3.18.79 \(\int \frac {7+5 e^2-12 x^2+5 \log (x)}{e} \, dx\)

Optimal. Leaf size=25 \[ \frac {x \left (2-\frac {3}{x}-4 x^2+5 \left (e^2+\log (x)\right )\right )}{e} \]

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Rubi [A]  time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.40, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 2295} \begin {gather*} -\frac {4 x^3}{e}+\frac {\left (7+5 e^2\right ) x}{e}-\frac {5 x}{e}+\frac {5 x \log (x)}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7 + 5*E^2 - 12*x^2 + 5*Log[x])/E,x]

[Out]

(-5*x)/E + ((7 + 5*E^2)*x)/E - (4*x^3)/E + (5*x*Log[x])/E

Rule 12

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

Rule 2295

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (7+5 e^2-12 x^2+5 \log (x)\right ) \, dx}{e}\\ &=\frac {\left (7+5 e^2\right ) x}{e}-\frac {4 x^3}{e}+\frac {5 \int \log (x) \, dx}{e}\\ &=-\frac {5 x}{e}+\frac {\left (7+5 e^2\right ) x}{e}-\frac {4 x^3}{e}+\frac {5 x \log (x)}{e}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 24, normalized size = 0.96 \begin {gather*} \frac {2 x+5 e^2 x-4 x^3+5 x \log (x)}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7 + 5*E^2 - 12*x^2 + 5*Log[x])/E,x]

[Out]

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

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fricas [A]  time = 0.76, size = 23, normalized size = 0.92 \begin {gather*} -{\left (4 \, x^{3} - 5 \, x e^{2} - 5 \, x \log \relax (x) - 2 \, x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)+5*exp(2)-12*x^2+7)/exp(1),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.23, size = 23, normalized size = 0.92 \begin {gather*} -{\left (4 \, x^{3} - 5 \, x e^{2} - 5 \, x \log \relax (x) - 2 \, x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)+5*exp(2)-12*x^2+7)/exp(1),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 25, normalized size = 1.00

method	result	size
default	\({\mathrm e}^{-1} \left (-4 x^{3}+2 x +5 \,{\mathrm e}^{2} x +5 x \ln \relax (x )\right )\)	\(25\)
risch	\(5 x \,{\mathrm e}-4 \,{\mathrm e}^{-1} x^{3}+5 x \,{\mathrm e}^{-1} \ln \relax (x )+2 \,{\mathrm e}^{-1} x\)	\(26\)
norman	\(\left (2+5 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-1} x -4 \,{\mathrm e}^{-1} x^{3}+5 x \,{\mathrm e}^{-1} \ln \relax (x )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*ln(x)+5*exp(2)-12*x^2+7)/exp(1),x,method=_RETURNVERBOSE)

[Out]

1/exp(1)*(-4*x^3+2*x+5*exp(2)*x+5*x*ln(x))

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maxima [A]  time = 0.34, size = 23, normalized size = 0.92 \begin {gather*} -{\left (4 \, x^{3} - 5 \, x e^{2} - 5 \, x \log \relax (x) - 2 \, x\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*log(x)+5*exp(2)-12*x^2+7)/exp(1),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.13, size = 19, normalized size = 0.76 \begin {gather*} x\,{\mathrm {e}}^{-1}\,\left (5\,{\mathrm {e}}^2+5\,\ln \relax (x)-4\,x^2+2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-1)*(5*exp(2) + 5*log(x) - 12*x^2 + 7),x)

[Out]

x*exp(-1)*(5*exp(2) + 5*log(x) - 4*x^2 + 2)

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sympy [A]  time = 0.10, size = 29, normalized size = 1.16 \begin {gather*} - \frac {4 x^{3}}{e} + \frac {5 x \log {\relax (x )}}{e} + \frac {x \left (2 + 5 e^{2}\right )}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*ln(x)+5*exp(2)-12*x**2+7)/exp(1),x)

[Out]

-4*x**3*exp(-1) + 5*x*exp(-1)*log(x) + x*(2 + 5*exp(2))*exp(-1)

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