3.8.94 \(\int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx\)

Optimal. Leaf size=14 \[ \frac {2}{3} (-4-x) x (7+\log (x)) \]

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Rubi [B]  time = 0.01, antiderivative size = 33, normalized size of antiderivative = 2.36, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {12, 2313, 9} \begin {gather*} -5 x^2-\frac {2}{3} \left (x^2+4 x\right ) \log (x)-\frac {64 x}{3}+\frac {1}{3} (x+4)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-64 - 30*x + (-8 - 4*x)*Log[x])/3,x]

[Out]

(-64*x)/3 - 5*x^2 + (4 + x)^2/3 - (2*(4*x + x^2)*Log[x])/3

Rule 9

Int[(a_)*((b_) + (c_.)*(x_)), x_Symbol] :> Simp[(a*(b + c*x)^2)/(2*c), x] /; FreeQ[{a, b, c}, 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 2313

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{3} \int (-64-30 x+(-8-4 x) \log (x)) \, dx\\ &=-\frac {64 x}{3}-5 x^2+\frac {1}{3} \int (-8-4 x) \log (x) \, dx\\ &=-\frac {64 x}{3}-5 x^2-\frac {2}{3} \left (4 x+x^2\right ) \log (x)-\frac {1}{3} \int 2 (-4-x) \, dx\\ &=-\frac {64 x}{3}-5 x^2+\frac {1}{3} (4+x)^2-\frac {2}{3} \left (4 x+x^2\right ) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.00, size = 29, normalized size = 2.07 \begin {gather*} -\frac {56 x}{3}-\frac {14 x^2}{3}-\frac {8}{3} x \log (x)-\frac {2}{3} x^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-64 - 30*x + (-8 - 4*x)*Log[x])/3,x]

[Out]

(-56*x)/3 - (14*x^2)/3 - (8*x*Log[x])/3 - (2*x^2*Log[x])/3

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fricas [A]  time = 0.77, size = 20, normalized size = 1.43 \begin {gather*} -\frac {14}{3} \, x^{2} - \frac {2}{3} \, {\left (x^{2} + 4 \, x\right )} \log \relax (x) - \frac {56}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-4*x-8)*log(x)-10*x-64/3,x, algorithm="fricas")

[Out]

-14/3*x^2 - 2/3*(x^2 + 4*x)*log(x) - 56/3*x

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giac [B]  time = 0.31, size = 21, normalized size = 1.50 \begin {gather*} -\frac {2}{3} \, x^{2} \log \relax (x) - \frac {14}{3} \, x^{2} - \frac {8}{3} \, x \log \relax (x) - \frac {56}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-4*x-8)*log(x)-10*x-64/3,x, algorithm="giac")

[Out]

-2/3*x^2*log(x) - 14/3*x^2 - 8/3*x*log(x) - 56/3*x

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maple [B]  time = 0.02, size = 22, normalized size = 1.57

method	result	size
default	\(-\frac {56 x}{3}-\frac {14 x^{2}}{3}-\frac {8 x \ln \relax (x )}{3}-\frac {2 x^{2} \ln \relax (x )}{3}\)	\(22\)
norman	\(-\frac {56 x}{3}-\frac {14 x^{2}}{3}-\frac {8 x \ln \relax (x )}{3}-\frac {2 x^{2} \ln \relax (x )}{3}\)	\(22\)
risch	\(\frac {\left (-2 x^{2}-8 x \right ) \ln \relax (x )}{3}-\frac {14 x^{2}}{3}-\frac {56 x}{3}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3*(-4*x-8)*ln(x)-10*x-64/3,x,method=_RETURNVERBOSE)

[Out]

-56/3*x-14/3*x^2-8/3*x*ln(x)-2/3*x^2*ln(x)

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maxima [A]  time = 0.52, size = 20, normalized size = 1.43 \begin {gather*} -\frac {14}{3} \, x^{2} - \frac {2}{3} \, {\left (x^{2} + 4 \, x\right )} \log \relax (x) - \frac {56}{3} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-4*x-8)*log(x)-10*x-64/3,x, algorithm="maxima")

[Out]

-14/3*x^2 - 2/3*(x^2 + 4*x)*log(x) - 56/3*x

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mupad [B]  time = 0.64, size = 10, normalized size = 0.71 \begin {gather*} -\frac {2\,x\,\left (\ln \relax (x)+7\right )\,\left (x+4\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- 10*x - (log(x)*(4*x + 8))/3 - 64/3,x)

[Out]

-(2*x*(log(x) + 7)*(x + 4))/3

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sympy [A]  time = 0.10, size = 27, normalized size = 1.93 \begin {gather*} - \frac {14 x^{2}}{3} - \frac {56 x}{3} + \left (- \frac {2 x^{2}}{3} - \frac {8 x}{3}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3*(-4*x-8)*ln(x)-10*x-64/3,x)

[Out]

-14*x**2/3 - 56*x/3 + (-2*x**2/3 - 8*x/3)*log(x)

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