\(\int \frac {1296+e^4 (360+144 x)+e^8 (10 x+3 x^2)}{e^8} \, dx\) [2497]

Optimal result

Integrand size = 28, antiderivative size = 13 \[ \int \frac {1296+e^4 (360+144 x)+e^8 \left (10 x+3 x^2\right )}{e^8} \, dx=(5+x) \left (\frac {36}{e^4}+x\right )^2 \]

[Out]

(5+x)*(x+36/exp(1)^4)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(13)=26\).

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {12} \[ \int \frac {1296+e^4 (360+144 x)+e^8 \left (10 x+3 x^2\right )}{e^8} \, dx=x^3+5 x^2+\frac {1296 x}{e^8}+\frac {18 (2 x+5)^2}{e^4} \]

[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{align*} \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{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(29\) vs. \(2(13)=26\).

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.23 \[ \int \frac {1296+e^4 (360+144 x)+e^8 \left (10 x+3 x^2\right )}{e^8} \, dx=\frac {1296 x}{e^8}+\frac {360 x}{e^4}+5 x^2+\frac {72 x^2}{e^4}+x^3 \]

[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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08

method	result	size
risch	\(72 \,{\mathrm e}^{-4} x^{2}+5 x^{2}+x^{3}+1296 \,{\mathrm e}^{-8} x +360 \,{\mathrm e}^{-4} x\)	\(27\)
gosper	\(x \left (x^{2} {\mathrm e}^{8}+5 x \,{\mathrm e}^{8}+72 x \,{\mathrm e}^{4}+360 \,{\mathrm e}^{4}+1296\right ) {\mathrm e}^{-8}\)	\(37\)
parallelrisch	\({\mathrm e}^{-8} \left ({\mathrm e}^{8} x^{3}+5 x^{2} {\mathrm e}^{8}+72 x^{2} {\mathrm e}^{4}+360 x \,{\mathrm e}^{4}+1296 x \right )\)	\(43\)
norman	\(\left ({\mathrm e}^{7} x^{3}+{\mathrm e}^{3} \left (5 \,{\mathrm e}^{4}+72\right ) x^{2}+72 \left (5 \,{\mathrm e}^{4}+18\right ) {\mathrm e}^{-1} x \right ) {\mathrm e}^{-7}\)	\(46\)
default	\({\mathrm e}^{-8} \left ({\mathrm e}^{8} x^{3}+\frac {\left (\left (10 \,{\mathrm e}^{4}+36\right ) {\mathrm e}^{4}+108 \,{\mathrm e}^{4}\right ) x^{2}}{2}+360 x \,{\mathrm e}^{4}+1296 x \right )\)	\(50\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.31 \[ \int \frac {1296+e^4 (360+144 x)+e^8 \left (10 x+3 x^2\right )}{e^8} \, dx={\left ({\left (x^{3} + 5 \, x^{2}\right )} e^{8} + 72 \, {\left (x^{2} + 5 \, x\right )} e^{4} + 1296 \, x\right )} e^{\left (-8\right )} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08 \[ \int \frac {1296+e^4 (360+144 x)+e^8 \left (10 x+3 x^2\right )}{e^8} \, dx=x^{3} + \frac {x^{2} \cdot \left (72 + 5 e^{4}\right )}{e^{4}} + \frac {x \left (1296 + 360 e^{4}\right )}{e^{8}} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).

Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.31 \[ \int \frac {1296+e^4 (360+144 x)+e^8 \left (10 x+3 x^2\right )}{e^8} \, dx={\left ({\left (x^{3} + 5 \, x^{2}\right )} e^{8} + 72 \, {\left (x^{2} + 5 \, x\right )} e^{4} + 1296 \, x\right )} e^{\left (-8\right )} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.31 \[ \int \frac {1296+e^4 (360+144 x)+e^8 \left (10 x+3 x^2\right )}{e^8} \, dx={\left ({\left (x^{3} + 5 \, x^{2}\right )} e^{8} + 72 \, {\left (x^{2} + 5 \, x\right )} e^{4} + 1296 \, x\right )} e^{\left (-8\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 2.00 \[ \int \frac {1296+e^4 (360+144 x)+e^8 \left (10 x+3 x^2\right )}{e^8} \, dx=x^3+{\mathrm {e}}^{-4}\,\left (5\,{\mathrm {e}}^4+72\right )\,x^2+{\mathrm {e}}^{-8}\,\left (360\,{\mathrm {e}}^4+1296\right )\,x \]

[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)