3.52.85 \(\int \frac {-9-10 x^4+4 x^5+2 x^4 \log (2 x)}{2 x^4} \, dx\)

Optimal. Leaf size=28 \[ \frac {3}{2 x^3}-x+\frac {x+x^2 (-5+x+\log (2 x))}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 0.71, number of steps used = 6, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {12, 14, 2295} \begin {gather*} \frac {3}{2 x^3}+x^2-6 x+x \log (2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-9 - 10*x^4 + 4*x^5 + 2*x^4*Log[2*x])/(2*x^4),x]

[Out]

3/(2*x^3) - 6*x + x^2 + x*Log[2*x]

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]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {-9-10 x^4+4 x^5+2 x^4 \log (2 x)}{x^4} \, dx\\ &=\frac {1}{2} \int \left (\frac {-9-10 x^4+4 x^5}{x^4}+2 \log (2 x)\right ) \, dx\\ &=\frac {1}{2} \int \frac {-9-10 x^4+4 x^5}{x^4} \, dx+\int \log (2 x) \, dx\\ &=-x+x \log (2 x)+\frac {1}{2} \int \left (-10-\frac {9}{x^4}+4 x\right ) \, dx\\ &=\frac {3}{2 x^3}-6 x+x^2+x \log (2 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 0.71 \begin {gather*} \frac {3}{2 x^3}-6 x+x^2+x \log (2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9 - 10*x^4 + 4*x^5 + 2*x^4*Log[2*x])/(2*x^4),x]

[Out]

3/(2*x^3) - 6*x + x^2 + x*Log[2*x]

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fricas [A]  time = 0.48, size = 26, normalized size = 0.93 \begin {gather*} \frac {2 \, x^{5} + 2 \, x^{4} \log \left (2 \, x\right ) - 12 \, x^{4} + 3}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^4*log(2*x)+4*x^5-10*x^4-9)/x^4,x, algorithm="fricas")

[Out]

1/2*(2*x^5 + 2*x^4*log(2*x) - 12*x^4 + 3)/x^3

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giac [A]  time = 0.15, size = 18, normalized size = 0.64 \begin {gather*} x^{2} + x \log \left (2 \, x\right ) - 6 \, x + \frac {3}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^4*log(2*x)+4*x^5-10*x^4-9)/x^4,x, algorithm="giac")

[Out]

x^2 + x*log(2*x) - 6*x + 3/2/x^3

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maple [A]  time = 0.04, size = 19, normalized size = 0.68

method	result	size
derivativedivides	\(x \ln \left (2 x \right )-6 x +x^{2}+\frac {3}{2 x^{3}}\)	\(19\)
default	\(x \ln \left (2 x \right )-6 x +x^{2}+\frac {3}{2 x^{3}}\)	\(19\)
norman	\(\frac {\frac {3}{2}+x^{5}+x^{4} \ln \left (2 x \right )-6 x^{4}}{x^{3}}\)	\(23\)
risch	\(x \ln \left (2 x \right )+\frac {2 x^{5}-12 x^{4}+3}{2 x^{3}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(2*x^4*ln(2*x)+4*x^5-10*x^4-9)/x^4,x,method=_RETURNVERBOSE)

[Out]

x*ln(2*x)-6*x+x^2+3/2/x^3

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maxima [A]  time = 0.34, size = 18, normalized size = 0.64 \begin {gather*} x^{2} + x \log \left (2 \, x\right ) - 6 \, x + \frac {3}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x^4*log(2*x)+4*x^5-10*x^4-9)/x^4,x, algorithm="maxima")

[Out]

x^2 + x*log(2*x) - 6*x + 3/2/x^3

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mupad [B]  time = 3.26, size = 17, normalized size = 0.61 \begin {gather*} x\,\left (\ln \left (2\,x\right )-6\right )+x^2+\frac {3}{2\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*log(2*x) - 5*x^4 + 2*x^5 - 9/2)/x^4,x)

[Out]

x*(log(2*x) - 6) + x^2 + 3/(2*x^3)

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sympy [A]  time = 0.11, size = 19, normalized size = 0.68 \begin {gather*} x^{2} + x \log {\left (2 x \right )} - 6 x + \frac {3}{2 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*x**4*ln(2*x)+4*x**5-10*x**4-9)/x**4,x)

[Out]

x**2 + x*log(2*x) - 6*x + 3/(2*x**3)

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