3.48.64 \(\int (2+e^7+x+(e^7+2 x) \log (4 x)) \, dx\)

Optimal. Leaf size=16 \[ 9+2 x+x \left (e^7+x\right ) \log (4 x) \]

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.75, number of steps used = 3, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2313} \begin {gather*} \left (x^2+e^7 x\right ) \log (4 x)+\left (2+e^7\right ) x-e^7 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2 + E^7 + x + (E^7 + 2*x)*Log[4*x],x]

[Out]

-(E^7*x) + (2 + E^7)*x + (E^7*x + x^2)*Log[4*x]

Rule 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a,
b, c, d, e, n, r}, x] && IGtQ[q, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\left (2+e^7\right ) x+\frac {x^2}{2}+\int \left (e^7+2 x\right ) \log (4 x) \, dx\\ &=\left (2+e^7\right ) x+\frac {x^2}{2}+\left (e^7 x+x^2\right ) \log (4 x)-\int \left (e^7+x\right ) \, dx\\ &=-e^7 x+\left (2+e^7\right ) x+\left (e^7 x+x^2\right ) \log (4 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.31 \begin {gather*} 2 x+e^7 x \log (4 x)+x^2 \log (4 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2 + E^7 + x + (E^7 + 2*x)*Log[4*x],x]

[Out]

2*x + E^7*x*Log[4*x] + x^2*Log[4*x]

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fricas [A]  time = 0.89, size = 17, normalized size = 1.06 \begin {gather*} {\left (x^{2} + x e^{7}\right )} \log \left (4 \, x\right ) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(7)+2*x)*log(4*x)+exp(7)+2+x,x, algorithm="fricas")

[Out]

(x^2 + x*e^7)*log(4*x) + 2*x

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giac [A]  time = 0.15, size = 20, normalized size = 1.25 \begin {gather*} x^{2} \log \left (4 \, x\right ) + x e^{7} \log \left (4 \, x\right ) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(7)+2*x)*log(4*x)+exp(7)+2+x,x, algorithm="giac")

[Out]

x^2*log(4*x) + x*e^7*log(4*x) + 2*x

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maple [A]  time = 0.05, size = 18, normalized size = 1.12

method	result	size
risch	\(\left (x \,{\mathrm e}^{7}+x^{2}\right ) \ln \left (4 x \right )+2 x\)	\(18\)
derivativedivides	\(2 x +{\mathrm e}^{7} \ln \left (4 x \right ) x +x^{2} \ln \left (4 x \right )\)	\(21\)
default	\(2 x +{\mathrm e}^{7} \ln \left (4 x \right ) x +x^{2} \ln \left (4 x \right )\)	\(21\)
norman	\(2 x +{\mathrm e}^{7} \ln \left (4 x \right ) x +x^{2} \ln \left (4 x \right )\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(7)+2*x)*ln(4*x)+exp(7)+2+x,x,method=_RETURNVERBOSE)

[Out]

(x*exp(7)+x^2)*ln(4*x)+2*x

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maxima [A]  time = 0.37, size = 17, normalized size = 1.06 \begin {gather*} {\left (x^{2} + x e^{7}\right )} \log \left (4 \, x\right ) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(7)+2*x)*log(4*x)+exp(7)+2+x,x, algorithm="maxima")

[Out]

(x^2 + x*e^7)*log(4*x) + 2*x

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mupad [B]  time = 3.29, size = 20, normalized size = 1.25 \begin {gather*} x\,\left (\ln \left (4\,x\right )\,{\mathrm {e}}^7+2\right )+x^2\,\ln \left (4\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x + exp(7) + log(4*x)*(2*x + exp(7)) + 2,x)

[Out]

x*(log(4*x)*exp(7) + 2) + x^2*log(4*x)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.94 \begin {gather*} 2 x + \left (x^{2} + x e^{7}\right ) \log {\left (4 x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(7)+2*x)*ln(4*x)+exp(7)+2+x,x)

[Out]

2*x + (x**2 + x*exp(7))*log(4*x)

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