3.53.47 \(\int (-e^x+x^3+4 x^3 \log (x)) \, dx\)

Optimal. Leaf size=29 \[ -e^x+\left (-4+e^{\frac {3}{-5+e^{e^4}}}\right )^2+x^4 \log (x) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.41, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2194, 2304} \begin {gather*} x^4 \log (x)-e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-E^x + x^3 + 4*x^3*Log[x],x]

[Out]

-E^x + x^4*Log[x]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.41 \begin {gather*} -e^x+x^4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-E^x + x^3 + 4*x^3*Log[x],x]

[Out]

-E^x + x^4*Log[x]

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fricas [A]  time = 0.54, size = 11, normalized size = 0.38 \begin {gather*} x^{4} \log \relax (x) - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*log(x) - e^x

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giac [A]  time = 0.13, size = 11, normalized size = 0.38 \begin {gather*} x^{4} \log \relax (x) - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*log(x) - e^x

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maple [A]  time = 0.03, size = 12, normalized size = 0.41

method	result	size
default	\(x^{4} \ln \relax (x )-{\mathrm e}^{x}\)	\(12\)
norman	\(x^{4} \ln \relax (x )-{\mathrm e}^{x}\)	\(12\)
risch	\(x^{4} \ln \relax (x )-{\mathrm e}^{x}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.36, size = 11, normalized size = 0.38 \begin {gather*} x^{4} \log \relax (x) - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4*log(x) - e^x

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mupad [B]  time = 3.51, size = 11, normalized size = 0.38 \begin {gather*} x^4\,\ln \relax (x)-{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*x^3*log(x) - exp(x) + x^3,x)

[Out]

x^4*log(x) - exp(x)

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sympy [A]  time = 0.23, size = 8, normalized size = 0.28 \begin {gather*} x^{4} \log {\relax (x )} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*x**3*ln(x)+x**3-exp(x),x)

[Out]

x**4*log(x) - exp(x)

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