3.99.31 \(\int \frac {-18 x^2+60 x^3+e^{e^x} (12 x-30 x^2+e^x (6 x^2-30 x^3))}{1-10 x+25 x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac {6 \left (e^{e^x}-x\right ) x}{-5+\frac {1}{x}} \]

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Rubi [F]  time = 0.83, 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 {-18 x^2+60 x^3+e^{e^x} \left (12 x-30 x^2+e^x \left (6 x^2-30 x^3\right )\right )}{1-10 x+25 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-18*x^2 + 60*x^3 + E^E^x*(12*x - 30*x^2 + E^x*(6*x^2 - 30*x^3)))/(1 - 10*x + 25*x^2),x]

[Out]

(-6*E^E^x)/25 - 6/(125*(1 - 5*x)) + (6*x)/25 + (6*x^2)/5 - (6*ExpIntegralEi[E^x])/5 - (6*Defer[Int][E^(E^x + x
)*x, x])/5 + (6*Defer[Int][E^E^x/(-1 + 5*x)^2, x])/5 - (6*Defer[Int][E^(E^x + x)/(-1 + 5*x), x])/25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-18 x^2+60 x^3+e^{e^x} \left (12 x-30 x^2+e^x \left (6 x^2-30 x^3\right )\right )}{(-1+5 x)^2} \, dx\\ &=\int \left (\frac {12 e^{e^x} x}{(-1+5 x)^2}-\frac {18 x^2}{(-1+5 x)^2}-\frac {30 e^{e^x} x^2}{(-1+5 x)^2}+\frac {60 x^3}{(-1+5 x)^2}-\frac {6 e^{e^x+x} x^2}{-1+5 x}\right ) \, dx\\ &=-\left (6 \int \frac {e^{e^x+x} x^2}{-1+5 x} \, dx\right )+12 \int \frac {e^{e^x} x}{(-1+5 x)^2} \, dx-18 \int \frac {x^2}{(-1+5 x)^2} \, dx-30 \int \frac {e^{e^x} x^2}{(-1+5 x)^2} \, dx+60 \int \frac {x^3}{(-1+5 x)^2} \, dx\\ &=-\left (6 \int \left (\frac {e^{e^x+x}}{25}+\frac {1}{5} e^{e^x+x} x+\frac {e^{e^x+x}}{25 (-1+5 x)}\right ) \, dx\right )+12 \int \left (\frac {e^{e^x}}{5 (-1+5 x)^2}+\frac {e^{e^x}}{5 (-1+5 x)}\right ) \, dx-18 \int \left (\frac {1}{25}+\frac {1}{25 (-1+5 x)^2}+\frac {2}{25 (-1+5 x)}\right ) \, dx-30 \int \left (\frac {e^{e^x}}{25}+\frac {e^{e^x}}{25 (-1+5 x)^2}+\frac {2 e^{e^x}}{25 (-1+5 x)}\right ) \, dx+60 \int \left (\frac {2}{125}+\frac {x}{25}+\frac {1}{125 (-1+5 x)^2}+\frac {3}{125 (-1+5 x)}\right ) \, dx\\ &=-\frac {6}{125 (1-5 x)}+\frac {6 x}{25}+\frac {6 x^2}{5}-\frac {6}{25} \int e^{e^x+x} \, dx-\frac {6}{25} \int \frac {e^{e^x+x}}{-1+5 x} \, dx-\frac {6}{5} \int e^{e^x} \, dx-\frac {6}{5} \int e^{e^x+x} x \, dx-\frac {6}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx+\frac {12}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx\\ &=-\frac {6}{125 (1-5 x)}+\frac {6 x}{25}+\frac {6 x^2}{5}-\frac {6}{25} \int \frac {e^{e^x+x}}{-1+5 x} \, dx-\frac {6}{25} \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )-\frac {6}{5} \int e^{e^x+x} x \, dx-\frac {6}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx-\frac {6}{5} \operatorname {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )+\frac {12}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx\\ &=-\frac {6 e^{e^x}}{25}-\frac {6}{125 (1-5 x)}+\frac {6 x}{25}+\frac {6 x^2}{5}-\frac {6 \text {Ei}\left (e^x\right )}{5}-\frac {6}{25} \int \frac {e^{e^x+x}}{-1+5 x} \, dx-\frac {6}{5} \int e^{e^x+x} x \, dx-\frac {6}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx+\frac {12}{5} \int \frac {e^{e^x}}{(-1+5 x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 29, normalized size = 1.53 \begin {gather*} -\frac {6 \left (1-5 x-125 e^{e^x} x^2+125 x^3\right )}{125-625 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18*x^2 + 60*x^3 + E^E^x*(12*x - 30*x^2 + E^x*(6*x^2 - 30*x^3)))/(1 - 10*x + 25*x^2),x]

[Out]

(-6*(1 - 5*x - 125*E^E^x*x^2 + 125*x^3))/(125 - 625*x)

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fricas [A]  time = 0.61, size = 27, normalized size = 1.42 \begin {gather*} \frac {6 \, {\left (125 \, x^{3} - 125 \, x^{2} e^{\left (e^{x}\right )} - 5 \, x + 1\right )}}{125 \, {\left (5 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x^3+6*x^2)*exp(x)-30*x^2+12*x)*exp(exp(x))+60*x^3-18*x^2)/(25*x^2-10*x+1),x, algorithm="frica
s")

[Out]

6/125*(125*x^3 - 125*x^2*e^(e^x) - 5*x + 1)/(5*x - 1)

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giac [B]  time = 0.14, size = 39, normalized size = 2.05 \begin {gather*} \frac {6 \, {\left (125 \, x^{3} e^{x} - 125 \, x^{2} e^{\left (x + e^{x}\right )} - 5 \, x e^{x} + e^{x}\right )}}{125 \, {\left (5 \, x e^{x} - e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x^3+6*x^2)*exp(x)-30*x^2+12*x)*exp(exp(x))+60*x^3-18*x^2)/(25*x^2-10*x+1),x, algorithm="giac"
)

[Out]

6/125*(125*x^3*e^x - 125*x^2*e^(x + e^x) - 5*x*e^x + e^x)/(5*x*e^x - e^x)

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maple [A]  time = 0.14, size = 23, normalized size = 1.21

method	result	size
norman	\(\frac {6 x^{3}-6 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2}}{5 x -1}\)	\(23\)
risch	\(\frac {6 x^{2}}{5}+\frac {6 x}{25}+\frac {6}{625 \left (x -\frac {1}{5}\right )}-\frac {6 x^{2} {\mathrm e}^{{\mathrm e}^{x}}}{5 x -1}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-30*x^3+6*x^2)*exp(x)-30*x^2+12*x)*exp(exp(x))+60*x^3-18*x^2)/(25*x^2-10*x+1),x,method=_RETURNVERBOSE)

[Out]

(6*x^3-6*exp(exp(x))*x^2)/(5*x-1)

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maxima [A]  time = 0.45, size = 33, normalized size = 1.74 \begin {gather*} \frac {6}{5} \, x^{2} - \frac {6 \, x^{2} e^{\left (e^{x}\right )}}{5 \, x - 1} + \frac {6}{25} \, x + \frac {6}{125 \, {\left (5 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x^3+6*x^2)*exp(x)-30*x^2+12*x)*exp(exp(x))+60*x^3-18*x^2)/(25*x^2-10*x+1),x, algorithm="maxim
a")

[Out]

6/5*x^2 - 6*x^2*e^(e^x)/(5*x - 1) + 6/25*x + 6/125/(5*x - 1)

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mupad [B]  time = 5.83, size = 19, normalized size = 1.00 \begin {gather*} \frac {6\,x^2\,\left (x-{\mathrm {e}}^{{\mathrm {e}}^x}\right )}{5\,x-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x))*(12*x + exp(x)*(6*x^2 - 30*x^3) - 30*x^2) - 18*x^2 + 60*x^3)/(25*x^2 - 10*x + 1),x)

[Out]

(6*x^2*(x - exp(exp(x))))/(5*x - 1)

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sympy [B]  time = 0.17, size = 32, normalized size = 1.68 \begin {gather*} \frac {6 x^{2}}{5} - \frac {6 x^{2} e^{e^{x}}}{5 x - 1} + \frac {6 x}{25} + \frac {6}{625 x - 125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x**3+6*x**2)*exp(x)-30*x**2+12*x)*exp(exp(x))+60*x**3-18*x**2)/(25*x**2-10*x+1),x)

[Out]

6*x**2/5 - 6*x**2*exp(exp(x))/(5*x - 1) + 6*x/25 + 6/(625*x - 125)

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