∫ 40x+ee5∕x+x (−20e5∕x+8x+4x2)+e25x2−50x3+25x4 (−4−200x2+600x3−400x4)_________ e50x2−100x3+50x4x2−10e25x2−50x3+25x4x3+25x4+e2e5∕x+2xx4+ee5∕x+x(−2e25x2−50x3+25x4x3+10x4) dx

3.98.55 \(\int \frac {40 x+e^{e^{5/x}+x} (-20 e^{5/x}+8 x+4 x^2)+e^{25 x^2-50 x^3+25 x^4} (-4-200 x^2+600 x^3-400 x^4)}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4)} \, dx\)

Optimal. Leaf size=36 \[ \frac {4}{x \left (e^{25 (-1+x)^2 x^2}-\left (5+e^{e^{5/x}+x}\right ) x\right )} \]

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Rubi [F] time = 164.86, 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 {40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )}{e^{50 x^2-100 x^3+50 x^4} x^2-10 e^{25 x^2-50 x^3+25 x^4} x^3+25 x^4+e^{2 e^{5/x}+2 x} x^4+e^{e^{5/x}+x} \left (-2 e^{25 x^2-50 x^3+25 x^4} x^3+10 x^4\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(40*x + E^(E^(5/x) + x)*(-20*E^(5/x) + 8*x + 4*x^2) + E^(25*x^2 - 50*x^3 + 25*x^4)*(-4 - 200*x^2 + 600*x^3

- 400*x^4))/(E^(50*x^2 - 100*x^3 + 50*x^4)*x^2 - 10*E^(25*x^2 - 50*x^3 + 25*x^4)*x^3 + 25*x^4 + E^(2*E^(5/x)

+ 2*x)*x^4 + E^(E^(5/x) + x)*(-2*E^(25*x^2 - 50*x^3 + 25*x^4)*x^3 + 10*x^4)),x]

[Out]

-200*Defer[Int][E^(50*x^3)/(E^(25*x^2 + 25*x^4) - 5*E^(50*x^3)*x - E^(E^(5/x) + x + 50*x^3)*x), x] + 4*Defer[I

nt][E^(E^(5/x) + x + 100*x^3)/(-E^(25*x^2 + 25*x^4) + 5*E^(50*x^3)*x + E^(E^(5/x) + x + 50*x^3)*x)^2, x] - 20*

Defer[Int][E^(E^(5/x) + 5/x + x + 100*x^3)/(x^2*(-E^(25*x^2 + 25*x^4) + 5*E^(50*x^3)*x + E^(E^(5/x) + x + 50*x

^3)*x)^2), x] + 20*Defer[Int][E^(100*x^3)/(x*(-E^(25*x^2 + 25*x^4) + 5*E^(50*x^3)*x + E^(E^(5/x) + x + 50*x^3)

*x)^2), x] + 4*Defer[Int][E^(E^(5/x) + x + 100*x^3)/(x*(-E^(25*x^2 + 25*x^4) + 5*E^(50*x^3)*x + E^(E^(5/x) + x

+ 50*x^3)*x)^2), x] - 1000*Defer[Int][(E^(100*x^3)*x)/(-E^(25*x^2 + 25*x^4) + 5*E^(50*x^3)*x + E^(E^(5/x) + x

+ 50*x^3)*x)^2, x] - 200*Defer[Int][(E^(E^(5/x) + x + 100*x^3)*x)/(-E^(25*x^2 + 25*x^4) + 5*E^(50*x^3)*x + E^

(E^(5/x) + x + 50*x^3)*x)^2, x] + 3000*Defer[Int][(E^(100*x^3)*x^2)/(-E^(25*x^2 + 25*x^4) + 5*E^(50*x^3)*x + E

^(E^(5/x) + x + 50*x^3)*x)^2, x] + 600*Defer[Int][(E^(E^(5/x) + x + 100*x^3)*x^2)/(-E^(25*x^2 + 25*x^4) + 5*E^

(50*x^3)*x + E^(E^(5/x) + x + 50*x^3)*x)^2, x] - 2000*Defer[Int][(E^(100*x^3)*x^3)/(-E^(25*x^2 + 25*x^4) + 5*E

^(50*x^3)*x + E^(E^(5/x) + x + 50*x^3)*x)^2, x] - 400*Defer[Int][(E^(E^(5/x) + x + 100*x^3)*x^3)/(-E^(25*x^2 +

25*x^4) + 5*E^(50*x^3)*x + E^(E^(5/x) + x + 50*x^3)*x)^2, x] + 4*Defer[Int][E^(50*x^3)/(x^2*(-E^(25*x^2 + 25*

x^4) + 5*E^(50*x^3)*x + E^(E^(5/x) + x + 50*x^3)*x)), x] - 600*Defer[Int][(E^(50*x^3)*x)/(-E^(25*x^2 + 25*x^4)

+ 5*E^(50*x^3)*x + E^(E^(5/x) + x + 50*x^3)*x), x] + 400*Defer[Int][(E^(50*x^3)*x^2)/(-E^(25*x^2 + 25*x^4) +

5*E^(50*x^3)*x + E^(E^(5/x) + x + 50*x^3)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{100 x^3} \left (40 x+e^{e^{5/x}+x} \left (-20 e^{5/x}+8 x+4 x^2\right )+e^{25 x^2-50 x^3+25 x^4} \left (-4-200 x^2+600 x^3-400 x^4\right )\right )}{x^2 \left (e^{25 x^2+25 x^4}-5 e^{50 x^3} x-e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx\\ &=\int \left (\frac {4 e^{50 x^3} \left (1+50 x^2-150 x^3+100 x^4\right )}{x^2 \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )}-\frac {4 e^{100 x^3} \left (5 e^{e^{5/x}+\frac {5}{x}+x}-5 x-e^{e^{5/x}+x} x-e^{e^{5/x}+x} x^2+250 x^3+50 e^{e^{5/x}+x} x^3-750 x^4-150 e^{e^{5/x}+x} x^4+500 x^5+100 e^{e^{5/x}+x} x^5\right )}{x^2 \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}\right ) \, dx\\ &=4 \int \frac {e^{50 x^3} \left (1+50 x^2-150 x^3+100 x^4\right )}{x^2 \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )} \, dx-4 \int \frac {e^{100 x^3} \left (5 e^{e^{5/x}+\frac {5}{x}+x}-5 x-e^{e^{5/x}+x} x-e^{e^{5/x}+x} x^2+250 x^3+50 e^{e^{5/x}+x} x^3-750 x^4-150 e^{e^{5/x}+x} x^4+500 x^5+100 e^{e^{5/x}+x} x^5\right )}{x^2 \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx\\ &=4 \int \left (-\frac {50 e^{50 x^3}}{e^{25 x^2+25 x^4}-5 e^{50 x^3} x-e^{e^{5/x}+x+50 x^3} x}+\frac {e^{50 x^3}}{x^2 \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )}-\frac {150 e^{50 x^3} x}{-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x}+\frac {100 e^{50 x^3} x^2}{-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x}\right ) \, dx-4 \int \frac {e^{100 x^3} \left (5 e^{e^{5/x}+\frac {5}{x}+x}+5 x \left (-1+50 x^2-150 x^3+100 x^4\right )+e^{e^{5/x}+x} x \left (-1-x+50 x^2-150 x^3+100 x^4\right )\right )}{x^2 \left (e^{25 \left (x^2+x^4\right )}-5 e^{50 x^3} x-e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx\\ &=4 \int \frac {e^{50 x^3}}{x^2 \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )} \, dx-4 \int \left (-\frac {e^{e^{5/x}+x+100 x^3}}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}+\frac {5 e^{e^{5/x}+\frac {5}{x}+x+100 x^3}}{x^2 \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}-\frac {5 e^{100 x^3}}{x \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}-\frac {e^{e^{5/x}+x+100 x^3}}{x \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}+\frac {250 e^{100 x^3} x}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}+\frac {50 e^{e^{5/x}+x+100 x^3} x}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}-\frac {750 e^{100 x^3} x^2}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}-\frac {150 e^{e^{5/x}+x+100 x^3} x^2}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}+\frac {500 e^{100 x^3} x^3}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}+\frac {100 e^{e^{5/x}+x+100 x^3} x^3}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2}\right ) \, dx-200 \int \frac {e^{50 x^3}}{e^{25 x^2+25 x^4}-5 e^{50 x^3} x-e^{e^{5/x}+x+50 x^3} x} \, dx+400 \int \frac {e^{50 x^3} x^2}{-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x} \, dx-600 \int \frac {e^{50 x^3} x}{-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x} \, dx\\ &=4 \int \frac {e^{e^{5/x}+x+100 x^3}}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx+4 \int \frac {e^{e^{5/x}+x+100 x^3}}{x \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx+4 \int \frac {e^{50 x^3}}{x^2 \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )} \, dx-20 \int \frac {e^{e^{5/x}+\frac {5}{x}+x+100 x^3}}{x^2 \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx+20 \int \frac {e^{100 x^3}}{x \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx-200 \int \frac {e^{50 x^3}}{e^{25 x^2+25 x^4}-5 e^{50 x^3} x-e^{e^{5/x}+x+50 x^3} x} \, dx-200 \int \frac {e^{e^{5/x}+x+100 x^3} x}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx-400 \int \frac {e^{e^{5/x}+x+100 x^3} x^3}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx+400 \int \frac {e^{50 x^3} x^2}{-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x} \, dx+600 \int \frac {e^{e^{5/x}+x+100 x^3} x^2}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx-600 \int \frac {e^{50 x^3} x}{-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x} \, dx-1000 \int \frac {e^{100 x^3} x}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx-2000 \int \frac {e^{100 x^3} x^3}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx+3000 \int \frac {e^{100 x^3} x^2}{\left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.21, size = 58, normalized size = 1.61 \begin {gather*} -\frac {4 e^{50 x^3}}{x \left (-e^{25 x^2+25 x^4}+5 e^{50 x^3} x+e^{e^{5/x}+x+50 x^3} x\right )} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(40*x + E^(E^(5/x) + x)*(-20*E^(5/x) + 8*x + 4*x^2) + E^(25*x^2 - 50*x^3 + 25*x^4)*(-4 - 200*x^2 + 6

00*x^3 - 400*x^4))/(E^(50*x^2 - 100*x^3 + 50*x^4)*x^2 - 10*E^(25*x^2 - 50*x^3 + 25*x^4)*x^3 + 25*x^4 + E^(2*E^

(5/x) + 2*x)*x^4 + E^(E^(5/x) + x)*(-2*E^(25*x^2 - 50*x^3 + 25*x^4)*x^3 + 10*x^4)),x]

[Out]

(-4*E^(50*x^3))/(x*(-E^(25*x^2 + 25*x^4) + 5*E^(50*x^3)*x + E^(E^(5/x) + x + 50*x^3)*x))

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fricas [A] time = 0.50, size = 43, normalized size = 1.19 \begin {gather*} -\frac {4}{x^{2} e^{\left (x + e^{\frac {5}{x}}\right )} + 5 \, x^{2} - x e^{\left (25 \, x^{4} - 50 \, x^{3} + 25 \, x^{2}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200*x^2-4)*exp(25*x^4-50*x^3+25*x^2)+40*

x)/(x^4*exp(exp(5/x)+x)^2+(-2*x^3*exp(25*x^4-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x

^2)^2-10*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x, algorithm="fricas")

[Out]

-4/(x^2*e^(x + e^(5/x)) + 5*x^2 - x*e^(25*x^4 - 50*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(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200*x^2-4)*exp(25*x^4-50*x^3+25*x^2)+40*

x)/(x^4*exp(exp(5/x)+x)^2+(-2*x^3*exp(25*x^4-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x

^2)^2-10*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x, algorithm="giac")

[Out]

Timed out

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maple [A] time = 0.31, size = 36, normalized size = 1.00


method	result	size

risch	\(-\frac {4}{x \left (x \,{\mathrm e}^{{\mathrm e}^{\frac {5}{x}}+x}-{\mathrm e}^{25 x^{2} \left (x -1\right )^{2}}+5 x \right )}\)	\(36\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200*x^2-4)*exp(25*x^4-50*x^3+25*x^2)+40*x)/(x^

4*exp(exp(5/x)+x)^2+(-2*x^3*exp(25*x^4-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x^2)^2-

10*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x,method=_RETURNVERBOSE)

[Out]

-4/x/(x*exp(exp(5/x)+x)-exp(25*x^2*(x-1)^2)+5*x)

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maxima [A] time = 0.48, size = 55, normalized size = 1.53 \begin {gather*} -\frac {4 \, e^{\left (50 \, x^{3}\right )}}{5 \, x^{2} e^{\left (50 \, x^{3}\right )} + x^{2} e^{\left (50 \, x^{3} + x + e^{\frac {5}{x}}\right )} - x e^{\left (25 \, x^{4} + 25 \, x^{2}\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(5/x)+4*x^2+8*x)*exp(exp(5/x)+x)+(-400*x^4+600*x^3-200*x^2-4)*exp(25*x^4-50*x^3+25*x^2)+40*

x)/(x^4*exp(exp(5/x)+x)^2+(-2*x^3*exp(25*x^4-50*x^3+25*x^2)+10*x^4)*exp(exp(5/x)+x)+x^2*exp(25*x^4-50*x^3+25*x

^2)^2-10*x^3*exp(25*x^4-50*x^3+25*x^2)+25*x^4),x, algorithm="maxima")

[Out]

-4*e^(50*x^3)/(5*x^2*e^(50*x^3) + x^2*e^(50*x^3 + x + e^(5/x)) - x*e^(25*x^4 + 25*x^2))

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mupad [B] time = 6.18, size = 274, normalized size = 7.61 \begin {gather*} -\frac {20\,{\mathrm {e}}^{\frac {5}{x}+25\,x^2-50\,x^3+25\,x^4}-100\,x\,{\mathrm {e}}^{5/x}+20\,x^3-{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}\,\left (-400\,x^5+600\,x^4-200\,x^3+4\,x^2+4\,x\right )}{\left (5\,x-{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}+x\,{\mathrm {e}}^{x+{\mathrm {e}}^{5/x}}\right )\,\left (5\,x\,{\mathrm {e}}^{\frac {5}{x}+25\,x^2-50\,x^3+25\,x^4}-x^2\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}-x^3\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}+50\,x^4\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}-150\,x^5\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}+100\,x^6\,{\mathrm {e}}^{25\,x^4-50\,x^3+25\,x^2}-25\,x^2\,{\mathrm {e}}^{5/x}+5\,x^4\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((40*x + exp(x + exp(5/x))*(8*x - 20*exp(5/x) + 4*x^2) - exp(25*x^2 - 50*x^3 + 25*x^4)*(200*x^2 - 600*x^3 +

400*x^4 + 4))/(x^2*exp(50*x^2 - 100*x^3 + 50*x^4) - 10*x^3*exp(25*x^2 - 50*x^3 + 25*x^4) - exp(x + exp(5/x))*

(2*x^3*exp(25*x^2 - 50*x^3 + 25*x^4) - 10*x^4) + x^4*exp(2*x + 2*exp(5/x)) + 25*x^4),x)

[Out]

-(20*exp(5/x + 25*x^2 - 50*x^3 + 25*x^4) - 100*x*exp(5/x) + 20*x^3 - exp(25*x^2 - 50*x^3 + 25*x^4)*(4*x + 4*x^

2 - 200*x^3 + 600*x^4 - 400*x^5))/((5*x - exp(25*x^2 - 50*x^3 + 25*x^4) + x*exp(x + exp(5/x)))*(5*x*exp(5/x +

25*x^2 - 50*x^3 + 25*x^4) - x^2*exp(25*x^2 - 50*x^3 + 25*x^4) - x^3*exp(25*x^2 - 50*x^3 + 25*x^4) + 50*x^4*exp

(25*x^2 - 50*x^3 + 25*x^4) - 150*x^5*exp(25*x^2 - 50*x^3 + 25*x^4) + 100*x^6*exp(25*x^2 - 50*x^3 + 25*x^4) - 2

5*x^2*exp(5/x) + 5*x^4))

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sympy [A] time = 0.67, size = 37, normalized size = 1.03 \begin {gather*} - \frac {4}{x^{2} e^{x + e^{\frac {5}{x}}} + 5 x^{2} - x e^{25 x^{4} - 50 x^{3} + 25 x^{2}}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-20*exp(5/x)+4*x**2+8*x)*exp(exp(5/x)+x)+(-400*x**4+600*x**3-200*x**2-4)*exp(25*x**4-50*x**3+25*x*

*2)+40*x)/(x**4*exp(exp(5/x)+x)**2+(-2*x**3*exp(25*x**4-50*x**3+25*x**2)+10*x**4)*exp(exp(5/x)+x)+x**2*exp(25*

x**4-50*x**3+25*x**2)**2-10*x**3*exp(25*x**4-50*x**3+25*x**2)+25*x**4),x)

[Out]

-4/(x**2*exp(x + exp(5/x)) + 5*x**2 - x*exp(25*x**4 - 50*x**3 + 25*x**2))

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