3.2.13 \(\int \frac {2+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx\)

Optimal. Leaf size=23 \[ \log \left (3+25 x^2 \left (e^3+3 x^2\right )^2+2 \log (x)\right ) \]

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Rubi [F]  time = 0.68, 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+50 e^6 x^2+600 e^3 x^4+1350 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 + 50*E^6*x^2 + 600*E^3*x^4 + 1350*x^6)/(3*x + 25*E^6*x^3 + 150*E^3*x^5 + 225*x^7 + 2*x*Log[x]),x]

[Out]

2*Defer[Int][1/(x*(3 + 25*E^6*x^2 + 150*E^3*x^4 + 225*x^6 + 2*Log[x])), x] + 50*E^6*Defer[Int][x/(3 + 25*E^6*x
^2 + 150*E^3*x^4 + 225*x^6 + 2*Log[x]), x] + 600*E^3*Defer[Int][x^3/(3 + 25*E^6*x^2 + 150*E^3*x^4 + 225*x^6 +
2*Log[x]), x] + 1350*Defer[Int][x^5/(3 + 25*E^6*x^2 + 150*E^3*x^4 + 225*x^6 + 2*Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (1+25 e^6 x^2+300 e^3 x^4+675 x^6\right )}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx\\ &=2 \int \frac {1+25 e^6 x^2+300 e^3 x^4+675 x^6}{3 x+25 e^6 x^3+150 e^3 x^5+225 x^7+2 x \log (x)} \, dx\\ &=2 \int \left (\frac {1}{x \left (3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)\right )}+\frac {25 e^6 x}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)}+\frac {300 e^3 x^3}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)}+\frac {675 x^5}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)}\right ) \, dx\\ &=2 \int \frac {1}{x \left (3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)\right )} \, dx+1350 \int \frac {x^5}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)} \, dx+\left (600 e^3\right ) \int \frac {x^3}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)} \, dx+\left (50 e^6\right ) \int \frac {x}{3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.46, size = 28, normalized size = 1.22 \begin {gather*} \log \left (3+25 e^6 x^2+150 e^3 x^4+225 x^6+2 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 50*E^6*x^2 + 600*E^3*x^4 + 1350*x^6)/(3*x + 25*E^6*x^3 + 150*E^3*x^5 + 225*x^7 + 2*x*Log[x]),x]

[Out]

Log[3 + 25*E^6*x^2 + 150*E^3*x^4 + 225*x^6 + 2*Log[x]]

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fricas [A]  time = 1.11, size = 26, normalized size = 1.13 \begin {gather*} \log \left (225 \, x^{6} + 150 \, x^{4} e^{3} + 25 \, x^{2} e^{6} + 2 \, \log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^2*exp(3)^2+600*x^4*exp(3)+1350*x^6+2)/(2*x*log(x)+25*x^3*exp(3)^2+150*x^5*exp(3)+225*x^7+3*x),
x, algorithm="fricas")

[Out]

log(225*x^6 + 150*x^4*e^3 + 25*x^2*e^6 + 2*log(x) + 3)

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giac [A]  time = 0.26, size = 26, normalized size = 1.13 \begin {gather*} \log \left (225 \, x^{6} + 150 \, x^{4} e^{3} + 25 \, x^{2} e^{6} + 2 \, \log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^2*exp(3)^2+600*x^4*exp(3)+1350*x^6+2)/(2*x*log(x)+25*x^3*exp(3)^2+150*x^5*exp(3)+225*x^7+3*x),
x, algorithm="giac")

[Out]

log(225*x^6 + 150*x^4*e^3 + 25*x^2*e^6 + 2*log(x) + 3)

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maple [A]  time = 0.06, size = 25, normalized size = 1.09

method	result	size
risch	\(\ln \left (\frac {225 x^{6}}{2}+75 x^{4} {\mathrm e}^{3}+\frac {25 x^{2} {\mathrm e}^{6}}{2}+\ln \relax (x )+\frac {3}{2}\right )\)	\(25\)
norman	\(\ln \left (225 x^{6}+150 x^{4} {\mathrm e}^{3}+25 x^{2} {\mathrm e}^{6}+2 \ln \relax (x )+3\right )\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x^2*exp(3)^2+600*x^4*exp(3)+1350*x^6+2)/(2*x*ln(x)+25*x^3*exp(3)^2+150*x^5*exp(3)+225*x^7+3*x),x,metho
d=_RETURNVERBOSE)

[Out]

ln(225/2*x^6+75*x^4*exp(3)+25/2*x^2*exp(6)+ln(x)+3/2)

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maxima [A]  time = 0.48, size = 24, normalized size = 1.04 \begin {gather*} \log \left (\frac {225}{2} \, x^{6} + 75 \, x^{4} e^{3} + \frac {25}{2} \, x^{2} e^{6} + \log \relax (x) + \frac {3}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x^2*exp(3)^2+600*x^4*exp(3)+1350*x^6+2)/(2*x*log(x)+25*x^3*exp(3)^2+150*x^5*exp(3)+225*x^7+3*x),
x, algorithm="maxima")

[Out]

log(225/2*x^6 + 75*x^4*e^3 + 25/2*x^2*e^6 + log(x) + 3/2)

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mupad [B]  time = 0.47, size = 24, normalized size = 1.04 \begin {gather*} \ln \left (\ln \relax (x)+75\,x^4\,{\mathrm {e}}^3+\frac {25\,x^2\,{\mathrm {e}}^6}{2}+\frac {225\,x^6}{2}+\frac {3}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((600*x^4*exp(3) + 50*x^2*exp(6) + 1350*x^6 + 2)/(3*x + 150*x^5*exp(3) + 25*x^3*exp(6) + 2*x*log(x) + 225*x
^7),x)

[Out]

log(log(x) + 75*x^4*exp(3) + (25*x^2*exp(6))/2 + (225*x^6)/2 + 3/2)

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sympy [A]  time = 0.18, size = 32, normalized size = 1.39 \begin {gather*} \log {\left (\frac {225 x^{6}}{2} + 75 x^{4} e^{3} + \frac {25 x^{2} e^{6}}{2} + \log {\relax (x )} + \frac {3}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((50*x**2*exp(3)**2+600*x**4*exp(3)+1350*x**6+2)/(2*x*ln(x)+25*x**3*exp(3)**2+150*x**5*exp(3)+225*x**
7+3*x),x)

[Out]

log(225*x**6/2 + 75*x**4*exp(3) + 25*x**2*exp(6)/2 + log(x) + 3/2)

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