3.102.51 \(\int \frac {64 x^3+e^2 (3+192 x^3-40 x^4+6 x^5)}{4 e^2} \, dx\)

Optimal. Leaf size=29 \[ \frac {1}{4} x \left (3+x^3 \left (x^2-4 \left (-4+2 \left (-4-\frac {2}{e^2}+x\right )\right )\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 1, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {12} \begin {gather*} \frac {x^6}{4}-2 x^5+\frac {4 x^4}{e^2}+12 x^4+\frac {3 x}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(64*x^3 + E^2*(3 + 192*x^3 - 40*x^4 + 6*x^5))/(4*E^2),x]

[Out]

(3*x)/4 + 12*x^4 + (4*x^4)/E^2 - 2*x^5 + x^6/4

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 (64 x^3+e^2 \left (3+192 x^3-40 x^4+6 x^5\right )\right ) \, dx}{4 e^2}\\ &=\frac {4 x^4}{e^2}+\frac {1}{4} \int \left (3+192 x^3-40 x^4+6 x^5\right ) \, dx\\ &=\frac {3 x}{4}+12 x^4+\frac {4 x^4}{e^2}-2 x^5+\frac {x^6}{4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 33, normalized size = 1.14 \begin {gather*} \frac {3 x}{4}+\frac {4 \left (1+3 e^2\right ) x^4}{e^2}-2 x^5+\frac {x^6}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64*x^3 + E^2*(3 + 192*x^3 - 40*x^4 + 6*x^5))/(4*E^2),x]

[Out]

(3*x)/4 + (4*(1 + 3*E^2)*x^4)/E^2 - 2*x^5 + x^6/4

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fricas [A]  time = 0.55, size = 30, normalized size = 1.03 \begin {gather*} \frac {1}{4} \, {\left (16 \, x^{4} + {\left (x^{6} - 8 \, x^{5} + 48 \, x^{4} + 3 \, x\right )} e^{2}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((6*x^5-40*x^4+192*x^3+3)*exp(2)+64*x^3)/exp(2),x, algorithm="fricas")

[Out]

1/4*(16*x^4 + (x^6 - 8*x^5 + 48*x^4 + 3*x)*e^2)*e^(-2)

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giac [A]  time = 0.12, size = 30, normalized size = 1.03 \begin {gather*} \frac {1}{4} \, {\left (16 \, x^{4} + {\left (x^{6} - 8 \, x^{5} + 48 \, x^{4} + 3 \, x\right )} e^{2}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((6*x^5-40*x^4+192*x^3+3)*exp(2)+64*x^3)/exp(2),x, algorithm="giac")

[Out]

1/4*(16*x^4 + (x^6 - 8*x^5 + 48*x^4 + 3*x)*e^2)*e^(-2)

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maple [A]  time = 0.03, size = 27, normalized size = 0.93

method	result	size
risch	\(\frac {x^{6}}{4}-2 x^{5}+12 x^{4}+4 \,{\mathrm e}^{-2} x^{4}+\frac {3 x}{4}\)	\(27\)
norman	\(\frac {3 x}{4}-2 x^{5}+\frac {x^{6}}{4}+4 \left (3 \,{\mathrm e}^{2}+1\right ) {\mathrm e}^{-2} x^{4}\)	\(30\)
default	\(\frac {{\mathrm e}^{-2} \left ({\mathrm e}^{2} \left (x^{6}-8 x^{5}+48 x^{4}+3 x \right )+16 x^{4}\right )}{4}\)	\(33\)
gosper	\(\frac {x \left ({\mathrm e}^{2} x^{5}-8 x^{4} {\mathrm e}^{2}+48 x^{3} {\mathrm e}^{2}+16 x^{3}+3 \,{\mathrm e}^{2}\right ) {\mathrm e}^{-2}}{4}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((6*x^5-40*x^4+192*x^3+3)*exp(2)+64*x^3)/exp(2),x,method=_RETURNVERBOSE)

[Out]

1/4*x^6-2*x^5+12*x^4+4*exp(-2)*x^4+3/4*x

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maxima [A]  time = 0.35, size = 30, normalized size = 1.03 \begin {gather*} \frac {1}{4} \, {\left (16 \, x^{4} + {\left (x^{6} - 8 \, x^{5} + 48 \, x^{4} + 3 \, x\right )} e^{2}\right )} e^{\left (-2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((6*x^5-40*x^4+192*x^3+3)*exp(2)+64*x^3)/exp(2),x, algorithm="maxima")

[Out]

1/4*(16*x^4 + (x^6 - 8*x^5 + 48*x^4 + 3*x)*e^2)*e^(-2)

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mupad [B]  time = 0.08, size = 24, normalized size = 0.83 \begin {gather*} \frac {x^6}{4}-2\,x^5+\left (4\,{\mathrm {e}}^{-2}+12\right )\,x^4+\frac {3\,x}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-2)*((exp(2)*(192*x^3 - 40*x^4 + 6*x^5 + 3))/4 + 16*x^3),x)

[Out]

(3*x)/4 + x^4*(4*exp(-2) + 12) - 2*x^5 + x^6/4

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sympy [A]  time = 0.06, size = 27, normalized size = 0.93 \begin {gather*} \frac {x^{6}}{4} - 2 x^{5} + \frac {x^{4} \left (4 + 12 e^{2}\right )}{e^{2}} + \frac {3 x}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((6*x**5-40*x**4+192*x**3+3)*exp(2)+64*x**3)/exp(2),x)

[Out]

x**6/4 - 2*x**5 + x**4*(4 + 12*exp(2))*exp(-2) + 3*x/4

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