3.73.78 \(\int \frac {-1250-1250 e^x}{e^{3+2 x}+625 x+e^3 x^2+e^x (625+2 e^3 x)} \, dx\)

Optimal. Leaf size=18 \[ 2+\log \left (\left (e^3+\frac {625}{e^x+x}\right )^2\right ) \]

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Rubi [F]  time = 0.43, 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 {-1250-1250 e^x}{e^{3+2 x}+625 x+e^3 x^2+e^x \left (625+2 e^3 x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1250 - 1250*E^x)/(E^(3 + 2*x) + 625*x + E^3*x^2 + E^x*(625 + 2*E^3*x)),x]

[Out]

-2*Defer[Int][(E^x + x)^(-1), x] + 2*Defer[Int][x/(E^x + x), x] - 2*(625 - E^3)*Defer[Int][(625 + E^(3 + x) +
E^3*x)^(-1), x] - 2*E^3*Defer[Int][x/(625 + E^(3 + x) + E^3*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1250 \left (-1-e^x\right )}{e^{3+2 x}+625 x+e^3 x^2+e^x \left (625+2 e^3 x\right )} \, dx\\ &=1250 \int \frac {-1-e^x}{e^{3+2 x}+625 x+e^3 x^2+e^x \left (625+2 e^3 x\right )} \, dx\\ &=1250 \int \left (\frac {-1+x}{625 \left (e^x+x\right )}-\frac {625-e^3+e^3 x}{625 \left (625+e^{3+x}+e^3 x\right )}\right ) \, dx\\ &=2 \int \frac {-1+x}{e^x+x} \, dx-2 \int \frac {625-e^3+e^3 x}{625+e^{3+x}+e^3 x} \, dx\\ &=2 \int \left (-\frac {1}{e^x+x}+\frac {x}{e^x+x}\right ) \, dx-2 \int \left (\frac {625 \left (1-\frac {e^3}{625}\right )}{625+e^{3+x}+e^3 x}+\frac {e^3 x}{625+e^{3+x}+e^3 x}\right ) \, dx\\ &=-\left (2 \int \frac {1}{e^x+x} \, dx\right )+2 \int \frac {x}{e^x+x} \, dx-\left (2 e^3\right ) \int \frac {x}{625+e^{3+x}+e^3 x} \, dx-\left (2 \left (625-e^3\right )\right ) \int \frac {1}{625+e^{3+x}+e^3 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.14, size = 30, normalized size = 1.67 \begin {gather*} -1250 \left (\frac {1}{625} \log \left (e^x+x\right )-\frac {1}{625} \log \left (625+e^{3+x}+e^3 x\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1250 - 1250*E^x)/(E^(3 + 2*x) + 625*x + E^3*x^2 + E^x*(625 + 2*E^3*x)),x]

[Out]

-1250*(Log[E^x + x]/625 - Log[625 + E^(3 + x) + E^3*x]/625)

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fricas [A]  time = 0.65, size = 21, normalized size = 1.17 \begin {gather*} 2 \, \log \left (x e^{3} + e^{\left (x + 3\right )} + 625\right ) - 2 \, \log \left (x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*exp(x)-1250)/(exp(3)*exp(x)^2+(2*x*exp(3)+625)*exp(x)+x^2*exp(3)+625*x),x, algorithm="fricas"
)

[Out]

2*log(x*e^3 + e^(x + 3) + 625) - 2*log(x + e^x)

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giac [A]  time = 0.21, size = 24, normalized size = 1.33 \begin {gather*} 2 \, \log \left (-x e^{3} - e^{\left (x + 3\right )} - 625\right ) - 2 \, \log \left (x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*exp(x)-1250)/(exp(3)*exp(x)^2+(2*x*exp(3)+625)*exp(x)+x^2*exp(3)+625*x),x, algorithm="giac")

[Out]

2*log(-x*e^3 - e^(x + 3) - 625) - 2*log(x + e^x)

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maple [A]  time = 0.11, size = 23, normalized size = 1.28

method	result	size
norman	\(-2 \ln \left ({\mathrm e}^{x}+x \right )+2 \ln \left ({\mathrm e}^{x} {\mathrm e}^{3}+x \,{\mathrm e}^{3}+625\right )\)	\(23\)
risch	\(2 \ln \left ({\mathrm e}^{x}+\left (x \,{\mathrm e}^{3}+625\right ) {\mathrm e}^{-3}\right )-2 \ln \left ({\mathrm e}^{x}+x \right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1250*exp(x)-1250)/(exp(3)*exp(x)^2+(2*x*exp(3)+625)*exp(x)+x^2*exp(3)+625*x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(exp(x)+x)+2*ln(exp(x)*exp(3)+x*exp(3)+625)

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maxima [A]  time = 0.40, size = 24, normalized size = 1.33 \begin {gather*} 2 \, \log \left ({\left (x e^{3} + e^{\left (x + 3\right )} + 625\right )} e^{\left (-3\right )}\right ) - 2 \, \log \left (x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*exp(x)-1250)/(exp(3)*exp(x)^2+(2*x*exp(3)+625)*exp(x)+x^2*exp(3)+625*x),x, algorithm="maxima"
)

[Out]

2*log((x*e^3 + e^(x + 3) + 625)*e^(-3)) - 2*log(x + e^x)

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mupad [B]  time = 0.20, size = 19, normalized size = 1.06 \begin {gather*} 2\,\ln \left (x+625\,{\mathrm {e}}^{-3}+{\mathrm {e}}^x\right )-2\,\ln \left (x+{\mathrm {e}}^x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(1250*exp(x) + 1250)/(625*x + exp(2*x)*exp(3) + x^2*exp(3) + exp(x)*(2*x*exp(3) + 625)),x)

[Out]

2*log(x + 625*exp(-3) + exp(x)) - 2*log(x + exp(x))

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sympy [A]  time = 0.27, size = 27, normalized size = 1.50 \begin {gather*} - 2 \log {\left (x + e^{x} \right )} + 2 \log {\left (\frac {4 x e^{3} + 2500}{4 e^{3}} + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1250*exp(x)-1250)/(exp(3)*exp(x)**2+(2*x*exp(3)+625)*exp(x)+x**2*exp(3)+625*x),x)

[Out]

-2*log(x + exp(x)) + 2*log((4*x*exp(3) + 2500)*exp(-3)/4 + exp(x))

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