3.103.46 \(\int \frac {-1+2 x^2+4 x^3}{x^2} \, dx\)

Optimal. Leaf size=16 \[ -5+\frac {1}{x}+2 \left (4+e^{16}+x+x^2\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 12, normalized size of antiderivative = 0.75, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \begin {gather*} 2 x^2+2 x+\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + 2*x^2 + 4*x^3)/x^2,x]

[Out]

x^(-1) + 2*x + 2*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]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2-\frac {1}{x^2}+4 x\right ) \, dx\\ &=\frac {1}{x}+2 x+2 x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{x}+2 x+2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 2*x^2 + 4*x^3)/x^2,x]

[Out]

x^(-1) + 2*x + 2*x^2

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fricas [A]  time = 0.74, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \, x^{3} + 2 \, x^{2} + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+2*x^2-1)/x^2,x, algorithm="fricas")

[Out]

(2*x^3 + 2*x^2 + 1)/x

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giac [A]  time = 0.17, size = 12, normalized size = 0.75 \begin {gather*} 2 \, x^{2} + 2 \, x + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+2*x^2-1)/x^2,x, algorithm="giac")

[Out]

2*x^2 + 2*x + 1/x

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maple [A]  time = 0.02, size = 13, normalized size = 0.81

method	result	size
default	\(2 x^{2}+2 x +\frac {1}{x}\)	\(13\)
risch	\(2 x^{2}+2 x +\frac {1}{x}\)	\(13\)
gosper	\(\frac {2 x^{3}+2 x^{2}+1}{x}\)	\(17\)
norman	\(\frac {2 x^{3}+2 x^{2}+1}{x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^3+2*x^2-1)/x^2,x,method=_RETURNVERBOSE)

[Out]

2*x^2+2*x+1/x

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maxima [A]  time = 0.36, size = 12, normalized size = 0.75 \begin {gather*} 2 \, x^{2} + 2 \, x + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^3+2*x^2-1)/x^2,x, algorithm="maxima")

[Out]

2*x^2 + 2*x + 1/x

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mupad [B]  time = 0.03, size = 12, normalized size = 0.75 \begin {gather*} 2\,x+\frac {1}{x}+2\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 + 4*x^3 - 1)/x^2,x)

[Out]

2*x + 1/x + 2*x^2

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sympy [A]  time = 0.06, size = 10, normalized size = 0.62 \begin {gather*} 2 x^{2} + 2 x + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**3+2*x**2-1)/x**2,x)

[Out]

2*x**2 + 2*x + 1/x

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