3.67.23 \(\int \frac {e^x (-1600 x^2-64 x^5)+e^{5 e^{-x}} (e^x (3200-256 x^3)+(16000 x+640 x^4) \log (\frac {25+x^3}{x}))}{e^x (25 x^7+x^{10})+e^{5 e^{-x}+x} (300 x^5+12 x^8) \log (\frac {25+x^3}{x})+e^{10 e^{-x}+x} (1200 x^3+48 x^6) \log ^2(\frac {25+x^3}{x})+e^{15 e^{-x}+x} (1600 x+64 x^4) \log ^3(\frac {25+x^3}{x})} \, dx\)

Optimal. Leaf size=29 \[ \frac {16}{\left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25}{x}+x^2\right )\right )^2} \]

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Rubi [F]  time = 20.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^x \left (-1600 x^2-64 x^5\right )+e^{5 e^{-x}} \left (e^x \left (3200-256 x^3\right )+\left (16000 x+640 x^4\right ) \log \left (\frac {25+x^3}{x}\right )\right )}{e^x \left (25 x^7+x^{10}\right )+e^{5 e^{-x}+x} \left (300 x^5+12 x^8\right ) \log \left (\frac {25+x^3}{x}\right )+e^{10 e^{-x}+x} \left (1200 x^3+48 x^6\right ) \log ^2\left (\frac {25+x^3}{x}\right )+e^{15 e^{-x}+x} \left (1600 x+64 x^4\right ) \log ^3\left (\frac {25+x^3}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-1600*x^2 - 64*x^5) + E^(5/E^x)*(E^x*(3200 - 256*x^3) + (16000*x + 640*x^4)*Log[(25 + x^3)/x]))/(E^x
*(25*x^7 + x^10) + E^(5/E^x + x)*(300*x^5 + 12*x^8)*Log[(25 + x^3)/x] + E^(10/E^x + x)*(1200*x^3 + 48*x^6)*Log
[(25 + x^3)/x]^2 + E^(15/E^x + x)*(1600*x + 64*x^4)*Log[(25 + x^3)/x]^3),x]

[Out]

-64*Defer[Int][x/(x^2 + 4*E^(5/E^x)*Log[(25 + x^3)/x])^3, x] + 64*Defer[Int][x/(Log[(25 + x^3)/x]*(x^2 + 4*E^(
5/E^x)*Log[(25 + x^3)/x])^3), x] + 160*5^(1/3)*Defer[Int][1/((5^(2/3) + x)*Log[(25 + x^3)/x]*(x^2 + 4*E^(5/E^x
)*Log[(25 + x^3)/x])^3), x] + 160*(-1)^(2/3)*5^(1/3)*Defer[Int][1/((5^(2/3) - (-1)^(1/3)*x)*Log[(25 + x^3)/x]*
(x^2 + 4*E^(5/E^x)*Log[(25 + x^3)/x])^3), x] - 160*(-5)^(1/3)*Defer[Int][1/((5^(2/3) + (-1)^(2/3)*x)*Log[(25 +
x^3)/x]*(x^2 + 4*E^(5/E^x)*Log[(25 + x^3)/x])^3), x] + 640*Defer[Int][(E^(5/E^x - x)*Log[(25 + x^3)/x])/(x^2
+ 4*E^(5/E^x)*Log[(25 + x^3)/x])^3, x] + 32*Defer[Int][1/(x*Log[(25 + x^3)/x]*(x^2 + 4*E^(5/E^x)*Log[(25 + x^3
)/x])^2), x] - 32*Defer[Int][1/(((-5)^(2/3) + x)*Log[(25 + x^3)/x]*(x^2 + 4*E^(5/E^x)*Log[(25 + x^3)/x])^2), x
] - 32*Defer[Int][1/((5^(2/3) + x)*Log[(25 + x^3)/x]*(x^2 + 4*E^(5/E^x)*Log[(25 + x^3)/x])^2), x] - 32*Defer[I
nt][1/((-((-1)^(1/3)*5^(2/3)) + x)*Log[(25 + x^3)/x]*(x^2 + 4*E^(5/E^x)*Log[(25 + x^3)/x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (-64 e^x \left (x^2 \left (25+x^3\right )+e^{5 e^{-x}} \left (-50+4 x^3\right )\right )+640 e^{5 e^{-x}} x \left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right )\right )}{x \left (25+x^3\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx\\ &=\int \left (-\frac {64 \left (-50 e^{5 e^{-x}}+25 x^2+4 e^{5 e^{-x}} x^3+x^5\right )}{x \left (25+x^3\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3}+\frac {640 e^{5 e^{-x}-x} \log \left (\frac {25+x^3}{x}\right )}{\left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3}\right ) \, dx\\ &=-\left (64 \int \frac {-50 e^{5 e^{-x}}+25 x^2+4 e^{5 e^{-x}} x^3+x^5}{x \left (25+x^3\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx\right )+640 \int \frac {e^{5 e^{-x}-x} \log \left (\frac {25+x^3}{x}\right )}{\left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx\\ &=-\left (64 \int \left (\frac {-25+2 x^3}{2 x \left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2}+\frac {x \left (25-2 x^3+50 \log \left (\frac {25+x^3}{x}\right )+2 x^3 \log \left (\frac {25+x^3}{x}\right )\right )}{2 \left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3}\right ) \, dx\right )+640 \int \frac {e^{5 e^{-x}-x} \log \left (\frac {25+x^3}{x}\right )}{\left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx\\ &=-\left (32 \int \frac {-25+2 x^3}{x \left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2} \, dx\right )-32 \int \frac {x \left (25-2 x^3+50 \log \left (\frac {25+x^3}{x}\right )+2 x^3 \log \left (\frac {25+x^3}{x}\right )\right )}{\left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx+640 \int \frac {e^{5 e^{-x}-x} \log \left (\frac {25+x^3}{x}\right )}{\left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx\\ &=-\left (32 \int \frac {x \left (25-2 x^3+2 \left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right )\right )}{\left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx\right )-32 \int \left (-\frac {1}{x \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2}+\frac {3 x^2}{\left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2}\right ) \, dx+640 \int \frac {e^{5 e^{-x}-x} \log \left (\frac {25+x^3}{x}\right )}{\left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx\\ &=32 \int \frac {1}{x \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2} \, dx-32 \int \left (\frac {50 x}{\left (25+x^3\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3}+\frac {2 x^4}{\left (25+x^3\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3}+\frac {25 x}{\left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3}-\frac {2 x^4}{\left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3}\right ) \, dx-96 \int \frac {x^2}{\left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2} \, dx+640 \int \frac {e^{5 e^{-x}-x} \log \left (\frac {25+x^3}{x}\right )}{\left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx\\ &=32 \int \frac {1}{x \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2} \, dx-64 \int \frac {x^4}{\left (25+x^3\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx+64 \int \frac {x^4}{\left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx-96 \int \left (\frac {1}{3 \left ((-5)^{2/3}+x\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2}+\frac {1}{3 \left (5^{2/3}+x\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2}+\frac {1}{3 \left (-\sqrt [3]{-1} 5^{2/3}+x\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2}\right ) \, dx+640 \int \frac {e^{5 e^{-x}-x} \log \left (\frac {25+x^3}{x}\right )}{\left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx-800 \int \frac {x}{\left (25+x^3\right ) \log \left (\frac {25+x^3}{x}\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx-1600 \int \frac {x}{\left (25+x^3\right ) \left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 29, normalized size = 1.00 \begin {gather*} \frac {16}{\left (x^2+4 e^{5 e^{-x}} \log \left (\frac {25+x^3}{x}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-1600*x^2 - 64*x^5) + E^(5/E^x)*(E^x*(3200 - 256*x^3) + (16000*x + 640*x^4)*Log[(25 + x^3)/x])
)/(E^x*(25*x^7 + x^10) + E^(5/E^x + x)*(300*x^5 + 12*x^8)*Log[(25 + x^3)/x] + E^(10/E^x + x)*(1200*x^3 + 48*x^
6)*Log[(25 + x^3)/x]^2 + E^(15/E^x + x)*(1600*x + 64*x^4)*Log[(25 + x^3)/x]^3),x]

[Out]

16/(x^2 + 4*E^(5/E^x)*Log[(25 + x^3)/x])^2

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fricas [B]  time = 0.48, size = 73, normalized size = 2.52 \begin {gather*} \frac {16 \, e^{\left (2 \, x\right )}}{x^{4} e^{\left (2 \, x\right )} + 8 \, x^{2} e^{\left ({\left (x e^{x} + 5\right )} e^{\left (-x\right )} + x\right )} \log \left (\frac {x^{3} + 25}{x}\right ) + 16 \, e^{\left (2 \, {\left (x e^{x} + 5\right )} e^{\left (-x\right )}\right )} \log \left (\frac {x^{3} + 25}{x}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((640*x^4+16000*x)*log((x^3+25)/x)+(-256*x^3+3200)*exp(x))*exp(5/exp(x))+(-64*x^5-1600*x^2)*exp(x))
/((64*x^4+1600*x)*exp(x)*log((x^3+25)/x)^3*exp(5/exp(x))^3+(48*x^6+1200*x^3)*exp(x)*log((x^3+25)/x)^2*exp(5/ex
p(x))^2+(12*x^8+300*x^5)*exp(x)*log((x^3+25)/x)*exp(5/exp(x))+(x^10+25*x^7)*exp(x)),x, algorithm="fricas")

[Out]

16*e^(2*x)/(x^4*e^(2*x) + 8*x^2*e^((x*e^x + 5)*e^(-x) + x)*log((x^3 + 25)/x) + 16*e^(2*(x*e^x + 5)*e^(-x))*log
((x^3 + 25)/x)^2)

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giac [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((((640*x^4+16000*x)*log((x^3+25)/x)+(-256*x^3+3200)*exp(x))*exp(5/exp(x))+(-64*x^5-1600*x^2)*exp(x))
/((64*x^4+1600*x)*exp(x)*log((x^3+25)/x)^3*exp(5/exp(x))^3+(48*x^6+1200*x^3)*exp(x)*log((x^3+25)/x)^2*exp(5/ex
p(x))^2+(12*x^8+300*x^5)*exp(x)*log((x^3+25)/x)*exp(5/exp(x))+(x^10+25*x^7)*exp(x)),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (640 x^{4}+16000 x \right ) \ln \left (\frac {x^{3}+25}{x}\right )+\left (-256 x^{3}+3200\right ) {\mathrm e}^{x}\right ) {\mathrm e}^{5 \,{\mathrm e}^{-x}}+\left (-64 x^{5}-1600 x^{2}\right ) {\mathrm e}^{x}}{\left (64 x^{4}+1600 x \right ) {\mathrm e}^{x} \ln \left (\frac {x^{3}+25}{x}\right )^{3} {\mathrm e}^{15 \,{\mathrm e}^{-x}}+\left (48 x^{6}+1200 x^{3}\right ) {\mathrm e}^{x} \ln \left (\frac {x^{3}+25}{x}\right )^{2} {\mathrm e}^{10 \,{\mathrm e}^{-x}}+\left (12 x^{8}+300 x^{5}\right ) {\mathrm e}^{x} \ln \left (\frac {x^{3}+25}{x}\right ) {\mathrm e}^{5 \,{\mathrm e}^{-x}}+\left (x^{10}+25 x^{7}\right ) {\mathrm e}^{x}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((640*x^4+16000*x)*ln((x^3+25)/x)+(-256*x^3+3200)*exp(x))*exp(5/exp(x))+(-64*x^5-1600*x^2)*exp(x))/((64*x
^4+1600*x)*exp(x)*ln((x^3+25)/x)^3*exp(5/exp(x))^3+(48*x^6+1200*x^3)*exp(x)*ln((x^3+25)/x)^2*exp(5/exp(x))^2+(
12*x^8+300*x^5)*exp(x)*ln((x^3+25)/x)*exp(5/exp(x))+(x^10+25*x^7)*exp(x)),x)

[Out]

int((((640*x^4+16000*x)*ln((x^3+25)/x)+(-256*x^3+3200)*exp(x))*exp(5/exp(x))+(-64*x^5-1600*x^2)*exp(x))/((64*x
^4+1600*x)*exp(x)*ln((x^3+25)/x)^3*exp(5/exp(x))^3+(48*x^6+1200*x^3)*exp(x)*ln((x^3+25)/x)^2*exp(5/exp(x))^2+(
12*x^8+300*x^5)*exp(x)*ln((x^3+25)/x)*exp(5/exp(x))+(x^10+25*x^7)*exp(x)),x)

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maxima [B]  time = 1.84, size = 67, normalized size = 2.31 \begin {gather*} \frac {16}{x^{4} + 16 \, {\left (\log \left (x^{3} + 25\right )^{2} - 2 \, \log \left (x^{3} + 25\right ) \log \relax (x) + \log \relax (x)^{2}\right )} e^{\left (10 \, e^{\left (-x\right )}\right )} + 8 \, {\left (x^{2} \log \left (x^{3} + 25\right ) - x^{2} \log \relax (x)\right )} e^{\left (5 \, e^{\left (-x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((640*x^4+16000*x)*log((x^3+25)/x)+(-256*x^3+3200)*exp(x))*exp(5/exp(x))+(-64*x^5-1600*x^2)*exp(x))
/((64*x^4+1600*x)*exp(x)*log((x^3+25)/x)^3*exp(5/exp(x))^3+(48*x^6+1200*x^3)*exp(x)*log((x^3+25)/x)^2*exp(5/ex
p(x))^2+(12*x^8+300*x^5)*exp(x)*log((x^3+25)/x)*exp(5/exp(x))+(x^10+25*x^7)*exp(x)),x, algorithm="maxima")

[Out]

16/(x^4 + 16*(log(x^3 + 25)^2 - 2*log(x^3 + 25)*log(x) + log(x)^2)*e^(10*e^(-x)) + 8*(x^2*log(x^3 + 25) - x^2*
log(x))*e^(5*e^(-x)))

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mupad [B]  time = 4.78, size = 51, normalized size = 1.76 \begin {gather*} \frac {16}{16\,{\mathrm {e}}^{10\,{\mathrm {e}}^{-x}}\,{\ln \left (\frac {x^3+25}{x}\right )}^2+x^4+8\,x^2\,{\mathrm {e}}^{5\,{\mathrm {e}}^{-x}}\,\ln \left (\frac {x^3+25}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(1600*x^2 + 64*x^5) - exp(5*exp(-x))*(log((x^3 + 25)/x)*(16000*x + 640*x^4) - exp(x)*(256*x^3 - 3
200)))/(exp(x)*(25*x^7 + x^10) + exp(10*exp(-x))*exp(x)*log((x^3 + 25)/x)^2*(1200*x^3 + 48*x^6) + exp(5*exp(-x
))*exp(x)*log((x^3 + 25)/x)*(300*x^5 + 12*x^8) + exp(15*exp(-x))*exp(x)*log((x^3 + 25)/x)^3*(1600*x + 64*x^4))
,x)

[Out]

16/(16*exp(10*exp(-x))*log((x^3 + 25)/x)^2 + x^4 + 8*x^2*exp(5*exp(-x))*log((x^3 + 25)/x))

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sympy [A]  time = 0.71, size = 42, normalized size = 1.45 \begin {gather*} \frac {16}{x^{4} + 8 x^{2} e^{5 e^{- x}} \log {\left (\frac {x^{3} + 25}{x} \right )} + 16 e^{10 e^{- x}} \log {\left (\frac {x^{3} + 25}{x} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((640*x**4+16000*x)*ln((x**3+25)/x)+(-256*x**3+3200)*exp(x))*exp(5/exp(x))+(-64*x**5-1600*x**2)*exp
(x))/((64*x**4+1600*x)*exp(x)*ln((x**3+25)/x)**3*exp(5/exp(x))**3+(48*x**6+1200*x**3)*exp(x)*ln((x**3+25)/x)**
2*exp(5/exp(x))**2+(12*x**8+300*x**5)*exp(x)*ln((x**3+25)/x)*exp(5/exp(x))+(x**10+25*x**7)*exp(x)),x)

[Out]

16/(x**4 + 8*x**2*exp(5*exp(-x))*log((x**3 + 25)/x) + 16*exp(10*exp(-x))*log((x**3 + 25)/x)**2)

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