3.81.72 \(\int \frac {2+20 x}{45+2 x+10 x^2} \, dx\)

Optimal. Leaf size=12 \[ \log \left (9+x \left (\frac {2}{5}+2 x\right )\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 11, normalized size of antiderivative = 0.92, 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 (10 x^2+2 x+45\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 20*x)/(45 + 2*x + 10*x^2),x]

[Out]

Log[45 + 2*x + 10*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 {gather*} \begin {aligned} \text {integral} &=\log \left (45+2 x+10 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 0.92 \begin {gather*} \log \left (45+2 x+10 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 20*x)/(45 + 2*x + 10*x^2),x]

[Out]

Log[45 + 2*x + 10*x^2]

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fricas [A]  time = 1.01, size = 11, normalized size = 0.92 \begin {gather*} \log \left (10 \, x^{2} + 2 \, x + 45\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x+2)/(10*x^2+2*x+45),x, algorithm="fricas")

[Out]

log(10*x^2 + 2*x + 45)

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giac [A]  time = 0.20, size = 11, normalized size = 0.92 \begin {gather*} \log \left (10 \, x^{2} + 2 \, x + 45\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x+2)/(10*x^2+2*x+45),x, algorithm="giac")

[Out]

log(10*x^2 + 2*x + 45)

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maple [A]  time = 0.66, size = 12, normalized size = 1.00

method	result	size
derivativedivides	\(\ln \left (10 x^{2}+2 x +45\right )\)	\(12\)
default	\(\ln \left (10 x^{2}+2 x +45\right )\)	\(12\)
norman	\(\ln \left (10 x^{2}+2 x +45\right )\)	\(12\)
risch	\(\ln \left (10 x^{2}+2 x +45\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x+2)/(10*x^2+2*x+45),x,method=_RETURNVERBOSE)

[Out]

ln(10*x^2+2*x+45)

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maxima [A]  time = 0.35, size = 11, normalized size = 0.92 \begin {gather*} \log \left (10 \, x^{2} + 2 \, x + 45\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x+2)/(10*x^2+2*x+45),x, algorithm="maxima")

[Out]

log(10*x^2 + 2*x + 45)

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mupad [B]  time = 0.05, size = 9, normalized size = 0.75 \begin {gather*} \ln \left (x^2+\frac {x}{5}+\frac {9}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x + 2)/(2*x + 10*x^2 + 45),x)

[Out]

log(x/5 + x^2 + 9/2)

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sympy [A]  time = 0.08, size = 10, normalized size = 0.83 \begin {gather*} \log {\left (10 x^{2} + 2 x + 45 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x+2)/(10*x**2+2*x+45),x)

[Out]

log(10*x**2 + 2*x + 45)

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