3.20.53 \(\int \frac {48 x^3+e^6 (4+12 x-36 x^2+48 x^3)}{e^6} \, dx\)

Optimal. Leaf size=24 \[ x \left (4+6 \left (x+2 x \left (-x+x \left (x+\frac {x}{e^6}\right )\right )\right )\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {12} \begin {gather*} \frac {12 x^4}{e^6}+12 x^4-12 x^3+6 x^2+4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(48*x^3 + E^6*(4 + 12*x - 36*x^2 + 48*x^3))/E^6,x]

[Out]

4*x + 6*x^2 - 12*x^3 + 12*x^4 + (12*x^4)/E^6

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 (48 x^3+e^6 \left (4+12 x-36 x^2+48 x^3\right )\right ) \, dx}{e^6}\\ &=\frac {12 x^4}{e^6}+\int \left (4+12 x-36 x^2+48 x^3\right ) \, dx\\ &=4 x+6 x^2-12 x^3+12 x^4+\frac {12 x^4}{e^6}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.12 \begin {gather*} 4 x+6 x^2-12 x^3+\frac {12 \left (1+e^6\right ) x^4}{e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48*x^3 + E^6*(4 + 12*x - 36*x^2 + 48*x^3))/E^6,x]

[Out]

4*x + 6*x^2 - 12*x^3 + (12*(1 + E^6)*x^4)/E^6

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fricas [A]  time = 0.83, size = 32, normalized size = 1.33 \begin {gather*} 2 \, {\left (6 \, x^{4} + {\left (6 \, x^{4} - 6 \, x^{3} + 3 \, x^{2} + 2 \, x\right )} e^{6}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^3-36*x^2+12*x+4)*exp(6)+48*x^3)/exp(6),x, algorithm="fricas")

[Out]

2*(6*x^4 + (6*x^4 - 6*x^3 + 3*x^2 + 2*x)*e^6)*e^(-6)

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giac [A]  time = 0.56, size = 32, normalized size = 1.33 \begin {gather*} 2 \, {\left (6 \, x^{4} + {\left (6 \, x^{4} - 6 \, x^{3} + 3 \, x^{2} + 2 \, x\right )} e^{6}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^3-36*x^2+12*x+4)*exp(6)+48*x^3)/exp(6),x, algorithm="giac")

[Out]

2*(6*x^4 + (6*x^4 - 6*x^3 + 3*x^2 + 2*x)*e^6)*e^(-6)

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maple [A]  time = 0.03, size = 27, normalized size = 1.12

method	result	size
risch	\(12 x^{4}-12 x^{3}+12 \,{\mathrm e}^{-6} x^{4}+6 x^{2}+4 x\)	\(27\)
norman	\(4 x +6 x^{2}-12 x^{3}+12 \left (1+{\mathrm e}^{6}\right ) {\mathrm e}^{-6} x^{4}\)	\(28\)
default	\({\mathrm e}^{-6} \left ({\mathrm e}^{6} \left (12 x^{4}-12 x^{3}+6 x^{2}+4 x \right )+12 x^{4}\right )\)	\(34\)
gosper	\(2 x \left (6 x^{3} {\mathrm e}^{6}-6 x^{2} {\mathrm e}^{6}+6 x^{3}+3 x \,{\mathrm e}^{6}+2 \,{\mathrm e}^{6}\right ) {\mathrm e}^{-6}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((48*x^3-36*x^2+12*x+4)*exp(6)+48*x^3)/exp(6),x,method=_RETURNVERBOSE)

[Out]

12*x^4-12*x^3+12*exp(-6)*x^4+6*x^2+4*x

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maxima [A]  time = 0.59, size = 32, normalized size = 1.33 \begin {gather*} 2 \, {\left (6 \, x^{4} + {\left (6 \, x^{4} - 6 \, x^{3} + 3 \, x^{2} + 2 \, x\right )} e^{6}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x^3-36*x^2+12*x+4)*exp(6)+48*x^3)/exp(6),x, algorithm="maxima")

[Out]

2*(6*x^4 + (6*x^4 - 6*x^3 + 3*x^2 + 2*x)*e^6)*e^(-6)

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mupad [B]  time = 0.04, size = 24, normalized size = 1.00 \begin {gather*} \left (12\,{\mathrm {e}}^{-6}+12\right )\,x^4-12\,x^3+6\,x^2+4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-6)*(exp(6)*(12*x - 36*x^2 + 48*x^3 + 4) + 48*x^3),x)

[Out]

4*x + x^4*(12*exp(-6) + 12) + 6*x^2 - 12*x^3

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sympy [A]  time = 0.06, size = 26, normalized size = 1.08 \begin {gather*} \frac {x^{4} \left (12 + 12 e^{6}\right )}{e^{6}} - 12 x^{3} + 6 x^{2} + 4 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((48*x**3-36*x**2+12*x+4)*exp(6)+48*x**3)/exp(6),x)

[Out]

x**4*(12 + 12*exp(6))*exp(-6) - 12*x**3 + 6*x**2 + 4*x

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