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

Optimal. Leaf size=56 \[ \frac {(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt {4 x^2+12 x+9}}+\frac {1}{4} e \sqrt {4 x^2+12 x+9} \]

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Rubi [A]  time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {640, 608, 31} \begin {gather*} \frac {(2 x+3) (2 d-3 e) \log (2 x+3)}{4 \sqrt {4 x^2+12 x+9}}+\frac {1}{4} e \sqrt {4 x^2+12 x+9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*Sqrt[9 + 12*x + 4*x^2])/4 + ((2*d - 3*e)*(3 + 2*x)*Log[3 + 2*x])/(4*Sqrt[9 + 12*x + 4*x^2])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 608

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin {align*} \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx &=\frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {1}{2} (2 d-3 e) \int \frac {1}{\sqrt {9+12 x+4 x^2}} \, dx\\ &=\frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {((2 d-3 e) (6+4 x)) \int \frac {1}{6+4 x} \, dx}{2 \sqrt {9+12 x+4 x^2}}\\ &=\frac {1}{4} e \sqrt {9+12 x+4 x^2}+\frac {(2 d-3 e) (3+2 x) \log (3+2 x)}{4 \sqrt {9+12 x+4 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 42, normalized size = 0.75 \begin {gather*} \frac {(2 x+3) ((2 d-3 e) \log (2 x+3)+e (2 x+3))}{4 \sqrt {(2 x+3)^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{\sqrt {9+12 x+4 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 20, normalized size = 0.36 \begin {gather*} \frac {1}{2} \, e x + \frac {1}{4} \, {\left (2 \, d - 3 \, e\right )} \log \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)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.20, size = 46, normalized size = 0.82 \begin {gather*} -\frac {1}{4} \, {\left (2 \, d - 3 \, e\right )} \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} + 12 \, x + 9} - 3 \right |}\right ) + \frac {1}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(1/2),x, algorithm="giac")

[Out]

-1/4*(2*d - 3*e)*log(abs(-2*x + sqrt(4*x^2 + 12*x + 9) - 3)) + 1/4*sqrt(4*x^2 + 12*x + 9)*e

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maple [A]  time = 0.05, size = 40, normalized size = 0.71 \begin {gather*} \frac {\left (2 x +3\right ) \left (2 d \ln \left (2 x +3\right )+2 e x -3 e \ln \left (2 x +3\right )\right )}{4 \sqrt {\left (2 x +3\right )^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/(4*x^2+12*x+9)^(1/2),x)

[Out]

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

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maxima [A]  time = 2.26, size = 30, normalized size = 0.54 \begin {gather*} \frac {1}{2} \, d \log \left (x + \frac {3}{2}\right ) - \frac {3}{4} \, e \log \left (x + \frac {3}{2}\right ) + \frac {1}{4} \, \sqrt {4 \, x^{2} + 12 \, x + 9} e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x^2+12*x+9)^(1/2),x, algorithm="maxima")

[Out]

1/2*d*log(x + 3/2) - 3/4*e*log(x + 3/2) + 1/4*sqrt(4*x^2 + 12*x + 9)*e

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mupad [B]  time = 1.50, size = 46, normalized size = 0.82 \begin {gather*} \frac {e\,\sqrt {4\,x^2+12\,x+9}}{4}-\frac {3\,e\,\ln \left (x+\frac {\left |2\,x+3\right |}{2}+\frac {3}{2}\right )}{4}+\frac {d\,\ln \left (4\,x+6\right )\,\mathrm {sign}\left (8\,x+12\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(12*x + 4*x^2 + 9)^(1/2),x)

[Out]

(e*(12*x + 4*x^2 + 9)^(1/2))/4 - (3*e*log(x + abs(2*x + 3)/2 + 3/2))/4 + (d*log(4*x + 6)*sign(8*x + 12))/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d + e x}{\sqrt {\left (2 x + 3\right )^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/(4*x**2+12*x+9)**(1/2),x)

[Out]

Integral((d + e*x)/sqrt((2*x + 3)**2), x)

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