3.59.78 \(\int \frac {81 x^6+36 x^7+3 x^8+e^{\frac {2 (16+8 x+x^2)}{x^4}} (-3200-1200 x-100 x^2-25 x^4)+(90 x^6+20 x^7) \log (25)+25 x^6 \log ^2(25)+e^{\frac {16+8 x+x^2}{x^4}} (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+(-3200 x-1200 x^2-100 x^3) \log (25))}{x^6} \, dx\)

Optimal. Leaf size=27 \[ x \left (9+x+5 \left (\frac {e^{\frac {(4+x)^2}{x^4}}}{x}+\log (25)\right )\right )^2 \]

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Rubi [B]  time = 0.33, antiderivative size = 141, normalized size of antiderivative = 5.22, number of steps used = 5, number of rules used = 3, integrand size = 130, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6, 14, 2288} \begin {gather*} x^3+2 x^2 (9+5 \log (25))-\frac {25 e^{\frac {2 (x+4)^2}{x^4}} \left (x^2+12 x+32\right )}{\left (\frac {x+4}{x^4}-\frac {2 (x+4)^2}{x^5}\right ) x^6}-\frac {10 e^{\frac {(x+4)^2}{x^4}} \left (x^3+x^2 (21+10 \log (5))+20 x (7+6 \log (5))+32 (9+5 \log (25))\right )}{\left (\frac {x+4}{x^4}-\frac {2 (x+4)^2}{x^5}\right ) x^5}+x (81+10 (9+5 \log (5)) \log (25)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(81*x^6 + 36*x^7 + 3*x^8 + E^((2*(16 + 8*x + x^2))/x^4)*(-3200 - 1200*x - 100*x^2 - 25*x^4) + (90*x^6 + 20
*x^7)*Log[25] + 25*x^6*Log[25]^2 + E^((16 + 8*x + x^2)/x^4)*(-5760*x - 2800*x^2 - 420*x^3 - 20*x^4 + 10*x^6 +
(-3200*x - 1200*x^2 - 100*x^3)*Log[25]))/x^6,x]

[Out]

x^3 - (25*E^((2*(4 + x)^2)/x^4)*(32 + 12*x + x^2))/(x^6*((4 + x)/x^4 - (2*(4 + x)^2)/x^5)) + 2*x^2*(9 + 5*Log[
25]) + x*(81 + 10*(9 + 5*Log[5])*Log[25]) - (10*E^((4 + x)^2/x^4)*(x^3 + 20*x*(7 + 6*Log[5]) + x^2*(21 + 10*Lo
g[5]) + 32*(9 + 5*Log[25])))/(x^5*((4 + x)/x^4 - (2*(4 + x)^2)/x^5))

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]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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} &=\int \frac {36 x^7+3 x^8+e^{\frac {2 \left (16+8 x+x^2\right )}{x^4}} \left (-3200-1200 x-100 x^2-25 x^4\right )+\left (90 x^6+20 x^7\right ) \log (25)+e^{\frac {16+8 x+x^2}{x^4}} \left (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+\left (-3200 x-1200 x^2-100 x^3\right ) \log (25)\right )+x^6 \left (81+25 \log ^2(25)\right )}{x^6} \, dx\\ &=\int \left (3 x^2-\frac {25 e^{\frac {2 (4+x)^2}{x^4}} \left (128+48 x+4 x^2+x^4\right )}{x^6}+\frac {10 e^{\frac {(4+x)^2}{x^4}} \left (-2 x^3+x^5-42 x^2 \left (1+\frac {10 \log (5)}{21}\right )-280 x \left (1+\frac {6 \log (5)}{7}\right )-576 \left (1+\frac {5 \log (25)}{9}\right )\right )}{x^5}+36 x \left (1+\frac {5 \log (25)}{9}\right )+81 \left (1+\frac {10}{81} (9+5 \log (5)) \log (25)\right )\right ) \, dx\\ &=x^3+2 x^2 (9+5 \log (25))+x (81+10 (9+5 \log (5)) \log (25))+10 \int \frac {e^{\frac {(4+x)^2}{x^4}} \left (-2 x^3+x^5-42 x^2 \left (1+\frac {10 \log (5)}{21}\right )-280 x \left (1+\frac {6 \log (5)}{7}\right )-576 \left (1+\frac {5 \log (25)}{9}\right )\right )}{x^5} \, dx-25 \int \frac {e^{\frac {2 (4+x)^2}{x^4}} \left (128+48 x+4 x^2+x^4\right )}{x^6} \, dx\\ &=x^3-\frac {25 e^{\frac {2 (4+x)^2}{x^4}} \left (32+12 x+x^2\right )}{x^6 \left (\frac {4+x}{x^4}-\frac {2 (4+x)^2}{x^5}\right )}+2 x^2 (9+5 \log (25))+x (81+10 (9+5 \log (5)) \log (25))-\frac {10 e^{\frac {(4+x)^2}{x^4}} \left (x^3+20 x (7+6 \log (5))+x^2 (21+10 \log (5))+32 (9+5 \log (25))\right )}{x^5 \left (\frac {4+x}{x^4}-\frac {2 (4+x)^2}{x^5}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 29, normalized size = 1.07 \begin {gather*} \frac {\left (5 e^{\frac {(4+x)^2}{x^4}}+x (9+x+5 \log (25))\right )^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(81*x^6 + 36*x^7 + 3*x^8 + E^((2*(16 + 8*x + x^2))/x^4)*(-3200 - 1200*x - 100*x^2 - 25*x^4) + (90*x^
6 + 20*x^7)*Log[25] + 25*x^6*Log[25]^2 + E^((16 + 8*x + x^2)/x^4)*(-5760*x - 2800*x^2 - 420*x^3 - 20*x^4 + 10*
x^6 + (-3200*x - 1200*x^2 - 100*x^3)*Log[25]))/x^6,x]

[Out]

(5*E^((4 + x)^2/x^4) + x*(9 + x + 5*Log[25]))^2/x

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fricas [B]  time = 0.94, size = 83, normalized size = 3.07 \begin {gather*} \frac {x^{4} + 100 \, x^{2} \log \relax (5)^{2} + 18 \, x^{3} + 81 \, x^{2} + 10 \, {\left (x^{2} + 10 \, x \log \relax (5) + 9 \, x\right )} e^{\left (\frac {x^{2} + 8 \, x + 16}{x^{4}}\right )} + 20 \, {\left (x^{3} + 9 \, x^{2}\right )} \log \relax (5) + 25 \, e^{\left (\frac {2 \, {\left (x^{2} + 8 \, x + 16\right )}}{x^{4}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^4-100*x^2-1200*x-3200)*exp((x^2+8*x+16)/x^4)^2+(2*(-100*x^3-1200*x^2-3200*x)*log(5)+10*x^6-2
0*x^4-420*x^3-2800*x^2-5760*x)*exp((x^2+8*x+16)/x^4)+100*x^6*log(5)^2+2*(20*x^7+90*x^6)*log(5)+3*x^8+36*x^7+81
*x^6)/x^6,x, algorithm="fricas")

[Out]

(x^4 + 100*x^2*log(5)^2 + 18*x^3 + 81*x^2 + 10*(x^2 + 10*x*log(5) + 9*x)*e^((x^2 + 8*x + 16)/x^4) + 20*(x^3 +
9*x^2)*log(5) + 25*e^(2*(x^2 + 8*x + 16)/x^4))/x

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giac [B]  time = 0.18, size = 109, normalized size = 4.04 \begin {gather*} \frac {x^{4} + 20 \, x^{3} \log \relax (5) + 100 \, x^{2} \log \relax (5)^{2} + 18 \, x^{3} + 10 \, x^{2} e^{\left (\frac {x^{2} + 8 \, x + 16}{x^{4}}\right )} + 180 \, x^{2} \log \relax (5) + 100 \, x e^{\left (\frac {x^{2} + 8 \, x + 16}{x^{4}}\right )} \log \relax (5) + 81 \, x^{2} + 90 \, x e^{\left (\frac {x^{2} + 8 \, x + 16}{x^{4}}\right )} + 25 \, e^{\left (\frac {2 \, {\left (x^{2} + 8 \, x + 16\right )}}{x^{4}}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^4-100*x^2-1200*x-3200)*exp((x^2+8*x+16)/x^4)^2+(2*(-100*x^3-1200*x^2-3200*x)*log(5)+10*x^6-2
0*x^4-420*x^3-2800*x^2-5760*x)*exp((x^2+8*x+16)/x^4)+100*x^6*log(5)^2+2*(20*x^7+90*x^6)*log(5)+3*x^8+36*x^7+81
*x^6)/x^6,x, algorithm="giac")

[Out]

(x^4 + 20*x^3*log(5) + 100*x^2*log(5)^2 + 18*x^3 + 10*x^2*e^((x^2 + 8*x + 16)/x^4) + 180*x^2*log(5) + 100*x*e^
((x^2 + 8*x + 16)/x^4)*log(5) + 81*x^2 + 90*x*e^((x^2 + 8*x + 16)/x^4) + 25*e^(2*(x^2 + 8*x + 16)/x^4))/x

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maple [B]  time = 0.24, size = 68, normalized size = 2.52

method	result	size
risch	\(100 x \ln \relax (5)^{2}+20 x^{2} \ln \relax (5)+x^{3}+180 x \ln \relax (5)+18 x^{2}+81 x +\frac {25 \,{\mathrm e}^{\frac {2 \left (4+x \right )^{2}}{x^{4}}}}{x}+\left (90+100 \ln \relax (5)+10 x \right ) {\mathrm e}^{\frac {\left (4+x \right )^{2}}{x^{4}}}\)	\(68\)
norman	\(\frac {x^{8}+\left (20 \ln \relax (5)+18\right ) x^{7}+\left (100 \ln \relax (5)^{2}+180 \ln \relax (5)+81\right ) x^{6}+\left (90+100 \ln \relax (5)\right ) x^{5} {\mathrm e}^{\frac {x^{2}+8 x +16}{x^{4}}}+10 \,{\mathrm e}^{\frac {x^{2}+8 x +16}{x^{4}}} x^{6}+25 \,{\mathrm e}^{\frac {2 x^{2}+16 x +32}{x^{4}}} x^{4}}{x^{5}}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-25*x^4-100*x^2-1200*x-3200)*exp((x^2+8*x+16)/x^4)^2+(2*(-100*x^3-1200*x^2-3200*x)*ln(5)+10*x^6-20*x^4-4
20*x^3-2800*x^2-5760*x)*exp((x^2+8*x+16)/x^4)+100*x^6*ln(5)^2+2*(20*x^7+90*x^6)*ln(5)+3*x^8+36*x^7+81*x^6)/x^6
,x,method=_RETURNVERBOSE)

[Out]

100*x*ln(5)^2+20*x^2*ln(5)+x^3+180*x*ln(5)+18*x^2+81*x+25/x*exp(2*(4+x)^2/x^4)+(90+100*ln(5)+10*x)*exp((4+x)^2
/x^4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} x^{3} + 20 \, x^{2} \log \relax (5) + 100 \, x \log \relax (5)^{2} + 18 \, x^{2} + 180 \, x \log \relax (5) + 81 \, x + \frac {25 \, e^{\left (\frac {2}{x^{2}} + \frac {16}{x^{3}} + \frac {32}{x^{4}}\right )}}{x} + \int \frac {10 \, {\left (x^{5} - 2 \, x^{3} - 2 \, x^{2} {\left (10 \, \log \relax (5) + 21\right )} - 40 \, x {\left (6 \, \log \relax (5) + 7\right )} - 640 \, \log \relax (5) - 576\right )} e^{\left (\frac {1}{x^{2}} + \frac {8}{x^{3}} + \frac {16}{x^{4}}\right )}}{x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x^4-100*x^2-1200*x-3200)*exp((x^2+8*x+16)/x^4)^2+(2*(-100*x^3-1200*x^2-3200*x)*log(5)+10*x^6-2
0*x^4-420*x^3-2800*x^2-5760*x)*exp((x^2+8*x+16)/x^4)+100*x^6*log(5)^2+2*(20*x^7+90*x^6)*log(5)+3*x^8+36*x^7+81
*x^6)/x^6,x, algorithm="maxima")

[Out]

x^3 + 20*x^2*log(5) + 100*x*log(5)^2 + 18*x^2 + 180*x*log(5) + 81*x + 25*e^(2/x^2 + 16/x^3 + 32/x^4)/x + integ
rate(10*(x^5 - 2*x^3 - 2*x^2*(10*log(5) + 21) - 40*x*(6*log(5) + 7) - 640*log(5) - 576)*e^(1/x^2 + 8/x^3 + 16/
x^4)/x^5, x)

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mupad [B]  time = 4.47, size = 33, normalized size = 1.22 \begin {gather*} \frac {{\left (9\,x+5\,{\mathrm {e}}^{\frac {x^2+8\,x+16}{x^4}}+10\,x\,\ln \relax (5)+x^2\right )}^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*x^6*log(5)^2 - exp((8*x + x^2 + 16)/x^4)*(5760*x + 2*log(5)*(3200*x + 1200*x^2 + 100*x^3) + 2800*x^2
+ 420*x^3 + 20*x^4 - 10*x^6) + 2*log(5)*(90*x^6 + 20*x^7) - exp((2*(8*x + x^2 + 16))/x^4)*(1200*x + 100*x^2 +
25*x^4 + 3200) + 81*x^6 + 36*x^7 + 3*x^8)/x^6,x)

[Out]

(9*x + 5*exp((8*x + x^2 + 16)/x^4) + 10*x*log(5) + x^2)^2/x

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sympy [B]  time = 0.25, size = 75, normalized size = 2.78 \begin {gather*} x^{3} + x^{2} \left (18 + 20 \log {\relax (5 )}\right ) + x \left (81 + 100 \log {\relax (5 )}^{2} + 180 \log {\relax (5 )}\right ) + \frac {\left (10 x^{2} + 90 x + 100 x \log {\relax (5 )}\right ) e^{\frac {x^{2} + 8 x + 16}{x^{4}}} + 25 e^{\frac {2 \left (x^{2} + 8 x + 16\right )}{x^{4}}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-25*x**4-100*x**2-1200*x-3200)*exp((x**2+8*x+16)/x**4)**2+(2*(-100*x**3-1200*x**2-3200*x)*ln(5)+10
*x**6-20*x**4-420*x**3-2800*x**2-5760*x)*exp((x**2+8*x+16)/x**4)+100*x**6*ln(5)**2+2*(20*x**7+90*x**6)*ln(5)+3
*x**8+36*x**7+81*x**6)/x**6,x)

[Out]

x**3 + x**2*(18 + 20*log(5)) + x*(81 + 100*log(5)**2 + 180*log(5)) + ((10*x**2 + 90*x + 100*x*log(5))*exp((x**
2 + 8*x + 16)/x**4) + 25*exp(2*(x**2 + 8*x + 16)/x**4))/x

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