3.35.89 \(\int \frac {e^{\frac {2 (9 x^2+200 e^x \log ^2(x))}{9 x^2}} (32 e^x \log (x)+e^x (-32+16 x) \log ^2(x))}{27 x^3} \, dx\)

Optimal. Leaf size=22 \[ \frac {1}{75} e^{2+\frac {400 e^x \log ^2(x)}{9 x^2}} \]

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Rubi [F]  time = 2.74, 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 {e^{\frac {2 \left (9 x^2+200 e^x \log ^2(x)\right )}{9 x^2}} \left (32 e^x \log (x)+e^x (-32+16 x) \log ^2(x)\right )}{27 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*(9*x^2 + 200*E^x*Log[x]^2))/(9*x^2))*(32*E^x*Log[x] + E^x*(-32 + 16*x)*Log[x]^2))/(27*x^3),x]

[Out]

(32*Defer[Int][(E^(2 + x + (400*E^x*Log[x]^2)/(9*x^2))*Log[x])/x^3, x])/27 - (32*Defer[Int][(E^(2 + x + (400*E
^x*Log[x]^2)/(9*x^2))*Log[x]^2)/x^3, x])/27 + (16*Defer[Int][(E^(2 + x + (400*E^x*Log[x]^2)/(9*x^2))*Log[x]^2)
/x^2, x])/27

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{27} \int \frac {e^{\frac {2 \left (9 x^2+200 e^x \log ^2(x)\right )}{9 x^2}} \left (32 e^x \log (x)+e^x (-32+16 x) \log ^2(x)\right )}{x^3} \, dx\\ &=\frac {1}{27} \int \frac {16 e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log (x) (2+(-2+x) \log (x))}{x^3} \, dx\\ &=\frac {16}{27} \int \frac {e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log (x) (2+(-2+x) \log (x))}{x^3} \, dx\\ &=\frac {16}{27} \int \left (\frac {2 e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log (x)}{x^3}+\frac {e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} (-2+x) \log ^2(x)}{x^3}\right ) \, dx\\ &=\frac {16}{27} \int \frac {e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} (-2+x) \log ^2(x)}{x^3} \, dx+\frac {32}{27} \int \frac {e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log (x)}{x^3} \, dx\\ &=\frac {16}{27} \int \left (-\frac {2 e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log ^2(x)}{x^3}+\frac {e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log ^2(x)}{x^2}\right ) \, dx+\frac {32}{27} \int \frac {e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log (x)}{x^3} \, dx\\ &=\frac {16}{27} \int \frac {e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log ^2(x)}{x^2} \, dx+\frac {32}{27} \int \frac {e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log (x)}{x^3} \, dx-\frac {32}{27} \int \frac {e^{2+x+\frac {400 e^x \log ^2(x)}{9 x^2}} \log ^2(x)}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 22, normalized size = 1.00 \begin {gather*} \frac {1}{75} e^{2+\frac {400 e^x \log ^2(x)}{9 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(9*x^2 + 200*E^x*Log[x]^2))/(9*x^2))*(32*E^x*Log[x] + E^x*(-32 + 16*x)*Log[x]^2))/(27*x^3),x]

[Out]

E^(2 + (400*E^x*Log[x]^2)/(9*x^2))/75

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fricas [A]  time = 1.13, size = 22, normalized size = 1.00 \begin {gather*} \frac {1}{75} \, e^{\left (\frac {2 \, {\left (200 \, e^{x} \log \relax (x)^{2} + 9 \, x^{2}\right )}}{9 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*((16*x-32)*exp(x)*log(x)^2+32*exp(x)*log(x))*exp(1/9*(200*exp(x)*log(x)^2+9*x^2)/x^2)^2/x^3,x,
algorithm="fricas")

[Out]

1/75*e^(2/9*(200*e^x*log(x)^2 + 9*x^2)/x^2)

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giac [A]  time = 0.15, size = 16, normalized size = 0.73 \begin {gather*} \frac {1}{75} \, e^{\left (\frac {400 \, e^{x} \log \relax (x)^{2}}{9 \, x^{2}} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*((16*x-32)*exp(x)*log(x)^2+32*exp(x)*log(x))*exp(1/9*(200*exp(x)*log(x)^2+9*x^2)/x^2)^2/x^3,x,
algorithm="giac")

[Out]

1/75*e^(400/9*e^x*log(x)^2/x^2 + 2)

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maple [A]  time = 0.03, size = 23, normalized size = 1.05

method	result	size
risch	\(\frac {{\mathrm e}^{\frac {\frac {400 \,{\mathrm e}^{x} \ln \relax (x )^{2}}{9}+2 x^{2}}{x^{2}}}}{75}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/27*((16*x-32)*exp(x)*ln(x)^2+32*exp(x)*ln(x))*exp(1/9*(200*exp(x)*ln(x)^2+9*x^2)/x^2)^2/x^3,x,method=_RE
TURNVERBOSE)

[Out]

1/75*exp(2/9*(200*exp(x)*ln(x)^2+9*x^2)/x^2)

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maxima [A]  time = 0.82, size = 16, normalized size = 0.73 \begin {gather*} \frac {1}{75} \, e^{\left (\frac {400 \, e^{x} \log \relax (x)^{2}}{9 \, x^{2}} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*((16*x-32)*exp(x)*log(x)^2+32*exp(x)*log(x))*exp(1/9*(200*exp(x)*log(x)^2+9*x^2)/x^2)^2/x^3,x,
algorithm="maxima")

[Out]

1/75*e^(400/9*e^x*log(x)^2/x^2 + 2)

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mupad [B]  time = 2.22, size = 16, normalized size = 0.73 \begin {gather*} \frac {{\mathrm {e}}^2\,{\mathrm {e}}^{\frac {400\,{\mathrm {e}}^x\,{\ln \relax (x)}^2}{9\,x^2}}}{75} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*((200*exp(x)*log(x)^2)/9 + x^2))/x^2)*(32*exp(x)*log(x) + exp(x)*log(x)^2*(16*x - 32)))/(27*x^3),x
)

[Out]

(exp(2)*exp((400*exp(x)*log(x)^2)/(9*x^2)))/75

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sympy [A]  time = 0.49, size = 22, normalized size = 1.00 \begin {gather*} \frac {e^{\frac {2 \left (x^{2} + \frac {200 e^{x} \log {\relax (x )}^{2}}{9}\right )}{x^{2}}}}{75} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/27*((16*x-32)*exp(x)*ln(x)**2+32*exp(x)*ln(x))*exp(1/9*(200*exp(x)*ln(x)**2+9*x**2)/x**2)**2/x**3,
x)

[Out]

exp(2*(x**2 + 200*exp(x)*log(x)**2/9)/x**2)/75

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