3.41.46 \(\int \frac {-8+22 x+13 x^2-13 x \log (\frac {3 e^x x^2}{2})+2 x \log ^2(\frac {3 e^x x^2}{2})}{x \log ^2(\frac {3 e^x x^2}{2})} \, dx\)

Optimal. Leaf size=23 \[ 2 x+\frac {4-13 x}{\log \left (\frac {3 e^x x^2}{2}\right )} \]

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Rubi [F]  time = 0.38, 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 {-8+22 x+13 x^2-13 x \log \left (\frac {3 e^x x^2}{2}\right )+2 x \log ^2\left (\frac {3 e^x x^2}{2}\right )}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8 + 22*x + 13*x^2 - 13*x*Log[(3*E^x*x^2)/2] + 2*x*Log[(3*E^x*x^2)/2]^2)/(x*Log[(3*E^x*x^2)/2]^2),x]

[Out]

2*x + 22*Defer[Int][Log[(3*E^x*x^2)/2]^(-2), x] - 8*Defer[Int][1/(x*Log[(3*E^x*x^2)/2]^2), x] + 13*Defer[Int][
x/Log[(3*E^x*x^2)/2]^2, x] - 13*Defer[Int][Log[(3*E^x*x^2)/2]^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+\frac {-8+22 x+13 x^2}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )}-\frac {13}{\log \left (\frac {3 e^x x^2}{2}\right )}\right ) \, dx\\ &=2 x-13 \int \frac {1}{\log \left (\frac {3 e^x x^2}{2}\right )} \, dx+\int \frac {-8+22 x+13 x^2}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx\\ &=2 x-13 \int \frac {1}{\log \left (\frac {3 e^x x^2}{2}\right )} \, dx+\int \left (\frac {22}{\log ^2\left (\frac {3 e^x x^2}{2}\right )}-\frac {8}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )}+\frac {13 x}{\log ^2\left (\frac {3 e^x x^2}{2}\right )}\right ) \, dx\\ &=2 x-8 \int \frac {1}{x \log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx+13 \int \frac {x}{\log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx-13 \int \frac {1}{\log \left (\frac {3 e^x x^2}{2}\right )} \, dx+22 \int \frac {1}{\log ^2\left (\frac {3 e^x x^2}{2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.67, size = 23, normalized size = 1.00 \begin {gather*} 2 x+\frac {4-13 x}{\log \left (\frac {3 e^x x^2}{2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-8 + 22*x + 13*x^2 - 13*x*Log[(3*E^x*x^2)/2] + 2*x*Log[(3*E^x*x^2)/2]^2)/(x*Log[(3*E^x*x^2)/2]^2),x
]

[Out]

2*x + (4 - 13*x)/Log[(3*E^x*x^2)/2]

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fricas [A]  time = 1.29, size = 27, normalized size = 1.17 \begin {gather*} \frac {2 \, x \log \left (\frac {3}{2} \, x^{2} e^{x}\right ) - 13 \, x + 4}{\log \left (\frac {3}{2} \, x^{2} e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/log(3/2*exp(x)*x^2)^2,x, algori
thm="fricas")

[Out]

(2*x*log(3/2*x^2*e^x) - 13*x + 4)/log(3/2*x^2*e^x)

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giac [A]  time = 0.22, size = 21, normalized size = 0.91 \begin {gather*} 2 \, x - \frac {13 \, x - 4}{x + \log \left (\frac {3}{2} \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/log(3/2*exp(x)*x^2)^2,x, algori
thm="giac")

[Out]

2*x - (13*x - 4)/(x + log(3/2*x^2))

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maple [A]  time = 0.25, size = 28, normalized size = 1.22

method	result	size
norman	\(\frac {4+2 x \ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )-13 x}{\ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )}\)	\(28\)
default	\(\frac {\left (-4 \ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )+8 \ln \relax (x )+4 x +26\right ) \ln \relax (x )+4 x \ln \relax (x )+2 x^{2}+4-2 \left (\ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )-2 \ln \relax (x )-x \right )^{2}+13 \ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )-26 \ln \relax (x )-13 x}{\ln \left (\frac {3 \,{\mathrm e}^{x} x^{2}}{2}\right )}\)	\(83\)
risch	\(2 x -\frac {2 i \left (13 x -4\right )}{\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{3}-2 i \ln \relax (2)+2 i \ln \relax (3)+4 i \ln \relax (x )+2 i \ln \left ({\mathrm e}^{x}\right )}\)	\(159\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*ln(3/2*exp(x)*x^2)^2-13*x*ln(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/ln(3/2*exp(x)*x^2)^2,x,method=_RETURNVE
RBOSE)

[Out]

(4+2*x*ln(3/2*exp(x)*x^2)-13*x)/ln(3/2*exp(x)*x^2)

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maxima [A]  time = 0.49, size = 25, normalized size = 1.09 \begin {gather*} 2 \, x - \frac {13 \, x - 4}{x + \log \relax (3) - \log \relax (2) + 2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*log(3/2*exp(x)*x^2)^2-13*x*log(3/2*exp(x)*x^2)+13*x^2+22*x-8)/x/log(3/2*exp(x)*x^2)^2,x, algori
thm="maxima")

[Out]

2*x - (13*x - 4)/(x + log(3) - log(2) + 2*log(x))

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mupad [B]  time = 3.08, size = 21, normalized size = 0.91 \begin {gather*} 2\,x-\frac {13\,x-4}{\ln \left (\frac {3\,x^2\,{\mathrm {e}}^x}{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((22*x - 13*x*log((3*x^2*exp(x))/2) + 2*x*log((3*x^2*exp(x))/2)^2 + 13*x^2 - 8)/(x*log((3*x^2*exp(x))/2)^2)
,x)

[Out]

2*x - (13*x - 4)/log((3*x^2*exp(x))/2)

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sympy [A]  time = 0.13, size = 19, normalized size = 0.83 \begin {gather*} 2 x + \frac {4 - 13 x}{\log {\left (\frac {3 x^{2} e^{x}}{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*ln(3/2*exp(x)*x**2)**2-13*x*ln(3/2*exp(x)*x**2)+13*x**2+22*x-8)/x/ln(3/2*exp(x)*x**2)**2,x)

[Out]

2*x + (4 - 13*x)/log(3*x**2*exp(x)/2)

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