3.54.28 \(\int \frac {24 x+16 e^4 x^{15}}{e^4} \, dx\)

Optimal. Leaf size=19 \[ x^{16}+12 \left (-e^2+\frac {x^2}{e^4}\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 0.63, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {12} \begin {gather*} x^{16}+\frac {12 x^2}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24*x + 16*E^4*x^15)/E^4,x]

[Out]

(12*x^2)/E^4 + x^16

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 (24 x+16 e^4 x^{15}\right ) \, dx}{e^4}\\ &=\frac {12 x^2}{e^4}+x^{16}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.63 \begin {gather*} \frac {12 x^2}{e^4}+x^{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*x + 16*E^4*x^15)/E^4,x]

[Out]

(12*x^2)/E^4 + x^16

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fricas [A]  time = 0.90, size = 15, normalized size = 0.79 \begin {gather*} {\left (x^{16} e^{4} + 12 \, x^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^15*exp(4)+24*x)/exp(4),x, algorithm="fricas")

[Out]

(x^16*e^4 + 12*x^2)*e^(-4)

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giac [A]  time = 1.97, size = 15, normalized size = 0.79 \begin {gather*} {\left (x^{16} e^{4} + 12 \, x^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^15*exp(4)+24*x)/exp(4),x, algorithm="giac")

[Out]

(x^16*e^4 + 12*x^2)*e^(-4)

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maple [A]  time = 0.05, size = 12, normalized size = 0.63

method	result	size
risch	\(x^{16}+12 x^{2} {\mathrm e}^{-4}\)	\(12\)
norman	\(x^{16}+12 x^{2} {\mathrm e}^{-4}\)	\(14\)
gosper	\(\left (x^{14} {\mathrm e}^{4}+12\right ) x^{2} {\mathrm e}^{-4}\)	\(17\)
default	\({\mathrm e}^{-4} \left ({\mathrm e}^{4} x^{16}+12 x^{2}\right )\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^15*exp(4)+24*x)/exp(4),x,method=_RETURNVERBOSE)

[Out]

x^16+12*x^2*exp(-4)

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maxima [A]  time = 0.50, size = 15, normalized size = 0.79 \begin {gather*} {\left (x^{16} e^{4} + 12 \, x^{2}\right )} e^{\left (-4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^15*exp(4)+24*x)/exp(4),x, algorithm="maxima")

[Out]

(x^16*e^4 + 12*x^2)*e^(-4)

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mupad [B]  time = 3.46, size = 11, normalized size = 0.58 \begin {gather*} x^{16}+12\,{\mathrm {e}}^{-4}\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-4)*(24*x + 16*x^15*exp(4)),x)

[Out]

12*x^2*exp(-4) + x^16

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sympy [A]  time = 0.05, size = 10, normalized size = 0.53 \begin {gather*} x^{16} + \frac {12 x^{2}}{e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x**15*exp(4)+24*x)/exp(4),x)

[Out]

x**16 + 12*x**2*exp(-4)

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