3.24.10 \(\int \frac {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} (10 x^2+34 x^4)+e^{12+x^2} (40 x^2+72 x^4)+(-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4) \log (x)+x^4 \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {4}{5+\left (2+e^{12+x^2}\right )^2+\frac {5}{x^2}-\log (x)} \]

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Rubi [F]  time = 4.13, 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 {40 x+4 x^3-32 e^{12+x^2} x^5-16 e^{24+2 x^2} x^5}{25+90 x^2+81 x^4+8 e^{36+3 x^2} x^4+e^{48+4 x^2} x^4+e^{24+2 x^2} \left (10 x^2+34 x^4\right )+e^{12+x^2} \left (40 x^2+72 x^4\right )+\left (-10 x^2-18 x^4-8 e^{12+x^2} x^4-2 e^{24+2 x^2} x^4\right ) \log (x)+x^4 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(40*x + 4*x^3 - 32*E^(12 + x^2)*x^5 - 16*E^(24 + 2*x^2)*x^5)/(25 + 90*x^2 + 81*x^4 + 8*E^(36 + 3*x^2)*x^4
+ E^(48 + 4*x^2)*x^4 + E^(24 + 2*x^2)*(10*x^2 + 34*x^4) + E^(12 + x^2)*(40*x^2 + 72*x^4) + (-10*x^2 - 18*x^4 -
8*E^(12 + x^2)*x^4 - 2*E^(24 + 2*x^2)*x^4)*Log[x] + x^4*Log[x]^2),x]

[Out]

40*Defer[Int][x/(5 + 9*x^2 + 4*E^(12 + x^2)*x^2 + E^(24 + 2*x^2)*x^2 - x^2*Log[x])^2, x] + 84*Defer[Int][x^3/(
5 + 9*x^2 + 4*E^(12 + x^2)*x^2 + E^(24 + 2*x^2)*x^2 - x^2*Log[x])^2, x] + 144*Defer[Int][x^5/(5 + 9*x^2 + 4*E^
(12 + x^2)*x^2 + E^(24 + 2*x^2)*x^2 - x^2*Log[x])^2, x] + 32*Defer[Int][(E^(12 + x^2)*x^5)/(5 + 9*x^2 + 4*E^(1
2 + x^2)*x^2 + E^(24 + 2*x^2)*x^2 - x^2*Log[x])^2, x] - 16*Defer[Int][x^3/(5 + 9*x^2 + 4*E^(12 + x^2)*x^2 + E^
(24 + 2*x^2)*x^2 - x^2*Log[x]), x] - 16*Defer[Int][(x^5*Log[x])/(-5 - 9*x^2 - 4*E^(12 + x^2)*x^2 - E^(24 + 2*x
^2)*x^2 + x^2*Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x \left (10+x^2-4 e^{12+x^2} \left (2+e^{12+x^2}\right ) x^4\right )}{\left (5+\left (9+4 e^{12+x^2}+e^{24+2 x^2}\right ) x^2-x^2 \log (x)\right )^2} \, dx\\ &=4 \int \frac {x \left (10+x^2-4 e^{12+x^2} \left (2+e^{12+x^2}\right ) x^4\right )}{\left (5+\left (9+4 e^{12+x^2}+e^{24+2 x^2}\right ) x^2-x^2 \log (x)\right )^2} \, dx\\ &=4 \int \left (-\frac {4 x^3}{5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)}+\frac {x \left (10+21 x^2+36 x^4+8 e^{12+x^2} x^4-4 x^4 \log (x)\right )}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}\right ) \, dx\\ &=4 \int \frac {x \left (10+21 x^2+36 x^4+8 e^{12+x^2} x^4-4 x^4 \log (x)\right )}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx-16 \int \frac {x^3}{5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)} \, dx\\ &=4 \int \left (\frac {10 x}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}+\frac {21 x^3}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}+\frac {36 x^5}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}+\frac {8 e^{12+x^2} x^5}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2}-\frac {4 x^5 \log (x)}{\left (-5-9 x^2-4 e^{12+x^2} x^2-e^{24+2 x^2} x^2+x^2 \log (x)\right )^2}\right ) \, dx-16 \int \frac {x^3}{5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)} \, dx\\ &=-\left (16 \int \frac {x^3}{5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)} \, dx\right )-16 \int \frac {x^5 \log (x)}{\left (-5-9 x^2-4 e^{12+x^2} x^2-e^{24+2 x^2} x^2+x^2 \log (x)\right )^2} \, dx+32 \int \frac {e^{12+x^2} x^5}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx+40 \int \frac {x}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx+84 \int \frac {x^3}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx+144 \int \frac {x^5}{\left (5+9 x^2+4 e^{12+x^2} x^2+e^{24+2 x^2} x^2-x^2 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.68, size = 40, normalized size = 1.54 \begin {gather*} \frac {4 x^2}{5+\left (9+4 e^{12+x^2}+e^{24+2 x^2}\right ) x^2-x^2 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(40*x + 4*x^3 - 32*E^(12 + x^2)*x^5 - 16*E^(24 + 2*x^2)*x^5)/(25 + 90*x^2 + 81*x^4 + 8*E^(36 + 3*x^2
)*x^4 + E^(48 + 4*x^2)*x^4 + E^(24 + 2*x^2)*(10*x^2 + 34*x^4) + E^(12 + x^2)*(40*x^2 + 72*x^4) + (-10*x^2 - 18
*x^4 - 8*E^(12 + x^2)*x^4 - 2*E^(24 + 2*x^2)*x^4)*Log[x] + x^4*Log[x]^2),x]

[Out]

(4*x^2)/(5 + (9 + 4*E^(12 + x^2) + E^(24 + 2*x^2))*x^2 - x^2*Log[x])

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fricas [A]  time = 0.85, size = 44, normalized size = 1.69 \begin {gather*} \frac {4 \, x^{2}}{x^{2} e^{\left (2 \, x^{2} + 24\right )} + 4 \, x^{2} e^{\left (x^{2} + 12\right )} - x^{2} \log \relax (x) + 9 \, x^{2} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^5*exp(x^2+12)^2-32*x^5*exp(x^2+12)+4*x^3+40*x)/(x^4*log(x)^2+(-2*x^4*exp(x^2+12)^2-8*x^4*exp(
x^2+12)-18*x^4-10*x^2)*log(x)+x^4*exp(x^2+12)^4+8*x^4*exp(x^2+12)^3+(34*x^4+10*x^2)*exp(x^2+12)^2+(72*x^4+40*x
^2)*exp(x^2+12)+81*x^4+90*x^2+25),x, algorithm="fricas")

[Out]

4*x^2/(x^2*e^(2*x^2 + 24) + 4*x^2*e^(x^2 + 12) - x^2*log(x) + 9*x^2 + 5)

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giac [A]  time = 2.49, size = 44, normalized size = 1.69 \begin {gather*} \frac {8 \, x^{2}}{x^{2} e^{\left (2 \, x^{2} + 24\right )} + 4 \, x^{2} e^{\left (x^{2} + 12\right )} - x^{2} \log \relax (x) + 9 \, x^{2} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^5*exp(x^2+12)^2-32*x^5*exp(x^2+12)+4*x^3+40*x)/(x^4*log(x)^2+(-2*x^4*exp(x^2+12)^2-8*x^4*exp(
x^2+12)-18*x^4-10*x^2)*log(x)+x^4*exp(x^2+12)^4+8*x^4*exp(x^2+12)^3+(34*x^4+10*x^2)*exp(x^2+12)^2+(72*x^4+40*x
^2)*exp(x^2+12)+81*x^4+90*x^2+25),x, algorithm="giac")

[Out]

8*x^2/(x^2*e^(2*x^2 + 24) + 4*x^2*e^(x^2 + 12) - x^2*log(x) + 9*x^2 + 5)

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maple [A]  time = 0.06, size = 45, normalized size = 1.73

method	result	size
risch	\(\frac {4 x^{2}}{{\mathrm e}^{2 x^{2}+24} x^{2}+4 \,{\mathrm e}^{x^{2}+12} x^{2}-x^{2} \ln \relax (x )+9 x^{2}+5}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*x^5*exp(x^2+12)^2-32*x^5*exp(x^2+12)+4*x^3+40*x)/(x^4*ln(x)^2+(-2*x^4*exp(x^2+12)^2-8*x^4*exp(x^2+12)
-18*x^4-10*x^2)*ln(x)+x^4*exp(x^2+12)^4+8*x^4*exp(x^2+12)^3+(34*x^4+10*x^2)*exp(x^2+12)^2+(72*x^4+40*x^2)*exp(
x^2+12)+81*x^4+90*x^2+25),x,method=_RETURNVERBOSE)

[Out]

4*x^2/(exp(2*x^2+24)*x^2+4*exp(x^2+12)*x^2-x^2*ln(x)+9*x^2+5)

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maxima [A]  time = 0.66, size = 44, normalized size = 1.69 \begin {gather*} \frac {4 \, x^{2}}{x^{2} e^{\left (2 \, x^{2} + 24\right )} + 4 \, x^{2} e^{\left (x^{2} + 12\right )} - x^{2} \log \relax (x) + 9 \, x^{2} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x^5*exp(x^2+12)^2-32*x^5*exp(x^2+12)+4*x^3+40*x)/(x^4*log(x)^2+(-2*x^4*exp(x^2+12)^2-8*x^4*exp(
x^2+12)-18*x^4-10*x^2)*log(x)+x^4*exp(x^2+12)^4+8*x^4*exp(x^2+12)^3+(34*x^4+10*x^2)*exp(x^2+12)^2+(72*x^4+40*x
^2)*exp(x^2+12)+81*x^4+90*x^2+25),x, algorithm="maxima")

[Out]

4*x^2/(x^2*e^(2*x^2 + 24) + 4*x^2*e^(x^2 + 12) - x^2*log(x) + 9*x^2 + 5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {40\,x-16\,x^5\,{\mathrm {e}}^{2\,x^2+24}-32\,x^5\,{\mathrm {e}}^{x^2+12}+4\,x^3}{{\mathrm {e}}^{2\,x^2+24}\,\left (34\,x^4+10\,x^2\right )-\ln \relax (x)\,\left (2\,x^4\,{\mathrm {e}}^{2\,x^2+24}+8\,x^4\,{\mathrm {e}}^{x^2+12}+10\,x^2+18\,x^4\right )+8\,x^4\,{\mathrm {e}}^{3\,x^2+36}+x^4\,{\mathrm {e}}^{4\,x^2+48}+x^4\,{\ln \relax (x)}^2+{\mathrm {e}}^{x^2+12}\,\left (72\,x^4+40\,x^2\right )+90\,x^2+81\,x^4+25} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x - 16*x^5*exp(2*x^2 + 24) - 32*x^5*exp(x^2 + 12) + 4*x^3)/(exp(2*x^2 + 24)*(10*x^2 + 34*x^4) - log(x)
*(2*x^4*exp(2*x^2 + 24) + 8*x^4*exp(x^2 + 12) + 10*x^2 + 18*x^4) + 8*x^4*exp(3*x^2 + 36) + x^4*exp(4*x^2 + 48)
+ x^4*log(x)^2 + exp(x^2 + 12)*(40*x^2 + 72*x^4) + 90*x^2 + 81*x^4 + 25),x)

[Out]

int((40*x - 16*x^5*exp(2*x^2 + 24) - 32*x^5*exp(x^2 + 12) + 4*x^3)/(exp(2*x^2 + 24)*(10*x^2 + 34*x^4) - log(x)
*(2*x^4*exp(2*x^2 + 24) + 8*x^4*exp(x^2 + 12) + 10*x^2 + 18*x^4) + 8*x^4*exp(3*x^2 + 36) + x^4*exp(4*x^2 + 48)
+ x^4*log(x)^2 + exp(x^2 + 12)*(40*x^2 + 72*x^4) + 90*x^2 + 81*x^4 + 25), x)

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sympy [B]  time = 0.39, size = 41, normalized size = 1.58 \begin {gather*} \frac {4 x^{2}}{4 x^{2} e^{x^{2} + 12} + x^{2} e^{2 x^{2} + 24} - x^{2} \log {\relax (x )} + 9 x^{2} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*x**5*exp(x**2+12)**2-32*x**5*exp(x**2+12)+4*x**3+40*x)/(x**4*ln(x)**2+(-2*x**4*exp(x**2+12)**2-
8*x**4*exp(x**2+12)-18*x**4-10*x**2)*ln(x)+x**4*exp(x**2+12)**4+8*x**4*exp(x**2+12)**3+(34*x**4+10*x**2)*exp(x
**2+12)**2+(72*x**4+40*x**2)*exp(x**2+12)+81*x**4+90*x**2+25),x)

[Out]

4*x**2/(4*x**2*exp(x**2 + 12) + x**2*exp(2*x**2 + 24) - x**2*log(x) + 9*x**2 + 5)

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