3.5.99 \(\int (e^{x^2} (600+4090 x+1200 x^2+90 x^3)+e^{x^2} (400+60 x) \log (x)+e^{x^2} (200+60 x+400 x^2+60 x^3) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} (10 x+10 x^3) \log ^4(x)) \, dx\)

Optimal. Leaf size=23 \[ 5 e^{x^2} \left (20+x-x \left (-2-\log ^2(x)\right )\right )^2 \]

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Rubi [F]  time = 0.79, 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 \left (e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right )+e^{x^2} (400+60 x) \log (x)+e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x)+20 e^{x^2} x \log ^3(x)+e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^x^2*(600 + 4090*x + 1200*x^2 + 90*x^3) + E^x^2*(400 + 60*x)*Log[x] + E^x^2*(200 + 60*x + 400*x^2 + 60*x^
3)*Log[x]^2 + 20*E^x^2*x*Log[x]^3 + E^x^2*(10*x + 10*x^3)*Log[x]^4,x]

[Out]

2000*E^x^2 + 600*E^x^2*x + 45*E^x^2*x^2 - 15*ExpIntegralEi[x^2] - 400*x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/
2}, x^2] + 30*E^x^2*Log[x] + 200*Sqrt[Pi]*Erfi[x]*Log[x] + 200*Defer[Int][E^x^2*Log[x]^2, x] + 60*Defer[Int][E
^x^2*x*Log[x]^2, x] + 400*Defer[Int][E^x^2*x^2*Log[x]^2, x] + 60*Defer[Int][E^x^2*x^3*Log[x]^2, x] + 20*Defer[
Int][E^x^2*x*Log[x]^3, x] + 10*Defer[Int][E^x^2*x*Log[x]^4, x] + 10*Defer[Int][E^x^2*x^3*Log[x]^4, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=20 \int e^{x^2} x \log ^3(x) \, dx+\int e^{x^2} \left (600+4090 x+1200 x^2+90 x^3\right ) \, dx+\int e^{x^2} (400+60 x) \log (x) \, dx+\int e^{x^2} \left (200+60 x+400 x^2+60 x^3\right ) \log ^2(x) \, dx+\int e^{x^2} \left (10 x+10 x^3\right ) \log ^4(x) \, dx\\ &=30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)+20 \int e^{x^2} x \log ^3(x) \, dx+\int \left (600 e^{x^2}+4090 e^{x^2} x+1200 e^{x^2} x^2+90 e^{x^2} x^3\right ) \, dx-\int \frac {10 \left (3 e^{x^2}+20 \sqrt {\pi } \text {erfi}(x)\right )}{x} \, dx+\int e^{x^2} x \left (10+10 x^2\right ) \log ^4(x) \, dx+\int \left (200 e^{x^2} \log ^2(x)+60 e^{x^2} x \log ^2(x)+400 e^{x^2} x^2 \log ^2(x)+60 e^{x^2} x^3 \log ^2(x)\right ) \, dx\\ &=30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)-10 \int \frac {3 e^{x^2}+20 \sqrt {\pi } \text {erfi}(x)}{x} \, dx+20 \int e^{x^2} x \log ^3(x) \, dx+60 \int e^{x^2} x \log ^2(x) \, dx+60 \int e^{x^2} x^3 \log ^2(x) \, dx+90 \int e^{x^2} x^3 \, dx+200 \int e^{x^2} \log ^2(x) \, dx+400 \int e^{x^2} x^2 \log ^2(x) \, dx+600 \int e^{x^2} \, dx+1200 \int e^{x^2} x^2 \, dx+4090 \int e^{x^2} x \, dx+\int \left (10 e^{x^2} x \log ^4(x)+10 e^{x^2} x^3 \log ^4(x)\right ) \, dx\\ &=2045 e^{x^2}+600 e^{x^2} x+45 e^{x^2} x^2+300 \sqrt {\pi } \text {erfi}(x)+30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)-10 \int \left (\frac {3 e^{x^2}}{x}+\frac {20 \sqrt {\pi } \text {erfi}(x)}{x}\right ) \, dx+10 \int e^{x^2} x \log ^4(x) \, dx+10 \int e^{x^2} x^3 \log ^4(x) \, dx+20 \int e^{x^2} x \log ^3(x) \, dx+60 \int e^{x^2} x \log ^2(x) \, dx+60 \int e^{x^2} x^3 \log ^2(x) \, dx-90 \int e^{x^2} x \, dx+200 \int e^{x^2} \log ^2(x) \, dx+400 \int e^{x^2} x^2 \log ^2(x) \, dx-600 \int e^{x^2} \, dx\\ &=2000 e^{x^2}+600 e^{x^2} x+45 e^{x^2} x^2+30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)+10 \int e^{x^2} x \log ^4(x) \, dx+10 \int e^{x^2} x^3 \log ^4(x) \, dx+20 \int e^{x^2} x \log ^3(x) \, dx-30 \int \frac {e^{x^2}}{x} \, dx+60 \int e^{x^2} x \log ^2(x) \, dx+60 \int e^{x^2} x^3 \log ^2(x) \, dx+200 \int e^{x^2} \log ^2(x) \, dx+400 \int e^{x^2} x^2 \log ^2(x) \, dx-\left (200 \sqrt {\pi }\right ) \int \frac {\text {erfi}(x)}{x} \, dx\\ &=2000 e^{x^2}+600 e^{x^2} x+45 e^{x^2} x^2-15 \text {Ei}\left (x^2\right )-400 x \, _2F_2\left (\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};x^2\right )+30 e^{x^2} \log (x)+200 \sqrt {\pi } \text {erfi}(x) \log (x)+10 \int e^{x^2} x \log ^4(x) \, dx+10 \int e^{x^2} x^3 \log ^4(x) \, dx+20 \int e^{x^2} x \log ^3(x) \, dx+60 \int e^{x^2} x \log ^2(x) \, dx+60 \int e^{x^2} x^3 \log ^2(x) \, dx+200 \int e^{x^2} \log ^2(x) \, dx+400 \int e^{x^2} x^2 \log ^2(x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.76, size = 20, normalized size = 0.87 \begin {gather*} 5 e^{x^2} \left (20+3 x+x \log ^2(x)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^x^2*(600 + 4090*x + 1200*x^2 + 90*x^3) + E^x^2*(400 + 60*x)*Log[x] + E^x^2*(200 + 60*x + 400*x^2 +
60*x^3)*Log[x]^2 + 20*E^x^2*x*Log[x]^3 + E^x^2*(10*x + 10*x^3)*Log[x]^4,x]

[Out]

5*E^x^2*(20 + 3*x + x*Log[x]^2)^2

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fricas [B]  time = 0.70, size = 49, normalized size = 2.13 \begin {gather*} 5 \, x^{2} e^{\left (x^{2}\right )} \log \relax (x)^{4} + 10 \, {\left (3 \, x^{2} + 20 \, x\right )} e^{\left (x^{2}\right )} \log \relax (x)^{2} + 5 \, {\left (9 \, x^{2} + 120 \, x + 400\right )} e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^3+10*x)*exp(x^2)*log(x)^4+20*x*exp(x^2)*log(x)^3+(60*x^3+400*x^2+60*x+200)*exp(x^2)*log(x)^2+(
60*x+400)*exp(x^2)*log(x)+(90*x^3+1200*x^2+4090*x+600)*exp(x^2),x, algorithm="fricas")

[Out]

5*x^2*e^(x^2)*log(x)^4 + 10*(3*x^2 + 20*x)*e^(x^2)*log(x)^2 + 5*(9*x^2 + 120*x + 400)*e^(x^2)

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giac [B]  time = 0.38, size = 57, normalized size = 2.48 \begin {gather*} 5 \, {\left ({\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} + e^{\left (x^{2}\right )}\right )} \log \relax (x)^{4} + 10 \, {\left (3 \, x^{2} + 20 \, x\right )} e^{\left (x^{2}\right )} \log \relax (x)^{2} + 5 \, {\left (9 \, x^{2} + 120 \, x + 400\right )} e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^3+10*x)*exp(x^2)*log(x)^4+20*x*exp(x^2)*log(x)^3+(60*x^3+400*x^2+60*x+200)*exp(x^2)*log(x)^2+(
60*x+400)*exp(x^2)*log(x)+(90*x^3+1200*x^2+4090*x+600)*exp(x^2),x, algorithm="giac")

[Out]

5*((x^2 - 1)*e^(x^2) + e^(x^2))*log(x)^4 + 10*(3*x^2 + 20*x)*e^(x^2)*log(x)^2 + 5*(9*x^2 + 120*x + 400)*e^(x^2
)

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maple [B]  time = 0.13, size = 46, normalized size = 2.00

method	result	size
risch	\(\left (45 x^{2}+600 x +2000\right ) {\mathrm e}^{x^{2}}+5 x^{2} {\mathrm e}^{x^{2}} \ln \relax (x )^{4}+10 x \left (3 x +20\right ) {\mathrm e}^{x^{2}} \ln \relax (x )^{2}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x^3+10*x)*exp(x^2)*ln(x)^4+20*x*exp(x^2)*ln(x)^3+(60*x^3+400*x^2+60*x+200)*exp(x^2)*ln(x)^2+(60*x+400)
*exp(x^2)*ln(x)+(90*x^3+1200*x^2+4090*x+600)*exp(x^2),x,method=_RETURNVERBOSE)

[Out]

(45*x^2+600*x+2000)*exp(x^2)+5*x^2*exp(x^2)*ln(x)^4+10*x*(3*x+20)*exp(x^2)*ln(x)^2

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maxima [B]  time = 0.50, size = 55, normalized size = 2.39 \begin {gather*} 5 \, {\left (x^{2} \log \relax (x)^{4} + 2 \, {\left (3 \, x^{2} + 20 \, x\right )} \log \relax (x)^{2}\right )} e^{\left (x^{2}\right )} + 45 \, {\left (x^{2} - 1\right )} e^{\left (x^{2}\right )} + 600 \, x e^{\left (x^{2}\right )} + 2045 \, e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x^3+10*x)*exp(x^2)*log(x)^4+20*x*exp(x^2)*log(x)^3+(60*x^3+400*x^2+60*x+200)*exp(x^2)*log(x)^2+(
60*x+400)*exp(x^2)*log(x)+(90*x^3+1200*x^2+4090*x+600)*exp(x^2),x, algorithm="maxima")

[Out]

5*(x^2*log(x)^4 + 2*(3*x^2 + 20*x)*log(x)^2)*e^(x^2) + 45*(x^2 - 1)*e^(x^2) + 600*x*e^(x^2) + 2045*e^(x^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\mathrm {e}}^{x^2}\,\left (10\,x^3+10\,x\right )\,{\ln \relax (x)}^4+20\,x\,{\mathrm {e}}^{x^2}\,{\ln \relax (x)}^3+{\mathrm {e}}^{x^2}\,\left (60\,x^3+400\,x^2+60\,x+200\right )\,{\ln \relax (x)}^2+{\mathrm {e}}^{x^2}\,\left (60\,x+400\right )\,\ln \relax (x)+{\mathrm {e}}^{x^2}\,\left (90\,x^3+1200\,x^2+4090\,x+600\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x^2)*(4090*x + 1200*x^2 + 90*x^3 + 600) + exp(x^2)*log(x)^4*(10*x + 10*x^3) + 20*x*exp(x^2)*log(x)^3 +
exp(x^2)*log(x)^2*(60*x + 400*x^2 + 60*x^3 + 200) + exp(x^2)*log(x)*(60*x + 400),x)

[Out]

int(exp(x^2)*(4090*x + 1200*x^2 + 90*x^3 + 600) + exp(x^2)*log(x)^4*(10*x + 10*x^3) + 20*x*exp(x^2)*log(x)^3 +
exp(x^2)*log(x)^2*(60*x + 400*x^2 + 60*x^3 + 200) + exp(x^2)*log(x)*(60*x + 400), x)

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sympy [B]  time = 0.45, size = 42, normalized size = 1.83 \begin {gather*} \left (5 x^{2} \log {\relax (x )}^{4} + 30 x^{2} \log {\relax (x )}^{2} + 45 x^{2} + 200 x \log {\relax (x )}^{2} + 600 x + 2000\right ) e^{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((10*x**3+10*x)*exp(x**2)*ln(x)**4+20*x*exp(x**2)*ln(x)**3+(60*x**3+400*x**2+60*x+200)*exp(x**2)*ln(x
)**2+(60*x+400)*exp(x**2)*ln(x)+(90*x**3+1200*x**2+4090*x+600)*exp(x**2),x)

[Out]

(5*x**2*log(x)**4 + 30*x**2*log(x)**2 + 45*x**2 + 200*x*log(x)**2 + 600*x + 2000)*exp(x**2)

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