3.51.8 \(\int (-1-2 x+e^{2 x} (2 x+2 x^2)+8 x^7 \log ^4(2)) \, dx\)

Optimal. Leaf size=27 \[ -4-x-x^2+e^{2 x} x^2+x^8 \log ^4(2) \]

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Rubi [A]  time = 0.06, antiderivative size = 26, normalized size of antiderivative = 0.96, number of steps used = 9, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1593, 2196, 2176, 2194} \begin {gather*} x^8 \log ^4(2)+e^{2 x} x^2-x^2-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 - 2*x + E^(2*x)*(2*x + 2*x^2) + 8*x^7*Log[2]^4,x]

[Out]

-x - x^2 + E^(2*x)*x^2 + x^8*Log[2]^4

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.96 \begin {gather*} -x-x^2+e^{2 x} x^2+x^8 \log ^4(2) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 - 2*x + E^(2*x)*(2*x + 2*x^2) + 8*x^7*Log[2]^4,x]

[Out]

-x - x^2 + E^(2*x)*x^2 + x^8*Log[2]^4

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fricas [A]  time = 0.74, size = 25, normalized size = 0.93 \begin {gather*} x^{8} \log \relax (2)^{4} + x^{2} e^{\left (2 \, x\right )} - x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+2*x)*exp(x)^2+8*x^7*log(2)^4-2*x-1,x, algorithm="fricas")

[Out]

x^8*log(2)^4 + x^2*e^(2*x) - x^2 - x

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giac [A]  time = 0.16, size = 25, normalized size = 0.93 \begin {gather*} x^{8} \log \relax (2)^{4} + x^{2} e^{\left (2 \, x\right )} - x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+2*x)*exp(x)^2+8*x^7*log(2)^4-2*x-1,x, algorithm="giac")

[Out]

x^8*log(2)^4 + x^2*e^(2*x) - x^2 - x

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maple [A]  time = 0.05, size = 26, normalized size = 0.96

method	result	size
default	\(-x +{\mathrm e}^{2 x} x^{2}-x^{2}+x^{8} \ln \relax (2)^{4}\)	\(26\)
norman	\(-x +{\mathrm e}^{2 x} x^{2}-x^{2}+x^{8} \ln \relax (2)^{4}\)	\(26\)
risch	\(-x +{\mathrm e}^{2 x} x^{2}-x^{2}+x^{8} \ln \relax (2)^{4}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2+2*x)*exp(x)^2+8*x^7*ln(2)^4-2*x-1,x,method=_RETURNVERBOSE)

[Out]

-x+exp(x)^2*x^2-x^2+x^8*ln(2)^4

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maxima [A]  time = 0.36, size = 25, normalized size = 0.93 \begin {gather*} x^{8} \log \relax (2)^{4} + x^{2} e^{\left (2 \, x\right )} - x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2+2*x)*exp(x)^2+8*x^7*log(2)^4-2*x-1,x, algorithm="maxima")

[Out]

x^8*log(2)^4 + x^2*e^(2*x) - x^2 - x

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mupad [B]  time = 3.18, size = 25, normalized size = 0.93 \begin {gather*} x^8\,{\ln \relax (2)}^4-x+x^2\,{\mathrm {e}}^{2\,x}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*x^7*log(2)^4 - 2*x + exp(2*x)*(2*x + 2*x^2) - 1,x)

[Out]

x^8*log(2)^4 - x + x^2*exp(2*x) - x^2

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sympy [A]  time = 0.09, size = 20, normalized size = 0.74 \begin {gather*} x^{8} \log {\relax (2 )}^{4} + x^{2} e^{2 x} - x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2+2*x)*exp(x)**2+8*x**7*ln(2)**4-2*x-1,x)

[Out]

x**8*log(2)**4 + x**2*exp(2*x) - x**2 - x

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