3.61.88 \(\int \frac {-4+5 x^2+8 x^3-3 x^4-4 x^5+(4+x^2) \log (4 x)}{4 x^2} \, dx\)

Optimal. Leaf size=27 \[ \left (1-\frac {x^2}{4}\right ) \left (x-\frac {-x^3+\log (4 x)}{x}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.33, number of steps used = 7, number of rules used = 3, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 14, 2334} \begin {gather*} -\frac {x^4}{4}-\frac {x^3}{4}+x^2+x-\frac {1}{4} \left (\frac {4}{x}-x\right ) \log (4 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + 5*x^2 + 8*x^3 - 3*x^4 - 4*x^5 + (4 + x^2)*Log[4*x])/(4*x^2),x]

[Out]

x + x^2 - x^3/4 - x^4/4 - ((4/x - x)*Log[4*x])/4

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 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]
] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {-4+5 x^2+8 x^3-3 x^4-4 x^5+\left (4+x^2\right ) \log (4 x)}{x^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {-4+5 x^2+8 x^3-3 x^4-4 x^5}{x^2}+\frac {\left (4+x^2\right ) \log (4 x)}{x^2}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-4+5 x^2+8 x^3-3 x^4-4 x^5}{x^2} \, dx+\frac {1}{4} \int \frac {\left (4+x^2\right ) \log (4 x)}{x^2} \, dx\\ &=-\frac {1}{4} \left (\frac {4}{x}-x\right ) \log (4 x)-\frac {1}{4} \int \left (1-\frac {4}{x^2}\right ) \, dx+\frac {1}{4} \int \left (5-\frac {4}{x^2}+8 x-3 x^2-4 x^3\right ) \, dx\\ &=x+x^2-\frac {x^3}{4}-\frac {x^4}{4}-\frac {1}{4} \left (\frac {4}{x}-x\right ) \log (4 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 37, normalized size = 1.37 \begin {gather*} x+x^2-\frac {x^3}{4}-\frac {x^4}{4}-\frac {\log (4 x)}{x}+\frac {1}{4} x \log (4 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 5*x^2 + 8*x^3 - 3*x^4 - 4*x^5 + (4 + x^2)*Log[4*x])/(4*x^2),x]

[Out]

x + x^2 - x^3/4 - x^4/4 - Log[4*x]/x + (x*Log[4*x])/4

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fricas [A]  time = 0.56, size = 33, normalized size = 1.22 \begin {gather*} -\frac {x^{5} + x^{4} - 4 \, x^{3} - 4 \, x^{2} - {\left (x^{2} - 4\right )} \log \left (4 \, x\right )}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x^2+4)*log(4*x)-4*x^5-3*x^4+8*x^3+5*x^2-4)/x^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 28, normalized size = 1.04 \begin {gather*} -\frac {1}{4} \, x^{4} - \frac {1}{4} \, x^{3} + x^{2} + \frac {1}{4} \, {\left (x - \frac {4}{x}\right )} \log \left (4 \, x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x^2+4)*log(4*x)-4*x^5-3*x^4+8*x^3+5*x^2-4)/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.04, size = 30, normalized size = 1.11

method	result	size
risch	\(\frac {\left (x^{2}-4\right ) \ln \left (4 x \right )}{4 x}-\frac {x^{4}}{4}-\frac {x^{3}}{4}+x^{2}+x\)	\(30\)
derivativedivides	\(-\frac {x^{4}}{4}-\frac {x^{3}}{4}+\frac {x \ln \left (4 x \right )}{4}+x +x^{2}-\frac {\ln \left (4 x \right )}{x}\)	\(32\)
default	\(-\frac {x^{4}}{4}-\frac {x^{3}}{4}+\frac {x \ln \left (4 x \right )}{4}+x +x^{2}-\frac {\ln \left (4 x \right )}{x}\)	\(32\)
norman	\(\frac {x^{2}+x^{3}-\frac {x^{4}}{4}-\frac {x^{5}}{4}+\frac {x^{2} \ln \left (4 x \right )}{4}-\ln \left (4 x \right )}{x}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((x^2+4)*ln(4*x)-4*x^5-3*x^4+8*x^3+5*x^2-4)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/4*(x^2-4)/x*ln(4*x)-1/4*x^4-1/4*x^3+x^2+x

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maxima [A]  time = 0.35, size = 31, normalized size = 1.15 \begin {gather*} -\frac {1}{4} \, x^{4} - \frac {1}{4} \, x^{3} + x^{2} + \frac {1}{4} \, x \log \left (4 \, x\right ) + x - \frac {\log \left (4 \, x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x^2+4)*log(4*x)-4*x^5-3*x^4+8*x^3+5*x^2-4)/x^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.44, size = 23, normalized size = 0.85 \begin {gather*} -\frac {\left (x^2-4\right )\,\left (x^2-\ln \left (4\,x\right )+x^3\right )}{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.12, size = 27, normalized size = 1.00 \begin {gather*} - \frac {x^{4}}{4} - \frac {x^{3}}{4} + x^{2} + x + \frac {\left (x^{2} - 4\right ) \log {\left (4 x \right )}}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((x**2+4)*ln(4*x)-4*x**5-3*x**4+8*x**3+5*x**2-4)/x**2,x)

[Out]

-x**4/4 - x**3/4 + x**2 + x + (x**2 - 4)*log(4*x)/(4*x)

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