3.85.26 \(\int (24 x-12 e^4 x+(4 x-2 e^4 x) \log (25)+e^e (-12+6 e^4+(-2+e^4) \log (25))) \, dx\)

Optimal. Leaf size=20 \[ \left (2-e^4\right ) x \left (-e^e+x\right ) (6+\log (25)) \]

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Rubi [B]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 2.15, number of steps used = 2, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6} \begin {gather*} 6 \left (2-e^4\right ) x^2+\left (2-e^4\right ) x^2 \log (25)-e^e \left (2-e^4\right ) x (6+\log (25)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[24*x - 12*E^4*x + (4*x - 2*E^4*x)*Log[25] + E^E*(-12 + 6*E^4 + (-2 + E^4)*Log[25]),x]

[Out]

6*(2 - E^4)*x^2 + (2 - E^4)*x^2*Log[25] - E^E*(2 - E^4)*x*(6 + Log[25])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\left (24-12 e^4\right ) x+\left (4 x-2 e^4 x\right ) \log (25)+e^e \left (-12+6 e^4+\left (-2+e^4\right ) \log (25)\right )\right ) \, dx\\ &=6 \left (2-e^4\right ) x^2+\left (2-e^4\right ) x^2 \log (25)-e^e \left (2-e^4\right ) x (6+\log (25))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.05 \begin {gather*} \left (-2+e^4\right ) \left (e^e x-x^2\right ) (6+\log (25)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[24*x - 12*E^4*x + (4*x - 2*E^4*x)*Log[25] + E^E*(-12 + 6*E^4 + (-2 + E^4)*Log[25]),x]

[Out]

(-2 + E^4)*(E^E*x - x^2)*(6 + Log[25])

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fricas [B]  time = 0.77, size = 54, normalized size = 2.70 \begin {gather*} -6 \, x^{2} e^{4} + 12 \, x^{2} + 2 \, {\left (3 \, x e^{4} + {\left (x e^{4} - 2 \, x\right )} \log \relax (5) - 6 \, x\right )} e^{e} - 2 \, {\left (x^{2} e^{4} - 2 \, x^{2}\right )} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(exp(4)-2)*log(5)+6*exp(4)-12)*exp(exp(1))+2*(-2*x*exp(4)+4*x)*log(5)-12*x*exp(4)+24*x,x, algorit
hm="fricas")

[Out]

-6*x^2*e^4 + 12*x^2 + 2*(3*x*e^4 + (x*e^4 - 2*x)*log(5) - 6*x)*e^e - 2*(x^2*e^4 - 2*x^2)*log(5)

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giac [B]  time = 0.13, size = 48, normalized size = 2.40 \begin {gather*} -6 \, x^{2} e^{4} + 2 \, {\left ({\left (e^{4} - 2\right )} \log \relax (5) + 3 \, e^{4} - 6\right )} x e^{e} + 12 \, x^{2} - 2 \, {\left (x^{2} e^{4} - 2 \, x^{2}\right )} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(exp(4)-2)*log(5)+6*exp(4)-12)*exp(exp(1))+2*(-2*x*exp(4)+4*x)*log(5)-12*x*exp(4)+24*x,x, algorit
hm="giac")

[Out]

-6*x^2*e^4 + 2*((e^4 - 2)*log(5) + 3*e^4 - 6)*x*e^e + 12*x^2 - 2*(x^2*e^4 - 2*x^2)*log(5)

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maple [A]  time = 0.03, size = 26, normalized size = 1.30

method	result	size
gosper	\(2 \left ({\mathrm e}^{4} \ln \relax (5)-2 \ln \relax (5)+3 \,{\mathrm e}^{4}-6\right ) x \left ({\mathrm e}^{{\mathrm e}}-x \right )\)	\(26\)
default	\(\left (2 \left ({\mathrm e}^{4}-2\right ) \ln \relax (5)+6 \,{\mathrm e}^{4}-12\right ) {\mathrm e}^{{\mathrm e}} x +2 \ln \relax (5) \left (-x^{2} {\mathrm e}^{4}+2 x^{2}\right )-6 x^{2} {\mathrm e}^{4}+12 x^{2}\)	\(50\)
norman	\(\left (-2 \,{\mathrm e}^{4} \ln \relax (5)+4 \ln \relax (5)-6 \,{\mathrm e}^{4}+12\right ) x^{2}+\left (2 \ln \relax (5) {\mathrm e}^{4} {\mathrm e}^{{\mathrm e}}-4 \ln \relax (5) {\mathrm e}^{{\mathrm e}}+6 \,{\mathrm e}^{4} {\mathrm e}^{{\mathrm e}}-12 \,{\mathrm e}^{{\mathrm e}}\right ) x\)	\(53\)
risch	\(2 \ln \relax (5) {\mathrm e}^{4} {\mathrm e}^{{\mathrm e}} x -4 \ln \relax (5) {\mathrm e}^{{\mathrm e}} x +6 \,{\mathrm e}^{4} {\mathrm e}^{{\mathrm e}} x -12 x \,{\mathrm e}^{{\mathrm e}}-2 \ln \relax (5) x^{2} {\mathrm e}^{4}+4 x^{2} \ln \relax (5)-6 x^{2} {\mathrm e}^{4}+12 x^{2}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(exp(4)-2)*ln(5)+6*exp(4)-12)*exp(exp(1))+2*(-2*x*exp(4)+4*x)*ln(5)-12*x*exp(4)+24*x,x,method=_RETURNVE
RBOSE)

[Out]

2*(exp(4)*ln(5)-2*ln(5)+3*exp(4)-6)*x*(exp(exp(1))-x)

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maxima [B]  time = 0.45, size = 48, normalized size = 2.40 \begin {gather*} -6 \, x^{2} e^{4} + 2 \, {\left ({\left (e^{4} - 2\right )} \log \relax (5) + 3 \, e^{4} - 6\right )} x e^{e} + 12 \, x^{2} - 2 \, {\left (x^{2} e^{4} - 2 \, x^{2}\right )} \log \relax (5) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(exp(4)-2)*log(5)+6*exp(4)-12)*exp(exp(1))+2*(-2*x*exp(4)+4*x)*log(5)-12*x*exp(4)+24*x,x, algorit
hm="maxima")

[Out]

-6*x^2*e^4 + 2*((e^4 - 2)*log(5) + 3*e^4 - 6)*x*e^e + 12*x^2 - 2*(x^2*e^4 - 2*x^2)*log(5)

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mupad [B]  time = 0.12, size = 38, normalized size = 1.90 \begin {gather*} 2\,x\,\left ({\mathrm {e}}^{\mathrm {e}+4}-2\,{\mathrm {e}}^{\mathrm {e}}\right )\,\left (\ln \relax (5)+3\right )-x^2\,\left (6\,{\mathrm {e}}^4+2\,\ln \relax (5)\,\left ({\mathrm {e}}^4-2\right )-12\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(24*x - 12*x*exp(4) + 2*log(5)*(4*x - 2*x*exp(4)) + exp(exp(1))*(6*exp(4) + 2*log(5)*(exp(4) - 2) - 12),x)

[Out]

2*x*(exp(exp(1) + 4) - 2*exp(exp(1)))*(log(5) + 3) - x^2*(6*exp(4) + 2*log(5)*(exp(4) - 2) - 12)

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sympy [B]  time = 0.07, size = 65, normalized size = 3.25 \begin {gather*} x^{2} \left (- 6 e^{4} - 2 e^{4} \log {\relax (5 )} + 4 \log {\relax (5 )} + 12\right ) + x \left (- 12 e^{e} - 4 e^{e} \log {\relax (5 )} + 2 e^{4} e^{e} \log {\relax (5 )} + 6 e^{4} e^{e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(exp(4)-2)*ln(5)+6*exp(4)-12)*exp(exp(1))+2*(-2*x*exp(4)+4*x)*ln(5)-12*x*exp(4)+24*x,x)

[Out]

x**2*(-6*exp(4) - 2*exp(4)*log(5) + 4*log(5) + 12) + x*(-12*exp(E) - 4*exp(E)*log(5) + 2*exp(4)*exp(E)*log(5)
+ 6*exp(4)*exp(E))

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