3.72.78 \(\int (3 x^2-16 x^3+20 x^4+5 e^{16} x^4+e^8 (-8 x^3+20 x^4)) \, dx\)

Optimal. Leaf size=16 \[ x \left (x-\left (2+e^8\right ) x^2\right )^2 \]

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Rubi [B]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 2.12, number of steps used = 3, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6} \begin {gather*} \left (4+e^{16}\right ) x^5+4 e^8 x^5-2 e^8 x^4-4 x^4+x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[3*x^2 - 16*x^3 + 20*x^4 + 5*E^16*x^4 + E^8*(-8*x^3 + 20*x^4),x]

[Out]

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

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (3 x^2-16 x^3+\left (20+5 e^{16}\right ) x^4+e^8 \left (-8 x^3+20 x^4\right )\right ) \, dx\\ &=x^3-4 x^4+\left (4+e^{16}\right ) x^5+e^8 \int \left (-8 x^3+20 x^4\right ) \, dx\\ &=x^3-4 x^4-2 e^8 x^4+4 e^8 x^5+\left (4+e^{16}\right ) x^5\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.94 \begin {gather*} x^3 \left (-1+\left (2+e^8\right ) x\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[3*x^2 - 16*x^3 + 20*x^4 + 5*E^16*x^4 + E^8*(-8*x^3 + 20*x^4),x]

[Out]

x^3*(-1 + (2 + E^8)*x)^2

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fricas [B]  time = 0.50, size = 35, normalized size = 2.19 \begin {gather*} x^{5} e^{16} + 4 \, x^{5} - 4 \, x^{4} + x^{3} + 2 \, {\left (2 \, x^{5} - x^{4}\right )} e^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*x^4*exp(4)^4+(20*x^4-8*x^3)*exp(4)^2+20*x^4-16*x^3+3*x^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.12, size = 35, normalized size = 2.19 \begin {gather*} x^{5} e^{16} + 4 \, x^{5} - 4 \, x^{4} + x^{3} + 2 \, {\left (2 \, x^{5} - x^{4}\right )} e^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*x^4*exp(4)^4+(20*x^4-8*x^3)*exp(4)^2+20*x^4-16*x^3+3*x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.02, size = 33, normalized size = 2.06

method	result	size
norman	\(\left ({\mathrm e}^{16}+4 \,{\mathrm e}^{8}+4\right ) x^{5}+\left (-2 \,{\mathrm e}^{8}-4\right ) x^{4}+x^{3}\)	\(33\)
risch	\(x^{5} {\mathrm e}^{16}+4 \,{\mathrm e}^{8} x^{5}-2 x^{4} {\mathrm e}^{8}+4 x^{5}-4 x^{4}+x^{3}\)	\(35\)
gosper	\(x^{3} \left (x^{2} {\mathrm e}^{16}+4 x^{2} {\mathrm e}^{8}-2 x \,{\mathrm e}^{8}+4 x^{2}-4 x +1\right )\)	\(39\)
default	\(x^{5} {\mathrm e}^{16}+{\mathrm e}^{8} \left (4 x^{5}-2 x^{4}\right )+4 x^{5}-4 x^{4}+x^{3}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5*x^4*exp(4)^4+(20*x^4-8*x^3)*exp(4)^2+20*x^4-16*x^3+3*x^2,x,method=_RETURNVERBOSE)

[Out]

(exp(4)^4+4*exp(4)^2+4)*x^5+(-2*exp(4)^2-4)*x^4+x^3

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maxima [B]  time = 0.35, size = 35, normalized size = 2.19 \begin {gather*} x^{5} e^{16} + 4 \, x^{5} - 4 \, x^{4} + x^{3} + 2 \, {\left (2 \, x^{5} - x^{4}\right )} e^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*x^4*exp(4)^4+(20*x^4-8*x^3)*exp(4)^2+20*x^4-16*x^3+3*x^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.27, size = 15, normalized size = 0.94 \begin {gather*} x^3\,{\left (2\,x+x\,{\mathrm {e}}^8-1\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(5*x^4*exp(16) - exp(8)*(8*x^3 - 20*x^4) + 3*x^2 - 16*x^3 + 20*x^4,x)

[Out]

x^3*(2*x + x*exp(8) - 1)^2

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sympy [B]  time = 0.06, size = 27, normalized size = 1.69 \begin {gather*} x^{5} \left (4 + 4 e^{8} + e^{16}\right ) + x^{4} \left (- 2 e^{8} - 4\right ) + x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(5*x**4*exp(4)**4+(20*x**4-8*x**3)*exp(4)**2+20*x**4-16*x**3+3*x**2,x)

[Out]

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

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