3.87.38 \(\int \frac {54 e x+(6-36 e x) \log (5)+(-2+6 e x) \log ^2(5)}{3 e \log ^2(5)} \, dx\)

Optimal. Leaf size=20 \[ \left (\frac {1}{3 e}-x+\frac {3 x}{\log (5)}\right )^2 \]

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Rubi [B]  time = 0.02, antiderivative size = 44, normalized size of antiderivative = 2.20, number of steps used = 2, number of rules used = 1, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {12} \begin {gather*} \frac {9 x^2}{\log ^2(5)}+\frac {(1-3 e x)^2}{9 e^2}-\frac {(1-6 e x)^2}{6 e^2 \log (5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(54*E*x + (6 - 36*E*x)*Log[5] + (-2 + 6*E*x)*Log[5]^2)/(3*E*Log[5]^2),x]

[Out]

(1 - 3*E*x)^2/(9*E^2) + (9*x^2)/Log[5]^2 - (1 - 6*E*x)^2/(6*E^2*Log[5])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (54 e x+(6-36 e x) \log (5)+(-2+6 e x) \log ^2(5)\right ) \, dx}{3 e \log ^2(5)}\\ &=\frac {(1-3 e x)^2}{9 e^2}+\frac {9 x^2}{\log ^2(5)}-\frac {(1-6 e x)^2}{6 e^2 \log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.65 \begin {gather*} \frac {2 (-3+\log (5)) \left (\frac {3}{2} e x^2 (-3+\log (5))-x \log (5)\right )}{3 e \log ^2(5)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(54*E*x + (6 - 36*E*x)*Log[5] + (-2 + 6*E*x)*Log[5]^2)/(3*E*Log[5]^2),x]

[Out]

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

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fricas [B]  time = 0.85, size = 47, normalized size = 2.35 \begin {gather*} \frac {{\left (27 \, x^{2} e + {\left (3 \, x^{2} e - 2 \, x\right )} \log \relax (5)^{2} - 6 \, {\left (3 \, x^{2} e - x\right )} \log \relax (5)\right )} e^{\left (-1\right )}}{3 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((6*x*exp(1)-2)*log(5)^2+(-36*x*exp(1)+6)*log(5)+54*x*exp(1))/exp(1)/log(5)^2,x, algorithm="fric
as")

[Out]

1/3*(27*x^2*e + (3*x^2*e - 2*x)*log(5)^2 - 6*(3*x^2*e - x)*log(5))*e^(-1)/log(5)^2

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giac [B]  time = 0.13, size = 47, normalized size = 2.35 \begin {gather*} \frac {{\left (27 \, x^{2} e + {\left (3 \, x^{2} e - 2 \, x\right )} \log \relax (5)^{2} - 6 \, {\left (3 \, x^{2} e - x\right )} \log \relax (5)\right )} e^{\left (-1\right )}}{3 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((6*x*exp(1)-2)*log(5)^2+(-36*x*exp(1)+6)*log(5)+54*x*exp(1))/exp(1)/log(5)^2,x, algorithm="giac
")

[Out]

1/3*(27*x^2*e + (3*x^2*e - 2*x)*log(5)^2 - 6*(3*x^2*e - x)*log(5))*e^(-1)/log(5)^2

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maple [A]  time = 0.04, size = 33, normalized size = 1.65

method	result	size
gosper	\(\frac {\left (\ln \relax (5)-3\right ) x \left (3 x \,{\mathrm e} \ln \relax (5)-9 x \,{\mathrm e}-2 \ln \relax (5)\right ) {\mathrm e}^{-1}}{3 \ln \relax (5)^{2}}\)	\(33\)
norman	\(\frac {\frac {\left (\ln \relax (5)^{2}-6 \ln \relax (5)+9\right ) x^{2}}{\ln \relax (5)}-\frac {2 \,{\mathrm e}^{-1} \left (\ln \relax (5)-3\right ) x}{3}}{\ln \relax (5)}\)	\(36\)
risch	\(x^{2}-\frac {6 x^{2}}{\ln \relax (5)}-\frac {2 \,{\mathrm e}^{-1} x}{3}+\frac {9 x^{2}}{\ln \relax (5)^{2}}+\frac {2 \,{\mathrm e}^{-1} x}{\ln \relax (5)}\)	\(37\)
default	\(\frac {{\mathrm e}^{-1} \left (\ln \relax (5)^{2} \left (3 x^{2} {\mathrm e}-2 x \right )+\ln \relax (5) \left (-18 x^{2} {\mathrm e}+6 x \right )+27 x^{2} {\mathrm e}\right )}{3 \ln \relax (5)^{2}}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*((6*x*exp(1)-2)*ln(5)^2+(-36*x*exp(1)+6)*ln(5)+54*x*exp(1))/exp(1)/ln(5)^2,x,method=_RETURNVERBOSE)

[Out]

1/3*(ln(5)-3)*x*(3*x*exp(1)*ln(5)-9*x*exp(1)-2*ln(5))/exp(1)/ln(5)^2

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maxima [B]  time = 0.37, size = 47, normalized size = 2.35 \begin {gather*} \frac {{\left (27 \, x^{2} e + {\left (3 \, x^{2} e - 2 \, x\right )} \log \relax (5)^{2} - 6 \, {\left (3 \, x^{2} e - x\right )} \log \relax (5)\right )} e^{\left (-1\right )}}{3 \, \log \relax (5)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((6*x*exp(1)-2)*log(5)^2+(-36*x*exp(1)+6)*log(5)+54*x*exp(1))/exp(1)/log(5)^2,x, algorithm="maxi
ma")

[Out]

1/3*(27*x^2*e + (3*x^2*e - 2*x)*log(5)^2 - 6*(3*x^2*e - x)*log(5))*e^(-1)/log(5)^2

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mupad [B]  time = 0.09, size = 30, normalized size = 1.50 \begin {gather*} -\frac {x\,{\mathrm {e}}^{-1}\,\left (\ln \relax (5)-3\right )\,\left (2\,\ln \relax (5)+9\,x\,\mathrm {e}-3\,x\,\mathrm {e}\,\ln \relax (5)\right )}{3\,{\ln \relax (5)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-1)*(18*x*exp(1) - (log(5)*(36*x*exp(1) - 6))/3 + (log(5)^2*(6*x*exp(1) - 2))/3))/log(5)^2,x)

[Out]

-(x*exp(-1)*(log(5) - 3)*(2*log(5) + 9*x*exp(1) - 3*x*exp(1)*log(5)))/(3*log(5)^2)

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sympy [B]  time = 0.07, size = 36, normalized size = 1.80 \begin {gather*} \frac {x^{2} \left (- 6 \log {\relax (5 )} + \log {\relax (5 )}^{2} + 9\right )}{\log {\relax (5 )}^{2}} + \frac {x \left (6 - 2 \log {\relax (5 )}\right )}{3 e \log {\relax (5 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*((6*x*exp(1)-2)*ln(5)**2+(-36*x*exp(1)+6)*ln(5)+54*x*exp(1))/exp(1)/ln(5)**2,x)

[Out]

x**2*(-6*log(5) + log(5)**2 + 9)/log(5)**2 + x*(6 - 2*log(5))*exp(-1)/(3*log(5))

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