3.61.76 \(\int \frac {-5-8 x^3}{8 x^2} \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{2} \left (\frac {5}{4 x}-x^2+\log (2)\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 14} \begin {gather*} \frac {5}{8 x}-\frac {x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 - 8*x^3)/(8*x^2),x]

[Out]

5/(8*x) - x^2/2

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}{8} \int \frac {-5-8 x^3}{x^2} \, dx\\ &=\frac {1}{8} \int \left (-\frac {5}{x^2}-8 x\right ) \, dx\\ &=\frac {5}{8 x}-\frac {x^2}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.79 \begin {gather*} \frac {5}{8 x}-\frac {x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 - 8*x^3)/(8*x^2),x]

[Out]

5/(8*x) - x^2/2

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fricas [A]  time = 1.07, size = 12, normalized size = 0.63 \begin {gather*} -\frac {4 \, x^{3} - 5}{8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-8*x^3-5)/x^2,x, algorithm="fricas")

[Out]

-1/8*(4*x^3 - 5)/x

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giac [A]  time = 0.31, size = 11, normalized size = 0.58 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {5}{8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-8*x^3-5)/x^2,x, algorithm="giac")

[Out]

-1/2*x^2 + 5/8/x

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maple [A]  time = 0.03, size = 12, normalized size = 0.63

method	result	size
default	\(-\frac {x^{2}}{2}+\frac {5}{8 x}\)	\(12\)
norman	\(\frac {\frac {5}{8}-\frac {x^{3}}{2}}{x}\)	\(12\)
risch	\(-\frac {x^{2}}{2}+\frac {5}{8 x}\)	\(12\)
gosper	\(-\frac {4 x^{3}-5}{8 x}\)	\(13\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*(-8*x^3-5)/x^2,x,method=_RETURNVERBOSE)

[Out]

-1/2*x^2+5/8/x

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maxima [A]  time = 0.34, size = 11, normalized size = 0.58 \begin {gather*} -\frac {1}{2} \, x^{2} + \frac {5}{8 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-8*x^3-5)/x^2,x, algorithm="maxima")

[Out]

-1/2*x^2 + 5/8/x

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mupad [B]  time = 0.02, size = 12, normalized size = 0.63 \begin {gather*} -\frac {4\,x^3-5}{8\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^3 + 5/8)/x^2,x)

[Out]

-(4*x^3 - 5)/(8*x)

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sympy [A]  time = 0.06, size = 8, normalized size = 0.42 \begin {gather*} - \frac {x^{2}}{2} + \frac {5}{8 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(-8*x**3-5)/x**2,x)

[Out]

-x**2/2 + 5/(8*x)

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