3.77.29 \(\int \frac {-25 \log (5)+e^{\frac {e^x x+2 x^3 \log (5)}{\log (5)}} (e^x (-1-x)-6 x^2 \log (5))}{25 \log (5)} \, dx\)

Optimal. Leaf size=26 \[ 2-\frac {1}{25} e^{2 x^3+\frac {e^x x}{\log (5)}}-x \]

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Rubi [A]  time = 0.13, antiderivative size = 25, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 2, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {12, 6706} \begin {gather*} -\frac {1}{25} e^{2 x^3+\frac {e^x x}{\log (5)}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-25*Log[5] + E^((E^x*x + 2*x^3*Log[5])/Log[5])*(E^x*(-1 - x) - 6*x^2*Log[5]))/(25*Log[5]),x]

[Out]

-1/25*E^(2*x^3 + (E^x*x)/Log[5]) - x

Rule 12

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (-25 \log (5)+e^{\frac {e^x x+2 x^3 \log (5)}{\log (5)}} \left (e^x (-1-x)-6 x^2 \log (5)\right )\right ) \, dx}{25 \log (5)}\\ &=-x+\frac {\int e^{\frac {e^x x+2 x^3 \log (5)}{\log (5)}} \left (e^x (-1-x)-6 x^2 \log (5)\right ) \, dx}{25 \log (5)}\\ &=-\frac {1}{25} e^{2 x^3+\frac {e^x x}{\log (5)}}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.46, size = 25, normalized size = 0.96 \begin {gather*} -\frac {1}{25} e^{2 x^3+\frac {e^x x}{\log (5)}}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25*Log[5] + E^((E^x*x + 2*x^3*Log[5])/Log[5])*(E^x*(-1 - x) - 6*x^2*Log[5]))/(25*Log[5]),x]

[Out]

-1/25*E^(2*x^3 + (E^x*x)/Log[5]) - x

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fricas [A]  time = 0.54, size = 24, normalized size = 0.92 \begin {gather*} -x - \frac {1}{25} \, e^{\left (\frac {2 \, x^{3} \log \relax (5) + x e^{x}}{\log \relax (5)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(((-x-1)*exp(x)-6*x^2*log(5))*exp((exp(x)*x+2*x^3*log(5))/log(5))-25*log(5))/log(5),x, algorith
m="fricas")

[Out]

-x - 1/25*e^((2*x^3*log(5) + x*e^x)/log(5))

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giac [A]  time = 0.23, size = 30, normalized size = 1.15 \begin {gather*} -\frac {25 \, x \log \relax (5) + e^{\left (2 \, x^{3} + \frac {x e^{x}}{\log \relax (5)}\right )} \log \relax (5)}{25 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(((-x-1)*exp(x)-6*x^2*log(5))*exp((exp(x)*x+2*x^3*log(5))/log(5))-25*log(5))/log(5),x, algorith
m="giac")

[Out]

-1/25*(25*x*log(5) + e^(2*x^3 + x*e^x/log(5))*log(5))/log(5)

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maple [A]  time = 0.12, size = 24, normalized size = 0.92

method	result	size
risch	\(-x -\frac {{\mathrm e}^{\frac {x \left (2 x^{2} \ln \relax (5)+{\mathrm e}^{x}\right )}{\ln \relax (5)}}}{25}\)	\(24\)
norman	\(-x -\frac {{\mathrm e}^{\frac {{\mathrm e}^{x} x +2 x^{3} \ln \relax (5)}{\ln \relax (5)}}}{25}\)	\(25\)
default	\(\frac {-\ln \relax (5) {\mathrm e}^{\frac {{\mathrm e}^{x} x +2 x^{3} \ln \relax (5)}{\ln \relax (5)}}-25 x \ln \relax (5)}{25 \ln \relax (5)}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*(((-x-1)*exp(x)-6*x^2*ln(5))*exp((exp(x)*x+2*x^3*ln(5))/ln(5))-25*ln(5))/ln(5),x,method=_RETURNVERBOS
E)

[Out]

-x-1/25*exp(x*(2*x^2*ln(5)+exp(x))/ln(5))

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maxima [A]  time = 0.52, size = 30, normalized size = 1.15 \begin {gather*} -\frac {25 \, x \log \relax (5) + e^{\left (2 \, x^{3} + \frac {x e^{x}}{\log \relax (5)}\right )} \log \relax (5)}{25 \, \log \relax (5)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(((-x-1)*exp(x)-6*x^2*log(5))*exp((exp(x)*x+2*x^3*log(5))/log(5))-25*log(5))/log(5),x, algorith
m="maxima")

[Out]

-1/25*(25*x*log(5) + e^(2*x^3 + x*e^x/log(5))*log(5))/log(5)

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mupad [B]  time = 0.17, size = 21, normalized size = 0.81 \begin {gather*} -x-\frac {{\mathrm {e}}^{2\,x^3}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^x}{\ln \relax (5)}}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5) + (exp((2*x^3*log(5) + x*exp(x))/log(5))*(exp(x)*(x + 1) + 6*x^2*log(5)))/25)/log(5),x)

[Out]

- x - (exp(2*x^3)*exp((x*exp(x))/log(5)))/25

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sympy [A]  time = 0.24, size = 22, normalized size = 0.85 \begin {gather*} - x - \frac {e^{\frac {2 x^{3} \log {\relax (5 )} + x e^{x}}{\log {\relax (5 )}}}}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*(((-x-1)*exp(x)-6*x**2*ln(5))*exp((exp(x)*x+2*x**3*ln(5))/ln(5))-25*ln(5))/ln(5),x)

[Out]

-x - exp((2*x**3*log(5) + x*exp(x))/log(5))/25

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