3.87.27 \(\int \frac {216 e^x x^3+e^x (-864 x^3-216 x^4) \log (x)+e^x \log ^2(x)}{\log ^2(x)} \, dx\)

Optimal. Leaf size=16 \[ e^x-\frac {216 e^x x^4}{\log (x)} \]

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Rubi [F]  time = 0.48, 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 {216 e^x x^3+e^x \left (-864 x^3-216 x^4\right ) \log (x)+e^x \log ^2(x)}{\log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(216*E^x*x^3 + E^x*(-864*x^3 - 216*x^4)*Log[x] + E^x*Log[x]^2)/Log[x]^2,x]

[Out]

E^x + 216*Defer[Int][(E^x*x^3)/Log[x]^2, x] - 864*Defer[Int][(E^x*x^3)/Log[x], x] - 216*Defer[Int][(E^x*x^4)/L
og[x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (216 x^3-216 x^3 (4+x) \log (x)+\log ^2(x)\right )}{\log ^2(x)} \, dx\\ &=\int \left (e^x+\frac {216 e^x x^3}{\log ^2(x)}-\frac {216 e^x x^3 (4+x)}{\log (x)}\right ) \, dx\\ &=216 \int \frac {e^x x^3}{\log ^2(x)} \, dx-216 \int \frac {e^x x^3 (4+x)}{\log (x)} \, dx+\int e^x \, dx\\ &=e^x-216 \int \left (\frac {4 e^x x^3}{\log (x)}+\frac {e^x x^4}{\log (x)}\right ) \, dx+216 \int \frac {e^x x^3}{\log ^2(x)} \, dx\\ &=e^x+216 \int \frac {e^x x^3}{\log ^2(x)} \, dx-216 \int \frac {e^x x^4}{\log (x)} \, dx-864 \int \frac {e^x x^3}{\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 16, normalized size = 1.00 \begin {gather*} e^x-\frac {216 e^x x^4}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(216*E^x*x^3 + E^x*(-864*x^3 - 216*x^4)*Log[x] + E^x*Log[x]^2)/Log[x]^2,x]

[Out]

E^x - (216*E^x*x^4)/Log[x]

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fricas [A]  time = 0.89, size = 20, normalized size = 1.25 \begin {gather*} -\frac {216 \, x^{4} e^{x} - e^{x} \log \relax (x)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*log(x)^2+(-216*x^4-864*x^3)*exp(x)*log(x)+216*exp(x)*x^3)/log(x)^2,x, algorithm="fricas")

[Out]

-(216*x^4*e^x - e^x*log(x))/log(x)

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giac [A]  time = 0.12, size = 20, normalized size = 1.25 \begin {gather*} -\frac {216 \, x^{4} e^{x} - e^{x} \log \relax (x)}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*log(x)^2+(-216*x^4-864*x^3)*exp(x)*log(x)+216*exp(x)*x^3)/log(x)^2,x, algorithm="giac")

[Out]

-(216*x^4*e^x - e^x*log(x))/log(x)

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maple [A]  time = 0.02, size = 15, normalized size = 0.94

method	result	size
risch	\({\mathrm e}^{x}-\frac {216 \,{\mathrm e}^{x} x^{4}}{\ln \relax (x )}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*ln(x)^2+(-216*x^4-864*x^3)*exp(x)*ln(x)+216*exp(x)*x^3)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

exp(x)-216*exp(x)*x^4/ln(x)

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maxima [A]  time = 0.40, size = 14, normalized size = 0.88 \begin {gather*} -\frac {216 \, x^{4} e^{x}}{\log \relax (x)} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*log(x)^2+(-216*x^4-864*x^3)*exp(x)*log(x)+216*exp(x)*x^3)/log(x)^2,x, algorithm="maxima")

[Out]

-216*x^4*e^x/log(x) + e^x

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mupad [B]  time = 5.22, size = 14, normalized size = 0.88 \begin {gather*} {\mathrm {e}}^x-\frac {216\,x^4\,{\mathrm {e}}^x}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((216*x^3*exp(x) + exp(x)*log(x)^2 - exp(x)*log(x)*(864*x^3 + 216*x^4))/log(x)^2,x)

[Out]

exp(x) - (216*x^4*exp(x))/log(x)

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sympy [A]  time = 0.25, size = 14, normalized size = 0.88 \begin {gather*} \frac {\left (- 216 x^{4} + \log {\relax (x )}\right ) e^{x}}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x)*ln(x)**2+(-216*x**4-864*x**3)*exp(x)*ln(x)+216*exp(x)*x**3)/ln(x)**2,x)

[Out]

(-216*x**4 + log(x))*exp(x)/log(x)

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