3.10.86 \(\int (450 e^5 x+900 e^5 x \log (\frac {x}{2})) \, dx\)

Optimal. Leaf size=14 \[ 450 e^5 x^2 \log \left (\frac {x}{2}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2304} \begin {gather*} 450 e^5 x^2 \log \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[450*E^5*x + 900*E^5*x*Log[x/2],x]

[Out]

450*E^5*x^2*Log[x/2]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=225 e^5 x^2+\left (900 e^5\right ) \int x \log \left (\frac {x}{2}\right ) \, dx\\ &=450 e^5 x^2 \log \left (\frac {x}{2}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} 450 e^5 x^2 \log \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[450*E^5*x + 900*E^5*x*Log[x/2],x]

[Out]

450*E^5*x^2*Log[x/2]

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fricas [A]  time = 0.84, size = 11, normalized size = 0.79 \begin {gather*} 450 \, x^{2} e^{5} \log \left (\frac {1}{2} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(900*x*exp(5)*log(1/2*x)+450*x*exp(5),x, algorithm="fricas")

[Out]

450*x^2*e^5*log(1/2*x)

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giac [B]  time = 0.39, size = 27, normalized size = 1.93 \begin {gather*} 225 \, x^{2} e^{5} + 225 \, {\left (2 \, x^{2} \log \left (\frac {1}{2} \, x\right ) - x^{2}\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(900*x*exp(5)*log(1/2*x)+450*x*exp(5),x, algorithm="giac")

[Out]

225*x^2*e^5 + 225*(2*x^2*log(1/2*x) - x^2)*e^5

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maple [A]  time = 0.02, size = 12, normalized size = 0.86

method	result	size
derivativedivides	\(450 x^{2} {\mathrm e}^{5} \ln \left (\frac {x}{2}\right )\)	\(12\)
default	\(450 x^{2} {\mathrm e}^{5} \ln \left (\frac {x}{2}\right )\)	\(12\)
norman	\(450 x^{2} {\mathrm e}^{5} \ln \left (\frac {x}{2}\right )\)	\(12\)
risch	\(450 x^{2} {\mathrm e}^{5} \ln \left (\frac {x}{2}\right )\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(900*x*exp(5)*ln(1/2*x)+450*x*exp(5),x,method=_RETURNVERBOSE)

[Out]

450*x^2*exp(5)*ln(1/2*x)

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maxima [B]  time = 0.35, size = 27, normalized size = 1.93 \begin {gather*} 225 \, x^{2} e^{5} + 225 \, {\left (2 \, x^{2} \log \left (\frac {1}{2} \, x\right ) - x^{2}\right )} e^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(900*x*exp(5)*log(1/2*x)+450*x*exp(5),x, algorithm="maxima")

[Out]

225*x^2*e^5 + 225*(2*x^2*log(1/2*x) - x^2)*e^5

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mupad [B]  time = 0.75, size = 11, normalized size = 0.79 \begin {gather*} 450\,x^2\,\ln \left (\frac {x}{2}\right )\,{\mathrm {e}}^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(450*x*exp(5) + 900*x*log(x/2)*exp(5),x)

[Out]

450*x^2*log(x/2)*exp(5)

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sympy [A]  time = 0.10, size = 12, normalized size = 0.86 \begin {gather*} 450 x^{2} e^{5} \log {\left (\frac {x}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(900*x*exp(5)*ln(1/2*x)+450*x*exp(5),x)

[Out]

450*x**2*exp(5)*log(x/2)

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