3.25.19 \(\int (e^x+6 x+3 x^2-18 e^4 x^3 \log (x)-36 e^4 x^3 \log ^2(x)) \, dx\)

Optimal. Leaf size=24 \[ -1+e^x+x^2 \left (3+x-9 e^4 x^2 \log ^2(x)\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2194, 2304, 2305} \begin {gather*} -9 e^4 x^4 \log ^2(x)+x^3+3 x^2+e^x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^x + 6*x + 3*x^2 - 18*E^4*x^3*Log[x] - 36*E^4*x^3*Log[x]^2,x]

[Out]

E^x + 3*x^2 + x^3 - 9*E^4*x^4*Log[x]^2

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
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=3 x^2+x^3-\left (18 e^4\right ) \int x^3 \log (x) \, dx-\left (36 e^4\right ) \int x^3 \log ^2(x) \, dx+\int e^x \, dx\\ &=e^x+3 x^2+x^3+\frac {9 e^4 x^4}{8}-\frac {9}{2} e^4 x^4 \log (x)-9 e^4 x^4 \log ^2(x)+\left (18 e^4\right ) \int x^3 \log (x) \, dx\\ &=e^x+3 x^2+x^3-9 e^4 x^4 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} e^x+3 x^2+x^3-9 e^4 x^4 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x + 6*x + 3*x^2 - 18*E^4*x^3*Log[x] - 36*E^4*x^3*Log[x]^2,x]

[Out]

E^x + 3*x^2 + x^3 - 9*E^4*x^4*Log[x]^2

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fricas [A]  time = 1.03, size = 22, normalized size = 0.92 \begin {gather*} -9 \, x^{4} e^{4} \log \relax (x)^{2} + x^{3} + 3 \, x^{2} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-36*x^3*exp(2)^2*log(x)^2-18*x^3*exp(2)^2*log(x)+exp(x)+3*x^2+6*x,x, algorithm="fricas")

[Out]

-9*x^4*e^4*log(x)^2 + x^3 + 3*x^2 + e^x

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giac [B]  time = 0.36, size = 52, normalized size = 2.17 \begin {gather*} x^{3} + 3 \, x^{2} - \frac {9}{8} \, {\left (8 \, x^{4} \log \relax (x)^{2} - 4 \, x^{4} \log \relax (x) + x^{4}\right )} e^{4} - \frac {9}{8} \, {\left (4 \, x^{4} \log \relax (x) - x^{4}\right )} e^{4} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-36*x^3*exp(2)^2*log(x)^2-18*x^3*exp(2)^2*log(x)+exp(x)+3*x^2+6*x,x, algorithm="giac")

[Out]

x^3 + 3*x^2 - 9/8*(8*x^4*log(x)^2 - 4*x^4*log(x) + x^4)*e^4 - 9/8*(4*x^4*log(x) - x^4)*e^4 + e^x

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maple [A]  time = 0.02, size = 23, normalized size = 0.96

method	result	size
risch	\(x^{3}+3 x^{2}-9 \,{\mathrm e}^{4} x^{4} \ln \relax (x )^{2}+{\mathrm e}^{x}\)	\(23\)
default	\(x^{3}+3 x^{2}-9 \,{\mathrm e}^{4} x^{4} \ln \relax (x )^{2}+{\mathrm e}^{x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-36*x^3*exp(2)^2*ln(x)^2-18*x^3*exp(2)^2*ln(x)+exp(x)+3*x^2+6*x,x,method=_RETURNVERBOSE)

[Out]

x^3+3*x^2-9*exp(4)*x^4*ln(x)^2+exp(x)

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maxima [A]  time = 0.57, size = 47, normalized size = 1.96 \begin {gather*} -\frac {9}{8} \, {\left (8 \, \log \relax (x)^{2} - 4 \, \log \relax (x) + 1\right )} x^{4} e^{4} + x^{3} + 3 \, x^{2} - \frac {9}{8} \, {\left (4 \, x^{4} \log \relax (x) - x^{4}\right )} e^{4} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-36*x^3*exp(2)^2*log(x)^2-18*x^3*exp(2)^2*log(x)+exp(x)+3*x^2+6*x,x, algorithm="maxima")

[Out]

-9/8*(8*log(x)^2 - 4*log(x) + 1)*x^4*e^4 + x^3 + 3*x^2 - 9/8*(4*x^4*log(x) - x^4)*e^4 + e^x

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mupad [B]  time = 1.38, size = 22, normalized size = 0.92 \begin {gather*} {\mathrm {e}}^x+3\,x^2+x^3-9\,x^4\,{\mathrm {e}}^4\,{\ln \relax (x)}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(6*x + exp(x) + 3*x^2 - 18*x^3*exp(4)*log(x) - 36*x^3*exp(4)*log(x)^2,x)

[Out]

exp(x) + 3*x^2 + x^3 - 9*x^4*exp(4)*log(x)^2

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sympy [A]  time = 0.28, size = 24, normalized size = 1.00 \begin {gather*} - 9 x^{4} e^{4} \log {\relax (x )}^{2} + x^{3} + 3 x^{2} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-36*x**3*exp(2)**2*ln(x)**2-18*x**3*exp(2)**2*ln(x)+exp(x)+3*x**2+6*x,x)

[Out]

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

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