3.61.69 \(\int \frac {184 x+e^{x+x^2} (-184 x-368 x^2)+e^{\frac {1}{16} (144-192 x+64 x^2+(-24+16 x) \log (x)+\log ^2(x))} (-276-2024 x+1472 x^2+(23+184 x) \log (x))}{32 x} \, dx\)

Optimal. Leaf size=30 \[ \frac {23}{4} \left (-e^{x (1+x)}+e^{\left (-3+2 x+\frac {\log (x)}{4}\right )^2}+x\right ) \]

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Rubi [F]  time = 2.04, 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 {184 x+e^{x+x^2} \left (-184 x-368 x^2\right )+\exp \left (\frac {1}{16} \left (144-192 x+64 x^2+(-24+16 x) \log (x)+\log ^2(x)\right )\right ) \left (-276-2024 x+1472 x^2+(23+184 x) \log (x)\right )}{32 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(184*x + E^(x + x^2)*(-184*x - 368*x^2) + E^((144 - 192*x + 64*x^2 + (-24 + 16*x)*Log[x] + Log[x]^2)/16)*(
-276 - 2024*x + 1472*x^2 + (23 + 184*x)*Log[x]))/(32*x),x]

[Out]

(-23*E^(x + x^2))/4 + (23*x)/4 - (69*Defer[Int][E^((144 - 192*x + 64*x^2 + Log[x]^2)/16)*x^(-5/2 + x), x])/8 -
(253*Defer[Int][E^((144 - 192*x + 64*x^2 + Log[x]^2)/16)*x^(-3/2 + x), x])/4 + 46*Defer[Int][E^((144 - 192*x
+ 64*x^2 + Log[x]^2)/16)*x^(-1/2 + x), x] + (23*Defer[Int][E^((144 - 192*x + 64*x^2 + Log[x]^2)/16)*x^(-5/2 +
x)*Log[x], x])/32 + (23*Defer[Int][E^((144 - 192*x + 64*x^2 + Log[x]^2)/16)*x^(-3/2 + x)*Log[x], x])/4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{32} \int \frac {184 x+e^{x+x^2} \left (-184 x-368 x^2\right )+\exp \left (\frac {1}{16} \left (144-192 x+64 x^2+(-24+16 x) \log (x)+\log ^2(x)\right )\right ) \left (-276-2024 x+1472 x^2+(23+184 x) \log (x)\right )}{x} \, dx\\ &=\frac {1}{32} \int \left (-184 \left (-1+e^{x+x^2}+2 e^{x+x^2} x\right )+23 e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x} (1+8 x) (-12+8 x+\log (x))\right ) \, dx\\ &=\frac {23}{32} \int e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x} (1+8 x) (-12+8 x+\log (x)) \, dx-\frac {23}{4} \int \left (-1+e^{x+x^2}+2 e^{x+x^2} x\right ) \, dx\\ &=\frac {23 x}{4}+\frac {23}{32} \int \left (4 e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x} \left (-3-22 x+16 x^2\right )+e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x} (1+8 x) \log (x)\right ) \, dx-\frac {23}{4} \int e^{x+x^2} \, dx-\frac {23}{2} \int e^{x+x^2} x \, dx\\ &=-\frac {23}{4} e^{x+x^2}+\frac {23 x}{4}+\frac {23}{32} \int e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x} (1+8 x) \log (x) \, dx+\frac {23}{8} \int e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x} \left (-3-22 x+16 x^2\right ) \, dx+\frac {23}{4} \int e^{x+x^2} \, dx-\frac {23 \int e^{\frac {1}{4} (1+2 x)^2} \, dx}{4 \sqrt [4]{e}}\\ &=-\frac {23}{4} e^{x+x^2}+\frac {23 x}{4}-\frac {23 \sqrt {\pi } \text {erfi}\left (\frac {1}{2} (1+2 x)\right )}{8 \sqrt [4]{e}}+\frac {23}{32} \int \left (e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x} \log (x)+8 e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {3}{2}+x} \log (x)\right ) \, dx+\frac {23}{8} \int \left (-3 e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x}-22 e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {3}{2}+x}+16 e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {1}{2}+x}\right ) \, dx+\frac {23 \int e^{\frac {1}{4} (1+2 x)^2} \, dx}{4 \sqrt [4]{e}}\\ &=-\frac {23}{4} e^{x+x^2}+\frac {23 x}{4}+\frac {23}{32} \int e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x} \log (x) \, dx+\frac {23}{4} \int e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {3}{2}+x} \log (x) \, dx-\frac {69}{8} \int e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {5}{2}+x} \, dx+46 \int e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {1}{2}+x} \, dx-\frac {253}{4} \int e^{\frac {1}{16} \left (144-192 x+64 x^2+\log ^2(x)\right )} x^{-\frac {3}{2}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.78, size = 52, normalized size = 1.73 \begin {gather*} \frac {23}{32} \left (-8 e^{x+x^2}+8 x+8 e^{9-12 x+4 x^2+\frac {\log ^2(x)}{16}} x^{1+\frac {1}{2} (-5+2 x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(184*x + E^(x + x^2)*(-184*x - 368*x^2) + E^((144 - 192*x + 64*x^2 + (-24 + 16*x)*Log[x] + Log[x]^2)
/16)*(-276 - 2024*x + 1472*x^2 + (23 + 184*x)*Log[x]))/(32*x),x]

[Out]

(23*(-8*E^(x + x^2) + 8*x + 8*E^(9 - 12*x + 4*x^2 + Log[x]^2/16)*x^(1 + (-5 + 2*x)/2)))/32

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fricas [A]  time = 0.60, size = 40, normalized size = 1.33 \begin {gather*} \frac {23}{4} \, x + \frac {23}{4} \, e^{\left (4 \, x^{2} + \frac {1}{2} \, {\left (2 \, x - 3\right )} \log \relax (x) + \frac {1}{16} \, \log \relax (x)^{2} - 12 \, x + 9\right )} - \frac {23}{4} \, e^{\left (x^{2} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(((184*x+23)*log(x)+1472*x^2-2024*x-276)*exp(1/16*log(x)^2+1/16*(16*x-24)*log(x)+4*x^2-12*x+9)+
(-368*x^2-184*x)*exp(x^2+x)+184*x)/x,x, algorithm="fricas")

[Out]

23/4*x + 23/4*e^(4*x^2 + 1/2*(2*x - 3)*log(x) + 1/16*log(x)^2 - 12*x + 9) - 23/4*e^(x^2 + x)

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giac [A]  time = 0.23, size = 39, normalized size = 1.30 \begin {gather*} \frac {23}{4} \, x + \frac {23}{4} \, e^{\left (4 \, x^{2} + x \log \relax (x) + \frac {1}{16} \, \log \relax (x)^{2} - 12 \, x - \frac {3}{2} \, \log \relax (x) + 9\right )} - \frac {23}{4} \, e^{\left (x^{2} + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(((184*x+23)*log(x)+1472*x^2-2024*x-276)*exp(1/16*log(x)^2+1/16*(16*x-24)*log(x)+4*x^2-12*x+9)+
(-368*x^2-184*x)*exp(x^2+x)+184*x)/x,x, algorithm="giac")

[Out]

23/4*x + 23/4*e^(4*x^2 + x*log(x) + 1/16*log(x)^2 - 12*x - 3/2*log(x) + 9) - 23/4*e^(x^2 + x)

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maple [A]  time = 0.05, size = 37, normalized size = 1.23

method	result	size
risch	\(\frac {23 x}{4}-\frac {23 \,{\mathrm e}^{\left (x +1\right ) x}}{4}+\frac {23 x^{x -\frac {3}{2}} {\mathrm e}^{\frac {\ln \relax (x )^{2}}{16}+9+4 x^{2}-12 x}}{4}\)	\(37\)
default	\(\frac {23 x}{4}-\frac {23 \,{\mathrm e}^{x^{2}+x}}{4}+\frac {23 \,{\mathrm e}^{\frac {\ln \relax (x )^{2}}{16}+\frac {\left (16 x -24\right ) \ln \relax (x )}{16}+4 x^{2}-12 x +9}}{4}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/32*(((184*x+23)*ln(x)+1472*x^2-2024*x-276)*exp(1/16*ln(x)^2+1/16*(16*x-24)*ln(x)+4*x^2-12*x+9)+(-368*x^2
-184*x)*exp(x^2+x)+184*x)/x,x,method=_RETURNVERBOSE)

[Out]

23/4*x-23/4*exp((x+1)*x)+23/4*x^(x-3/2)*exp(1/16*ln(x)^2+9+4*x^2-12*x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {23}{8} i \, \sqrt {\pi } \operatorname {erf}\left (i \, x + \frac {1}{2} i\right ) e^{\left (-\frac {1}{4}\right )} + \frac {23}{8} \, {\left (\frac {\sqrt {\pi } {\left (2 \, x + 1\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (2 \, x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {-{\left (2 \, x + 1\right )}^{2}}} - 2 \, e^{\left (\frac {1}{4} \, {\left (2 \, x + 1\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{4}\right )} + \frac {23}{4} \, x + \frac {23}{32} \, \int \frac {{\left (64 \, x^{2} e^{\left (4 \, x^{2} + x \log \relax (x) + 9\right )} + 8 \, {\left (e^{9} \log \relax (x) - 11 \, e^{9}\right )} x e^{\left (4 \, x^{2} + x \log \relax (x)\right )} + {\left (e^{9} \log \relax (x) - 12 \, e^{9}\right )} e^{\left (4 \, x^{2} + x \log \relax (x)\right )}\right )} e^{\left (\frac {1}{16} \, \log \relax (x)^{2} - 12 \, x\right )}}{x^{\frac {5}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(((184*x+23)*log(x)+1472*x^2-2024*x-276)*exp(1/16*log(x)^2+1/16*(16*x-24)*log(x)+4*x^2-12*x+9)+
(-368*x^2-184*x)*exp(x^2+x)+184*x)/x,x, algorithm="maxima")

[Out]

23/8*I*sqrt(pi)*erf(I*x + 1/2*I)*e^(-1/4) + 23/8*(sqrt(pi)*(2*x + 1)*(erf(1/2*sqrt(-(2*x + 1)^2)) - 1)/sqrt(-(
2*x + 1)^2) - 2*e^(1/4*(2*x + 1)^2))*e^(-1/4) + 23/4*x + 23/32*integrate((64*x^2*e^(4*x^2 + x*log(x) + 9) + 8*
(e^9*log(x) - 11*e^9)*x*e^(4*x^2 + x*log(x)) + (e^9*log(x) - 12*e^9)*e^(4*x^2 + x*log(x)))*e^(1/16*log(x)^2 -
12*x)/x^(5/2), x)

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mupad [B]  time = 4.79, size = 39, normalized size = 1.30 \begin {gather*} \frac {23\,x}{4}-\frac {23\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^x}{4}+\frac {23\,x^x\,{\mathrm {e}}^{\frac {{\ln \relax (x)}^2}{16}}\,{\mathrm {e}}^{-12\,x}\,{\mathrm {e}}^9\,{\mathrm {e}}^{4\,x^2}}{4\,x^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((exp(log(x)^2/16 - 12*x + (log(x)*(16*x - 24))/16 + 4*x^2 + 9)*(2024*x - log(x)*(184*x + 23) - 1472*x^2
+ 276))/32 - (23*x)/4 + (exp(x + x^2)*(184*x + 368*x^2))/32)/x,x)

[Out]

(23*x)/4 - (23*exp(x^2)*exp(x))/4 + (23*x^x*exp(log(x)^2/16)*exp(-12*x)*exp(9)*exp(4*x^2))/(4*x^(3/2))

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sympy [A]  time = 0.62, size = 44, normalized size = 1.47 \begin {gather*} \frac {23 x}{4} - \frac {23 e^{x^{2} + x}}{4} + \frac {23 e^{4 x^{2} - 12 x + \left (x - \frac {3}{2}\right ) \log {\relax (x )} + \frac {\log {\relax (x )}^{2}}{16} + 9}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/32*(((184*x+23)*ln(x)+1472*x**2-2024*x-276)*exp(1/16*ln(x)**2+1/16*(16*x-24)*ln(x)+4*x**2-12*x+9)+
(-368*x**2-184*x)*exp(x**2+x)+184*x)/x,x)

[Out]

23*x/4 - 23*exp(x**2 + x)/4 + 23*exp(4*x**2 - 12*x + (x - 3/2)*log(x) + log(x)**2/16 + 9)/4

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