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

Optimal. Leaf size=26 \[ 5-e^2+x-x^2+\frac {-2+\frac {x^2}{2}}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.62, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {12, 14} \begin {gather*} -x^2+\frac {3 x}{2}-\frac {2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + 3*x^2 - 4*x^3)/(2*x^2),x]

[Out]

-2/x + (3*x)/2 - 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]]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.62 \begin {gather*} -\frac {2}{x}+\frac {3 x}{2}-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-2/x + (3*x)/2 - x^2

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fricas [A]  time = 1.09, size = 17, normalized size = 0.65 \begin {gather*} -\frac {2 \, x^{3} - 3 \, x^{2} + 4}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*x^3 - 3*x^2 + 4)/x

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giac [A]  time = 0.23, size = 14, normalized size = 0.54 \begin {gather*} -x^{2} + \frac {3}{2} \, x - \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 + 3/2*x - 2/x

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maple [A]  time = 0.01, size = 15, normalized size = 0.58

method	result	size
default	\(\frac {3 x}{2}-x^{2}-\frac {2}{x}\)	\(15\)
risch	\(\frac {3 x}{2}-x^{2}-\frac {2}{x}\)	\(15\)
norman	\(\frac {-2+\frac {3}{2} x^{2}-x^{3}}{x}\)	\(17\)
gosper	\(-\frac {2 x^{3}-3 x^{2}+4}{2 x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*x-x^2-2/x

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maxima [A]  time = 0.42, size = 14, normalized size = 0.54 \begin {gather*} -x^{2} + \frac {3}{2} \, x - \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x^2 + 3/2*x - 2/x

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mupad [B]  time = 0.03, size = 15, normalized size = 0.58 \begin {gather*} -\frac {x^3-\frac {3\,x^2}{2}+2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2)/2 - 2*x^3 + 2)/x^2,x)

[Out]

-(x^3 - (3*x^2)/2 + 2)/x

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sympy [A]  time = 0.06, size = 10, normalized size = 0.38 \begin {gather*} - x^{2} + \frac {3 x}{2} - \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**2 + 3*x/2 - 2/x

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