3.102.38 \(\int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} (-40+9 x^2)}{9 x^2} \, dx\)

Optimal. Leaf size=15 \[ x \left (e^{1+\frac {20}{9 x^2}}+x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 17, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {12, 14, 2288} \begin {gather*} x^2+e^{\frac {20}{9 x^2}+1} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(18*x^3 + E^((20 + 9*x^2)/(9*x^2))*(-40 + 9*x^2))/(9*x^2),x]

[Out]

E^(1 + 20/(9*x^2))*x + x^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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} &=\frac {1}{9} \int \frac {18 x^3+e^{\frac {20+9 x^2}{9 x^2}} \left (-40+9 x^2\right )}{x^2} \, dx\\ &=\frac {1}{9} \int \left (18 x+\frac {e^{1+\frac {20}{9 x^2}} \left (-40+9 x^2\right )}{x^2}\right ) \, dx\\ &=x^2+\frac {1}{9} \int \frac {e^{1+\frac {20}{9 x^2}} \left (-40+9 x^2\right )}{x^2} \, dx\\ &=e^{1+\frac {20}{9 x^2}} x+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 17, normalized size = 1.13 \begin {gather*} e^{1+\frac {20}{9 x^2}} x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18*x^3 + E^((20 + 9*x^2)/(9*x^2))*(-40 + 9*x^2))/(9*x^2),x]

[Out]

E^(1 + 20/(9*x^2))*x + x^2

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fricas [A]  time = 0.50, size = 19, normalized size = 1.27 \begin {gather*} x^{2} + x e^{\left (\frac {9 \, x^{2} + 20}{9 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((9*x^2-40)*exp(1/9*(9*x^2+20)/x^2)+18*x^3)/x^2,x, algorithm="fricas")

[Out]

x^2 + x*e^(1/9*(9*x^2 + 20)/x^2)

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giac [A]  time = 0.20, size = 19, normalized size = 1.27 \begin {gather*} x^{2} + x e^{\left (\frac {9 \, x^{2} + 20}{9 \, x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((9*x^2-40)*exp(1/9*(9*x^2+20)/x^2)+18*x^3)/x^2,x, algorithm="giac")

[Out]

x^2 + x*e^(1/9*(9*x^2 + 20)/x^2)

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maple [A]  time = 0.06, size = 20, normalized size = 1.33

method	result	size
risch	\(x^{2}+x \,{\mathrm e}^{\frac {9 x^{2}+20}{9 x^{2}}}\)	\(20\)
norman	\(\frac {x^{3}+{\mathrm e}^{\frac {9 x^{2}+20}{9 x^{2}}} x^{2}}{x}\)	\(26\)
derivativedivides	\(x^{2}-\frac {2 i {\mathrm e} \sqrt {\pi }\, \sqrt {5}\, \erf \left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}-{\mathrm e} \left (-{\mathrm e}^{\frac {20}{9 x^{2}}} x -\frac {2 i \sqrt {\pi }\, \sqrt {5}\, \erf \left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}\right )\)	\(59\)
default	\(x^{2}-\frac {2 i {\mathrm e} \sqrt {\pi }\, \sqrt {5}\, \erf \left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}-{\mathrm e} \left (-{\mathrm e}^{\frac {20}{9 x^{2}}} x -\frac {2 i \sqrt {\pi }\, \sqrt {5}\, \erf \left (\frac {2 i \sqrt {5}}{3 x}\right )}{3}\right )\)	\(59\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/9*((9*x^2-40)*exp(1/9*(9*x^2+20)/x^2)+18*x^3)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^2+x*exp(1/9*(9*x^2+20)/x^2)

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maxima [C]  time = 0.37, size = 61, normalized size = 4.07 \begin {gather*} \frac {1}{3} \, \sqrt {5} x \sqrt {-\frac {1}{x^{2}}} e \Gamma \left (-\frac {1}{2}, -\frac {20}{9 \, x^{2}}\right ) + x^{2} + \frac {2 \, \sqrt {5} \sqrt {\pi } {\left (\operatorname {erf}\left (\frac {2}{3} \, \sqrt {5} \sqrt {-\frac {1}{x^{2}}}\right ) - 1\right )} e}{3 \, x \sqrt {-\frac {1}{x^{2}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((9*x^2-40)*exp(1/9*(9*x^2+20)/x^2)+18*x^3)/x^2,x, algorithm="maxima")

[Out]

1/3*sqrt(5)*x*sqrt(-1/x^2)*e*gamma(-1/2, -20/9/x^2) + x^2 + 2/3*sqrt(5)*sqrt(pi)*(erf(2/3*sqrt(5)*sqrt(-1/x^2)
) - 1)*e/(x*sqrt(-1/x^2))

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mupad [B]  time = 8.07, size = 12, normalized size = 0.80 \begin {gather*} x\,\left (x+{\mathrm {e}}^{\frac {20}{9\,x^2}+1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp((x^2 + 20/9)/x^2)*(9*x^2 - 40))/9 + 2*x^3)/x^2,x)

[Out]

x*(x + exp(20/(9*x^2) + 1))

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sympy [A]  time = 0.12, size = 15, normalized size = 1.00 \begin {gather*} x^{2} + x e^{\frac {x^{2} + \frac {20}{9}}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/9*((9*x**2-40)*exp(1/9*(9*x**2+20)/x**2)+18*x**3)/x**2,x)

[Out]

x**2 + x*exp((x**2 + 20/9)/x**2)

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