3.103.37 \(\int (-300 e^4 x-100 x^3+(150 e^4+150 x^2) \log (4)-50 x \log ^2(4)) \, dx\)

Optimal. Leaf size=32 \[ 4+3 e^{-e^2}-25 x^2 \left (-\frac {3 e^4}{x}-x+\log (4)\right )^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {6} \begin {gather*} -25 x^4+50 x^3 \log (4)-25 x^2 \left (6 e^4+\log ^2(4)\right )+150 e^4 x \log (4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-300*E^4*x - 100*x^3 + (150*E^4 + 150*x^2)*Log[4] - 50*x*Log[4]^2,x]

[Out]

-25*x^4 + 150*E^4*x*Log[4] + 50*x^3*Log[4] - 25*x^2*(6*E^4 + Log[4]^2)

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 (-100 x^3+\left (150 e^4+150 x^2\right ) \log (4)+x \left (-300 e^4-50 \log ^2(4)\right )\right ) \, dx\\ &=-25 x^4-25 x^2 \left (6 e^4+\log ^2(4)\right )+\log (4) \int \left (150 e^4+150 x^2\right ) \, dx\\ &=-25 x^4+150 e^4 x \log (4)+50 x^3 \log (4)-25 x^2 \left (6 e^4+\log ^2(4)\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 44, normalized size = 1.38 \begin {gather*} -50 \left (3 e^4 x^2+\frac {x^4}{2}-3 e^4 x \log (4)-x^3 \log (4)+\frac {1}{2} x^2 \log ^2(4)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-300*E^4*x - 100*x^3 + (150*E^4 + 150*x^2)*Log[4] - 50*x*Log[4]^2,x]

[Out]

-50*(3*E^4*x^2 + x^4/2 - 3*E^4*x*Log[4] - x^3*Log[4] + (x^2*Log[4]^2)/2)

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fricas [A]  time = 0.63, size = 35, normalized size = 1.09 \begin {gather*} -25 \, x^{4} - 100 \, x^{2} \log \relax (2)^{2} - 150 \, x^{2} e^{4} + 100 \, {\left (x^{3} + 3 \, x e^{4}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-200*x*log(2)^2+2*(150*exp(4)+150*x^2)*log(2)-300*x*exp(4)-100*x^3,x, algorithm="fricas")

[Out]

-25*x^4 - 100*x^2*log(2)^2 - 150*x^2*e^4 + 100*(x^3 + 3*x*e^4)*log(2)

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giac [A]  time = 0.13, size = 35, normalized size = 1.09 \begin {gather*} -25 \, x^{4} - 100 \, x^{2} \log \relax (2)^{2} - 150 \, x^{2} e^{4} + 100 \, {\left (x^{3} + 3 \, x e^{4}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-200*x*log(2)^2+2*(150*exp(4)+150*x^2)*log(2)-300*x*exp(4)-100*x^3,x, algorithm="giac")

[Out]

-25*x^4 - 100*x^2*log(2)^2 - 150*x^2*e^4 + 100*(x^3 + 3*x*e^4)*log(2)

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maple [A]  time = 0.03, size = 35, normalized size = 1.09

method	result	size
gosper	\(25 x \left (-4 x \ln \relax (2)^{2}+4 x^{2} \ln \relax (2)-x^{3}+12 \,{\mathrm e}^{4} \ln \relax (2)-6 x \,{\mathrm e}^{4}\right )\)	\(35\)
norman	\(\left (-100 \ln \relax (2)^{2}-150 \,{\mathrm e}^{4}\right ) x^{2}-25 x^{4}+100 x^{3} \ln \relax (2)+300 x \,{\mathrm e}^{4} \ln \relax (2)\)	\(36\)
risch	\(-100 x^{2} \ln \relax (2)^{2}+300 x \,{\mathrm e}^{4} \ln \relax (2)+100 x^{3} \ln \relax (2)-150 x^{2} {\mathrm e}^{4}-25 x^{4}\)	\(37\)
default	\(-100 x^{2} \ln \relax (2)^{2}+2 \ln \relax (2) \left (150 x \,{\mathrm e}^{4}+50 x^{3}\right )-150 x^{2} {\mathrm e}^{4}-25 x^{4}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-200*x*ln(2)^2+2*(150*exp(4)+150*x^2)*ln(2)-300*x*exp(4)-100*x^3,x,method=_RETURNVERBOSE)

[Out]

25*x*(-4*x*ln(2)^2+4*x^2*ln(2)-x^3+12*exp(4)*ln(2)-6*x*exp(4))

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maxima [A]  time = 0.37, size = 35, normalized size = 1.09 \begin {gather*} -25 \, x^{4} - 100 \, x^{2} \log \relax (2)^{2} - 150 \, x^{2} e^{4} + 100 \, {\left (x^{3} + 3 \, x e^{4}\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-200*x*log(2)^2+2*(150*exp(4)+150*x^2)*log(2)-300*x*exp(4)-100*x^3,x, algorithm="maxima")

[Out]

-25*x^4 - 100*x^2*log(2)^2 - 150*x^2*e^4 + 100*(x^3 + 3*x*e^4)*log(2)

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mupad [B]  time = 5.73, size = 36, normalized size = 1.12 \begin {gather*} -25\,x^4+100\,\ln \relax (2)\,x^3+\left (-150\,{\mathrm {e}}^4-100\,{\ln \relax (2)}^2\right )\,x^2+300\,{\mathrm {e}}^4\,\ln \relax (2)\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*log(2)*(150*exp(4) + 150*x^2) - 300*x*exp(4) - 200*x*log(2)^2 - 100*x^3,x)

[Out]

100*x^3*log(2) - x^2*(150*exp(4) + 100*log(2)^2) - 25*x^4 + 300*x*exp(4)*log(2)

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sympy [A]  time = 0.07, size = 39, normalized size = 1.22 \begin {gather*} - 25 x^{4} + 100 x^{3} \log {\relax (2 )} + x^{2} \left (- 150 e^{4} - 100 \log {\relax (2 )}^{2}\right ) + 300 x e^{4} \log {\relax (2 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-200*x*ln(2)**2+2*(150*exp(4)+150*x**2)*ln(2)-300*x*exp(4)-100*x**3,x)

[Out]

-25*x**4 + 100*x**3*log(2) + x**2*(-150*exp(4) - 100*log(2)**2) + 300*x*exp(4)*log(2)

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