3.38.51 \(\int (2+\log (\frac {(20 x^2+3 e x^2) \log (\log (5))}{e})) \, dx\)

Optimal. Leaf size=17 \[ x \log \left (\left (3+\frac {20}{e}\right ) x^2 \log (\log (5))\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2453, 2295} \begin {gather*} x \log \left (\frac {(20+3 e) x^2 \log (\log (5))}{e}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + Log[((20*x^2 + 3*E*x^2)*Log[Log[5]])/E],x]

[Out]

x*Log[((20 + 3*E)*x^2*Log[Log[5]])/E]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2453

Int[((a_.) + Log[(c_.)*(v_)^(p_.)]*(b_.))^(q_.), x_Symbol] :> Int[(a + b*Log[c*ExpandToSum[v, x]^p])^q, x] /;
FreeQ[{a, b, c, p, q}, x] && BinomialQ[v, x] &&  !BinomialMatchQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=2 x+\int \log \left (\frac {\left (20 x^2+3 e x^2\right ) \log (\log (5))}{e}\right ) \, dx\\ &=2 x+\int \log \left (\frac {(20+3 e) x^2 \log (\log (5))}{e}\right ) \, dx\\ &=x \log \left (\frac {(20+3 e) x^2 \log (\log (5))}{e}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.41 \begin {gather*} -x+x \log (20+3 e)+x \log \left (x^2\right )+x \log (\log (\log (5))) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 + Log[((20*x^2 + 3*E*x^2)*Log[Log[5]])/E],x]

[Out]

-x + x*Log[20 + 3*E] + x*Log[x^2] + x*Log[Log[Log[5]]]

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fricas [A]  time = 0.68, size = 22, normalized size = 1.29 \begin {gather*} x \log \left ({\left (3 \, x^{2} e + 20 \, x^{2}\right )} e^{\left (-1\right )} \log \left (\log \relax (5)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((3*x^2*exp(1)+20*x^2)*log(log(5))/exp(1))+2,x, algorithm="fricas")

[Out]

x*log((3*x^2*e + 20*x^2)*e^(-1)*log(log(5)))

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giac [A]  time = 0.11, size = 22, normalized size = 1.29 \begin {gather*} x \log \left ({\left (3 \, x^{2} e + 20 \, x^{2}\right )} e^{\left (-1\right )} \log \left (\log \relax (5)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((3*x^2*exp(1)+20*x^2)*log(log(5))/exp(1))+2,x, algorithm="giac")

[Out]

x*log((3*x^2*e + 20*x^2)*e^(-1)*log(log(5)))

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maple [A]  time = 0.04, size = 19, normalized size = 1.12

method	result	size
risch	\(x \ln \left (x^{2} \left (3 \,{\mathrm e}+20\right ) \ln \left (\ln \relax (5)\right ) {\mathrm e}^{-1}\right )\)	\(19\)
norman	\(x \ln \left (\left (3 x^{2} {\mathrm e}+20 x^{2}\right ) \ln \left (\ln \relax (5)\right ) {\mathrm e}^{-1}\right )\)	\(25\)
default	\(-x +x \ln \left (x^{2}\right )+\ln \left (3 \,{\mathrm e}+20\right ) x +\ln \left (\ln \left (\ln \relax (5)\right )\right ) x\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((3*x^2*exp(1)+20*x^2)*ln(ln(5))/exp(1))+2,x,method=_RETURNVERBOSE)

[Out]

x*ln(x^2*(3*exp(1)+20)*ln(ln(5))*exp(-1))

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maxima [A]  time = 0.35, size = 22, normalized size = 1.29 \begin {gather*} x \log \left ({\left (3 \, x^{2} e + 20 \, x^{2}\right )} e^{\left (-1\right )} \log \left (\log \relax (5)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log((3*x^2*exp(1)+20*x^2)*log(log(5))/exp(1))+2,x, algorithm="maxima")

[Out]

x*log((3*x^2*e + 20*x^2)*e^(-1)*log(log(5)))

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mupad [B]  time = 2.24, size = 22, normalized size = 1.29 \begin {gather*} x\,\left (\ln \left (\ln \left (\ln \relax (5)\right )\right )+\ln \left (3\,x^2\,\mathrm {e}+20\,x^2\right )-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(exp(-1)*log(log(5))*(3*x^2*exp(1) + 20*x^2)) + 2,x)

[Out]

x*(log(log(log(5))) + log(3*x^2*exp(1) + 20*x^2) - 1)

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sympy [A]  time = 0.10, size = 24, normalized size = 1.41 \begin {gather*} x \log {\left (\frac {\left (3 e x^{2} + 20 x^{2}\right ) \log {\left (\log {\relax (5 )} \right )}}{e} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln((3*x**2*exp(1)+20*x**2)*ln(ln(5))/exp(1))+2,x)

[Out]

x*log((3*E*x**2 + 20*x**2)*exp(-1)*log(log(5)))

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