3.61.74 \(\int \frac {-5+x^2+4 x^5+e^9 (x^2+2 x^3)}{x^2} \, dx\)

Optimal. Leaf size=22 \[ 5+\frac {5}{x}+x-e^9 (-1-x) x+x^4 \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.05, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {14} \begin {gather*} x^4+e^9 x^2+\left (1+e^9\right ) x+\frac {5}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + x^2 + 4*x^5 + E^9*(x^2 + 2*x^3))/x^2,x]

[Out]

5/x + (1 + E^9)*x + E^9*x^2 + x^4

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} &=\int \left (1+e^9-\frac {5}{x^2}+2 e^9 x+4 x^3\right ) \, dx\\ &=\frac {5}{x}+\left (1+e^9\right ) x+e^9 x^2+x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {5}{x}+x+e^9 x+e^9 x^2+x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + x^2 + 4*x^5 + E^9*(x^2 + 2*x^3))/x^2,x]

[Out]

5/x + x + E^9*x + E^9*x^2 + x^4

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fricas [A]  time = 0.55, size = 22, normalized size = 1.00 \begin {gather*} \frac {x^{5} + x^{2} + {\left (x^{3} + x^{2}\right )} e^{9} + 5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+x^2)*exp(9)+4*x^5+x^2-5)/x^2,x, algorithm="fricas")

[Out]

(x^5 + x^2 + (x^3 + x^2)*e^9 + 5)/x

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giac [A]  time = 0.14, size = 20, normalized size = 0.91 \begin {gather*} x^{4} + x^{2} e^{9} + x e^{9} + x + \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+x^2)*exp(9)+4*x^5+x^2-5)/x^2,x, algorithm="giac")

[Out]

x^4 + x^2*e^9 + x*e^9 + x + 5/x

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maple [A]  time = 0.05, size = 21, normalized size = 0.95

method	result	size
default	\(x^{4}+x^{2} {\mathrm e}^{9}+x \,{\mathrm e}^{9}+x +\frac {5}{x}\)	\(21\)
risch	\(x^{4}+x^{2} {\mathrm e}^{9}+x \,{\mathrm e}^{9}+x +\frac {5}{x}\)	\(21\)
norman	\(\frac {x^{5}+5+x^{3} {\mathrm e}^{9}+\left ({\mathrm e}^{9}+1\right ) x^{2}}{x}\)	\(24\)
gosper	\(\frac {x^{5}+x^{3} {\mathrm e}^{9}+x^{2} {\mathrm e}^{9}+x^{2}+5}{x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^4+x^2*exp(9)+x*exp(9)+x+5/x

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maxima [A]  time = 0.35, size = 21, normalized size = 0.95 \begin {gather*} x^{4} + x^{2} e^{9} + x {\left (e^{9} + 1\right )} + \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^3+x^2)*exp(9)+4*x^5+x^2-5)/x^2,x, algorithm="maxima")

[Out]

x^4 + x^2*e^9 + x*(e^9 + 1) + 5/x

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mupad [B]  time = 4.17, size = 21, normalized size = 0.95 \begin {gather*} x\,\left ({\mathrm {e}}^9+1\right )+x^2\,{\mathrm {e}}^9+\frac {5}{x}+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(exp(9) + 1) + x^2*exp(9) + 5/x + x^4

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sympy [A]  time = 0.08, size = 19, normalized size = 0.86 \begin {gather*} x^{4} + x^{2} e^{9} + x \left (1 + e^{9}\right ) + \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**3+x**2)*exp(9)+4*x**5+x**2-5)/x**2,x)

[Out]

x**4 + x**2*exp(9) + x*(1 + exp(9)) + 5/x

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