3.18.97 \(\int \frac {-1764-64 e^{\frac {32}{x^2}} x^2+210 x^3+50 x^6+e^{\frac {16}{x^2}} (-1344-84 x^2+320 x^3-10 x^5)}{x^5} \, dx\)

Optimal. Leaf size=24 \[ \left (-e^{\frac {16}{x^2}}+x+4 \left (-\frac {21}{4 x^2}+x\right )\right )^2 \]

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Rubi [A]  time = 0.13, antiderivative size = 42, normalized size of antiderivative = 1.75, number of steps used = 6, number of rules used = 3, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {14, 2209, 2288} \begin {gather*} \frac {441}{x^4}+25 x^2+e^{\frac {32}{x^2}}+\frac {2 e^{\frac {16}{x^2}} \left (21-5 x^3\right )}{x^2}-\frac {210}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1764 - 64*E^(32/x^2)*x^2 + 210*x^3 + 50*x^6 + E^(16/x^2)*(-1344 - 84*x^2 + 320*x^3 - 10*x^5))/x^5,x]

[Out]

E^(32/x^2) + 441/x^4 - 210/x + 25*x^2 + (2*E^(16/x^2)*(21 - 5*x^3))/x^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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 \left (-\frac {64 e^{\frac {32}{x^2}}}{x^3}-\frac {2 e^{\frac {16}{x^2}} \left (672+42 x^2-160 x^3+5 x^5\right )}{x^5}+\frac {2 \left (-882+105 x^3+25 x^6\right )}{x^5}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {16}{x^2}} \left (672+42 x^2-160 x^3+5 x^5\right )}{x^5} \, dx\right )+2 \int \frac {-882+105 x^3+25 x^6}{x^5} \, dx-64 \int \frac {e^{\frac {32}{x^2}}}{x^3} \, dx\\ &=e^{\frac {32}{x^2}}+\frac {2 e^{\frac {16}{x^2}} \left (21-5 x^3\right )}{x^2}+2 \int \left (-\frac {882}{x^5}+\frac {105}{x^2}+25 x\right ) \, dx\\ &=e^{\frac {32}{x^2}}+\frac {441}{x^4}-\frac {210}{x}+25 x^2+\frac {2 e^{\frac {16}{x^2}} \left (21-5 x^3\right )}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 24, normalized size = 1.00 \begin {gather*} \frac {\left (21+e^{\frac {16}{x^2}} x^2-5 x^3\right )^2}{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1764 - 64*E^(32/x^2)*x^2 + 210*x^3 + 50*x^6 + E^(16/x^2)*(-1344 - 84*x^2 + 320*x^3 - 10*x^5))/x^5,
x]

[Out]

(21 + E^(16/x^2)*x^2 - 5*x^3)^2/x^4

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fricas [B]  time = 0.70, size = 45, normalized size = 1.88 \begin {gather*} \frac {25 \, x^{6} + x^{4} e^{\left (\frac {32}{x^{2}}\right )} - 210 \, x^{3} - 2 \, {\left (5 \, x^{5} - 21 \, x^{2}\right )} e^{\left (\frac {16}{x^{2}}\right )} + 441}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^2*exp(16/x^2)^2+(-10*x^5+320*x^3-84*x^2-1344)*exp(16/x^2)+50*x^6+210*x^3-1764)/x^5,x, algorit
hm="fricas")

[Out]

(25*x^6 + x^4*e^(32/x^2) - 210*x^3 - 2*(5*x^5 - 21*x^2)*e^(16/x^2) + 441)/x^4

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giac [B]  time = 0.31, size = 45, normalized size = 1.88 \begin {gather*} \frac {25 \, x^{6} - 10 \, x^{5} e^{\left (\frac {16}{x^{2}}\right )} - 210 \, x^{3} + 42 \, x^{2} e^{\left (\frac {16}{x^{2}}\right )} + 441}{x^{4}} + e^{\left (\frac {32}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^2*exp(16/x^2)^2+(-10*x^5+320*x^3-84*x^2-1344)*exp(16/x^2)+50*x^6+210*x^3-1764)/x^5,x, algorit
hm="giac")

[Out]

(25*x^6 - 10*x^5*e^(16/x^2) - 210*x^3 + 42*x^2*e^(16/x^2) + 441)/x^4 + e^(32/x^2)

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maple [B]  time = 0.09, size = 42, normalized size = 1.75

method	result	size
risch	\(25 x^{2}+\frac {-210 x^{3}+441}{x^{4}}+{\mathrm e}^{\frac {32}{x^{2}}}-\frac {2 \left (5 x^{3}-21\right ) {\mathrm e}^{\frac {16}{x^{2}}}}{x^{2}}\)	\(42\)
derivativedivides	\(-\frac {210}{x}+25 x^{2}+\frac {441}{x^{4}}+{\mathrm e}^{\frac {32}{x^{2}}}+\frac {42 \,{\mathrm e}^{\frac {16}{x^{2}}}}{x^{2}}-10 \,{\mathrm e}^{\frac {16}{x^{2}}} x\)	\(43\)
default	\(-\frac {210}{x}+25 x^{2}+\frac {441}{x^{4}}+{\mathrm e}^{\frac {32}{x^{2}}}+\frac {42 \,{\mathrm e}^{\frac {16}{x^{2}}}}{x^{2}}-10 \,{\mathrm e}^{\frac {16}{x^{2}}} x\)	\(43\)
norman	\(\frac {441+x^{4} {\mathrm e}^{\frac {32}{x^{2}}}-210 x^{3}+25 x^{6}+42 \,{\mathrm e}^{\frac {16}{x^{2}}} x^{2}-10 \,{\mathrm e}^{\frac {16}{x^{2}}} x^{5}}{x^{4}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-64*x^2*exp(16/x^2)^2+(-10*x^5+320*x^3-84*x^2-1344)*exp(16/x^2)+50*x^6+210*x^3-1764)/x^5,x,method=_RETURN
VERBOSE)

[Out]

25*x^2+(-210*x^3+441)/x^4+exp(32/x^2)-2*(5*x^3-21)/x^2*exp(16/x^2)

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maxima [C]  time = 0.40, size = 83, normalized size = 3.46 \begin {gather*} -20 \, x \sqrt {-\frac {1}{x^{2}}} \Gamma \left (-\frac {1}{2}, -\frac {16}{x^{2}}\right ) + 25 \, x^{2} - \frac {40 \, \sqrt {\pi } {\left (\operatorname {erf}\left (4 \, \sqrt {-\frac {1}{x^{2}}}\right ) - 1\right )}}{x \sqrt {-\frac {1}{x^{2}}}} - \frac {210}{x} + \frac {441}{x^{4}} + e^{\left (\frac {32}{x^{2}}\right )} + \frac {21}{8} \, e^{\left (\frac {16}{x^{2}}\right )} - \frac {21}{8} \, \Gamma \left (2, -\frac {16}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x^2*exp(16/x^2)^2+(-10*x^5+320*x^3-84*x^2-1344)*exp(16/x^2)+50*x^6+210*x^3-1764)/x^5,x, algorit
hm="maxima")

[Out]

-20*x*sqrt(-1/x^2)*gamma(-1/2, -16/x^2) + 25*x^2 - 40*sqrt(pi)*(erf(4*sqrt(-1/x^2)) - 1)/(x*sqrt(-1/x^2)) - 21
0/x + 441/x^4 + e^(32/x^2) + 21/8*e^(16/x^2) - 21/8*gamma(2, -16/x^2)

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mupad [B]  time = 1.13, size = 23, normalized size = 0.96 \begin {gather*} \frac {{\left (x^2\,{\mathrm {e}}^{\frac {16}{x^2}}-5\,x^3+21\right )}^2}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(64*x^2*exp(32/x^2) - 210*x^3 - 50*x^6 + exp(16/x^2)*(84*x^2 - 320*x^3 + 10*x^5 + 1344) + 1764)/x^5,x)

[Out]

(x^2*exp(16/x^2) - 5*x^3 + 21)^2/x^4

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sympy [B]  time = 0.15, size = 41, normalized size = 1.71 \begin {gather*} 25 x^{2} + \frac {x^{2} e^{\frac {32}{x^{2}}} + \left (42 - 10 x^{3}\right ) e^{\frac {16}{x^{2}}}}{x^{2}} + \frac {441 - 210 x^{3}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-64*x**2*exp(16/x**2)**2+(-10*x**5+320*x**3-84*x**2-1344)*exp(16/x**2)+50*x**6+210*x**3-1764)/x**5,
x)

[Out]

25*x**2 + (x**2*exp(32/x**2) + (42 - 10*x**3)*exp(16/x**2))/x**2 + (441 - 210*x**3)/x**4

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