3.4.69 \(\int \frac {e^{-\frac {-50 x+25 x^2}{-56 x+20 x^2+4 x^3+(7+x) \log (5)}} (3136 x^2-2240 x^3-48 x^4+160 x^5+16 x^6+(-784 x+168 x^2+96 x^3+8 x^4) \log (5)+(49+14 x+x^2) \log ^2(5)+(400 x^3-400 x^4+100 x^5+(350 x-350 x^2-25 x^3) \log (5)) \log (x))}{3136 x^3-2240 x^4-48 x^5+160 x^6+16 x^7+(-784 x^2+168 x^3+96 x^4+8 x^5) \log (5)+(49 x+14 x^2+x^3) \log ^2(5)} \, dx\)

Optimal. Leaf size=36 \[ e^{-\frac {5 x}{4 \left (1+\frac {2+x}{5}\right ) \left (x+\frac {\log (5)}{4 (-2+x)}\right )}} \log (x) \]

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Rubi [B]  time = 0.43, antiderivative size = 222, normalized size of antiderivative = 6.17, number of steps used = 1, number of rules used = 1, integrand size = 202, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {2288} \begin {gather*} \frac {\left (4 x^5-16 x^4+16 x^3+\left (-x^3-14 x^2+14 x\right ) \log (5)\right ) \log (x) \exp \left (-\frac {25 \left (2 x-x^2\right )}{-4 x^3-20 x^2+56 x-(x+7) \log (5)}\right )}{\left (16 x^7+160 x^6-48 x^5-2240 x^4+3136 x^3+\left (x^3+14 x^2+49 x\right ) \log ^2(5)-8 \left (-x^5-12 x^4-21 x^3+98 x^2\right ) \log (5)\right ) \left (\frac {\left (2 x-x^2\right ) \left (-12 x^2-40 x+56-\log (5)\right )}{\left (-4 x^3-20 x^2+56 x-(x+7) \log (5)\right )^2}-\frac {2 (1-x)}{-4 x^3-20 x^2+56 x-(x+7) \log (5)}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3136*x^2 - 2240*x^3 - 48*x^4 + 160*x^5 + 16*x^6 + (-784*x + 168*x^2 + 96*x^3 + 8*x^4)*Log[5] + (49 + 14*x
+ x^2)*Log[5]^2 + (400*x^3 - 400*x^4 + 100*x^5 + (350*x - 350*x^2 - 25*x^3)*Log[5])*Log[x])/(E^((-50*x + 25*x
^2)/(-56*x + 20*x^2 + 4*x^3 + (7 + x)*Log[5]))*(3136*x^3 - 2240*x^4 - 48*x^5 + 160*x^6 + 16*x^7 + (-784*x^2 +
168*x^3 + 96*x^4 + 8*x^5)*Log[5] + (49*x + 14*x^2 + x^3)*Log[5]^2)),x]

[Out]

((16*x^3 - 16*x^4 + 4*x^5 + (14*x - 14*x^2 - x^3)*Log[5])*Log[x])/(E^((25*(2*x - x^2))/(56*x - 20*x^2 - 4*x^3
- (7 + x)*Log[5]))*(3136*x^3 - 2240*x^4 - 48*x^5 + 160*x^6 + 16*x^7 - 8*(98*x^2 - 21*x^3 - 12*x^4 - x^5)*Log[5
] + (49*x + 14*x^2 + x^3)*Log[5]^2)*(((2*x - x^2)*(56 - 40*x - 12*x^2 - Log[5]))/(56*x - 20*x^2 - 4*x^3 - (7 +
x)*Log[5])^2 - (2*(1 - x))/(56*x - 20*x^2 - 4*x^3 - (7 + x)*Log[5])))

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 {\exp \left (-\frac {25 \left (2 x-x^2\right )}{56 x-20 x^2-4 x^3-(7+x) \log (5)}\right ) \left (16 x^3-16 x^4+4 x^5+\left (14 x-14 x^2-x^3\right ) \log (5)\right ) \log (x)}{\left (3136 x^3-2240 x^4-48 x^5+160 x^6+16 x^7-8 \left (98 x^2-21 x^3-12 x^4-x^5\right ) \log (5)+\left (49 x+14 x^2+x^3\right ) \log ^2(5)\right ) \left (\frac {\left (2 x-x^2\right ) \left (56-40 x-12 x^2-\log (5)\right )}{\left (56 x-20 x^2-4 x^3-(7+x) \log (5)\right )^2}-\frac {2 (1-x)}{56 x-20 x^2-4 x^3-(7+x) \log (5)}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 44, normalized size = 1.22 \begin {gather*} 5^{-\frac {25 (-9+x)}{(252+\log (5)) \left (-8 x+4 x^2+\log (5)\right )}} e^{-\frac {1575}{(7+x) (252+\log (5))}} \log (x) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(3136*x^2 - 2240*x^3 - 48*x^4 + 160*x^5 + 16*x^6 + (-784*x + 168*x^2 + 96*x^3 + 8*x^4)*Log[5] + (49
+ 14*x + x^2)*Log[5]^2 + (400*x^3 - 400*x^4 + 100*x^5 + (350*x - 350*x^2 - 25*x^3)*Log[5])*Log[x])/(E^((-50*x
+ 25*x^2)/(-56*x + 20*x^2 + 4*x^3 + (7 + x)*Log[5]))*(3136*x^3 - 2240*x^4 - 48*x^5 + 160*x^6 + 16*x^7 + (-784*
x^2 + 168*x^3 + 96*x^4 + 8*x^5)*Log[5] + (49*x + 14*x^2 + x^3)*Log[5]^2)),x]

[Out]

Log[x]/(5^((25*(-9 + x))/((252 + Log[5])*(-8*x + 4*x^2 + Log[5])))*E^(1575/((7 + x)*(252 + Log[5]))))

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fricas [A]  time = 0.60, size = 35, normalized size = 0.97 \begin {gather*} e^{\left (-\frac {25 \, {\left (x^{2} - 2 \, x\right )}}{4 \, x^{3} + 20 \, x^{2} + {\left (x + 7\right )} \log \relax (5) - 56 \, x}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-25*x^3-350*x^2+350*x)*log(5)+100*x^5-400*x^4+400*x^3)*log(x)+(x^2+14*x+49)*log(5)^2+(8*x^4+96*x^
3+168*x^2-784*x)*log(5)+16*x^6+160*x^5-48*x^4-2240*x^3+3136*x^2)/((x^3+14*x^2+49*x)*log(5)^2+(8*x^5+96*x^4+168
*x^3-784*x^2)*log(5)+16*x^7+160*x^6-48*x^5-2240*x^4+3136*x^3)/exp((25*x^2-50*x)/((x+7)*log(5)+4*x^3+20*x^2-56*
x)),x, algorithm="fricas")

[Out]

e^(-25*(x^2 - 2*x)/(4*x^3 + 20*x^2 + (x + 7)*log(5) - 56*x))*log(x)

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giac [A]  time = 1.24, size = 37, normalized size = 1.03 \begin {gather*} e^{\left (-\frac {25 \, {\left (x^{2} - 2 \, x\right )}}{4 \, x^{3} + 20 \, x^{2} + x \log \relax (5) - 56 \, x + 7 \, \log \relax (5)}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-25*x^3-350*x^2+350*x)*log(5)+100*x^5-400*x^4+400*x^3)*log(x)+(x^2+14*x+49)*log(5)^2+(8*x^4+96*x^
3+168*x^2-784*x)*log(5)+16*x^6+160*x^5-48*x^4-2240*x^3+3136*x^2)/((x^3+14*x^2+49*x)*log(5)^2+(8*x^5+96*x^4+168
*x^3-784*x^2)*log(5)+16*x^7+160*x^6-48*x^5-2240*x^4+3136*x^3)/exp((25*x^2-50*x)/((x+7)*log(5)+4*x^3+20*x^2-56*
x)),x, algorithm="giac")

[Out]

e^(-25*(x^2 - 2*x)/(4*x^3 + 20*x^2 + x*log(5) - 56*x + 7*log(5)))*log(x)

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maple [A]  time = 0.06, size = 29, normalized size = 0.81

method	result	size
risch	\({\mathrm e}^{-\frac {25 \left (x -2\right ) x}{\left (x +7\right ) \left (4 x^{2}+\ln \relax (5)-8 x \right )}} \ln \relax (x )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-25*x^3-350*x^2+350*x)*ln(5)+100*x^5-400*x^4+400*x^3)*ln(x)+(x^2+14*x+49)*ln(5)^2+(8*x^4+96*x^3+168*x^2
-784*x)*ln(5)+16*x^6+160*x^5-48*x^4-2240*x^3+3136*x^2)/((x^3+14*x^2+49*x)*ln(5)^2+(8*x^5+96*x^4+168*x^3-784*x^
2)*ln(5)+16*x^7+160*x^6-48*x^5-2240*x^4+3136*x^3)/exp((25*x^2-50*x)/((x+7)*ln(5)+4*x^3+20*x^2-56*x)),x,method=
_RETURNVERBOSE)

[Out]

exp(-25*(x-2)*x/(x+7)/(4*x^2+ln(5)-8*x))*ln(x)

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maxima [B]  time = 1.25, size = 84, normalized size = 2.33 \begin {gather*} e^{\left (-\frac {25 \, x \log \relax (5)}{4 \, x^{2} {\left (\log \relax (5) + 252\right )} - 8 \, x {\left (\log \relax (5) + 252\right )} + \log \relax (5)^{2} + 252 \, \log \relax (5)} + \frac {225 \, \log \relax (5)}{4 \, x^{2} {\left (\log \relax (5) + 252\right )} - 8 \, x {\left (\log \relax (5) + 252\right )} + \log \relax (5)^{2} + 252 \, \log \relax (5)} - \frac {1575}{x {\left (\log \relax (5) + 252\right )} + 7 \, \log \relax (5) + 1764}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-25*x^3-350*x^2+350*x)*log(5)+100*x^5-400*x^4+400*x^3)*log(x)+(x^2+14*x+49)*log(5)^2+(8*x^4+96*x^
3+168*x^2-784*x)*log(5)+16*x^6+160*x^5-48*x^4-2240*x^3+3136*x^2)/((x^3+14*x^2+49*x)*log(5)^2+(8*x^5+96*x^4+168
*x^3-784*x^2)*log(5)+16*x^7+160*x^6-48*x^5-2240*x^4+3136*x^3)/exp((25*x^2-50*x)/((x+7)*log(5)+4*x^3+20*x^2-56*
x)),x, algorithm="maxima")

[Out]

e^(-25*x*log(5)/(4*x^2*(log(5) + 252) - 8*x*(log(5) + 252) + log(5)^2 + 252*log(5)) + 225*log(5)/(4*x^2*(log(5
) + 252) - 8*x*(log(5) + 252) + log(5)^2 + 252*log(5)) - 1575/(x*(log(5) + 252) + 7*log(5) + 1764))*log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {50\,x-25\,x^2}{\ln \relax (5)\,\left (x+7\right )-56\,x+20\,x^2+4\,x^3}}\,\left ({\ln \relax (5)}^2\,\left (x^2+14\,x+49\right )+\ln \relax (5)\,\left (8\,x^4+96\,x^3+168\,x^2-784\,x\right )-\ln \relax (x)\,\left (\ln \relax (5)\,\left (25\,x^3+350\,x^2-350\,x\right )-400\,x^3+400\,x^4-100\,x^5\right )+3136\,x^2-2240\,x^3-48\,x^4+160\,x^5+16\,x^6\right )}{\ln \relax (5)\,\left (8\,x^5+96\,x^4+168\,x^3-784\,x^2\right )+{\ln \relax (5)}^2\,\left (x^3+14\,x^2+49\,x\right )+3136\,x^3-2240\,x^4-48\,x^5+160\,x^6+16\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((50*x - 25*x^2)/(log(5)*(x + 7) - 56*x + 20*x^2 + 4*x^3))*(log(5)^2*(14*x + x^2 + 49) + log(5)*(168*x
^2 - 784*x + 96*x^3 + 8*x^4) - log(x)*(log(5)*(350*x^2 - 350*x + 25*x^3) - 400*x^3 + 400*x^4 - 100*x^5) + 3136
*x^2 - 2240*x^3 - 48*x^4 + 160*x^5 + 16*x^6))/(log(5)*(168*x^3 - 784*x^2 + 96*x^4 + 8*x^5) + log(5)^2*(49*x +
14*x^2 + x^3) + 3136*x^3 - 2240*x^4 - 48*x^5 + 160*x^6 + 16*x^7),x)

[Out]

int((exp((50*x - 25*x^2)/(log(5)*(x + 7) - 56*x + 20*x^2 + 4*x^3))*(log(5)^2*(14*x + x^2 + 49) + log(5)*(168*x
^2 - 784*x + 96*x^3 + 8*x^4) - log(x)*(log(5)*(350*x^2 - 350*x + 25*x^3) - 400*x^3 + 400*x^4 - 100*x^5) + 3136
*x^2 - 2240*x^3 - 48*x^4 + 160*x^5 + 16*x^6))/(log(5)*(168*x^3 - 784*x^2 + 96*x^4 + 8*x^5) + log(5)^2*(49*x +
14*x^2 + x^3) + 3136*x^3 - 2240*x^4 - 48*x^5 + 160*x^6 + 16*x^7), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-25*x**3-350*x**2+350*x)*ln(5)+100*x**5-400*x**4+400*x**3)*ln(x)+(x**2+14*x+49)*ln(5)**2+(8*x**4+
96*x**3+168*x**2-784*x)*ln(5)+16*x**6+160*x**5-48*x**4-2240*x**3+3136*x**2)/((x**3+14*x**2+49*x)*ln(5)**2+(8*x
**5+96*x**4+168*x**3-784*x**2)*ln(5)+16*x**7+160*x**6-48*x**5-2240*x**4+3136*x**3)/exp((25*x**2-50*x)/((x+7)*l
n(5)+4*x**3+20*x**2-56*x)),x)

[Out]

Timed out

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