {gather*} \frac {1}{4} e \log (2 x+3)-\frac {2 d-3 e}{4 (2 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)/(9 + 12*x + 4*x^2),x]

[Out]

-(2*d - 3*e)/(4*(3 + 2*x)) + (e*Log[3 + 2*x])/4

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int \frac {d+e x}{9+12 x+4 x^2} \, dx &=\int \frac {d+e x}{(3+2 x)^2} \, dx\\ &=\int \left (\frac {2 d-3 e}{2 (3+2 x)^2}+\frac {e}{2 (3+2 x)}\right ) \, dx\\ &=-\frac {2 d-3 e}{4 (3+2 x)}+\frac {1}{4} e \log (3+2 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 30, normalized size = 1.00 \begin {gather*} \frac {3 e-2 d}{4 (2 x+3)}+\frac {1}{4} e \log (2 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)/(9 + 12*x + 4*x^2),x]

[Out]

(-2*d + 3*e)/(4*(3 + 2*x)) + (e*Log[3 + 2*x])/4

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{9+12 x+4 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(d + e*x)/(9 + 12*x + 4*x^2),x]

[Out]

IntegrateAlgebraic[(d + e*x)/(9 + 12*x + 4*x^2), x]

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fricas [A]  time = 0.39, size = 31, normalized size = 1.03 \begin {gather*} \frac {{\left (2 \, e x + 3 \, e\right )} \log \left (2 \, x + 3\right ) - 2 \, d + 3 \, e}{4 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x^2+12*x+9),x, algorithm="fricas")

[Out]

1/4*((2*e*x + 3*e)*log(2*x + 3) - 2*d + 3*e)/(2*x + 3)

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giac [A]  time = 0.19, size = 29, normalized size = 0.97 \begin {gather*} \frac {1}{4} \, e \log \left ({\left | 2 \, x + 3 \right |}\right ) - \frac {2 \, d - 3 \, e}{4 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x^2+12*x+9),x, algorithm="giac")

[Out]

1/4*e*log(abs(2*x + 3)) - 1/4*(2*d - 3*e)/(2*x + 3)

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maple [A]  time = 0.05, size = 31, normalized size = 1.03 \begin {gather*} \frac {e \ln \left (2 x +3\right )}{4}-\frac {d}{2 \left (2 x +3\right )}+\frac {3 e}{4 \left (2 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(4*x^2+12*x+9),x)

[Out]

-1/2/(2*x+3)*d+3/4*e/(2*x+3)+1/4*e*ln(2*x+3)

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maxima [A]  time = 1.02, size = 26, normalized size = 0.87 \begin {gather*} \frac {1}{4} \, e \log \left (2 \, x + 3\right ) - \frac {2 \, d - 3 \, e}{4 \, {\left (2 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x^2+12*x+9),x, algorithm="maxima")

[Out]

1/4*e*log(2*x + 3) - 1/4*(2*d - 3*e)/(2*x + 3)

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mupad [B]  time = 0.04, size = 24, normalized size = 0.80 \begin {gather*} \frac {e\,\ln \left (x+\frac {3}{2}\right )}{4}-\frac {\frac {d}{2}-\frac {3\,e}{4}}{2\,x+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(12*x + 4*x^2 + 9),x)

[Out]

(e*log(x + 3/2))/4 - (d/2 - (3*e)/4)/(2*x + 3)

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sympy [A]  time = 0.14, size = 20, normalized size = 0.67 \begin {gather*} \frac {e \log {\left (2 x + 3 \right )}}{4} + \frac {- 2 d + 3 e}{8 x + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x**2+12*x+9),x)

[Out]

e*log(2*x + 3)/4 + (-2*d + 3*e)/(8*x + 12)

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