3.32.34 \(\int \frac {4 x^3+e^{\frac {2 x+3 x^3-x^4+(2+3 x^2-x^3) \log (x)}{2 x}} (-2-3 x^2-5 x^3+3 x^4+(2-3 x^2+2 x^3) \log (x))}{2 x^2} \, dx\)

Optimal. Leaf size=31 \[ -e^{\frac {1}{2} \left (\left (3+\frac {2}{x^2}\right ) x-x^2\right ) (x+\log (x))}+x^2 \]

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Rubi [F]  time = 4.67, 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 {4 x^3+\exp \left (\frac {2 x+3 x^3-x^4+\left (2+3 x^2-x^3\right ) \log (x)}{2 x}\right ) \left (-2-3 x^2-5 x^3+3 x^4+\left (2-3 x^2+2 x^3\right ) \log (x)\right )}{2 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x^3 + E^((2*x + 3*x^3 - x^4 + (2 + 3*x^2 - x^3)*Log[x])/(2*x))*(-2 - 3*x^2 - 5*x^3 + 3*x^4 + (2 - 3*x^2
+ 2*x^3)*Log[x]))/(2*x^2),x]

[Out]

x^2 - Defer[Int][E^(1 + (3*x^2)/2 - x^3/2)*x^(-2 + x^(-1) + (3*x)/2 - x^2/2), x] + Log[x]*Defer[Int][E^(1 + (3
*x^2)/2 - x^3/2)*x^(-2 + x^(-1) + (3*x)/2 - x^2/2), x] - (5*Defer[Int][E^(1 + (3*x^2)/2 - x^3/2)*x^(1 + x^(-1)
+ (3*x)/2 - x^2/2), x])/2 + Log[x]*Defer[Int][E^(1 + (3*x^2)/2 - x^3/2)*x^(1 + x^(-1) + (3*x)/2 - x^2/2), x]
+ (3*Defer[Int][E^(1 + (3*x^2)/2 - x^3/2)*x^(2 + x^(-1) + (3*x)/2 - x^2/2), x])/2 - (3*Defer[Int][E^(1 + (3*x^
2)/2 - x^3/2)*x^((1 + (3*x^2)/2 - x^3/2)/x), x])/2 - (3*Log[x]*Defer[Int][E^(1 + (3*x^2)/2 - x^3/2)*x^((1 + (3
*x^2)/2 - x^3/2)/x), x])/2 - Defer[Int][Defer[Int][E^(1 + (3*x^2)/2 - x^3/2)*x^(-2 + x^(-1) + (3*x)/2 - x^2/2)
, x]/x, x] - Defer[Int][Defer[Int][E^(1 + (3*x^2)/2 - x^3/2)*x^(1 + x^(-1) + (3*x)/2 - x^2/2), x]/x, x] + (3*D
efer[Int][Defer[Int][E^(1 + (3*x^2)/2 - x^3/2)*x^((1 + (3*x^2)/2 - x^3/2)/x), x]/x, x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {4 x^3+\exp \left (\frac {2 x+3 x^3-x^4+\left (2+3 x^2-x^3\right ) \log (x)}{2 x}\right ) \left (-2-3 x^2-5 x^3+3 x^4+\left (2-3 x^2+2 x^3\right ) \log (x)\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \left (4 x+e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \left (-2-3 x^2-5 x^3+3 x^4+2 \log (x)-3 x^2 \log (x)+2 x^3 \log (x)\right )\right ) \, dx\\ &=x^2+\frac {1}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \left (-2-3 x^2-5 x^3+3 x^4+2 \log (x)-3 x^2 \log (x)+2 x^3 \log (x)\right ) \, dx\\ &=x^2+\frac {1}{2} \int \left (-2 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}}-3 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}}-5 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}}+3 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}}+2 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x)-3 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x)+2 e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x)\right ) \, dx\\ &=x^2-\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx+\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x) \, dx-\frac {5}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx+\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x) \, dx+\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \log (x) \, dx\\ &=x^2+\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\frac {3}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1+\frac {3 x^2}{2}-\frac {x^3}{2}}{x}} \, dx+\frac {3}{2} \int \frac {\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1+\frac {3 x^2}{2}-\frac {x^3}{2}}{x}} \, dx}{x} \, dx-\frac {5}{2} \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx+\log (x) \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx+\log (x) \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\frac {1}{2} (3 \log (x)) \int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1+\frac {3 x^2}{2}-\frac {x^3}{2}}{x}} \, dx-\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx-\int \frac {\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{-2+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx}{x} \, dx-\int \frac {\int e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{1+\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \, dx}{x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.36, size = 42, normalized size = 1.35 \begin {gather*} x^2-e^{1+\frac {3 x^2}{2}-\frac {x^3}{2}} x^{\frac {1}{x}+\frac {3 x}{2}-\frac {x^2}{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^3 + E^((2*x + 3*x^3 - x^4 + (2 + 3*x^2 - x^3)*Log[x])/(2*x))*(-2 - 3*x^2 - 5*x^3 + 3*x^4 + (2 -
3*x^2 + 2*x^3)*Log[x]))/(2*x^2),x]

[Out]

x^2 - E^(1 + (3*x^2)/2 - x^3/2)*x^(x^(-1) + (3*x)/2 - x^2/2)

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fricas [A]  time = 0.50, size = 37, normalized size = 1.19 \begin {gather*} x^{2} - e^{\left (-\frac {x^{4} - 3 \, x^{3} + {\left (x^{3} - 3 \, x^{2} - 2\right )} \log \relax (x) - 2 \, x}{2 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - e^(-1/2*(x^4 - 3*x^3 + (x^3 - 3*x^2 - 2)*log(x) - 2*x)/x)

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giac [A]  time = 0.68, size = 37, normalized size = 1.19 \begin {gather*} x^{2} - e^{\left (-\frac {1}{2} \, x^{3} - \frac {1}{2} \, x^{2} \log \relax (x) + \frac {3}{2} \, x^{2} + \frac {3}{2} \, x \log \relax (x) + \frac {\log \relax (x)}{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - e^(-1/2*x^3 - 1/2*x^2*log(x) + 3/2*x^2 + 3/2*x*log(x) + log(x)/x + 1)

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maple [A]  time = 0.07, size = 27, normalized size = 0.87

method	result	size
risch	\(-{\mathrm e}^{-\frac {\left (x^{3}-3 x^{2}-2\right ) \left (x +\ln \relax (x )\right )}{2 x}}+x^{2}\)	\(27\)
default	\(-{\mathrm e}^{\frac {\left (-x^{3}+3 x^{2}+2\right ) \ln \relax (x )-x^{4}+3 x^{3}+2 x}{2 x}}+x^{2}\)	\(42\)
norman	\(\frac {x^{3}-x \,{\mathrm e}^{\frac {\left (-x^{3}+3 x^{2}+2\right ) \ln \relax (x )-x^{4}+3 x^{3}+2 x}{2 x}}}{x}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(((2*x^3-3*x^2+2)*ln(x)+3*x^4-5*x^3-3*x^2-2)*exp(1/2*((-x^3+3*x^2+2)*ln(x)-x^4+3*x^3+2*x)/x)+4*x^3)/x^
2,x,method=_RETURNVERBOSE)

[Out]

-exp(-1/2*(x^3-3*x^2-2)*(x+ln(x))/x)+x^2

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maxima [A]  time = 0.56, size = 37, normalized size = 1.19 \begin {gather*} x^{2} - e^{\left (-\frac {1}{2} \, x^{3} - \frac {1}{2} \, x^{2} \log \relax (x) + \frac {3}{2} \, x^{2} + \frac {3}{2} \, x \log \relax (x) + \frac {\log \relax (x)}{x} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2 - e^(-1/2*x^3 - 1/2*x^2*log(x) + 3/2*x^2 + 3/2*x*log(x) + log(x)/x + 1)

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mupad [B]  time = 2.10, size = 39, normalized size = 1.26 \begin {gather*} x^2-\frac {x^{1/x}\,{\left ({\mathrm {e}}^{x^2}\right )}^{3/2}\,{\mathrm {e}}^{\frac {3\,x\,\ln \relax (x)}{2}}\,\mathrm {e}\,{\mathrm {e}}^{-\frac {x^2\,\ln \relax (x)}{2}}}{\sqrt {{\mathrm {e}}^{x^3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp((x + (log(x)*(3*x^2 - x^3 + 2))/2 + (3*x^3)/2 - x^4/2)/x)*(3*x^2 - log(x)*(2*x^3 - 3*x^2 + 2) + 5*x
^3 - 3*x^4 + 2))/2 - 2*x^3)/x^2,x)

[Out]

x^2 - (x^(1/x)*exp(x^2)^(3/2)*exp((3*x*log(x))/2)*exp(1)*exp(-(x^2*log(x))/2))/exp(x^3)^(1/2)

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sympy [A]  time = 0.42, size = 34, normalized size = 1.10 \begin {gather*} x^{2} - e^{\frac {- \frac {x^{4}}{2} + \frac {3 x^{3}}{2} + x + \frac {\left (- x^{3} + 3 x^{2} + 2\right ) \log {\relax (x )}}{2}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((2*x**3-3*x**2+2)*ln(x)+3*x**4-5*x**3-3*x**2-2)*exp(1/2*((-x**3+3*x**2+2)*ln(x)-x**4+3*x**3+2*
x)/x)+4*x**3)/x**2,x)

[Out]

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

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