3.13.15 \(\int \frac {-25 e^4 x^2+e^{\frac {-7-x+x^2}{x}} (-112-16 x^2+16 e^4 x^2)}{-25 x^2+16 e^{\frac {-7-x+x^2}{x}} x^2} \, dx\)

Optimal. Leaf size=29 \[ e^4 x-\log \left (\frac {25}{16}-e^{\frac {-7-x+x^2}{x}}\right ) \]

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Rubi [F]  time = 1.11, 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 {-25 e^4 x^2+e^{\frac {-7-x+x^2}{x}} \left (-112-16 x^2+16 e^4 x^2\right )}{-25 x^2+16 e^{\frac {-7-x+x^2}{x}} x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-25*E^4*x^2 + E^((-7 - x + x^2)/x)*(-112 - 16*x^2 + 16*E^4*x^2))/(-25*x^2 + 16*E^((-7 - x + x^2)/x)*x^2),
x]

[Out]

7/x - (1 - E^4)*x - 25*Defer[Int][E^(1 + 7/x)/(-25*E^(1 + 7/x) + 16*E^x), x] - 175*Defer[Int][E^(1 + 7/x)/((-2
5*E^(1 + 7/x) + 16*E^x)*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {25 e^{1+\frac {7}{x}} \left (7+x^2\right )}{\left (25 e^{1+\frac {7}{x}}-16 e^x\right ) x^2}+\frac {-7-\left (1-e^4\right ) x^2}{x^2}\right ) \, dx\\ &=25 \int \frac {e^{1+\frac {7}{x}} \left (7+x^2\right )}{\left (25 e^{1+\frac {7}{x}}-16 e^x\right ) x^2} \, dx+\int \frac {-7-\left (1-e^4\right ) x^2}{x^2} \, dx\\ &=25 \int \left (-\frac {e^{1+\frac {7}{x}}}{-25 e^{1+\frac {7}{x}}+16 e^x}-\frac {7 e^{1+\frac {7}{x}}}{\left (-25 e^{1+\frac {7}{x}}+16 e^x\right ) x^2}\right ) \, dx+\int \left (-1+e^4-\frac {7}{x^2}\right ) \, dx\\ &=\frac {7}{x}-\left (1-e^4\right ) x-25 \int \frac {e^{1+\frac {7}{x}}}{-25 e^{1+\frac {7}{x}}+16 e^x} \, dx-175 \int \frac {e^{1+\frac {7}{x}}}{\left (-25 e^{1+\frac {7}{x}}+16 e^x\right ) x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 31, normalized size = 1.07 \begin {gather*} \frac {7}{x}+e^4 x-\log \left (-25 e^{1+\frac {7}{x}}+16 e^x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25*E^4*x^2 + E^((-7 - x + x^2)/x)*(-112 - 16*x^2 + 16*E^4*x^2))/(-25*x^2 + 16*E^((-7 - x + x^2)/x)
*x^2),x]

[Out]

7/x + E^4*x - Log[-25*E^(1 + 7/x) + 16*E^x]

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fricas [A]  time = 0.57, size = 25, normalized size = 0.86 \begin {gather*} x e^{4} - \log \left (16 \, e^{\left (\frac {x^{2} - x - 7}{x}\right )} - 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2*exp(4)-16*x^2-112)*exp((x^2-x-7)/x)-25*x^2*exp(4))/(16*x^2*exp((x^2-x-7)/x)-25*x^2),x, algo
rithm="fricas")

[Out]

x*e^4 - log(16*e^((x^2 - x - 7)/x) - 25)

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giac [A]  time = 0.32, size = 25, normalized size = 0.86 \begin {gather*} x e^{4} - \log \left (16 \, e^{\left (\frac {x^{2} - x - 7}{x}\right )} - 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2*exp(4)-16*x^2-112)*exp((x^2-x-7)/x)-25*x^2*exp(4))/(16*x^2*exp((x^2-x-7)/x)-25*x^2),x, algo
rithm="giac")

[Out]

x*e^4 - log(16*e^((x^2 - x - 7)/x) - 25)

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maple [A]  time = 0.09, size = 26, normalized size = 0.90

method	result	size
norman	\(x \,{\mathrm e}^{4}-\ln \left (16 \,{\mathrm e}^{\frac {x^{2}-x -7}{x}}-25\right )\)	\(26\)
risch	\(x \,{\mathrm e}^{4}-x +\frac {7}{x}+\frac {x^{2}-x -7}{x}-\ln \left ({\mathrm e}^{\frac {x^{2}-x -7}{x}}-\frac {25}{16}\right )\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^2*exp(4)-16*x^2-112)*exp((x^2-x-7)/x)-25*x^2*exp(4))/(16*x^2*exp((x^2-x-7)/x)-25*x^2),x,method=_RET
URNVERBOSE)

[Out]

x*exp(4)-ln(16*exp((x^2-x-7)/x)-25)

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maxima [A]  time = 0.54, size = 35, normalized size = 1.21 \begin {gather*} \frac {x^{2} e^{4} + 7}{x} - \log \left (-\frac {1}{25} \, {\left (16 \, e^{x} - 25 \, e^{\left (\frac {7}{x} + 1\right )}\right )} e^{\left (-1\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^2*exp(4)-16*x^2-112)*exp((x^2-x-7)/x)-25*x^2*exp(4))/(16*x^2*exp((x^2-x-7)/x)-25*x^2),x, algo
rithm="maxima")

[Out]

(x^2*e^4 + 7)/x - log(-1/25*(16*e^x - 25*e^(7/x + 1))*e^(-1))

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mupad [B]  time = 0.97, size = 22, normalized size = 0.76 \begin {gather*} x\,{\mathrm {e}}^4-\ln \left (16\,{\mathrm {e}}^{-1}\,{\mathrm {e}}^{-\frac {7}{x}}\,{\mathrm {e}}^x-25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(x - x^2 + 7)/x)*(16*x^2 - 16*x^2*exp(4) + 112) + 25*x^2*exp(4))/(25*x^2 - 16*x^2*exp(-(x - x^2 + 7)
/x)),x)

[Out]

x*exp(4) - log(16*exp(-1)*exp(-7/x)*exp(x) - 25)

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sympy [A]  time = 0.17, size = 19, normalized size = 0.66 \begin {gather*} x e^{4} - \log {\left (e^{\frac {x^{2} - x - 7}{x}} - \frac {25}{16} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**2*exp(4)-16*x**2-112)*exp((x**2-x-7)/x)-25*x**2*exp(4))/(16*x**2*exp((x**2-x-7)/x)-25*x**2),
x)

[Out]

x*exp(4) - log(exp((x**2 - x - 7)/x) - 25/16)

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