3.84.42 \(\int \frac {1+34 x+3 x^2+x^3+e^{2 x} (66 x^2+68 x^3+4 x^4+2 x^5)+(x+e^{2 x} (2 x^2+2 x^3)) \log (x)}{33 x+x^2+x^3+x \log (x)} \, dx\)

Optimal. Leaf size=20 \[ x+e^{2 x} x^2+\log \left (33+x+x^2+\log (x)\right ) \]

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Rubi [F]  time = 0.66, 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 {1+34 x+3 x^2+x^3+e^{2 x} \left (66 x^2+68 x^3+4 x^4+2 x^5\right )+\left (x+e^{2 x} \left (2 x^2+2 x^3\right )\right ) \log (x)}{33 x+x^2+x^3+x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + 34*x + 3*x^2 + x^3 + E^(2*x)*(66*x^2 + 68*x^3 + 4*x^4 + 2*x^5) + (x + E^(2*x)*(2*x^2 + 2*x^3))*Log[x]
)/(33*x + x^2 + x^3 + x*Log[x]),x]

[Out]

x + E^(2*x)*x^2 + Defer[Int][(33 + x + x^2 + Log[x])^(-1), x] + Defer[Int][1/(x*(33 + x + x^2 + Log[x])), x] +
2*Defer[Int][x/(33 + x + x^2 + Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 e^{2 x} x (1+x)+\frac {34}{33+x+x^2+\log (x)}+\frac {1}{x \left (33+x+x^2+\log (x)\right )}+\frac {3 x}{33+x+x^2+\log (x)}+\frac {x^2}{33+x+x^2+\log (x)}+\frac {\log (x)}{33+x+x^2+\log (x)}\right ) \, dx\\ &=2 \int e^{2 x} x (1+x) \, dx+3 \int \frac {x}{33+x+x^2+\log (x)} \, dx+34 \int \frac {1}{33+x+x^2+\log (x)} \, dx+\int \frac {1}{x \left (33+x+x^2+\log (x)\right )} \, dx+\int \frac {x^2}{33+x+x^2+\log (x)} \, dx+\int \frac {\log (x)}{33+x+x^2+\log (x)} \, dx\\ &=2 \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx+3 \int \frac {x}{33+x+x^2+\log (x)} \, dx+34 \int \frac {1}{33+x+x^2+\log (x)} \, dx+\int \frac {1}{x \left (33+x+x^2+\log (x)\right )} \, dx+\int \frac {x^2}{33+x+x^2+\log (x)} \, dx+\int \left (1+\frac {-33-x-x^2}{33+x+x^2+\log (x)}\right ) \, dx\\ &=x+2 \int e^{2 x} x \, dx+2 \int e^{2 x} x^2 \, dx+3 \int \frac {x}{33+x+x^2+\log (x)} \, dx+34 \int \frac {1}{33+x+x^2+\log (x)} \, dx+\int \frac {1}{x \left (33+x+x^2+\log (x)\right )} \, dx+\int \frac {x^2}{33+x+x^2+\log (x)} \, dx+\int \frac {-33-x-x^2}{33+x+x^2+\log (x)} \, dx\\ &=x+e^{2 x} x+e^{2 x} x^2-2 \int e^{2 x} x \, dx+3 \int \frac {x}{33+x+x^2+\log (x)} \, dx+34 \int \frac {1}{33+x+x^2+\log (x)} \, dx-\int e^{2 x} \, dx+\int \frac {1}{x \left (33+x+x^2+\log (x)\right )} \, dx+\int \frac {x^2}{33+x+x^2+\log (x)} \, dx+\int \left (-\frac {33}{33+x+x^2+\log (x)}-\frac {x}{33+x+x^2+\log (x)}-\frac {x^2}{33+x+x^2+\log (x)}\right ) \, dx\\ &=-\frac {e^{2 x}}{2}+x+e^{2 x} x^2+3 \int \frac {x}{33+x+x^2+\log (x)} \, dx-33 \int \frac {1}{33+x+x^2+\log (x)} \, dx+34 \int \frac {1}{33+x+x^2+\log (x)} \, dx+\int e^{2 x} \, dx+\int \frac {1}{x \left (33+x+x^2+\log (x)\right )} \, dx-\int \frac {x}{33+x+x^2+\log (x)} \, dx\\ &=x+e^{2 x} x^2+3 \int \frac {x}{33+x+x^2+\log (x)} \, dx-33 \int \frac {1}{33+x+x^2+\log (x)} \, dx+34 \int \frac {1}{33+x+x^2+\log (x)} \, dx+\int \frac {1}{x \left (33+x+x^2+\log (x)\right )} \, dx-\int \frac {x}{33+x+x^2+\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 20, normalized size = 1.00 \begin {gather*} x+e^{2 x} x^2+\log \left (33+x+x^2+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 34*x + 3*x^2 + x^3 + E^(2*x)*(66*x^2 + 68*x^3 + 4*x^4 + 2*x^5) + (x + E^(2*x)*(2*x^2 + 2*x^3))*
Log[x])/(33*x + x^2 + x^3 + x*Log[x]),x]

[Out]

x + E^(2*x)*x^2 + Log[33 + x + x^2 + Log[x]]

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fricas [A]  time = 0.65, size = 19, normalized size = 0.95 \begin {gather*} x^{2} e^{\left (2 \, x\right )} + x + \log \left (x^{2} + x + \log \relax (x) + 33\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+2*x^2)*exp(x)^2+x)*log(x)+(2*x^5+4*x^4+68*x^3+66*x^2)*exp(x)^2+x^3+3*x^2+34*x+1)/(x*log(x)+
x^3+x^2+33*x),x, algorithm="fricas")

[Out]

x^2*e^(2*x) + x + log(x^2 + x + log(x) + 33)

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giac [A]  time = 0.20, size = 19, normalized size = 0.95 \begin {gather*} x^{2} e^{\left (2 \, x\right )} + x + \log \left (x^{2} + x + \log \relax (x) + 33\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+2*x^2)*exp(x)^2+x)*log(x)+(2*x^5+4*x^4+68*x^3+66*x^2)*exp(x)^2+x^3+3*x^2+34*x+1)/(x*log(x)+
x^3+x^2+33*x),x, algorithm="giac")

[Out]

x^2*e^(2*x) + x + log(x^2 + x + log(x) + 33)

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maple [A]  time = 0.05, size = 20, normalized size = 1.00

method	result	size
risch	\(\ln \left (x +x^{2}+33+\ln \relax (x )\right )+{\mathrm e}^{2 x} x^{2}+x\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3+2*x^2)*exp(x)^2+x)*ln(x)+(2*x^5+4*x^4+68*x^3+66*x^2)*exp(x)^2+x^3+3*x^2+34*x+1)/(x*ln(x)+x^3+x^2+
33*x),x,method=_RETURNVERBOSE)

[Out]

ln(x+x^2+33+ln(x))+exp(2*x)*x^2+x

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maxima [A]  time = 0.39, size = 19, normalized size = 0.95 \begin {gather*} x^{2} e^{\left (2 \, x\right )} + x + \log \left (x^{2} + x + \log \relax (x) + 33\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+2*x^2)*exp(x)^2+x)*log(x)+(2*x^5+4*x^4+68*x^3+66*x^2)*exp(x)^2+x^3+3*x^2+34*x+1)/(x*log(x)+
x^3+x^2+33*x),x, algorithm="maxima")

[Out]

x^2*e^(2*x) + x + log(x^2 + x + log(x) + 33)

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mupad [B]  time = 5.17, size = 19, normalized size = 0.95 \begin {gather*} x+\ln \left (x+\ln \relax (x)+x^2+33\right )+x^2\,{\mathrm {e}}^{2\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((34*x + log(x)*(x + exp(2*x)*(2*x^2 + 2*x^3)) + exp(2*x)*(66*x^2 + 68*x^3 + 4*x^4 + 2*x^5) + 3*x^2 + x^3 +
1)/(33*x + x*log(x) + x^2 + x^3),x)

[Out]

x + log(x + log(x) + x^2 + 33) + x^2*exp(2*x)

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sympy [A]  time = 0.39, size = 20, normalized size = 1.00 \begin {gather*} x^{2} e^{2 x} + x + \log {\left (x^{2} + x + \log {\relax (x )} + 33 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**3+2*x**2)*exp(x)**2+x)*ln(x)+(2*x**5+4*x**4+68*x**3+66*x**2)*exp(x)**2+x**3+3*x**2+34*x+1)/(
x*ln(x)+x**3+x**2+33*x),x)

[Out]

x**2*exp(2*x) + x + log(x**2 + x + log(x) + 33)

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