3.95.39 \(\int \frac {-18 e^2+18 x^2+4 x^3-24 x^5+27 x^7}{18 x^2} \, dx\)

Optimal. Leaf size=31 \[ 7+\frac {e^2 (1-x)}{x}+x+\frac {1}{4} x^2 \left (-\frac {2}{3}+x^2\right )^2 \]

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Rubi [A]  time = 0.02, antiderivative size = 30, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {12, 14} \begin {gather*} \frac {x^6}{4}-\frac {x^4}{3}+\frac {x^2}{9}+x+\frac {e^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-18*E^2 + 18*x^2 + 4*x^3 - 24*x^5 + 27*x^7)/(18*x^2),x]

[Out]

E^2/x + x + x^2/9 - x^4/3 + x^6/4

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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{18} \int \frac {-18 e^2+18 x^2+4 x^3-24 x^5+27 x^7}{x^2} \, dx\\ &=\frac {1}{18} \int \left (18-\frac {18 e^2}{x^2}+4 x-24 x^3+27 x^5\right ) \, dx\\ &=\frac {e^2}{x}+x+\frac {x^2}{9}-\frac {x^4}{3}+\frac {x^6}{4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 0.97 \begin {gather*} \frac {e^2}{x}+x+\frac {x^2}{9}-\frac {x^4}{3}+\frac {x^6}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-18*E^2 + 18*x^2 + 4*x^3 - 24*x^5 + 27*x^7)/(18*x^2),x]

[Out]

E^2/x + x + x^2/9 - x^4/3 + x^6/4

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fricas [A]  time = 1.31, size = 30, normalized size = 0.97 \begin {gather*} \frac {9 \, x^{7} - 12 \, x^{5} + 4 \, x^{3} + 36 \, x^{2} + 36 \, e^{2}}{36 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/18*(-18*exp(2)+27*x^7-24*x^5+4*x^3+18*x^2)/x^2,x, algorithm="fricas")

[Out]

1/36*(9*x^7 - 12*x^5 + 4*x^3 + 36*x^2 + 36*e^2)/x

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giac [A]  time = 0.14, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{4} \, x^{6} - \frac {1}{3} \, x^{4} + \frac {1}{9} \, x^{2} + x + \frac {e^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/18*(-18*exp(2)+27*x^7-24*x^5+4*x^3+18*x^2)/x^2,x, algorithm="giac")

[Out]

1/4*x^6 - 1/3*x^4 + 1/9*x^2 + x + e^2/x

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maple [A]  time = 0.04, size = 24, normalized size = 0.77

method	result	size
default	\(\frac {x^{6}}{4}-\frac {x^{4}}{3}+\frac {x^{2}}{9}+x +\frac {{\mathrm e}^{2}}{x}\)	\(24\)
risch	\(\frac {x^{6}}{4}-\frac {x^{4}}{3}+\frac {x^{2}}{9}+x +\frac {{\mathrm e}^{2}}{x}\)	\(24\)
norman	\(\frac {x^{2}+\frac {x^{3}}{9}-\frac {x^{5}}{3}+\frac {x^{7}}{4}+{\mathrm e}^{2}}{x}\)	\(26\)
gosper	\(\frac {9 x^{7}-12 x^{5}+4 x^{3}+36 x^{2}+36 \,{\mathrm e}^{2}}{36 x}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/18*(-18*exp(2)+27*x^7-24*x^5+4*x^3+18*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/4*x^6-1/3*x^4+1/9*x^2+x+exp(2)/x

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maxima [A]  time = 0.35, size = 23, normalized size = 0.74 \begin {gather*} \frac {1}{4} \, x^{6} - \frac {1}{3} \, x^{4} + \frac {1}{9} \, x^{2} + x + \frac {e^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/18*(-18*exp(2)+27*x^7-24*x^5+4*x^3+18*x^2)/x^2,x, algorithm="maxima")

[Out]

1/4*x^6 - 1/3*x^4 + 1/9*x^2 + x + e^2/x

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mupad [B]  time = 0.04, size = 23, normalized size = 0.74 \begin {gather*} x+\frac {{\mathrm {e}}^2}{x}+\frac {x^2}{9}-\frac {x^4}{3}+\frac {x^6}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - exp(2) + (2*x^3)/9 - (4*x^5)/3 + (3*x^7)/2)/x^2,x)

[Out]

x + exp(2)/x + x^2/9 - x^4/3 + x^6/4

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sympy [A]  time = 0.09, size = 20, normalized size = 0.65 \begin {gather*} \frac {x^{6}}{4} - \frac {x^{4}}{3} + \frac {x^{2}}{9} + x + \frac {e^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/18*(-18*exp(2)+27*x**7-24*x**5+4*x**3+18*x**2)/x**2,x)

[Out]

x**6/4 - x**4/3 + x**2/9 + x + exp(2)/x

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