3.45.1 \(\int x^{-2 x} (-225-50 x+x^{2 x} (75-e^x+30 x+3 x^2)+(-250-50 x) \log (x)+x^x (-150-80 x-10 x^2+(-250-100 x-10 x^2) \log (x))) \, dx\)

Optimal. Leaf size=22 \[ -e^x+(5+x) \left (5+x+5 x^{-x}\right )^2 \]

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Rubi [F]  time = 1.76, 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 x^{-2 x} \left (-225-50 x+x^{2 x} \left (75-e^x+30 x+3 x^2\right )+(-250-50 x) \log (x)+x^x \left (-150-80 x-10 x^2+\left (-250-100 x-10 x^2\right ) \log (x)\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-225 - 50*x + x^(2*x)*(75 - E^x + 30*x + 3*x^2) + (-250 - 50*x)*Log[x] + x^x*(-150 - 80*x - 10*x^2 + (-25
0 - 100*x - 10*x^2)*Log[x]))/x^(2*x),x]

[Out]

-E^x + 75*x + 15*x^2 + x^3 + (150*x^(1 - 2*x)*Hypergeometric2F1[1, 1 - 2*x, 2 - 2*x, x])/(1 - 2*x) - (160*x^(1
- x)*Hypergeometric2F1[1, 1 - x, 2 - x, x/3])/(3*(1 - x)) + (350*x^(1 - x)*Hypergeometric2F1[1, 1 - x, 2 - x,
x/2])/(1 - x) - (25*x^(1 - 2*x)*(5 + x)*Log[x])/(1 - x) - (100*x^(1 - x)*(5 + x)*Log[x])/((2 - x)*(3 - x)) -
(10*x^(1 - x)*(5 + x)^2*Log[x])/(3 - x) - 50*Defer[Int][x^(1 - 2*x), x] - 90*Defer[Int][x^(1 - x), x] - 10*Def
er[Int][x^(2 - x), x] - 250*Defer[Int][x^(-2*x), x] - (125*Log[x]*Defer[Int][x^(-2*x), x])/(1 - x) - 280*Defer
[Int][x^(-x), x] - (500*Log[x]*Defer[Int][x^(-x), x])/((2 - x)*(3 - x)) - 125*Defer[Int][Defer[Int][x^(-2*x),
x]/(-1 + x), x] + 125*Defer[Int][Defer[Int][x^(-2*x), x]/x, x] + (500*Defer[Int][Defer[Int][x^(-x), x]/(-3 + x
), x])/3 - 250*Defer[Int][Defer[Int][x^(-x), x]/(-2 + x), x] + (250*Defer[Int][Defer[Int][x^(-x), x]/x, x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (75-e^x+30 x+3 x^2-50 x^{1-2 x}-225 x^{-2 x}-50 x^{-2 x} (5+x) \log (x)-10 x^{-x} (5+x) (3+x+5 \log (x)+x \log (x))\right ) \, dx\\ &=75 x+15 x^2+x^3-10 \int x^{-x} (5+x) (3+x+5 \log (x)+x \log (x)) \, dx-50 \int x^{1-2 x} \, dx-50 \int x^{-2 x} (5+x) \log (x) \, dx-225 \int x^{-2 x} \, dx-\int e^x \, dx\\ &=-e^x+75 x+15 x^2+x^3-\frac {25 x^{1-2 x} (5+x) \log (x)}{1-x}-10 \int \left (8 x^{1-x}+x^{2-x}+15 x^{-x}+x^{-x} (5+x)^2 \log (x)\right ) \, dx-50 \int x^{1-2 x} \, dx+50 \int \frac {x^{-2 x} (5+x)+\frac {5 \int x^{-2 x} \, dx}{x}}{2-2 x} \, dx-225 \int x^{-2 x} \, dx-\frac {(125 \log (x)) \int x^{-2 x} \, dx}{1-x}\\ &=-e^x+75 x+15 x^2+x^3-\frac {25 x^{1-2 x} (5+x) \log (x)}{1-x}-10 \int x^{2-x} \, dx-10 \int x^{-x} (5+x)^2 \log (x) \, dx-50 \int x^{1-2 x} \, dx+50 \int \left (-\frac {x^{-2 x} (5+x)}{2 (-1+x)}-\frac {5 \int x^{-2 x} \, dx}{2 (-1+x) x}\right ) \, dx-80 \int x^{1-x} \, dx-150 \int x^{-x} \, dx-225 \int x^{-2 x} \, dx-\frac {(125 \log (x)) \int x^{-2 x} \, dx}{1-x}\\ &=-e^x+75 x+15 x^2+x^3-\frac {25 x^{1-2 x} (5+x) \log (x)}{1-x}-\frac {100 x^{1-x} (5+x) \log (x)}{(2-x) (3-x)}-\frac {10 x^{1-x} (5+x)^2 \log (x)}{3-x}-10 \int x^{2-x} \, dx+10 \int \frac {x^{-1-x} \left (-x \left (-100-5 x+8 x^2+x^3\right )+50 x^x \int x^{-x} \, dx\right )}{(2-x) (3-x)} \, dx-25 \int \frac {x^{-2 x} (5+x)}{-1+x} \, dx-50 \int x^{1-2 x} \, dx-80 \int x^{1-x} \, dx-125 \int \frac {\int x^{-2 x} \, dx}{(-1+x) x} \, dx-150 \int x^{-x} \, dx-225 \int x^{-2 x} \, dx-\frac {(125 \log (x)) \int x^{-2 x} \, dx}{1-x}-\frac {(500 \log (x)) \int x^{-x} \, dx}{(2-x) (3-x)}\\ &=-e^x+75 x+15 x^2+x^3-\frac {25 x^{1-2 x} (5+x) \log (x)}{1-x}-\frac {100 x^{1-x} (5+x) \log (x)}{(2-x) (3-x)}-\frac {10 x^{1-x} (5+x)^2 \log (x)}{3-x}-10 \int x^{2-x} \, dx+10 \int \left (-\frac {x^{-x} (5+x) \left (-20+3 x+x^2\right )}{(-3+x) (-2+x)}+\frac {50 \int x^{-x} \, dx}{x \left (6-5 x+x^2\right )}\right ) \, dx-25 \int \left (x^{-2 x}+\frac {6 x^{-2 x}}{-1+x}\right ) \, dx-50 \int x^{1-2 x} \, dx-80 \int x^{1-x} \, dx-125 \int \left (\frac {\int x^{-2 x} \, dx}{-1+x}-\frac {\int x^{-2 x} \, dx}{x}\right ) \, dx-150 \int x^{-x} \, dx-225 \int x^{-2 x} \, dx-\frac {(125 \log (x)) \int x^{-2 x} \, dx}{1-x}-\frac {(500 \log (x)) \int x^{-x} \, dx}{(2-x) (3-x)}\\ &=-e^x+75 x+15 x^2+x^3-\frac {25 x^{1-2 x} (5+x) \log (x)}{1-x}-\frac {100 x^{1-x} (5+x) \log (x)}{(2-x) (3-x)}-\frac {10 x^{1-x} (5+x)^2 \log (x)}{3-x}-10 \int x^{2-x} \, dx-10 \int \frac {x^{-x} (5+x) \left (-20+3 x+x^2\right )}{(-3+x) (-2+x)} \, dx-25 \int x^{-2 x} \, dx-50 \int x^{1-2 x} \, dx-80 \int x^{1-x} \, dx-125 \int \frac {\int x^{-2 x} \, dx}{-1+x} \, dx+125 \int \frac {\int x^{-2 x} \, dx}{x} \, dx-150 \int \frac {x^{-2 x}}{-1+x} \, dx-150 \int x^{-x} \, dx-225 \int x^{-2 x} \, dx+500 \int \frac {\int x^{-x} \, dx}{x \left (6-5 x+x^2\right )} \, dx-\frac {(125 \log (x)) \int x^{-2 x} \, dx}{1-x}-\frac {(500 \log (x)) \int x^{-x} \, dx}{(2-x) (3-x)}\\ &=-e^x+75 x+15 x^2+x^3+\frac {150 x^{1-2 x} \, _2F_1(1,1-2 x;2-2 x;x)}{1-2 x}-\frac {25 x^{1-2 x} (5+x) \log (x)}{1-x}-\frac {100 x^{1-x} (5+x) \log (x)}{(2-x) (3-x)}-\frac {10 x^{1-x} (5+x)^2 \log (x)}{3-x}-10 \int x^{2-x} \, dx-10 \int \left (x^{1-x}+13 x^{-x}-\frac {16 x^{-x}}{-3+x}+\frac {70 x^{-x}}{-2+x}\right ) \, dx-25 \int x^{-2 x} \, dx-50 \int x^{1-2 x} \, dx-80 \int x^{1-x} \, dx-125 \int \frac {\int x^{-2 x} \, dx}{-1+x} \, dx+125 \int \frac {\int x^{-2 x} \, dx}{x} \, dx-150 \int x^{-x} \, dx-225 \int x^{-2 x} \, dx+500 \int \left (\frac {\int x^{-x} \, dx}{3 (-3+x)}-\frac {\int x^{-x} \, dx}{2 (-2+x)}+\frac {\int x^{-x} \, dx}{6 x}\right ) \, dx-\frac {(125 \log (x)) \int x^{-2 x} \, dx}{1-x}-\frac {(500 \log (x)) \int x^{-x} \, dx}{(2-x) (3-x)}\\ &=-e^x+75 x+15 x^2+x^3+\frac {150 x^{1-2 x} \, _2F_1(1,1-2 x;2-2 x;x)}{1-2 x}-\frac {25 x^{1-2 x} (5+x) \log (x)}{1-x}-\frac {100 x^{1-x} (5+x) \log (x)}{(2-x) (3-x)}-\frac {10 x^{1-x} (5+x)^2 \log (x)}{3-x}-10 \int x^{1-x} \, dx-10 \int x^{2-x} \, dx-25 \int x^{-2 x} \, dx-50 \int x^{1-2 x} \, dx-80 \int x^{1-x} \, dx+\frac {250}{3} \int \frac {\int x^{-x} \, dx}{x} \, dx-125 \int \frac {\int x^{-2 x} \, dx}{-1+x} \, dx+125 \int \frac {\int x^{-2 x} \, dx}{x} \, dx-130 \int x^{-x} \, dx-150 \int x^{-x} \, dx+160 \int \frac {x^{-x}}{-3+x} \, dx+\frac {500}{3} \int \frac {\int x^{-x} \, dx}{-3+x} \, dx-225 \int x^{-2 x} \, dx-250 \int \frac {\int x^{-x} \, dx}{-2+x} \, dx-700 \int \frac {x^{-x}}{-2+x} \, dx-\frac {(125 \log (x)) \int x^{-2 x} \, dx}{1-x}-\frac {(500 \log (x)) \int x^{-x} \, dx}{(2-x) (3-x)}\\ &=-e^x+75 x+15 x^2+x^3+\frac {150 x^{1-2 x} \, _2F_1(1,1-2 x;2-2 x;x)}{1-2 x}-\frac {160 x^{1-x} \, _2F_1\left (1,1-x;2-x;\frac {x}{3}\right )}{3 (1-x)}+\frac {350 x^{1-x} \, _2F_1\left (1,1-x;2-x;\frac {x}{2}\right )}{1-x}-\frac {25 x^{1-2 x} (5+x) \log (x)}{1-x}-\frac {100 x^{1-x} (5+x) \log (x)}{(2-x) (3-x)}-\frac {10 x^{1-x} (5+x)^2 \log (x)}{3-x}-10 \int x^{1-x} \, dx-10 \int x^{2-x} \, dx-25 \int x^{-2 x} \, dx-50 \int x^{1-2 x} \, dx-80 \int x^{1-x} \, dx+\frac {250}{3} \int \frac {\int x^{-x} \, dx}{x} \, dx-125 \int \frac {\int x^{-2 x} \, dx}{-1+x} \, dx+125 \int \frac {\int x^{-2 x} \, dx}{x} \, dx-130 \int x^{-x} \, dx-150 \int x^{-x} \, dx+\frac {500}{3} \int \frac {\int x^{-x} \, dx}{-3+x} \, dx-225 \int x^{-2 x} \, dx-250 \int \frac {\int x^{-x} \, dx}{-2+x} \, dx-\frac {(125 \log (x)) \int x^{-2 x} \, dx}{1-x}-\frac {(500 \log (x)) \int x^{-x} \, dx}{(2-x) (3-x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 39, normalized size = 1.77 \begin {gather*} -e^x+75 x+15 x^2+x^3+25 x^{-2 x} (5+x)+10 x^{-x} (5+x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-225 - 50*x + x^(2*x)*(75 - E^x + 30*x + 3*x^2) + (-250 - 50*x)*Log[x] + x^x*(-150 - 80*x - 10*x^2
+ (-250 - 100*x - 10*x^2)*Log[x]))/x^(2*x),x]

[Out]

-E^x + 75*x + 15*x^2 + x^3 + (25*(5 + x))/x^(2*x) + (10*(5 + x)^2)/x^x

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fricas [B]  time = 0.75, size = 48, normalized size = 2.18 \begin {gather*} \frac {{\left (x^{3} + 15 \, x^{2} + 75 \, x - e^{x}\right )} x^{2 \, x} + 10 \, {\left (x^{2} + 10 \, x + 25\right )} x^{x} + 25 \, x + 125}{x^{2 \, x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)+3*x^2+30*x+75)*exp(x*log(x))^2+((-10*x^2-100*x-250)*log(x)-10*x^2-80*x-150)*exp(x*log(x))+
(-50*x-250)*log(x)-50*x-225)/exp(x*log(x))^2,x, algorithm="fricas")

[Out]

((x^3 + 15*x^2 + 75*x - e^x)*x^(2*x) + 10*(x^2 + 10*x + 25)*x^x + 25*x + 125)/x^(2*x)

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giac [B]  time = 2.10, size = 60, normalized size = 2.73 \begin {gather*} x^{3} + 15 \, x^{2} + \frac {10 \, x^{2}}{x^{x}} + 75 \, x + \frac {25 \, x}{x^{2 \, x}} + \frac {100 \, x}{x^{x}} + \frac {125}{x^{2 \, x}} + \frac {250}{x^{x}} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)+3*x^2+30*x+75)*exp(x*log(x))^2+((-10*x^2-100*x-250)*log(x)-10*x^2-80*x-150)*exp(x*log(x))+
(-50*x-250)*log(x)-50*x-225)/exp(x*log(x))^2,x, algorithm="giac")

[Out]

x^3 + 15*x^2 + 10*x^2/x^x + 75*x + 25*x/x^(2*x) + 100*x/x^x + 125/x^(2*x) + 250/x^x - e^x

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maple [A]  time = 0.06, size = 44, normalized size = 2.00

method	result	size
risch	\(75 x +\left (125+25 x \right ) x^{-2 x}+\left (10 x^{2}+100 x +250\right ) x^{-x}+15 x^{2}+x^{3}-{\mathrm e}^{x}\)	\(44\)
default	\(75 x +\left (125+25 x \right ) {\mathrm e}^{-2 x \ln \relax (x )}+\left (10 x^{2}+100 x +250\right ) {\mathrm e}^{-x \ln \relax (x )}+15 x^{2}+x^{3}-{\mathrm e}^{x}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-exp(x)+3*x^2+30*x+75)*exp(x*ln(x))^2+((-10*x^2-100*x-250)*ln(x)-10*x^2-80*x-150)*exp(x*ln(x))+(-50*x-25
0)*ln(x)-50*x-225)/exp(x*ln(x))^2,x,method=_RETURNVERBOSE)

[Out]

75*x+(125+25*x)/(x^x)^2+(10*x^2+100*x+250)/(x^x)+15*x^2+x^3-exp(x)

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maxima [B]  time = 0.40, size = 43, normalized size = 1.95 \begin {gather*} x^{3} + 15 \, x^{2} + 75 \, x + \frac {25 \, {\left (x + 5\right )}}{x^{2 \, x}} + \frac {10 \, {\left (x^{2} + 10 \, x + 25\right )}}{x^{x}} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)+3*x^2+30*x+75)*exp(x*log(x))^2+((-10*x^2-100*x-250)*log(x)-10*x^2-80*x-150)*exp(x*log(x))+
(-50*x-250)*log(x)-50*x-225)/exp(x*log(x))^2,x, algorithm="maxima")

[Out]

x^3 + 15*x^2 + 75*x + 25*(x + 5)/x^(2*x) + 10*(x^2 + 10*x + 25)/x^x - e^x

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mupad [B]  time = 3.16, size = 45, normalized size = 2.05 \begin {gather*} 75\,x-{\mathrm {e}}^x+\frac {25\,x+125}{x^{2\,x}}+\frac {10\,x^2+100\,x+250}{x^x}+15\,x^2+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-2*x*log(x))*(50*x - exp(2*x*log(x))*(30*x - exp(x) + 3*x^2 + 75) + log(x)*(50*x + 250) + exp(x*log(x
))*(80*x + log(x)*(100*x + 10*x^2 + 250) + 10*x^2 + 150) + 225),x)

[Out]

75*x - exp(x) + (25*x + 125)/x^(2*x) + (100*x + 10*x^2 + 250)/x^x + 15*x^2 + x^3

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sympy [B]  time = 0.46, size = 44, normalized size = 2.00 \begin {gather*} x^{3} + 15 x^{2} + 75 x + \left (25 x + 125\right ) e^{- 2 x \log {\relax (x )}} + \left (10 x^{2} + 100 x + 250\right ) e^{- x \log {\relax (x )}} - e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(x)+3*x**2+30*x+75)*exp(x*ln(x))**2+((-10*x**2-100*x-250)*ln(x)-10*x**2-80*x-150)*exp(x*ln(x))
+(-50*x-250)*ln(x)-50*x-225)/exp(x*ln(x))**2,x)

[Out]

x**3 + 15*x**2 + 75*x + (25*x + 125)*exp(-2*x*log(x)) + (10*x**2 + 100*x + 250)*exp(-x*log(x)) - exp(x)

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