3.40.1 \(\int \frac {e^{\frac {16+6 x-x^2+24 x^3+9 x^6}{x}} (-16+x-x^2+48 x^3+45 x^6)}{x} \, dx\)

Optimal. Leaf size=24 \[ 2+e^{6-x+\frac {\left (4+3 x^3\right )^2}{x}} x \]

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Rubi [B]  time = 0.13, antiderivative size = 94, normalized size of antiderivative = 3.92, number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {2288} \begin {gather*} -\frac {e^{\frac {9 x^6+24 x^3-x^2+6 x+16}{x}} \left (-45 x^6-48 x^3+x^2+16\right )}{x \left (\frac {2 \left (27 x^5+36 x^2-x+3\right )}{x}-\frac {9 x^6+24 x^3-x^2+6 x+16}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((16 + 6*x - x^2 + 24*x^3 + 9*x^6)/x)*(-16 + x - x^2 + 48*x^3 + 45*x^6))/x,x]

[Out]

-((E^((16 + 6*x - x^2 + 24*x^3 + 9*x^6)/x)*(16 + x^2 - 48*x^3 - 45*x^6))/(x*((2*(3 - x + 36*x^2 + 27*x^5))/x -
(16 + 6*x - x^2 + 24*x^3 + 9*x^6)/x^2)))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {e^{\frac {16+6 x-x^2+24 x^3+9 x^6}{x}} \left (16+x^2-48 x^3-45 x^6\right )}{x \left (\frac {2 \left (3-x+36 x^2+27 x^5\right )}{x}-\frac {16+6 x-x^2+24 x^3+9 x^6}{x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 24, normalized size = 1.00 \begin {gather*} e^{6+\frac {16}{x}-x+24 x^2+9 x^5} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((16 + 6*x - x^2 + 24*x^3 + 9*x^6)/x)*(-16 + x - x^2 + 48*x^3 + 45*x^6))/x,x]

[Out]

E^(6 + 16/x - x + 24*x^2 + 9*x^5)*x

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fricas [A]  time = 0.48, size = 27, normalized size = 1.12 \begin {gather*} x e^{\left (\frac {9 \, x^{6} + 24 \, x^{3} - x^{2} + 6 \, x + 16}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((45*x^6+48*x^3-x^2+x-16)*exp((9*x^6+24*x^3-x^2+6*x+16)/x)/x,x, algorithm="fricas")

[Out]

x*e^((9*x^6 + 24*x^3 - x^2 + 6*x + 16)/x)

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giac [A]  time = 0.19, size = 27, normalized size = 1.12 \begin {gather*} x e^{\left (\frac {9 \, x^{6} + 24 \, x^{3} - x^{2} + 6 \, x + 16}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((45*x^6+48*x^3-x^2+x-16)*exp((9*x^6+24*x^3-x^2+6*x+16)/x)/x,x, algorithm="giac")

[Out]

x*e^((9*x^6 + 24*x^3 - x^2 + 6*x + 16)/x)

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maple [A]  time = 0.12, size = 28, normalized size = 1.17

method	result	size
gosper	\(x \,{\mathrm e}^{\frac {9 x^{6}+24 x^{3}-x^{2}+6 x +16}{x}}\)	\(28\)
norman	\(x \,{\mathrm e}^{\frac {9 x^{6}+24 x^{3}-x^{2}+6 x +16}{x}}\)	\(28\)
risch	\(x \,{\mathrm e}^{\frac {9 x^{6}+24 x^{3}-x^{2}+6 x +16}{x}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((45*x^6+48*x^3-x^2+x-16)*exp((9*x^6+24*x^3-x^2+6*x+16)/x)/x,x,method=_RETURNVERBOSE)

[Out]

x*exp((9*x^6+24*x^3-x^2+6*x+16)/x)

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maxima [A]  time = 0.43, size = 23, normalized size = 0.96 \begin {gather*} x e^{\left (9 \, x^{5} + 24 \, x^{2} - x + \frac {16}{x} + 6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((45*x^6+48*x^3-x^2+x-16)*exp((9*x^6+24*x^3-x^2+6*x+16)/x)/x,x, algorithm="maxima")

[Out]

x*e^(9*x^5 + 24*x^2 - x + 16/x + 6)

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mupad [B]  time = 2.27, size = 26, normalized size = 1.08 \begin {gather*} x\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^6\,{\mathrm {e}}^{9\,x^5}\,{\mathrm {e}}^{16/x}\,{\mathrm {e}}^{24\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((6*x - x^2 + 24*x^3 + 9*x^6 + 16)/x)*(x - x^2 + 48*x^3 + 45*x^6 - 16))/x,x)

[Out]

x*exp(-x)*exp(6)*exp(9*x^5)*exp(16/x)*exp(24*x^2)

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sympy [A]  time = 0.15, size = 22, normalized size = 0.92 \begin {gather*} x e^{\frac {9 x^{6} + 24 x^{3} - x^{2} + 6 x + 16}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((45*x**6+48*x**3-x**2+x-16)*exp((9*x**6+24*x**3-x**2+6*x+16)/x)/x,x)

[Out]

x*exp((9*x**6 + 24*x**3 - x**2 + 6*x + 16)/x)

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