3.101.70 \(\int \frac {6+x^2+2 x^3}{x^2} \, dx\)

Optimal. Leaf size=19 \[ -1-\frac {6}{x}+x+x^2-\frac {2}{\log \left (\frac {5}{3}\right )} \]

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 0.53, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {14} \begin {gather*} x^2+x-\frac {6}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 + x^2 + 2*x^3)/x^2,x]

[Out]

-6/x + 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 (1+\frac {6}{x^2}+2 x\right ) \, dx\\ &=-\frac {6}{x}+x+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.53 \begin {gather*} -\frac {6}{x}+x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 + x^2 + 2*x^3)/x^2,x]

[Out]

-6/x + x + x^2

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fricas [A]  time = 0.54, size = 12, normalized size = 0.63 \begin {gather*} \frac {x^{3} + x^{2} - 6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+x^2+6)/x^2,x, algorithm="fricas")

[Out]

(x^3 + x^2 - 6)/x

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giac [A]  time = 0.14, size = 10, normalized size = 0.53 \begin {gather*} x^{2} + x - \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+x^2+6)/x^2,x, algorithm="giac")

[Out]

x^2 + x - 6/x

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maple [A]  time = 0.01, size = 11, normalized size = 0.58

method	result	size
default	\(x^{2}+x -\frac {6}{x}\)	\(11\)
risch	\(x^{2}+x -\frac {6}{x}\)	\(11\)
gosper	\(\frac {x^{3}+x^{2}-6}{x}\)	\(13\)
norman	\(\frac {x^{3}+x^{2}-6}{x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^3+x^2+6)/x^2,x,method=_RETURNVERBOSE)

[Out]

x^2+x-6/x

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maxima [A]  time = 0.36, size = 10, normalized size = 0.53 \begin {gather*} x^{2} + x - \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^3+x^2+6)/x^2,x, algorithm="maxima")

[Out]

x^2 + x - 6/x

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mupad [B]  time = 0.04, size = 12, normalized size = 0.63 \begin {gather*} \frac {x^3+x^2-6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 2*x^3 + 6)/x^2,x)

[Out]

(x^2 + x^3 - 6)/x

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sympy [A]  time = 0.06, size = 7, normalized size = 0.37 \begin {gather*} x^{2} + x - \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**3+x**2+6)/x**2,x)

[Out]

x**2 + x - 6/x

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