3.95.43 \(\int \frac {100 x^2+100 x^3+25 x^4+e^{e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}+\frac {3+200 x+100 x^2}{50 x+25 x^2}} (6+6 x)}{100 x^2+100 x^3+25 x^4} \, dx\)

Optimal. Leaf size=22 \[ -e^{e^{4+\frac {3}{25 x (2+x)}}}+x \]

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Rubi [F]  time = 12.23, 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 {100 x^2+100 x^3+25 x^4+\exp \left (e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}+\frac {3+200 x+100 x^2}{50 x+25 x^2}\right ) (6+6 x)}{100 x^2+100 x^3+25 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(100*x^2 + 100*x^3 + 25*x^4 + E^(E^((3 + 200*x + 100*x^2)/(50*x + 25*x^2)) + (3 + 200*x + 100*x^2)/(50*x +
25*x^2))*(6 + 6*x))/(100*x^2 + 100*x^3 + 25*x^4),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {100 x^2+100 x^3+25 x^4+\exp \left (e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}+\frac {3+200 x+100 x^2}{50 x+25 x^2}\right ) (6+6 x)}{x^2 \left (100+100 x+25 x^2\right )} \, dx\\ &=\int \frac {100 x^2+100 x^3+25 x^4+\exp \left (e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}+\frac {3+200 x+100 x^2}{50 x+25 x^2}\right ) (6+6 x)}{25 x^2 (2+x)^2} \, dx\\ &=\frac {1}{25} \int \frac {100 x^2+100 x^3+25 x^4+\exp \left (e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}+\frac {3+200 x+100 x^2}{50 x+25 x^2}\right ) (6+6 x)}{x^2 (2+x)^2} \, dx\\ &=\frac {1}{25} \int \left (25+\frac {6 \exp \left (\frac {3+200 x+50 \exp \left (\frac {3}{50 x+25 x^2}+\frac {200 x}{50 x+25 x^2}+\frac {100 x^2}{50 x+25 x^2}\right ) x+100 x^2+25 \exp \left (\frac {3}{50 x+25 x^2}+\frac {200 x}{50 x+25 x^2}+\frac {100 x^2}{50 x+25 x^2}\right ) x^2}{25 x (2+x)}\right ) (1+x)}{x^2 (2+x)^2}\right ) \, dx\\ &=x+\frac {6}{25} \int \frac {\exp \left (\frac {3+200 x+50 \exp \left (\frac {3}{50 x+25 x^2}+\frac {200 x}{50 x+25 x^2}+\frac {100 x^2}{50 x+25 x^2}\right ) x+100 x^2+25 \exp \left (\frac {3}{50 x+25 x^2}+\frac {200 x}{50 x+25 x^2}+\frac {100 x^2}{50 x+25 x^2}\right ) x^2}{25 x (2+x)}\right ) (1+x)}{x^2 (2+x)^2} \, dx\\ &=x+\frac {6}{25} \int \frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right ) (1+x)}{x^2 (2+x)^2} \, dx\\ &=x+\frac {6}{25} \int \left (\frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right )}{4 x^2}-\frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right )}{4 (2+x)^2}\right ) \, dx\\ &=x+\frac {3}{50} \int \frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right )}{x^2} \, dx-\frac {3}{50} \int \frac {\exp \left (\frac {3+50 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x+25 \left (4+e^{\frac {3+200 x+100 x^2}{50 x+25 x^2}}\right ) x^2}{25 x (2+x)}\right )}{(2+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.21, size = 32, normalized size = 1.45 \begin {gather*} \frac {1}{25} \left (-25 e^{e^{4+\frac {3}{50 x}-\frac {3}{50 (2+x)}}}+25 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100*x^2 + 100*x^3 + 25*x^4 + E^(E^((3 + 200*x + 100*x^2)/(50*x + 25*x^2)) + (3 + 200*x + 100*x^2)/(
50*x + 25*x^2))*(6 + 6*x))/(100*x^2 + 100*x^3 + 25*x^4),x]

[Out]

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

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fricas [B]  time = 0.56, size = 103, normalized size = 4.68 \begin {gather*} {\left (x e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} - e^{\left (\frac {100 \, x^{2} + 25 \, {\left (x^{2} + 2 \, x\right )} e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )}\right )} e^{\left (-\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x+6)*exp((100*x^2+200*x+3)/(25*x^2+50*x))*exp(exp((100*x^2+200*x+3)/(25*x^2+50*x)))+25*x^4+100*x
^3+100*x^2)/(25*x^4+100*x^3+100*x^2),x, algorithm="fricas")

[Out]

(x*e^(1/25*(100*x^2 + 200*x + 3)/(x^2 + 2*x)) - e^(1/25*(100*x^2 + 25*(x^2 + 2*x)*e^(1/25*(100*x^2 + 200*x + 3
)/(x^2 + 2*x)) + 200*x + 3)/(x^2 + 2*x)))*e^(-1/25*(100*x^2 + 200*x + 3)/(x^2 + 2*x))

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giac [B]  time = 0.81, size = 124, normalized size = 5.64 \begin {gather*} {\left (x e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} - e^{\left (\frac {25 \, x^{2} e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} + 100 \, x^{2} + 50 \, x e^{\left (\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )}\right )} e^{\left (-\frac {100 \, x^{2} + 200 \, x + 3}{25 \, {\left (x^{2} + 2 \, x\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x+6)*exp((100*x^2+200*x+3)/(25*x^2+50*x))*exp(exp((100*x^2+200*x+3)/(25*x^2+50*x)))+25*x^4+100*x
^3+100*x^2)/(25*x^4+100*x^3+100*x^2),x, algorithm="giac")

[Out]

(x*e^(1/25*(100*x^2 + 200*x + 3)/(x^2 + 2*x)) - e^(1/25*(25*x^2*e^(1/25*(100*x^2 + 200*x + 3)/(x^2 + 2*x)) + 1
00*x^2 + 50*x*e^(1/25*(100*x^2 + 200*x + 3)/(x^2 + 2*x)) + 200*x + 3)/(x^2 + 2*x)))*e^(-1/25*(100*x^2 + 200*x
+ 3)/(x^2 + 2*x))

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maple [A]  time = 0.15, size = 27, normalized size = 1.23

method	result	size
risch	\(x -{\mathrm e}^{{\mathrm e}^{\frac {100 x^{2}+200 x +3}{25 x \left (2+x \right )}}}\)	\(27\)
norman	\(\frac {x^{3}-4 x -2 x \,{\mathrm e}^{{\mathrm e}^{\frac {100 x^{2}+200 x +3}{25 x^{2}+50 x}}}-x^{2} {\mathrm e}^{{\mathrm e}^{\frac {100 x^{2}+200 x +3}{25 x^{2}+50 x}}}}{x \left (2+x \right )}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x+6)*exp((100*x^2+200*x+3)/(25*x^2+50*x))*exp(exp((100*x^2+200*x+3)/(25*x^2+50*x)))+25*x^4+100*x^3+100
*x^2)/(25*x^4+100*x^3+100*x^2),x,method=_RETURNVERBOSE)

[Out]

x-exp(exp(1/25*(100*x^2+200*x+3)/x/(2+x)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} x + \frac {1}{25} \, \int \frac {6 \, {\left (x e^{4} + e^{4}\right )} e^{\left (-\frac {3}{50 \, {\left (x + 2\right )}} + \frac {3}{50 \, x} + e^{\left (-\frac {3}{50 \, {\left (x + 2\right )}} + \frac {3}{50 \, x} + 4\right )}\right )}}{x^{4} + 4 \, x^{3} + 4 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x+6)*exp((100*x^2+200*x+3)/(25*x^2+50*x))*exp(exp((100*x^2+200*x+3)/(25*x^2+50*x)))+25*x^4+100*x
^3+100*x^2)/(25*x^4+100*x^3+100*x^2),x, algorithm="maxima")

[Out]

x + 1/25*integrate(6*(x*e^4 + e^4)*e^(-3/50/(x + 2) + 3/50/x + e^(-3/50/(x + 2) + 3/50/x + 4))/(x^4 + 4*x^3 +
4*x^2), x)

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mupad [B]  time = 6.24, size = 37, normalized size = 1.68 \begin {gather*} x-{\mathrm {e}}^{{\mathrm {e}}^{\frac {4\,x}{x+2}}\,{\mathrm {e}}^{\frac {3}{25\,x^2+50\,x}}\,{\mathrm {e}}^{\frac {8}{x+2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((100*x^2 + 100*x^3 + 25*x^4 + exp(exp((200*x + 100*x^2 + 3)/(50*x + 25*x^2)))*exp((200*x + 100*x^2 + 3)/(5
0*x + 25*x^2))*(6*x + 6))/(100*x^2 + 100*x^3 + 25*x^4),x)

[Out]

x - exp(exp((4*x)/(x + 2))*exp(3/(50*x + 25*x^2))*exp(8/(x + 2)))

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sympy [A]  time = 0.46, size = 22, normalized size = 1.00 \begin {gather*} x - e^{e^{\frac {100 x^{2} + 200 x + 3}{25 x^{2} + 50 x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x+6)*exp((100*x**2+200*x+3)/(25*x**2+50*x))*exp(exp((100*x**2+200*x+3)/(25*x**2+50*x)))+25*x**4+
100*x**3+100*x**2)/(25*x**4+100*x**3+100*x**2),x)

[Out]

x - exp(exp((100*x**2 + 200*x + 3)/(25*x**2 + 50*x)))

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