3.34.19 \(\int (1+32 x+48 x^2+16 x^3+e^{2 e^4} (32 x+e^5 (-64-48 x^2)+e^{10} (64 x+16 x^3))+e^{e^4} (-64 x-48 x^2+e^5 (64+64 x+48 x^2+32 x^3))) \, dx\)

Optimal. Leaf size=31 \[ x+4 \left (x (2+x)+e^{e^4} \left (-2 x+e^5 \left (4+x^2\right )\right )\right )^2 \]

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Rubi [B]  time = 0.05, antiderivative size = 149, normalized size of antiderivative = 4.81, number of steps used = 6, number of rules used = 0, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} 4 e^{2 \left (5+e^4\right )} x^4+8 e^{5+e^4} x^4+4 x^4-16 e^{5+2 e^4} x^3+16 e^{5+e^4} x^3-16 e^{e^4} x^3+16 x^3+32 e^{2 \left (5+e^4\right )} x^2+32 e^{5+e^4} x^2+16 e^{2 e^4} x^2-32 e^{e^4} x^2+16 x^2-64 e^{5+2 e^4} x+64 e^{5+e^4} x+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + 32*x + 48*x^2 + 16*x^3 + E^(2*E^4)*(32*x + E^5*(-64 - 48*x^2) + E^10*(64*x + 16*x^3)) + E^E^4*(-64*x -
48*x^2 + E^5*(64 + 64*x + 48*x^2 + 32*x^3)),x]

[Out]

x + 64*E^(5 + E^4)*x - 64*E^(5 + 2*E^4)*x + 16*x^2 - 32*E^E^4*x^2 + 16*E^(2*E^4)*x^2 + 32*E^(5 + E^4)*x^2 + 32
*E^(2*(5 + E^4))*x^2 + 16*x^3 - 16*E^E^4*x^3 + 16*E^(5 + E^4)*x^3 - 16*E^(5 + 2*E^4)*x^3 + 4*x^4 + 8*E^(5 + E^
4)*x^4 + 4*E^(2*(5 + E^4))*x^4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=x+16 x^2+16 x^3+4 x^4+e^{e^4} \int \left (-64 x-48 x^2+e^5 \left (64+64 x+48 x^2+32 x^3\right )\right ) \, dx+e^{2 e^4} \int \left (32 x+e^5 \left (-64-48 x^2\right )+e^{10} \left (64 x+16 x^3\right )\right ) \, dx\\ &=x+16 x^2-32 e^{e^4} x^2+16 e^{2 e^4} x^2+16 x^3-16 e^{e^4} x^3+4 x^4+e^{5+e^4} \int \left (64+64 x+48 x^2+32 x^3\right ) \, dx+e^{2 \left (5+e^4\right )} \int \left (64 x+16 x^3\right ) \, dx+e^{5+2 e^4} \int \left (-64-48 x^2\right ) \, dx\\ &=x+64 e^{5+e^4} x-64 e^{5+2 e^4} x+16 x^2-32 e^{e^4} x^2+16 e^{2 e^4} x^2+32 e^{5+e^4} x^2+32 e^{2 \left (5+e^4\right )} x^2+16 x^3-16 e^{e^4} x^3+16 e^{5+e^4} x^3-16 e^{5+2 e^4} x^3+4 x^4+8 e^{5+e^4} x^4+4 e^{2 \left (5+e^4\right )} x^4\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.06, size = 93, normalized size = 3.00 \begin {gather*} x \left (1+16 x+16 e^{2 e^4} x+16 x^2+4 x^3-16 e^{e^4} x (2+x)-16 e^{5+2 e^4} \left (4+x^2\right )+4 e^{2 \left (5+e^4\right )} x \left (8+x^2\right )+8 e^{5+e^4} \left (8+4 x+2 x^2+x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + 32*x + 48*x^2 + 16*x^3 + E^(2*E^4)*(32*x + E^5*(-64 - 48*x^2) + E^10*(64*x + 16*x^3)) + E^E^4*(-
64*x - 48*x^2 + E^5*(64 + 64*x + 48*x^2 + 32*x^3)),x]

[Out]

x*(1 + 16*x + 16*E^(2*E^4)*x + 16*x^2 + 4*x^3 - 16*E^E^4*x*(2 + x) - 16*E^(5 + 2*E^4)*(4 + x^2) + 4*E^(2*(5 +
E^4))*x*(8 + x^2) + 8*E^(5 + E^4)*(8 + 4*x + 2*x^2 + x^3))

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fricas [B]  time = 0.57, size = 90, normalized size = 2.90 \begin {gather*} 4 \, x^{4} + 16 \, x^{3} + 16 \, x^{2} + 4 \, {\left (4 \, x^{2} + {\left (x^{4} + 8 \, x^{2}\right )} e^{10} - 4 \, {\left (x^{3} + 4 \, x\right )} e^{5}\right )} e^{\left (2 \, e^{4}\right )} - 8 \, {\left (2 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, x^{2} + 8 \, x\right )} e^{5}\right )} e^{\left (e^{4}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^3+64*x)*exp(5)^2+(-48*x^2-64)*exp(5)+32*x)*exp(exp(4))^2+((32*x^3+48*x^2+64*x+64)*exp(5)-48*x
^2-64*x)*exp(exp(4))+16*x^3+48*x^2+32*x+1,x, algorithm="fricas")

[Out]

4*x^4 + 16*x^3 + 16*x^2 + 4*(4*x^2 + (x^4 + 8*x^2)*e^10 - 4*(x^3 + 4*x)*e^5)*e^(2*e^4) - 8*(2*x^3 + 4*x^2 - (x
^4 + 2*x^3 + 4*x^2 + 8*x)*e^5)*e^(e^4) + x

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giac [B]  time = 0.18, size = 90, normalized size = 2.90 \begin {gather*} 4 \, x^{4} + 16 \, x^{3} + 16 \, x^{2} + 4 \, {\left (4 \, x^{2} + {\left (x^{4} + 8 \, x^{2}\right )} e^{10} - 4 \, {\left (x^{3} + 4 \, x\right )} e^{5}\right )} e^{\left (2 \, e^{4}\right )} - 8 \, {\left (2 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, x^{2} + 8 \, x\right )} e^{5}\right )} e^{\left (e^{4}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^3+64*x)*exp(5)^2+(-48*x^2-64)*exp(5)+32*x)*exp(exp(4))^2+((32*x^3+48*x^2+64*x+64)*exp(5)-48*x
^2-64*x)*exp(exp(4))+16*x^3+48*x^2+32*x+1,x, algorithm="giac")

[Out]

4*x^4 + 16*x^3 + 16*x^2 + 4*(4*x^2 + (x^4 + 8*x^2)*e^10 - 4*(x^3 + 4*x)*e^5)*e^(2*e^4) - 8*(2*x^3 + 4*x^2 - (x
^4 + 2*x^3 + 4*x^2 + 8*x)*e^5)*e^(e^4) + x

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maple [A]  time = 0.04, size = 95, normalized size = 3.06

method	result	size
default	\({\mathrm e}^{2 \,{\mathrm e}^{4}} \left ({\mathrm e}^{10} \left (4 x^{4}+32 x^{2}\right )+{\mathrm e}^{5} \left (-16 x^{3}-64 x \right )+16 x^{2}\right )+{\mathrm e}^{{\mathrm e}^{4}} \left ({\mathrm e}^{5} \left (8 x^{4}+16 x^{3}+32 x^{2}+64 x \right )-16 x^{3}-32 x^{2}\right )+4 x^{4}+16 x^{3}+16 x^{2}+x\)	\(95\)
norman	\(\left (4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}}+8 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+4\right ) x^{4}+\left (-16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}}+16 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-16 \,{\mathrm e}^{{\mathrm e}^{4}}+16\right ) x^{3}+\left (32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}}+32 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}}-32 \,{\mathrm e}^{{\mathrm e}^{4}}+16\right ) x^{2}+\left (-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}}+64 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+1\right ) x\)	\(109\)
gosper	\(x \left (4 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{3}+32 \,{\mathrm e}^{10} {\mathrm e}^{2 \,{\mathrm e}^{4}} x -16 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+8 x^{3} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 x^{2} {\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-64 \,{\mathrm e}^{5} {\mathrm e}^{2 \,{\mathrm e}^{4}}+32 x \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x -16 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{3}+64 \,{\mathrm e}^{5} {\mathrm e}^{{\mathrm e}^{4}}-32 x \,{\mathrm e}^{{\mathrm e}^{4}}+16 x^{2}+16 x +1\right )\)	\(122\)
risch	\(4 x^{4} {\mathrm e}^{2 \,{\mathrm e}^{4}+10}+32 x^{2} {\mathrm e}^{2 \,{\mathrm e}^{4}+10}-16 x^{3} {\mathrm e}^{2 \,{\mathrm e}^{4}+5}-64 x \,{\mathrm e}^{2 \,{\mathrm e}^{4}+5}+16 \,{\mathrm e}^{2 \,{\mathrm e}^{4}} x^{2}+8 x^{4} {\mathrm e}^{5+{\mathrm e}^{4}}+16 x^{3} {\mathrm e}^{5+{\mathrm e}^{4}}+32 x^{2} {\mathrm e}^{5+{\mathrm e}^{4}}+64 x \,{\mathrm e}^{5+{\mathrm e}^{4}}-16 \,{\mathrm e}^{{\mathrm e}^{4}} x^{3}-32 x^{2} {\mathrm e}^{{\mathrm e}^{4}}+4 x^{4}+16 x^{3}+16 x^{2}+x\)	\(128\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^3+64*x)*exp(5)^2+(-48*x^2-64)*exp(5)+32*x)*exp(exp(4))^2+((32*x^3+48*x^2+64*x+64)*exp(5)-48*x^2-64*
x)*exp(exp(4))+16*x^3+48*x^2+32*x+1,x,method=_RETURNVERBOSE)

[Out]

exp(exp(4))^2*(exp(5)^2*(4*x^4+32*x^2)+exp(5)*(-16*x^3-64*x)+16*x^2)+exp(exp(4))*(exp(5)*(8*x^4+16*x^3+32*x^2+
64*x)-16*x^3-32*x^2)+4*x^4+16*x^3+16*x^2+x

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maxima [B]  time = 0.59, size = 90, normalized size = 2.90 \begin {gather*} 4 \, x^{4} + 16 \, x^{3} + 16 \, x^{2} + 4 \, {\left (4 \, x^{2} + {\left (x^{4} + 8 \, x^{2}\right )} e^{10} - 4 \, {\left (x^{3} + 4 \, x\right )} e^{5}\right )} e^{\left (2 \, e^{4}\right )} - 8 \, {\left (2 \, x^{3} + 4 \, x^{2} - {\left (x^{4} + 2 \, x^{3} + 4 \, x^{2} + 8 \, x\right )} e^{5}\right )} e^{\left (e^{4}\right )} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^3+64*x)*exp(5)^2+(-48*x^2-64)*exp(5)+32*x)*exp(exp(4))^2+((32*x^3+48*x^2+64*x+64)*exp(5)-48*x
^2-64*x)*exp(exp(4))+16*x^3+48*x^2+32*x+1,x, algorithm="maxima")

[Out]

4*x^4 + 16*x^3 + 16*x^2 + 4*(4*x^2 + (x^4 + 8*x^2)*e^10 - 4*(x^3 + 4*x)*e^5)*e^(2*e^4) - 8*(2*x^3 + 4*x^2 - (x
^4 + 2*x^3 + 4*x^2 + 8*x)*e^5)*e^(e^4) + x

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mupad [B]  time = 1.97, size = 99, normalized size = 3.19 \begin {gather*} \left (8\,{\mathrm {e}}^{{\mathrm {e}}^4+5}+4\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+10}+4\right )\,x^4+\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^4}\,\left (48\,{\mathrm {e}}^5-48\right )}{3}-16\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+5}+16\right )\,x^3+\left (\frac {{\mathrm {e}}^{{\mathrm {e}}^4}\,\left (64\,{\mathrm {e}}^5-64\right )}{2}+\frac {{\mathrm {e}}^{2\,{\mathrm {e}}^4}\,\left (64\,{\mathrm {e}}^{10}+32\right )}{2}+16\right )\,x^2+\left (64\,{\mathrm {e}}^{{\mathrm {e}}^4+5}-64\,{\mathrm {e}}^{2\,{\mathrm {e}}^4+5}+1\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(32*x - exp(exp(4))*(64*x - exp(5)*(64*x + 48*x^2 + 32*x^3 + 64) + 48*x^2) + exp(2*exp(4))*(32*x + exp(10)*
(64*x + 16*x^3) - exp(5)*(48*x^2 + 64)) + 48*x^2 + 16*x^3 + 1,x)

[Out]

x^4*(8*exp(exp(4) + 5) + 4*exp(2*exp(4) + 10) + 4) + x^2*((exp(exp(4))*(64*exp(5) - 64))/2 + (exp(2*exp(4))*(6
4*exp(10) + 32))/2 + 16) + x*(64*exp(exp(4) + 5) - 64*exp(2*exp(4) + 5) + 1) + x^3*((exp(exp(4))*(48*exp(5) -
48))/3 - 16*exp(2*exp(4) + 5) + 16)

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sympy [B]  time = 0.09, size = 128, normalized size = 4.13 \begin {gather*} x^{4} \left (4 + 8 e^{5} e^{e^{4}} + 4 e^{10} e^{2 e^{4}}\right ) + x^{3} \left (- 16 e^{5} e^{2 e^{4}} - 16 e^{e^{4}} + 16 + 16 e^{5} e^{e^{4}}\right ) + x^{2} \left (- 32 e^{e^{4}} + 16 + 32 e^{5} e^{e^{4}} + 16 e^{2 e^{4}} + 32 e^{10} e^{2 e^{4}}\right ) + x \left (- 64 e^{5} e^{2 e^{4}} + 1 + 64 e^{5} e^{e^{4}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**3+64*x)*exp(5)**2+(-48*x**2-64)*exp(5)+32*x)*exp(exp(4))**2+((32*x**3+48*x**2+64*x+64)*exp(5
)-48*x**2-64*x)*exp(exp(4))+16*x**3+48*x**2+32*x+1,x)

[Out]

x**4*(4 + 8*exp(5)*exp(exp(4)) + 4*exp(10)*exp(2*exp(4))) + x**3*(-16*exp(5)*exp(2*exp(4)) - 16*exp(exp(4)) +
16 + 16*exp(5)*exp(exp(4))) + x**2*(-32*exp(exp(4)) + 16 + 32*exp(5)*exp(exp(4)) + 16*exp(2*exp(4)) + 32*exp(1
0)*exp(2*exp(4))) + x*(-64*exp(5)*exp(2*exp(4)) + 1 + 64*exp(5)*exp(exp(4)))

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