\(\int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx\) [794]

Optimal result

Integrand size = 17, antiderivative size = 14 \[ \int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx=\frac {2}{3} (-4-x) x (7+\log (x)) \]

[Out]

-2/3*(ln(x)+7)*x*(4+x)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(36\) vs. \(2(14)=28\).

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.57, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {12, 2350, 9} \[ \int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx=-5 x^2-\frac {2}{3} x^2 \log (x)-\frac {64 x}{3}+\frac {1}{3} (x+4)^2-\frac {8}{3} x \log (x) \]

[In]

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

[Out]

(-64*x)/3 - 5*x^2 + (4 + x)^2/3 - (8*x*Log[x])/3 - (2*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 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = IntHide[(d +
e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, 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{align*} \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 {8}{3} x \log (x)-\frac {2}{3} x^2 \log (x)-\frac {1}{3} \int 2 (-4-x) \, dx \\ & = -\frac {64 x}{3}-5 x^2+\frac {1}{3} (4+x)^2-\frac {8}{3} x \log (x)-\frac {2}{3} x^2 \log (x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(29\) vs. \(2(14)=28\).

Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 2.07 \[ \int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx=-\frac {56 x}{3}-\frac {14 x^2}{3}-\frac {8}{3} x \log (x)-\frac {2}{3} x^2 \log (x) \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(21\) vs. \(2(10)=20\).

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx=-\frac {14}{3} \, x^{2} - \frac {2}{3} \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right ) - \frac {56}{3} \, x \]

[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

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx=- \frac {14 x^{2}}{3} - \frac {56 x}{3} + \left (- \frac {2 x^{2}}{3} - \frac {8 x}{3}\right ) \log {\left (x \right )} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.43 \[ \int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx=-\frac {14}{3} \, x^{2} - \frac {2}{3} \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right ) - \frac {56}{3} \, x \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (10) = 20\).

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx=-\frac {2}{3} \, x^{2} \log \left (x\right ) - \frac {14}{3} \, x^{2} - \frac {8}{3} \, x \log \left (x\right ) - \frac {56}{3} \, x \]

[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

Mupad [B] (verification not implemented)

Time = 7.90 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \frac {1}{3} (-64-30 x+(-8-4 x) \log (x)) \, dx=-\frac {2\,x\,\left (\ln \left (x\right )+7\right )\,\left (x+4\right )}{3} \]

[In]

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

[Out]

-(2*x*(log(x) + 7)*(x + 4))/3