3.83.3 \(\int \frac {20+x^3+2 x^4}{x^3} \, dx\)

Optimal. Leaf size=25 \[ 5-\frac {10}{x^2}+x+x^2+\frac {5}{\log (4) (3-\log (5))} \]

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 0.40, 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-\frac {10}{x^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20 + x^3 + 2*x^4)/x^3,x]

[Out]

-10/x^2 + 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 {20}{x^3}+2 x\right ) \, dx\\ &=-\frac {10}{x^2}+x+x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.40 \begin {gather*} -\frac {10}{x^2}+x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20 + x^3 + 2*x^4)/x^3,x]

[Out]

-10/x^2 + x + x^2

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fricas [A]  time = 0.93, size = 12, normalized size = 0.48 \begin {gather*} \frac {x^{4} + x^{3} - 10}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4+x^3+20)/x^3,x, algorithm="fricas")

[Out]

(x^4 + x^3 - 10)/x^2

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giac [A]  time = 0.17, size = 10, normalized size = 0.40 \begin {gather*} x^{2} + x - \frac {10}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4+x^3+20)/x^3,x, algorithm="giac")

[Out]

x^2 + x - 10/x^2

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maple [A]  time = 0.02, size = 11, normalized size = 0.44

method	result	size
default	\(x^{2}+x -\frac {10}{x^{2}}\)	\(11\)
risch	\(x^{2}+x -\frac {10}{x^{2}}\)	\(11\)
gosper	\(\frac {x^{4}+x^{3}-10}{x^{2}}\)	\(13\)
norman	\(\frac {x^{4}+x^{3}-10}{x^{2}}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4+x^3+20)/x^3,x,method=_RETURNVERBOSE)

[Out]

x^2+x-10/x^2

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maxima [A]  time = 0.37, size = 10, normalized size = 0.40 \begin {gather*} x^{2} + x - \frac {10}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4+x^3+20)/x^3,x, algorithm="maxima")

[Out]

x^2 + x - 10/x^2

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mupad [B]  time = 0.03, size = 12, normalized size = 0.48 \begin {gather*} \frac {x^4+x^3-10}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 + 2*x^4 + 20)/x^3,x)

[Out]

(x^3 + x^4 - 10)/x^2

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sympy [A]  time = 0.06, size = 8, normalized size = 0.32 \begin {gather*} x^{2} + x - \frac {10}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**4+x**3+20)/x**3,x)

[Out]

x**2 + x - 10/x**2

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