3.103.40 \(\int \frac {2+2 e^{2 x}+e^x (1+4 x)+e^{3 x^2} (6 e^x x+12 x^2)}{e^x+2 x} \, dx\)

Optimal. Leaf size=22 \[ -1+2 e^x+e^{3 x^2}+\log \left (e^x+2 x\right ) \]

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Rubi [F]  time = 0.28, 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 {2+2 e^{2 x}+e^x (1+4 x)+e^{3 x^2} \left (6 e^x x+12 x^2\right )}{e^x+2 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + 2*E^(2*x) + E^x*(1 + 4*x) + E^(3*x^2)*(6*E^x*x + 12*x^2))/(E^x + 2*x),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.14, size = 21, normalized size = 0.95 \begin {gather*} 2 e^x+e^{3 x^2}+\log \left (e^x+2 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 2*E^(2*x) + E^x*(1 + 4*x) + E^(3*x^2)*(6*E^x*x + 12*x^2))/(E^x + 2*x),x]

[Out]

2*E^x + E^(3*x^2) + Log[E^x + 2*x]

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fricas [A]  time = 0.79, size = 18, normalized size = 0.82 \begin {gather*} e^{\left (3 \, x^{2}\right )} + 2 \, e^{x} + \log \left (2 \, x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(x)*x+12*x^2)*exp(3*x^2)+2*exp(x)^2+(4*x+1)*exp(x)+2)/(exp(x)+2*x),x, algorithm="fricas")

[Out]

e^(3*x^2) + 2*e^x + log(2*x + e^x)

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giac [A]  time = 0.18, size = 18, normalized size = 0.82 \begin {gather*} e^{\left (3 \, x^{2}\right )} + 2 \, e^{x} + \log \left (2 \, x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(x)*x+12*x^2)*exp(3*x^2)+2*exp(x)^2+(4*x+1)*exp(x)+2)/(exp(x)+2*x),x, algorithm="giac")

[Out]

e^(3*x^2) + 2*e^x + log(2*x + e^x)

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maple [A]  time = 0.04, size = 19, normalized size = 0.86

method	result	size
norman	\(2 \,{\mathrm e}^{x}+{\mathrm e}^{3 x^{2}}+\ln \left ({\mathrm e}^{x}+2 x \right )\)	\(19\)
risch	\(2 \,{\mathrm e}^{x}+{\mathrm e}^{3 x^{2}}+\ln \left ({\mathrm e}^{x}+2 x \right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*exp(x)*x+12*x^2)*exp(3*x^2)+2*exp(x)^2+(4*x+1)*exp(x)+2)/(exp(x)+2*x),x,method=_RETURNVERBOSE)

[Out]

2*exp(x)+exp(3*x^2)+ln(exp(x)+2*x)

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maxima [A]  time = 0.39, size = 18, normalized size = 0.82 \begin {gather*} e^{\left (3 \, x^{2}\right )} + 2 \, e^{x} + \log \left (2 \, x + e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(x)*x+12*x^2)*exp(3*x^2)+2*exp(x)^2+(4*x+1)*exp(x)+2)/(exp(x)+2*x),x, algorithm="maxima")

[Out]

e^(3*x^2) + 2*e^x + log(2*x + e^x)

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mupad [B]  time = 0.10, size = 18, normalized size = 0.82 \begin {gather*} \ln \left (2\,x+{\mathrm {e}}^x\right )+{\mathrm {e}}^{3\,x^2}+2\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*exp(2*x) + exp(3*x^2)*(6*x*exp(x) + 12*x^2) + exp(x)*(4*x + 1) + 2)/(2*x + exp(x)),x)

[Out]

log(2*x + exp(x)) + exp(3*x^2) + 2*exp(x)

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sympy [A]  time = 0.23, size = 19, normalized size = 0.86 \begin {gather*} 2 e^{x} + e^{3 x^{2}} + \log {\left (2 x + e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*exp(x)*x+12*x**2)*exp(3*x**2)+2*exp(x)**2+(4*x+1)*exp(x)+2)/(exp(x)+2*x),x)

[Out]

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

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