3.50.70 \(\int \frac {1}{80} 3^{5+x^2} x^{-10-x^2} (81+9 x^2+18 x^2 \log (\frac {x}{3})) \, dx\)

Optimal. Leaf size=20 \[ -\frac {1}{80} 3^{7+x^2} x^{-9-x^2} \]

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Rubi [F]  time = 0.44, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{80} 3^{5+x^2} x^{-10-x^2} \left (81+9 x^2+18 x^2 \log \left (\frac {x}{3}\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3^(5 + x^2)*x^(-10 - x^2)*(81 + 9*x^2 + 18*x^2*Log[x/3]))/80,x]

[Out]

Defer[Int][3^(9 + x^2)*x^(-10 - x^2), x]/80 + Defer[Int][3^(7 + x^2)*x^(-8 - x^2), x]/80 + (Log[x/3]*Defer[Int
][3^(7 + x^2)*x^(-8 - x^2), x])/40 - Defer[Int][Defer[Int][3^(7 + x^2)*x^(-8 - x^2), x]/x, x]/40

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{80} \int 3^{5+x^2} x^{-10-x^2} \left (81+9 x^2+18 x^2 \log \left (\frac {x}{3}\right )\right ) \, dx\\ &=\frac {1}{80} \int 3^{7+x^2} x^{-10-x^2} \left (9+x^2+2 x^2 \log \left (\frac {x}{3}\right )\right ) \, dx\\ &=\frac {1}{80} \int \left (3^{9+x^2} x^{-10-x^2}+3^{7+x^2} x^{-8-x^2}+2\ 3^{7+x^2} x^{-8-x^2} \log \left (\frac {x}{3}\right )\right ) \, dx\\ &=\frac {1}{80} \int 3^{9+x^2} x^{-10-x^2} \, dx+\frac {1}{80} \int 3^{7+x^2} x^{-8-x^2} \, dx+\frac {1}{40} \int 3^{7+x^2} x^{-8-x^2} \log \left (\frac {x}{3}\right ) \, dx\\ &=\frac {1}{80} \int 3^{9+x^2} x^{-10-x^2} \, dx+\frac {1}{80} \int 3^{7+x^2} x^{-8-x^2} \, dx-\frac {1}{40} \int \frac {\int 3^{7+x^2} x^{-8-x^2} \, dx}{x} \, dx+\frac {1}{40} \log \left (\frac {x}{3}\right ) \int 3^{7+x^2} x^{-8-x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 20, normalized size = 1.00 \begin {gather*} -\frac {1}{80} 3^{7+x^2} x^{-9-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3^(5 + x^2)*x^(-10 - x^2)*(81 + 9*x^2 + 18*x^2*Log[x/3]))/80,x]

[Out]

-1/80*(3^(7 + x^2)*x^(-9 - x^2))

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fricas [A]  time = 0.92, size = 16, normalized size = 0.80 \begin {gather*} -\frac {9}{80 \, \left (\frac {1}{3} \, x\right )^{x^{2} + 5} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*(18*x^2*log(1/3*x)+9*x^2+81)/x^5/exp((x^2+5)*log(1/3*x)),x, algorithm="fricas")

[Out]

-9/80/((1/3*x)^(x^2 + 5)*x^4)

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giac [A]  time = 0.18, size = 14, normalized size = 0.70 \begin {gather*} -\frac {2187}{80 \, \left (\frac {1}{3} \, x\right )^{\left (x^{2}\right )} x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*(18*x^2*log(1/3*x)+9*x^2+81)/x^5/exp((x^2+5)*log(1/3*x)),x, algorithm="giac")

[Out]

-2187/80/((1/3*x)^(x^2)*x^9)

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maple [A]  time = 0.04, size = 17, normalized size = 0.85

method	result	size
risch	\(-\frac {9 \left (\frac {x}{3}\right )^{-x^{2}-5}}{80 x^{4}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/80*(18*x^2*ln(1/3*x)+9*x^2+81)/x^5/exp((x^2+5)*ln(1/3*x)),x,method=_RETURNVERBOSE)

[Out]

-9/80/((1/3*x)^(x^2+5))/x^4

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maxima [A]  time = 0.51, size = 20, normalized size = 1.00 \begin {gather*} -\frac {2187 \, e^{\left (x^{2} \log \relax (3) - x^{2} \log \relax (x)\right )}}{80 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*(18*x^2*log(1/3*x)+9*x^2+81)/x^5/exp((x^2+5)*log(1/3*x)),x, algorithm="maxima")

[Out]

-2187/80*e^(x^2*log(3) - x^2*log(x))/x^9

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mupad [B]  time = 3.29, size = 16, normalized size = 0.80 \begin {gather*} -\frac {2187\,3^{x^2}}{80\,x^{x^2+9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-log(x/3)*(x^2 + 5))*((9*x^2*log(x/3))/40 + (9*x^2)/80 + 81/80))/x^5,x)

[Out]

-(2187*3^(x^2))/(80*x^(x^2 + 9))

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sympy [A]  time = 0.27, size = 19, normalized size = 0.95 \begin {gather*} - \frac {9 e^{- \left (x^{2} + 5\right ) \log {\left (\frac {x}{3} \right )}}}{80 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/80*(18*x**2*ln(1/3*x)+9*x**2+81)/x**5/exp((x**2+5)*ln(1/3*x)),x)

[Out]

-9*exp(-(x**2 + 5)*log(x/3))/(80*x**4)

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