3.17.91 \(\int (-16+16 x+96 x^2-112 x^3-80 x^4+96 x^5+e^{32} (2 x-6 x^2+4 x^3)+e^{16} (4-24 x+12 x^2+48 x^3-40 x^4)) \, dx\)

Optimal. Leaf size=27 \[ \left (2-e^{16}-\frac {2}{x}+4 x\right )^2 \left (x-x^2\right )^2 \]

________________________________________________________________________________________

Rubi [B]  time = 0.05, antiderivative size = 89, normalized size of antiderivative = 3.30, number of steps used = 3, number of rules used = 0, integrand size = 67, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} 16 x^6-8 e^{16} x^5-16 x^5+e^{32} x^4+12 e^{16} x^4-28 x^4-2 e^{32} x^3+4 e^{16} x^3+32 x^3+e^{32} x^2-12 e^{16} x^2+8 x^2+4 e^{16} x-16 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-16 + 16*x + 96*x^2 - 112*x^3 - 80*x^4 + 96*x^5 + E^32*(2*x - 6*x^2 + 4*x^3) + E^16*(4 - 24*x + 12*x^2 + 4
8*x^3 - 40*x^4),x]

[Out]

-16*x + 4*E^16*x + 8*x^2 - 12*E^16*x^2 + E^32*x^2 + 32*x^3 + 4*E^16*x^3 - 2*E^32*x^3 - 28*x^4 + 12*E^16*x^4 +
E^32*x^4 - 16*x^5 - 8*E^16*x^5 + 16*x^6

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-16 x+8 x^2+32 x^3-28 x^4-16 x^5+16 x^6+e^{16} \int \left (4-24 x+12 x^2+48 x^3-40 x^4\right ) \, dx+e^{32} \int \left (2 x-6 x^2+4 x^3\right ) \, dx\\ &=-16 x+4 e^{16} x+8 x^2-12 e^{16} x^2+e^{32} x^2+32 x^3+4 e^{16} x^3-2 e^{32} x^3-28 x^4+12 e^{16} x^4+e^{32} x^4-16 x^5-8 e^{16} x^5+16 x^6\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.01, size = 89, normalized size = 3.30 \begin {gather*} -16 x+4 e^{16} x+8 x^2-12 e^{16} x^2+e^{32} x^2+32 x^3+4 e^{16} x^3-2 e^{32} x^3-28 x^4+12 e^{16} x^4+e^{32} x^4-16 x^5-8 e^{16} x^5+16 x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-16 + 16*x + 96*x^2 - 112*x^3 - 80*x^4 + 96*x^5 + E^32*(2*x - 6*x^2 + 4*x^3) + E^16*(4 - 24*x + 12*x
^2 + 48*x^3 - 40*x^4),x]

[Out]

-16*x + 4*E^16*x + 8*x^2 - 12*E^16*x^2 + E^32*x^2 + 32*x^3 + 4*E^16*x^3 - 2*E^32*x^3 - 28*x^4 + 12*E^16*x^4 +
E^32*x^4 - 16*x^5 - 8*E^16*x^5 + 16*x^6

________________________________________________________________________________________

fricas [B]  time = 0.83, size = 72, normalized size = 2.67 \begin {gather*} 16 \, x^{6} - 16 \, x^{5} - 28 \, x^{4} + 32 \, x^{3} + 8 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{32} - 4 \, {\left (2 \, x^{5} - 3 \, x^{4} - x^{3} + 3 \, x^{2} - x\right )} e^{16} - 16 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-6*x^2+2*x)*exp(16)^2+(-40*x^4+48*x^3+12*x^2-24*x+4)*exp(16)+96*x^5-80*x^4-112*x^3+96*x^2+16*x
-16,x, algorithm="fricas")

[Out]

16*x^6 - 16*x^5 - 28*x^4 + 32*x^3 + 8*x^2 + (x^4 - 2*x^3 + x^2)*e^32 - 4*(2*x^5 - 3*x^4 - x^3 + 3*x^2 - x)*e^1
6 - 16*x

________________________________________________________________________________________

giac [B]  time = 0.33, size = 72, normalized size = 2.67 \begin {gather*} 16 \, x^{6} - 16 \, x^{5} - 28 \, x^{4} + 32 \, x^{3} + 8 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{32} - 4 \, {\left (2 \, x^{5} - 3 \, x^{4} - x^{3} + 3 \, x^{2} - x\right )} e^{16} - 16 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-6*x^2+2*x)*exp(16)^2+(-40*x^4+48*x^3+12*x^2-24*x+4)*exp(16)+96*x^5-80*x^4-112*x^3+96*x^2+16*x
-16,x, algorithm="giac")

[Out]

16*x^6 - 16*x^5 - 28*x^4 + 32*x^3 + 8*x^2 + (x^4 - 2*x^3 + x^2)*e^32 - 4*(2*x^5 - 3*x^4 - x^3 + 3*x^2 - x)*e^1
6 - 16*x

________________________________________________________________________________________

maple [B]  time = 0.03, size = 69, normalized size = 2.56

method	result	size
norman	\(\left (-8 \,{\mathrm e}^{16}-16\right ) x^{5}+\left (4 \,{\mathrm e}^{16}-16\right ) x +\left (-2 \,{\mathrm e}^{32}+4 \,{\mathrm e}^{16}+32\right ) x^{3}+\left ({\mathrm e}^{32}-12 \,{\mathrm e}^{16}+8\right ) x^{2}+\left ({\mathrm e}^{32}+12 \,{\mathrm e}^{16}-28\right ) x^{4}+16 x^{6}\)	\(69\)
default	\({\mathrm e}^{32} \left (x^{4}-2 x^{3}+x^{2}\right )+{\mathrm e}^{16} \left (-8 x^{5}+12 x^{4}+4 x^{3}-12 x^{2}+4 x \right )+16 x^{6}-16 x^{5}-28 x^{4}+32 x^{3}+8 x^{2}-16 x\)	\(74\)
gosper	\(x \left ({\mathrm e}^{32} x^{3}-8 \,{\mathrm e}^{16} x^{4}+16 x^{5}-2 \,{\mathrm e}^{32} x^{2}+12 x^{3} {\mathrm e}^{16}-16 x^{4}+x \,{\mathrm e}^{32}+4 x^{2} {\mathrm e}^{16}-28 x^{3}-12 x \,{\mathrm e}^{16}+32 x^{2}+4 \,{\mathrm e}^{16}+8 x -16\right )\)	\(81\)
risch	\(x^{4} {\mathrm e}^{32}-2 \,{\mathrm e}^{32} x^{3}+{\mathrm e}^{32} x^{2}-8 \,{\mathrm e}^{16} x^{5}+12 \,{\mathrm e}^{16} x^{4}+4 x^{3} {\mathrm e}^{16}-12 x^{2} {\mathrm e}^{16}+4 x \,{\mathrm e}^{16}+16 x^{6}-16 x^{5}-28 x^{4}+32 x^{3}+8 x^{2}-16 x\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3-6*x^2+2*x)*exp(16)^2+(-40*x^4+48*x^3+12*x^2-24*x+4)*exp(16)+96*x^5-80*x^4-112*x^3+96*x^2+16*x-16,x,
method=_RETURNVERBOSE)

[Out]

(-8*exp(16)-16)*x^5+(4*exp(16)-16)*x+(-2*exp(16)^2+4*exp(16)+32)*x^3+(exp(16)^2-12*exp(16)+8)*x^2+(exp(16)^2+1
2*exp(16)-28)*x^4+16*x^6

________________________________________________________________________________________

maxima [B]  time = 0.39, size = 72, normalized size = 2.67 \begin {gather*} 16 \, x^{6} - 16 \, x^{5} - 28 \, x^{4} + 32 \, x^{3} + 8 \, x^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{32} - 4 \, {\left (2 \, x^{5} - 3 \, x^{4} - x^{3} + 3 \, x^{2} - x\right )} e^{16} - 16 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3-6*x^2+2*x)*exp(16)^2+(-40*x^4+48*x^3+12*x^2-24*x+4)*exp(16)+96*x^5-80*x^4-112*x^3+96*x^2+16*x
-16,x, algorithm="maxima")

[Out]

16*x^6 - 16*x^5 - 28*x^4 + 32*x^3 + 8*x^2 + (x^4 - 2*x^3 + x^2)*e^32 - 4*(2*x^5 - 3*x^4 - x^3 + 3*x^2 - x)*e^1
6 - 16*x

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 63, normalized size = 2.33 \begin {gather*} 16\,x^6+\left (-8\,{\mathrm {e}}^{16}-16\right )\,x^5+\left (12\,{\mathrm {e}}^{16}+{\mathrm {e}}^{32}-28\right )\,x^4+\left (4\,{\mathrm {e}}^{16}-2\,{\mathrm {e}}^{32}+32\right )\,x^3+\left ({\mathrm {e}}^{32}-12\,{\mathrm {e}}^{16}+8\right )\,x^2+\left (4\,{\mathrm {e}}^{16}-16\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*x + exp(32)*(2*x - 6*x^2 + 4*x^3) + exp(16)*(12*x^2 - 24*x + 48*x^3 - 40*x^4 + 4) + 96*x^2 - 112*x^3 -
80*x^4 + 96*x^5 - 16,x)

[Out]

x^3*(4*exp(16) - 2*exp(32) + 32) - x^5*(8*exp(16) + 16) + x^2*(exp(32) - 12*exp(16) + 8) + x^4*(12*exp(16) + e
xp(32) - 28) + 16*x^6 + x*(4*exp(16) - 16)

________________________________________________________________________________________

sympy [B]  time = 0.07, size = 66, normalized size = 2.44 \begin {gather*} 16 x^{6} + x^{5} \left (- 8 e^{16} - 16\right ) + x^{4} \left (-28 + 12 e^{16} + e^{32}\right ) + x^{3} \left (- 2 e^{32} + 32 + 4 e^{16}\right ) + x^{2} \left (- 12 e^{16} + 8 + e^{32}\right ) + x \left (-16 + 4 e^{16}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3-6*x**2+2*x)*exp(16)**2+(-40*x**4+48*x**3+12*x**2-24*x+4)*exp(16)+96*x**5-80*x**4-112*x**3+96
*x**2+16*x-16,x)

[Out]

16*x**6 + x**5*(-8*exp(16) - 16) + x**4*(-28 + 12*exp(16) + exp(32)) + x**3*(-2*exp(32) + 32 + 4*exp(16)) + x*
*2*(-12*exp(16) + 8 + exp(32)) + x*(-16 + 4*exp(16))

________________________________________________________________________________________