3.15.39 \(\int \frac {2+300 x}{-5+e^{625}+x+75 x^2} \, dx\)

Optimal. Leaf size=20 \[ \log \left (16 \left (-2+e^{625}+x-3 \left (1-25 x^2\right )\right )^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {628} \begin {gather*} 2 \log \left (-75 x^2-x-e^{625}+5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 300*x)/(-5 + E^625 + x + 75*x^2),x]

[Out]

2*Log[5 - E^625 - x - 75*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} &=2 \log \left (5-e^{625}-x-75 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 0.70 \begin {gather*} 2 \log \left (-5+e^{625}+x+75 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 300*x)/(-5 + E^625 + x + 75*x^2),x]

[Out]

2*Log[-5 + E^625 + x + 75*x^2]

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fricas [A]  time = 0.73, size = 13, normalized size = 0.65 \begin {gather*} 2 \, \log \left (75 \, x^{2} + x + e^{625} - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((300*x+2)/(exp(625)+75*x^2+x-5),x, algorithm="fricas")

[Out]

2*log(75*x^2 + x + e^625 - 5)

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giac [A]  time = 0.24, size = 13, normalized size = 0.65 \begin {gather*} 2 \, \log \left (75 \, x^{2} + x + e^{625} - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((300*x+2)/(exp(625)+75*x^2+x-5),x, algorithm="giac")

[Out]

2*log(75*x^2 + x + e^625 - 5)

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maple [A]  time = 0.64, size = 14, normalized size = 0.70

method	result	size
default	\(2 \ln \left ({\mathrm e}^{625}+75 x^{2}+x -5\right )\)	\(14\)
norman	\(2 \ln \left ({\mathrm e}^{625}+75 x^{2}+x -5\right )\)	\(14\)
risch	\(2 \ln \left ({\mathrm e}^{625}+75 x^{2}+x -5\right )\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((300*x+2)/(exp(625)+75*x^2+x-5),x,method=_RETURNVERBOSE)

[Out]

2*ln(exp(625)+75*x^2+x-5)

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maxima [A]  time = 0.51, size = 13, normalized size = 0.65 \begin {gather*} 2 \, \log \left (75 \, x^{2} + x + e^{625} - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((300*x+2)/(exp(625)+75*x^2+x-5),x, algorithm="maxima")

[Out]

2*log(75*x^2 + x + e^625 - 5)

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mupad [B]  time = 0.98, size = 13, normalized size = 0.65 \begin {gather*} 2\,\ln \left (75\,x^2+x+{\mathrm {e}}^{625}-5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((300*x + 2)/(x + exp(625) + 75*x^2 - 5),x)

[Out]

2*log(x + exp(625) + 75*x^2 - 5)

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sympy [A]  time = 0.12, size = 14, normalized size = 0.70 \begin {gather*} 2 \log {\left (75 x^{2} + x - 5 + e^{625} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((300*x+2)/(exp(625)+75*x**2+x-5),x)

[Out]

2*log(75*x**2 + x - 5 + exp(625))

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