3.36.49 \(\int \frac {1+e^4+e^9 (-640+752 x-288 x^2+36 x^3)}{e^4} \, dx\)

Optimal. Leaf size=27 \[ x+\frac {x}{e^4}+e^5 (4-2 x)^2 \left (-5+\frac {3 x}{2}\right )^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {12} \begin {gather*} 9 e^5 x^4-96 e^5 x^3+376 e^5 x^2-640 e^5 x+\left (1+\frac {1}{e^4}\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + E^4 + E^9*(-640 + 752*x - 288*x^2 + 36*x^3))/E^4,x]

[Out]

(1 + E^(-4))*x - 640*E^5*x + 376*E^5*x^2 - 96*E^5*x^3 + 9*E^5*x^4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (1+e^4+e^9 \left (-640+752 x-288 x^2+36 x^3\right )\right ) \, dx}{e^4}\\ &=\left (1+\frac {1}{e^4}\right ) x+e^5 \int \left (-640+752 x-288 x^2+36 x^3\right ) \, dx\\ &=\left (1+\frac {1}{e^4}\right ) x-640 e^5 x+376 e^5 x^2-96 e^5 x^3+9 e^5 x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 37, normalized size = 1.37 \begin {gather*} x+\frac {x}{e^4}-640 e^5 x+376 e^5 x^2-96 e^5 x^3+9 e^5 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + E^4 + E^9*(-640 + 752*x - 288*x^2 + 36*x^3))/E^4,x]

[Out]

x + x/E^4 - 640*E^5*x + 376*E^5*x^2 - 96*E^5*x^3 + 9*E^5*x^4

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fricas [A]  time = 0.46, size = 31, normalized size = 1.15 \begin {gather*} {\left ({\left (9 \, x^{4} - 96 \, x^{3} + 376 \, x^{2} - 640 \, x\right )} e^{9} + x e^{4} + x\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x^3-288*x^2+752*x-640)*exp(4)*exp(5)+exp(4)+1)/exp(4),x, algorithm="fricas")

[Out]

((9*x^4 - 96*x^3 + 376*x^2 - 640*x)*e^9 + x*e^4 + x)*e^(-4)

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giac [A]  time = 0.18, size = 31, normalized size = 1.15 \begin {gather*} {\left ({\left (9 \, x^{4} - 96 \, x^{3} + 376 \, x^{2} - 640 \, x\right )} e^{9} + x e^{4} + x\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x^3-288*x^2+752*x-640)*exp(4)*exp(5)+exp(4)+1)/exp(4),x, algorithm="giac")

[Out]

((9*x^4 - 96*x^3 + 376*x^2 - 640*x)*e^9 + x*e^4 + x)*e^(-4)

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maple [A]  time = 0.04, size = 33, normalized size = 1.22

method	result	size
risch	\(9 x^{4} {\mathrm e}^{5}-96 x^{3} {\mathrm e}^{5}+376 x^{2} {\mathrm e}^{5}-640 x \,{\mathrm e}^{5}+x +{\mathrm e}^{-4} x\)	\(33\)
default	\({\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5} \left (9 x^{4}-96 x^{3}+376 x^{2}-640 x \right )+x \,{\mathrm e}^{4}+x \right )\)	\(36\)
gosper	\(x \left (9 x^{3} {\mathrm e}^{4} {\mathrm e}^{5}-96 x^{2} {\mathrm e}^{4} {\mathrm e}^{5}+376 x \,{\mathrm e}^{4} {\mathrm e}^{5}-640 \,{\mathrm e}^{4} {\mathrm e}^{5}+{\mathrm e}^{4}+1\right ) {\mathrm e}^{-4}\)	\(42\)
norman	\(376 x^{2} {\mathrm e}^{5}-96 x^{3} {\mathrm e}^{5}+9 x^{4} {\mathrm e}^{5}-{\mathrm e}^{-4} \left (640 \,{\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) x\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((36*x^3-288*x^2+752*x-640)*exp(4)*exp(5)+exp(4)+1)/exp(4),x,method=_RETURNVERBOSE)

[Out]

9*x^4*exp(5)-96*x^3*exp(5)+376*x^2*exp(5)-640*x*exp(5)+x+exp(-4)*x

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maxima [A]  time = 0.39, size = 31, normalized size = 1.15 \begin {gather*} {\left ({\left (9 \, x^{4} - 96 \, x^{3} + 376 \, x^{2} - 640 \, x\right )} e^{9} + x e^{4} + x\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x^3-288*x^2+752*x-640)*exp(4)*exp(5)+exp(4)+1)/exp(4),x, algorithm="maxima")

[Out]

((9*x^4 - 96*x^3 + 376*x^2 - 640*x)*e^9 + x*e^4 + x)*e^(-4)

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mupad [B]  time = 2.16, size = 34, normalized size = 1.26 \begin {gather*} 9\,{\mathrm {e}}^5\,x^4-96\,{\mathrm {e}}^5\,x^3+376\,{\mathrm {e}}^5\,x^2+{\mathrm {e}}^{-4}\,\left ({\mathrm {e}}^4-640\,{\mathrm {e}}^9+1\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(exp(4) + exp(9)*(752*x - 288*x^2 + 36*x^3 - 640) + 1),x)

[Out]

376*x^2*exp(5) - 96*x^3*exp(5) + 9*x^4*exp(5) + x*exp(-4)*(exp(4) - 640*exp(9) + 1)

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sympy [A]  time = 0.07, size = 39, normalized size = 1.44 \begin {gather*} 9 x^{4} e^{5} - 96 x^{3} e^{5} + 376 x^{2} e^{5} + \frac {x \left (- 640 e^{9} + 1 + e^{4}\right )}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((36*x**3-288*x**2+752*x-640)*exp(4)*exp(5)+exp(4)+1)/exp(4),x)

[Out]

9*x**4*exp(5) - 96*x**3*exp(5) + 376*x**2*exp(5) + x*(-640*exp(9) + 1 + exp(4))*exp(-4)

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