3.84.48 \(\int e^{\frac {50}{x}+2 x} (2400 x^6-384 x^7-96 x^8) \, dx\)

Optimal. Leaf size=18 \[ -2-48 e^{\frac {50}{x}+2 x} x^8 \]

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Rubi [A]  time = 0.06, antiderivative size = 32, normalized size of antiderivative = 1.78, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1594, 2288} \begin {gather*} \frac {48 e^{2 x+\frac {50}{x}} x^6 \left (25-x^2\right )}{1-\frac {25}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(50/x + 2*x)*(2400*x^6 - 384*x^7 - 96*x^8),x]

[Out]

(48*E^(50/x + 2*x)*x^6*(25 - x^2))/(1 - 25/x^2)

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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} &=\int e^{\frac {50}{x}+2 x} x^6 \left (2400-384 x-96 x^2\right ) \, dx\\ &=\frac {48 e^{\frac {50}{x}+2 x} x^6 \left (25-x^2\right )}{1-\frac {25}{x^2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 16, normalized size = 0.89 \begin {gather*} -48 e^{\frac {50}{x}+2 x} x^8 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(50/x + 2*x)*(2400*x^6 - 384*x^7 - 96*x^8),x]

[Out]

-48*E^(50/x + 2*x)*x^8

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fricas [A]  time = 0.58, size = 16, normalized size = 0.89 \begin {gather*} -48 \, x^{8} e^{\left (\frac {2 \, {\left (x^{2} + 25\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x^8-384*x^7+2400*x^6)*exp(25/x)^2*exp(x)^2,x, algorithm="fricas")

[Out]

-48*x^8*e^(2*(x^2 + 25)/x)

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giac [A]  time = 0.30, size = 16, normalized size = 0.89 \begin {gather*} -48 \, x^{8} e^{\left (\frac {2 \, {\left (x^{2} + 25\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x^8-384*x^7+2400*x^6)*exp(25/x)^2*exp(x)^2,x, algorithm="giac")

[Out]

-48*x^8*e^(2*(x^2 + 25)/x)

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maple [A]  time = 0.09, size = 17, normalized size = 0.94

method	result	size
risch	\(-48 x^{8} {\mathrm e}^{\frac {2 x^{2}+50}{x}}\)	\(17\)
gosper	\(-48 x^{8} {\mathrm e}^{2 x} {\mathrm e}^{\frac {50}{x}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-96*x^8-384*x^7+2400*x^6)*exp(25/x)^2*exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

-48*x^8*exp(2*(x^2+25)/x)

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maxima [A]  time = 0.43, size = 15, normalized size = 0.83 \begin {gather*} -48 \, x^{8} e^{\left (2 \, x + \frac {50}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x^8-384*x^7+2400*x^6)*exp(25/x)^2*exp(x)^2,x, algorithm="maxima")

[Out]

-48*x^8*e^(2*x + 50/x)

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mupad [B]  time = 5.65, size = 15, normalized size = 0.83 \begin {gather*} -48\,x^8\,{\mathrm {e}}^{2\,x+\frac {50}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(2*x)*exp(50/x)*(384*x^7 - 2400*x^6 + 96*x^8),x)

[Out]

-48*x^8*exp(2*x + 50/x)

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sympy [A]  time = 16.32, size = 15, normalized size = 0.83 \begin {gather*} - 48 x^{8} e^{\frac {50}{x}} e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-96*x**8-384*x**7+2400*x**6)*exp(25/x)**2*exp(x)**2,x)

[Out]

-48*x**8*exp(50/x)*exp(2*x)

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