3.36.83 \(\int \frac {25+x^3+2 x^4}{x^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac {25}{2 x^2}-x+(1+x)^2 \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 0.75, 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 {25}{2 x^2}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(25 + x^3 + 2*x^4)/x^3,x]

[Out]

-25/(2*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 {25}{x^3}+2 x\right ) \, dx\\ &=-\frac {25}{2 x^2}+x+x^2\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 12, normalized size = 0.75 \begin {gather*} -\frac {25}{2 x^2}+x+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 + x^3 + 2*x^4)/x^3,x]

[Out]

-25/(2*x^2) + x + x^2

________________________________________________________________________________________

fricas [A]  time = 1.01, size = 17, normalized size = 1.06 \begin {gather*} \frac {2 \, x^{4} + 2 \, x^{3} - 25}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4+x^3+25)/x^3,x, algorithm="fricas")

[Out]

1/2*(2*x^4 + 2*x^3 - 25)/x^2

________________________________________________________________________________________

giac [A]  time = 0.15, size = 10, normalized size = 0.62 \begin {gather*} x^{2} + x - \frac {25}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4+x^3+25)/x^3,x, algorithm="giac")

[Out]

x^2 + x - 25/2/x^2

________________________________________________________________________________________

maple [A]  time = 0.02, size = 11, normalized size = 0.69




method result size



default \(x^{2}+x -\frac {25}{2 x^{2}}\) \(11\)
risch \(x^{2}+x -\frac {25}{2 x^{2}}\) \(11\)
norman \(\frac {x^{4}+x^{3}-\frac {25}{2}}{x^{2}}\) \(13\)
gosper \(\frac {2 x^{4}+2 x^{3}-25}{2 x^{2}}\) \(18\)



Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4+x^3+25)/x^3,x,method=_RETURNVERBOSE)

[Out]

x^2+x-25/2/x^2

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 10, normalized size = 0.62 \begin {gather*} x^{2} + x - \frac {25}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^4+x^3+25)/x^3,x, algorithm="maxima")

[Out]

x^2 + x - 25/2/x^2

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 12, normalized size = 0.75 \begin {gather*} \frac {x^4+x^3-\frac {25}{2}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 + 2*x^4 + 25)/x^3,x)

[Out]

(x^3 + x^4 - 25/2)/x^2

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 10, normalized size = 0.62 \begin {gather*} x^{2} + x - \frac {25}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**4+x**3+25)/x**3,x)

[Out]

x**2 + x - 25/(2*x**2)

________________________________________________________________________________________