3.44.5 \(\int \frac {150 x^2+e^{\frac {e^x}{x^2}} (50 x^2+e^x (-50+25 x))}{2 x} \, dx\)

Optimal. Leaf size=20 \[ 1+\frac {25}{2} \left (3+e^{\frac {e^x}{x^2}}\right ) x^2 \]

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Rubi [F]  time = 0.26, 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 {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{2 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(150*x^2 + E^(E^x/x^2)*(50*x^2 + E^x*(-50 + 25*x)))/(2*x),x]

[Out]

(75*x^2)/2 + (25*Defer[Int][E^(E^x/x^2 + x), x])/2 - 25*Defer[Int][E^(E^x/x^2 + x)/x, x] + 25*Defer[Int][E^(E^
x/x^2)*x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {150 x^2+e^{\frac {e^x}{x^2}} \left (50 x^2+e^x (-50+25 x)\right )}{x} \, dx\\ &=\frac {1}{2} \int \left (\frac {25 e^{\frac {e^x}{x^2}+x} (-2+x)}{x}+50 \left (3+e^{\frac {e^x}{x^2}}\right ) x\right ) \, dx\\ &=\frac {25}{2} \int \frac {e^{\frac {e^x}{x^2}+x} (-2+x)}{x} \, dx+25 \int \left (3+e^{\frac {e^x}{x^2}}\right ) x \, dx\\ &=\frac {25}{2} \int \left (e^{\frac {e^x}{x^2}+x}-\frac {2 e^{\frac {e^x}{x^2}+x}}{x}\right ) \, dx+25 \int \left (3 x+e^{\frac {e^x}{x^2}} x\right ) \, dx\\ &=\frac {75 x^2}{2}+\frac {25}{2} \int e^{\frac {e^x}{x^2}+x} \, dx-25 \int \frac {e^{\frac {e^x}{x^2}+x}}{x} \, dx+25 \int e^{\frac {e^x}{x^2}} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 18, normalized size = 0.90 \begin {gather*} \frac {25}{2} \left (3+e^{\frac {e^x}{x^2}}\right ) x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(150*x^2 + E^(E^x/x^2)*(50*x^2 + E^x*(-50 + 25*x)))/(2*x),x]

[Out]

(25*(3 + E^(E^x/x^2))*x^2)/2

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fricas [A]  time = 1.19, size = 18, normalized size = 0.90 \begin {gather*} \frac {25}{2} \, x^{2} e^{\left (\frac {e^{x}}{x^{2}}\right )} + \frac {75}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((25*x-50)*exp(x)+50*x^2)*exp(exp(x)/x^2)+150*x^2)/x,x, algorithm="fricas")

[Out]

25/2*x^2*e^(e^x/x^2) + 75/2*x^2

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giac [A]  time = 0.17, size = 29, normalized size = 1.45 \begin {gather*} \frac {25}{2} \, {\left (3 \, x^{2} e^{x} + x^{2} e^{\left (\frac {x^{3} + e^{x}}{x^{2}}\right )}\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((25*x-50)*exp(x)+50*x^2)*exp(exp(x)/x^2)+150*x^2)/x,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.03, size = 19, normalized size = 0.95

method	result	size
norman	\(\frac {75 x^{2}}{2}+\frac {25 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{x^{2}}} x^{2}}{2}\)	\(19\)
risch	\(\frac {75 x^{2}}{2}+\frac {25 \,{\mathrm e}^{\frac {{\mathrm e}^{x}}{x^{2}}} x^{2}}{2}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(((25*x-50)*exp(x)+50*x^2)*exp(exp(x)/x^2)+150*x^2)/x,x,method=_RETURNVERBOSE)

[Out]

75/2*x^2+25/2*exp(exp(x)/x^2)*x^2

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maxima [A]  time = 0.39, size = 18, normalized size = 0.90 \begin {gather*} \frac {25}{2} \, x^{2} e^{\left (\frac {e^{x}}{x^{2}}\right )} + \frac {75}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((25*x-50)*exp(x)+50*x^2)*exp(exp(x)/x^2)+150*x^2)/x,x, algorithm="maxima")

[Out]

25/2*x^2*e^(e^x/x^2) + 75/2*x^2

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mupad [B]  time = 3.17, size = 14, normalized size = 0.70 \begin {gather*} \frac {25\,x^2\,\left ({\mathrm {e}}^{\frac {{\mathrm {e}}^x}{x^2}}+3\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((75*x^2 + (exp(exp(x)/x^2)*(exp(x)*(25*x - 50) + 50*x^2))/2)/x,x)

[Out]

(25*x^2*(exp(exp(x)/x^2) + 3))/2

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sympy [A]  time = 1.58, size = 20, normalized size = 1.00 \begin {gather*} \frac {25 x^{2} e^{\frac {e^{x}}{x^{2}}}}{2} + \frac {75 x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((25*x-50)*exp(x)+50*x**2)*exp(exp(x)/x**2)+150*x**2)/x,x)

[Out]

25*x**2*exp(exp(x)/x**2)/2 + 75*x**2/2

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