3.39.77 \(\int \frac {-96+x^2-2 x^3+x^2 \log (2)}{x^2} \, dx\)

Optimal. Leaf size=15 \[ \frac {96}{x}+x+x (-x+\log (2)) \]

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6, 14} \begin {gather*} -x^2+\frac {96}{x}+x (1+\log (2)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-96 + x^2 - 2*x^3 + x^2*Log[2])/x^2,x]

[Out]

96/x - x^2 + x*(1 + Log[2])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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} &=\int \frac {-96-2 x^3+x^2 (1+\log (2))}{x^2} \, dx\\ &=\int \left (1-\frac {96}{x^2}-2 x+\log (2)\right ) \, dx\\ &=\frac {96}{x}-x^2+x (1+\log (2))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.07 \begin {gather*} \frac {96}{x}+x-x^2+x \log (2) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-96 + x^2 - 2*x^3 + x^2*Log[2])/x^2,x]

[Out]

96/x + x - x^2 + x*Log[2]

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fricas [A]  time = 0.75, size = 22, normalized size = 1.47 \begin {gather*} -\frac {x^{3} - x^{2} \log \relax (2) - x^{2} - 96}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(2)-2*x^3+x^2-96)/x^2,x, algorithm="fricas")

[Out]

-(x^3 - x^2*log(2) - x^2 - 96)/x

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giac [A]  time = 0.13, size = 16, normalized size = 1.07 \begin {gather*} -x^{2} + x \log \relax (2) + x + \frac {96}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(2)-2*x^3+x^2-96)/x^2,x, algorithm="giac")

[Out]

-x^2 + x*log(2) + x + 96/x

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maple [A]  time = 0.02, size = 17, normalized size = 1.13

method	result	size
default	\(-x^{2}+x +x \ln \relax (2)+\frac {96}{x}\)	\(17\)
risch	\(-x^{2}+x +x \ln \relax (2)+\frac {96}{x}\)	\(17\)
norman	\(\frac {-x^{3}+\left (1+\ln \relax (2)\right ) x^{2}+96}{x}\)	\(20\)
gosper	\(\frac {x^{2} \ln \relax (2)-x^{3}+x^{2}+96}{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*ln(2)-2*x^3+x^2-96)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x^2+x+x*ln(2)+96/x

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maxima [A]  time = 0.34, size = 17, normalized size = 1.13 \begin {gather*} -x^{2} + x {\left (\log \relax (2) + 1\right )} + \frac {96}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(2)-2*x^3+x^2-96)/x^2,x, algorithm="maxima")

[Out]

-x^2 + x*(log(2) + 1) + 96/x

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mupad [B]  time = 0.04, size = 17, normalized size = 1.13 \begin {gather*} x\,\left (\ln \relax (2)+1\right )+\frac {96}{x}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*log(2) + x^2 - 2*x^3 - 96)/x^2,x)

[Out]

x*(log(2) + 1) + 96/x - x^2

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sympy [A]  time = 0.08, size = 14, normalized size = 0.93 \begin {gather*} - x^{2} - x \left (-1 - \log {\relax (2 )}\right ) + \frac {96}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*ln(2)-2*x**3+x**2-96)/x**2,x)

[Out]

-x**2 - x*(-1 - log(2)) + 96/x

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