3.49.91 \(\int (15 x^3+e^{2+x} (-16 x^3-4 x^4)+e^{4+2 x} (4 x^3+2 x^4)-4 x^3 \log (x)) \, dx\)

Optimal. Leaf size=18 \[ x^4 \left (\left (-2+e^{2+x}\right )^2-\log (x)\right ) \]

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Rubi [A]  time = 0.29, antiderivative size = 34, normalized size of antiderivative = 1.89, number of steps used = 26, number of rules used = 5, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.102, Rules used = {1593, 2196, 2176, 2194, 2304} \begin {gather*} -4 e^{x+2} x^4+e^{2 x+4} x^4+4 x^4-x^4 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[15*x^3 + E^(2 + x)*(-16*x^3 - 4*x^4) + E^(4 + 2*x)*(4*x^3 + 2*x^4) - 4*x^3*Log[x],x]

[Out]

4*x^4 - 4*E^(2 + x)*x^4 + E^(4 + 2*x)*x^4 - x^4*Log[x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

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 {15 x^4}{4}-4 \int x^3 \log (x) \, dx+\int e^{2+x} \left (-16 x^3-4 x^4\right ) \, dx+\int e^{4+2 x} \left (4 x^3+2 x^4\right ) \, dx\\ &=4 x^4-x^4 \log (x)+\int e^{2+x} (-16-4 x) x^3 \, dx+\int e^{4+2 x} x^3 (4+2 x) \, dx\\ &=4 x^4-x^4 \log (x)+\int \left (-16 e^{2+x} x^3-4 e^{2+x} x^4\right ) \, dx+\int \left (4 e^{4+2 x} x^3+2 e^{4+2 x} x^4\right ) \, dx\\ &=4 x^4-x^4 \log (x)+2 \int e^{4+2 x} x^4 \, dx+4 \int e^{4+2 x} x^3 \, dx-4 \int e^{2+x} x^4 \, dx-16 \int e^{2+x} x^3 \, dx\\ &=-16 e^{2+x} x^3+2 e^{4+2 x} x^3+4 x^4-4 e^{2+x} x^4+e^{4+2 x} x^4-x^4 \log (x)-4 \int e^{4+2 x} x^3 \, dx-6 \int e^{4+2 x} x^2 \, dx+16 \int e^{2+x} x^3 \, dx+48 \int e^{2+x} x^2 \, dx\\ &=48 e^{2+x} x^2-3 e^{4+2 x} x^2+4 x^4-4 e^{2+x} x^4+e^{4+2 x} x^4-x^4 \log (x)+6 \int e^{4+2 x} x \, dx+6 \int e^{4+2 x} x^2 \, dx-48 \int e^{2+x} x^2 \, dx-96 \int e^{2+x} x \, dx\\ &=-96 e^{2+x} x+3 e^{4+2 x} x+4 x^4-4 e^{2+x} x^4+e^{4+2 x} x^4-x^4 \log (x)-3 \int e^{4+2 x} \, dx-6 \int e^{4+2 x} x \, dx+96 \int e^{2+x} \, dx+96 \int e^{2+x} x \, dx\\ &=96 e^{2+x}-\frac {3}{2} e^{4+2 x}+4 x^4-4 e^{2+x} x^4+e^{4+2 x} x^4-x^4 \log (x)+3 \int e^{4+2 x} \, dx-96 \int e^{2+x} \, dx\\ &=4 x^4-4 e^{2+x} x^4+e^{4+2 x} x^4-x^4 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 18, normalized size = 1.00 \begin {gather*} x^4 \left (\left (-2+e^{2+x}\right )^2-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[15*x^3 + E^(2 + x)*(-16*x^3 - 4*x^4) + E^(4 + 2*x)*(4*x^3 + 2*x^4) - 4*x^3*Log[x],x]

[Out]

x^4*((-2 + E^(2 + x))^2 - Log[x])

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fricas [A]  time = 0.72, size = 32, normalized size = 1.78 \begin {gather*} x^{4} e^{\left (2 \, x + 4\right )} - 4 \, x^{4} e^{\left (x + 2\right )} - x^{4} \log \relax (x) + 4 \, x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x^3*log(x)+(2*x^4+4*x^3)*exp(2+x)^2+(-4*x^4-16*x^3)*exp(2+x)+15*x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.21, size = 32, normalized size = 1.78 \begin {gather*} x^{4} e^{\left (2 \, x + 4\right )} - 4 \, x^{4} e^{\left (x + 2\right )} - x^{4} \log \relax (x) + 4 \, x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x^3*log(x)+(2*x^4+4*x^3)*exp(2+x)^2+(-4*x^4-16*x^3)*exp(2+x)+15*x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 33, normalized size = 1.83

method	result	size
risch	\(-x^{4} \ln \relax (x )+4 x^{4}+x^{4} {\mathrm e}^{2 x +4}-4 x^{4} {\mathrm e}^{2+x}\)	\(33\)
default	\(128 \,{\mathrm e}^{2+x} \left (2+x \right )-64 \,{\mathrm e}^{2+x}+32 \,{\mathrm e}^{2+x} \left (2+x \right )^{3}-96 \,{\mathrm e}^{2+x} \left (2+x \right )^{2}-4 \,{\mathrm e}^{2+x} \left (2+x \right )^{4}+{\mathrm e}^{2 x +4} \left (2+x \right )^{4}-8 \left (2+x \right )^{3} {\mathrm e}^{2 x +4}+24 \left (2+x \right )^{2} {\mathrm e}^{2 x +4}-32 \left (2+x \right ) {\mathrm e}^{2 x +4}+16 \,{\mathrm e}^{2 x +4}+4 x^{4}-x^{4} \ln \relax (x )\)	\(119\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-4*x^3*ln(x)+(2*x^4+4*x^3)*exp(2+x)^2+(-4*x^4-16*x^3)*exp(2+x)+15*x^3,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.36, size = 84, normalized size = 4.67 \begin {gather*} x^{4} e^{\left (2 \, x + 4\right )} - x^{4} \log \relax (x) + 4 \, x^{4} - 4 \, {\left (x^{4} e^{2} - 4 \, x^{3} e^{2} + 12 \, x^{2} e^{2} - 24 \, x e^{2} + 24 \, e^{2}\right )} e^{x} - 16 \, {\left (x^{3} e^{2} - 3 \, x^{2} e^{2} + 6 \, x e^{2} - 6 \, e^{2}\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x^3*log(x)+(2*x^4+4*x^3)*exp(2+x)^2+(-4*x^4-16*x^3)*exp(2+x)+15*x^3,x, algorithm="maxima")

[Out]

x^4*e^(2*x + 4) - x^4*log(x) + 4*x^4 - 4*(x^4*e^2 - 4*x^3*e^2 + 12*x^2*e^2 - 24*x*e^2 + 24*e^2)*e^x - 16*(x^3*
e^2 - 3*x^2*e^2 + 6*x*e^2 - 6*e^2)*e^x

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mupad [B]  time = 3.63, size = 23, normalized size = 1.28 \begin {gather*} -x^4\,\left (4\,{\mathrm {e}}^{x+2}-{\mathrm {e}}^{2\,x+4}+\ln \relax (x)-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x + 4)*(4*x^3 + 2*x^4) - exp(x + 2)*(16*x^3 + 4*x^4) - 4*x^3*log(x) + 15*x^3,x)

[Out]

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

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sympy [B]  time = 0.33, size = 31, normalized size = 1.72 \begin {gather*} - 4 x^{4} e^{x + 2} + x^{4} e^{2 x + 4} - x^{4} \log {\relax (x )} + 4 x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x**3*ln(x)+(2*x**4+4*x**3)*exp(2+x)**2+(-4*x**4-16*x**3)*exp(2+x)+15*x**3,x)

[Out]

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

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