3.104.3 \(\int \frac {2 x^2-3 x^3-6 x^6+e^{\frac {5}{x^2}} (-10+2 x^2)}{x} \, dx\)

Optimal. Leaf size=21 \[ x^2 \left (1+e^{\frac {5}{x^2}}-x-x^4\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {14, 2288} \begin {gather*} -x^6-x^3+e^{\frac {5}{x^2}} x^2+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^2 - 3*x^3 - 6*x^6 + E^(5/x^2)*(-10 + 2*x^2))/x,x]

[Out]

x^2 + E^(5/x^2)*x^2 - x^3 - x^6

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.19 \begin {gather*} x^2+e^{\frac {5}{x^2}} x^2-x^3-x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2 - 3*x^3 - 6*x^6 + E^(5/x^2)*(-10 + 2*x^2))/x,x]

[Out]

x^2 + E^(5/x^2)*x^2 - x^3 - x^6

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fricas [A]  time = 0.60, size = 24, normalized size = 1.14 \begin {gather*} -x^{6} - x^{3} + x^{2} e^{\left (\frac {5}{x^{2}}\right )} + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-10)*exp(5/x^2)-6*x^6-3*x^3+2*x^2)/x,x, algorithm="fricas")

[Out]

-x^6 - x^3 + x^2*e^(5/x^2) + x^2

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giac [A]  time = 0.12, size = 24, normalized size = 1.14 \begin {gather*} -x^{6} - x^{3} + x^{2} e^{\left (\frac {5}{x^{2}}\right )} + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-10)*exp(5/x^2)-6*x^6-3*x^3+2*x^2)/x,x, algorithm="giac")

[Out]

-x^6 - x^3 + x^2*e^(5/x^2) + x^2

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maple [A]  time = 0.09, size = 25, normalized size = 1.19

method	result	size
derivativedivides	\(x^{2}+{\mathrm e}^{\frac {5}{x^{2}}} x^{2}-x^{3}-x^{6}\)	\(25\)
default	\(x^{2}+{\mathrm e}^{\frac {5}{x^{2}}} x^{2}-x^{3}-x^{6}\)	\(25\)
norman	\(x^{2}+{\mathrm e}^{\frac {5}{x^{2}}} x^{2}-x^{3}-x^{6}\)	\(25\)
risch	\(x^{2}+{\mathrm e}^{\frac {5}{x^{2}}} x^{2}-x^{3}-x^{6}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-10)*exp(5/x^2)-6*x^6-3*x^3+2*x^2)/x,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [C]  time = 0.36, size = 31, normalized size = 1.48 \begin {gather*} -x^{6} - x^{3} + x^{2} + 5 \, {\rm Ei}\left (\frac {5}{x^{2}}\right ) - 5 \, \Gamma \left (-1, -\frac {5}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-10)*exp(5/x^2)-6*x^6-3*x^3+2*x^2)/x,x, algorithm="maxima")

[Out]

-x^6 - x^3 + x^2 + 5*Ei(5/x^2) - 5*gamma(-1, -5/x^2)

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mupad [B]  time = 6.23, size = 19, normalized size = 0.90 \begin {gather*} -x^2\,\left (x-{\mathrm {e}}^{\frac {5}{x^2}}+x^4-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5/x^2)*(2*x^2 - 10) + 2*x^2 - 3*x^3 - 6*x^6)/x,x)

[Out]

-x^2*(x - exp(5/x^2) + x^4 - 1)

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sympy [A]  time = 0.13, size = 19, normalized size = 0.90 \begin {gather*} - x^{6} - x^{3} + x^{2} e^{\frac {5}{x^{2}}} + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-10)*exp(5/x**2)-6*x**6-3*x**3+2*x**2)/x,x)

[Out]

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

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