3.66.98 \(\int \frac {-4+10 x^2-2 x^3}{x^2} \, dx\)

Optimal. Leaf size=22 \[ -3+x+(8-x) x+\frac {4+x+x^2}{x}+\log (2) \]

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Rubi [A]  time = 0.01, antiderivative size = 14, normalized size of antiderivative = 0.64, 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*} -x^2+10 x+\frac {4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + 10*x^2 - 2*x^3)/x^2,x]

[Out]

4/x + 10*x - 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 (10-\frac {4}{x^2}-2 x\right ) \, dx\\ &=\frac {4}{x}+10 x-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 14, normalized size = 0.64 \begin {gather*} \frac {4}{x}+10 x-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 10*x^2 - 2*x^3)/x^2,x]

[Out]

4/x + 10*x - x^2

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fricas [A]  time = 0.54, size = 15, normalized size = 0.68 \begin {gather*} -\frac {x^{3} - 10 \, x^{2} - 4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+10*x^2-4)/x^2,x, algorithm="fricas")

[Out]

-(x^3 - 10*x^2 - 4)/x

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giac [A]  time = 0.24, size = 14, normalized size = 0.64 \begin {gather*} -x^{2} + 10 \, x + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+10*x^2-4)/x^2,x, algorithm="giac")

[Out]

-x^2 + 10*x + 4/x

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maple [A]  time = 0.03, size = 15, normalized size = 0.68

method	result	size
default	\(10 x -x^{2}+\frac {4}{x}\)	\(15\)
risch	\(10 x -x^{2}+\frac {4}{x}\)	\(15\)
gosper	\(-\frac {x^{3}-10 x^{2}-4}{x}\)	\(16\)
norman	\(\frac {-x^{3}+10 x^{2}+4}{x}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^3+10*x^2-4)/x^2,x,method=_RETURNVERBOSE)

[Out]

10*x-x^2+4/x

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maxima [A]  time = 0.50, size = 14, normalized size = 0.64 \begin {gather*} -x^{2} + 10 \, x + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3+10*x^2-4)/x^2,x, algorithm="maxima")

[Out]

-x^2 + 10*x + 4/x

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mupad [B]  time = 0.03, size = 16, normalized size = 0.73 \begin {gather*} \frac {-x^3+10\,x^2+4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x^3 - 10*x^2 + 4)/x^2,x)

[Out]

(10*x^2 - x^3 + 4)/x

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sympy [A]  time = 0.06, size = 8, normalized size = 0.36 \begin {gather*} - x^{2} + 10 x + \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**3+10*x**2-4)/x**2,x)

[Out]

-x**2 + 10*x + 4/x

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