3.25.97 \(\int \frac {1296+e^4 (360+144 x)+e^8 (10 x+3 x^2)}{e^8} \, dx\)

Optimal. Leaf size=13 \[ (5+x) \left (\frac {36}{e^4}+x\right )^2 \]

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Rubi [B]  time = 0.01, antiderivative size = 27, normalized size of antiderivative = 2.08, 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*} x^3+5 x^2+\frac {1296 x}{e^8}+\frac {18 (2 x+5)^2}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1296 + E^4*(360 + 144*x) + E^8*(10*x + 3*x^2))/E^8,x]

[Out]

(1296*x)/E^8 + 5*x^2 + x^3 + (18*(5 + 2*x)^2)/E^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 (1296+e^4 (360+144 x)+e^8 \left (10 x+3 x^2\right )\right ) \, dx}{e^8}\\ &=\frac {1296 x}{e^8}+\frac {18 (5+2 x)^2}{e^4}+\int \left (10 x+3 x^2\right ) \, dx\\ &=\frac {1296 x}{e^8}+5 x^2+x^3+\frac {18 (5+2 x)^2}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 29, normalized size = 2.23 \begin {gather*} \frac {1296 x}{e^8}+\frac {360 x}{e^4}+5 x^2+\frac {72 x^2}{e^4}+x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1296 + E^4*(360 + 144*x) + E^8*(10*x + 3*x^2))/E^8,x]

[Out]

(1296*x)/E^8 + (360*x)/E^4 + 5*x^2 + (72*x^2)/E^4 + x^3

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fricas [B]  time = 0.68, size = 30, normalized size = 2.31 \begin {gather*} {\left ({\left (x^{3} + 5 \, x^{2}\right )} e^{8} + 72 \, {\left (x^{2} + 5 \, x\right )} e^{4} + 1296 \, x\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+10*x)*exp(1)^8+(144*x+360)*exp(1)^4+1296)/exp(1)^8,x, algorithm="fricas")

[Out]

((x^3 + 5*x^2)*e^8 + 72*(x^2 + 5*x)*e^4 + 1296*x)*e^(-8)

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giac [B]  time = 0.29, size = 30, normalized size = 2.31 \begin {gather*} {\left ({\left (x^{3} + 5 \, x^{2}\right )} e^{8} + 72 \, {\left (x^{2} + 5 \, x\right )} e^{4} + 1296 \, x\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+10*x)*exp(1)^8+(144*x+360)*exp(1)^4+1296)/exp(1)^8,x, algorithm="giac")

[Out]

((x^3 + 5*x^2)*e^8 + 72*(x^2 + 5*x)*e^4 + 1296*x)*e^(-8)

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maple [A]  time = 0.02, size = 27, normalized size = 2.08

method	result	size
risch	\(x^{3}+5 x^{2}+72 x^{2} {\mathrm e}^{-4}+360 \,{\mathrm e}^{-4} x +1296 \,{\mathrm e}^{-8} x\)	\(27\)
gosper	\(x \left (x^{2} {\mathrm e}^{8}+5 x \,{\mathrm e}^{8}+72 x \,{\mathrm e}^{4}+360 \,{\mathrm e}^{4}+1296\right ) {\mathrm e}^{-8}\)	\(37\)
default	\({\mathrm e}^{-8} \left ({\mathrm e}^{8} \left (x^{3}+5 x^{2}\right )+{\mathrm e}^{4} \left (72 x^{2}+360 x \right )+1296 x \right )\)	\(38\)
norman	\(\left ({\mathrm e}^{7} x^{3}+{\mathrm e}^{3} \left (5 \,{\mathrm e}^{4}+72\right ) x^{2}+72 \left (5 \,{\mathrm e}^{4}+18\right ) {\mathrm e}^{-1} x \right ) {\mathrm e}^{-7}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2+10*x)*exp(1)^8+(144*x+360)*exp(1)^4+1296)/exp(1)^8,x,method=_RETURNVERBOSE)

[Out]

x^3+5*x^2+72*x^2*exp(-4)+360*exp(-4)*x+1296*exp(-8)*x

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maxima [B]  time = 0.38, size = 30, normalized size = 2.31 \begin {gather*} {\left ({\left (x^{3} + 5 \, x^{2}\right )} e^{8} + 72 \, {\left (x^{2} + 5 \, x\right )} e^{4} + 1296 \, x\right )} e^{\left (-8\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+10*x)*exp(1)^8+(144*x+360)*exp(1)^4+1296)/exp(1)^8,x, algorithm="maxima")

[Out]

((x^3 + 5*x^2)*e^8 + 72*(x^2 + 5*x)*e^4 + 1296*x)*e^(-8)

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mupad [B]  time = 0.07, size = 26, normalized size = 2.00 \begin {gather*} x^3+{\mathrm {e}}^{-4}\,\left (5\,{\mathrm {e}}^4+72\right )\,x^2+{\mathrm {e}}^{-8}\,\left (360\,{\mathrm {e}}^4+1296\right )\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-8)*(exp(8)*(10*x + 3*x^2) + exp(4)*(144*x + 360) + 1296),x)

[Out]

x^3 + x*exp(-8)*(360*exp(4) + 1296) + x^2*exp(-4)*(5*exp(4) + 72)

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sympy [B]  time = 0.07, size = 27, normalized size = 2.08 \begin {gather*} x^{3} + \frac {x^{2} \left (72 + 5 e^{4}\right )}{e^{4}} + \frac {x \left (1296 + 360 e^{4}\right )}{e^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2+10*x)*exp(1)**8+(144*x+360)*exp(1)**4+1296)/exp(1)**8,x)

[Out]

x**3 + x**2*(72 + 5*exp(4))*exp(-4) + x*(1296 + 360*exp(4))*exp(-8)

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