3.49.12 \(\int e^{\frac {2 (2+(450 x+525 x^2) \log (4))}{x}} (-4+2 x+1050 x^2 \log (4)) \, dx\)

Optimal. Leaf size=25 \[ e^{\frac {2 (2+75 x (x+6 (1+x)) \log (4))}{x}} x^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.06, antiderivative size = 69, normalized size of antiderivative = 2.76, number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2288} \begin {gather*} -\frac {4^{\frac {2 \left (525 x^2+450 x\right )}{x}} e^{4/x} \left (2-525 x^2 \log (4)\right )}{\frac {150 (7 x+3) \log (4)}{x}-\frac {75 \left (7 x^2+6 x\right ) \log (4)+2}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^((2*(2 + (450*x + 525*x^2)*Log[4]))/x)*(-4 + 2*x + 1050*x^2*Log[4]),x]

[Out]

-((4^((2*(450*x + 525*x^2))/x)*E^(4/x)*(2 - 525*x^2*Log[4]))/((150*(3 + 7*x)*Log[4])/x - (2 + 75*(6*x + 7*x^2)
*Log[4])/x^2))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {4^{\frac {2 \left (450 x+525 x^2\right )}{x}} e^{4/x} \left (2-525 x^2 \log (4)\right )}{\frac {150 (3+7 x) \log (4)}{x}-\frac {2+75 \left (6 x+7 x^2\right ) \log (4)}{x^2}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 18, normalized size = 0.72 \begin {gather*} 2^{1800+2100 x} e^{4/x} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^((2*(2 + (450*x + 525*x^2)*Log[4]))/x)*(-4 + 2*x + 1050*x^2*Log[4]),x]

[Out]

2^(1800 + 2100*x)*E^(4/x)*x^2

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 25, normalized size = 1.00 \begin {gather*} x^{2} e^{\left (\frac {4 \, {\left (75 \, {\left (7 \, x^{2} + 6 \, x\right )} \log \relax (2) + 1\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2100*x^2*log(2)+2*x-4)*exp((2*(525*x^2+450*x)*log(2)+2)/x)^2,x, algorithm="fricas")

[Out]

x^2*e^(4*(75*(7*x^2 + 6*x)*log(2) + 1)/x)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 24, normalized size = 0.96 \begin {gather*} x^{2} e^{\left (\frac {4 \, {\left (525 \, x^{2} \log \relax (2) + 450 \, x \log \relax (2) + 1\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2100*x^2*log(2)+2*x-4)*exp((2*(525*x^2+450*x)*log(2)+2)/x)^2,x, algorithm="giac")

[Out]

x^2*e^(4*(525*x^2*log(2) + 450*x*log(2) + 1)/x)

________________________________________________________________________________________

maple [A]  time = 0.54, size = 25, normalized size = 1.00

method	result	size
risch	\(x^{2} {\mathrm e}^{\frac {2100 x^{2} \ln \relax (2)+1800 x \ln \relax (2)+4}{x}}\)	\(25\)
gosper	\(x^{2} {\mathrm e}^{\frac {2100 x^{2} \ln \relax (2)+1800 x \ln \relax (2)+4}{x}}\)	\(27\)
norman	\(x^{2} {\mathrm e}^{\frac {4 \left (525 x^{2}+450 x \right ) \ln \relax (2)+4}{x}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2100*x^2*ln(2)+2*x-4)*exp((2*(525*x^2+450*x)*ln(2)+2)/x)^2,x,method=_RETURNVERBOSE)

[Out]

x^2*exp(4*(525*x^2*ln(2)+450*x*ln(2)+1)/x)

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 17, normalized size = 0.68 \begin {gather*} 71448348576730208360402604523024658663907311448489024669693316988935593287322878666163481950176220037593478347105937422686501991894419788796088422137966026262523598150372719976137911322484446114613284904383977643176193557817897027023063420124852033989626806764509137929914787205373413116077254242653423277386226627159120168223623660139965116969572411841665962582988716865792650075294655252525257343163566042824495509307872827973214736884381496689456792434150079470111661811761376161068055664012337698456291039551943299284254570579952324837376 \, x^{2} e^{\left (2100 \, x \log \relax (2) + \frac {4}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2100*x^2*log(2)+2*x-4)*exp((2*(525*x^2+450*x)*log(2)+2)/x)^2,x, algorithm="maxima")

[Out]

71448348576730208360402604523024658663907311448489024669693316988935593287322878666163481950176220037593478347
10593742268650199189441978879608842213796602626252359815037271997613791132248444611461328490438397764317619355
78178970270230634201248520339896268067645091379299147872053734131160772542426534232773862266271591201682236236
60139965116969572411841665962582988716865792650075294655252525257343163566042824495509307872827973214736884381
496689456792434150079470111661811761376161068055664012337698456291039551943299284254570579952324837376*x^2*e^(
2100*x*log(2) + 4/x)

________________________________________________________________________________________

mupad [B]  time = 4.11, size = 17, normalized size = 0.68 \begin {gather*} 2^{2100\,x+1800}\,x^2\,{\mathrm {e}}^{4/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((2*(2*log(2)*(450*x + 525*x^2) + 2))/x)*(2*x + 2100*x^2*log(2) - 4),x)

[Out]

2^(2100*x + 1800)*x^2*exp(4/x)

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 20, normalized size = 0.80 \begin {gather*} x^{2} e^{\frac {2 \left (\left (1050 x^{2} + 900 x\right ) \log {\relax (2 )} + 2\right )}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2100*x**2*ln(2)+2*x-4)*exp((2*(525*x**2+450*x)*ln(2)+2)/x)**2,x)

[Out]

x**2*exp(2*((1050*x**2 + 900*x)*log(2) + 2)/x)

________________________________________________________________________________________