3.12.4 \(\int \frac {12600 e^2+504 e^5+168 x-504 e^2 \log (5)}{e^4} \, dx\)

Optimal. Leaf size=21 \[ 84 \left (\frac {x}{e^2}-3 \left (-25-e^3+\log (5)\right )\right )^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.10, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {9} \begin {gather*} \frac {84 \left (x+3 e^2 \left (25+e^3-\log (5)\right )\right )^2}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(12600*E^2 + 504*E^5 + 168*x - 504*E^2*Log[5])/E^4,x]

[Out]

(84*(x + 3*E^2*(25 + E^3 - Log[5]))^2)/E^4

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {84 \left (x+3 e^2 \left (25+e^3-\log (5)\right )\right )^2}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.38 \begin {gather*} \frac {168 \left (3 e^5 x+\frac {x^2}{2}-3 e^2 x (-25+\log (5))\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12600*E^2 + 504*E^5 + 168*x - 504*E^2*Log[5])/E^4,x]

[Out]

(168*(3*E^5*x + x^2/2 - 3*E^2*x*(-25 + Log[5])))/E^4

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fricas [A]  time = 0.93, size = 27, normalized size = 1.29 \begin {gather*} -84 \, {\left (6 \, x e^{2} \log \relax (5) - x^{2} - 6 \, x e^{5} - 150 \, x e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-504*exp(2)*log(5)+504*exp(2)*exp(3)+12600*exp(2)+168*x)/exp(2)^2,x, algorithm="fricas")

[Out]

-84*(6*x*e^2*log(5) - x^2 - 6*x*e^5 - 150*x*e^2)*e^(-4)

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giac [A]  time = 0.33, size = 27, normalized size = 1.29 \begin {gather*} -84 \, {\left (6 \, x e^{2} \log \relax (5) - x^{2} - 6 \, x e^{5} - 150 \, x e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-504*exp(2)*log(5)+504*exp(2)*exp(3)+12600*exp(2)+168*x)/exp(2)^2,x, algorithm="giac")

[Out]

-84*(6*x*e^2*log(5) - x^2 - 6*x*e^5 - 150*x*e^2)*e^(-4)

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maple [A]  time = 0.02, size = 26, normalized size = 1.24

method	result	size
risch	\(504 x \,{\mathrm e}-504 \,{\mathrm e}^{-2} x \ln \relax (5)+12600 x \,{\mathrm e}^{-2}+84 x^{2} {\mathrm e}^{-4}\)	\(26\)
gosper	\(-84 x \left (6 \,{\mathrm e}^{2} \ln \relax (5)-6 \,{\mathrm e}^{2} {\mathrm e}^{3}-150 \,{\mathrm e}^{2}-x \right ) {\mathrm e}^{-4}\)	\(28\)
norman	\(\left (\left (-504 \ln \relax (5)+504 \,{\mathrm e}^{3}+12600\right ) x +84 x^{2} {\mathrm e}^{-2}\right ) {\mathrm e}^{-2}\)	\(28\)
default	\({\mathrm e}^{-4} \left (-504 x \,{\mathrm e}^{2} \ln \relax (5)+504 x \,{\mathrm e}^{2} {\mathrm e}^{3}+12600 \,{\mathrm e}^{2} x +84 x^{2}\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-504*exp(2)*ln(5)+504*exp(2)*exp(3)+12600*exp(2)+168*x)/exp(2)^2,x,method=_RETURNVERBOSE)

[Out]

504*x*exp(1)-504*exp(-2)*x*ln(5)+12600*x*exp(-2)+84*x^2*exp(-4)

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maxima [A]  time = 0.62, size = 27, normalized size = 1.29 \begin {gather*} -84 \, {\left (6 \, x e^{2} \log \relax (5) - x^{2} - 6 \, x e^{5} - 150 \, x e^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-504*exp(2)*log(5)+504*exp(2)*exp(3)+12600*exp(2)+168*x)/exp(2)^2,x, algorithm="maxima")

[Out]

-84*(6*x*e^2*log(5) - x^2 - 6*x*e^5 - 150*x*e^2)*e^(-4)

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mupad [B]  time = 0.26, size = 24, normalized size = 1.14 \begin {gather*} \frac {{\mathrm {e}}^{-4}\,{\left (168\,x+12600\,{\mathrm {e}}^2+504\,{\mathrm {e}}^5-504\,{\mathrm {e}}^2\,\ln \relax (5)\right )}^2}{336} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(168*x + 12600*exp(2) + 504*exp(5) - 504*exp(2)*log(5)),x)

[Out]

(exp(-4)*(168*x + 12600*exp(2) + 504*exp(5) - 504*exp(2)*log(5))^2)/336

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sympy [A]  time = 0.06, size = 24, normalized size = 1.14 \begin {gather*} \frac {84 x^{2}}{e^{4}} + \frac {x \left (- 504 \log {\relax (5 )} + 504 e^{3} + 12600\right )}{e^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-504*exp(2)*ln(5)+504*exp(2)*exp(3)+12600*exp(2)+168*x)/exp(2)**2,x)

[Out]

84*x**2*exp(-4) + x*(-504*log(5) + 504*exp(3) + 12600)*exp(-2)

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