3.22.36 \(\int \frac {-1-2 x^3-2 x^2 \log (7)-2 e^x x^2 \log (7)}{2 x^2 \log (7)} \, dx\)

Optimal. Leaf size=26 \[ -7-e^x-x+\frac {\left (\frac {1}{x^2}-x\right ) x}{2 \log (7)} \]

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Rubi [A]  time = 0.05, antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps used = 6, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 14, 2194} \begin {gather*} -\frac {x^2}{2 \log (7)}-x-e^x+\frac {1}{2 x \log (7)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - 2*x^3 - 2*x^2*Log[7] - 2*E^x*x^2*Log[7])/(2*x^2*Log[7]),x]

[Out]

-E^x - x + 1/(2*x*Log[7]) - x^2/(2*Log[7])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-1-2 x^3-2 x^2 \log (7)-2 e^x x^2 \log (7)}{x^2} \, dx}{2 \log (7)}\\ &=\frac {\int \left (-2 e^x \log (7)+\frac {-1-2 x^3-2 x^2 \log (7)}{x^2}\right ) \, dx}{2 \log (7)}\\ &=\frac {\int \frac {-1-2 x^3-2 x^2 \log (7)}{x^2} \, dx}{2 \log (7)}-\int e^x \, dx\\ &=-e^x+\frac {\int \left (-\frac {1}{x^2}-2 x-2 \log (7)\right ) \, dx}{2 \log (7)}\\ &=-e^x-x+\frac {1}{2 x \log (7)}-\frac {x^2}{2 \log (7)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 25, normalized size = 0.96 \begin {gather*} -\frac {-\frac {1}{x}+x^2+e^x \log (49)+x \log (49)}{\log (49)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - 2*x^3 - 2*x^2*Log[7] - 2*E^x*x^2*Log[7])/(2*x^2*Log[7]),x]

[Out]

-((-x^(-1) + x^2 + E^x*Log[49] + x*Log[49])/Log[49])

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fricas [A]  time = 1.53, size = 28, normalized size = 1.08 \begin {gather*} -\frac {x^{3} + 2 \, x^{2} \log \relax (7) + 2 \, x e^{x} \log \relax (7) - 1}{2 \, x \log \relax (7)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*x^2*log(7)*exp(x)-2*x^2*log(7)-2*x^3-1)/x^2/log(7),x, algorithm="fricas")

[Out]

-1/2*(x^3 + 2*x^2*log(7) + 2*x*e^x*log(7) - 1)/(x*log(7))

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giac [A]  time = 0.17, size = 28, normalized size = 1.08 \begin {gather*} -\frac {x^{3} + 2 \, x^{2} \log \relax (7) + 2 \, x e^{x} \log \relax (7) - 1}{2 \, x \log \relax (7)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*x^2*log(7)*exp(x)-2*x^2*log(7)-2*x^3-1)/x^2/log(7),x, algorithm="giac")

[Out]

-1/2*(x^3 + 2*x^2*log(7) + 2*x*e^x*log(7) - 1)/(x*log(7))

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maple [A]  time = 0.05, size = 27, normalized size = 1.04

method	result	size
default	\(\frac {-x^{2}+\frac {1}{x}-2 \ln \relax (7) {\mathrm e}^{x}-2 x \ln \relax (7)}{2 \ln \relax (7)}\)	\(27\)
risch	\(-\frac {x^{2}}{2 \ln \relax (7)}+\frac {1}{2 \ln \relax (7) x}-{\mathrm e}^{x}-x\)	\(27\)
norman	\(\frac {-x^{2}+\frac {1}{2 \ln \relax (7)}-{\mathrm e}^{x} x -\frac {x^{3}}{2 \ln \relax (7)}}{x}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-2*x^2*ln(7)*exp(x)-2*x^2*ln(7)-2*x^3-1)/x^2/ln(7),x,method=_RETURNVERBOSE)

[Out]

1/2/ln(7)*(-x^2+1/x-2*ln(7)*exp(x)-2*x*ln(7))

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maxima [A]  time = 0.34, size = 26, normalized size = 1.00 \begin {gather*} -\frac {x^{2} + 2 \, x \log \relax (7) + 2 \, e^{x} \log \relax (7) - \frac {1}{x}}{2 \, \log \relax (7)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*x^2*log(7)*exp(x)-2*x^2*log(7)-2*x^3-1)/x^2/log(7),x, algorithm="maxima")

[Out]

-1/2*(x^2 + 2*x*log(7) + 2*e^x*log(7) - 1/x)/log(7)

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mupad [B]  time = 1.19, size = 28, normalized size = 1.08 \begin {gather*} -\frac {2\,x^2\,\ln \relax (7)+x^3+2\,x\,{\mathrm {e}}^x\,\ln \relax (7)-1}{2\,x\,\ln \relax (7)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^2*log(7) + x^3 + x^2*exp(x)*log(7) + 1/2)/(x^2*log(7)),x)

[Out]

-(2*x^2*log(7) + x^3 + 2*x*exp(x)*log(7) - 1)/(2*x*log(7))

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sympy [A]  time = 0.12, size = 20, normalized size = 0.77 \begin {gather*} \frac {- x^{2} - 2 x \log {\relax (7 )} + \frac {1}{x}}{2 \log {\relax (7 )}} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-2*x**2*ln(7)*exp(x)-2*x**2*ln(7)-2*x**3-1)/x**2/ln(7),x)

[Out]

(-x**2 - 2*x*log(7) + 1/x)/(2*log(7)) - exp(x)

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