∫ 1_ 9(7 − 90x2 − 180x3 + (−90x − 270x2) log(x) − 90x log2(x)) dx

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Rubi [B] time = 0.05, antiderivative size = 40, normalized size of antiderivative = 2.22, number of steps used = 9, number of rules used = 6, integrand size = 35,

\frac{number of rules}{integrand size}

= 0.171, Rules used = {12, 1593, 43, 2334, 2305, 2304}

\begin{matrix} - 5 x^{4} - 5 x^{2} \log^{2} (x) + 5 x^{2} \log (x) - 5 (2 x^{3} + x^{2}) \log (x) + \frac{7 x}{9} \end{matrix}

Int[(7 - 90*x^2 - 180*x^3 + (-90*x - 270*x^2)*Log[x] - 90*x*Log[x]^2)/9,x]

(7*x)/9 - 5*x^4 + 5*x^2*Log[x] - 5*(x^2 + 2*x^3)*Log[x] - 5*x^2*Log[x]^2

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d

*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]

&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F

reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^

n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo

g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{

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])

\begin{matrix} \begin{aligned} integral & = \frac{1}{9} \int (7 - 90 x^{2} - 180 x^{3} + (- 90 x - 270 x^{2}) \log (x) - 90 x \log^{2} (x)) d x \\ = \frac{7 x}{9} - \frac{10 x^{3}}{3} - 5 x^{4} + \frac{1}{9} \int (- 90 x - 270 x^{2}) \log (x) d x - 10 \int x \log^{2} (x) d x \\ = \frac{7 x}{9} - \frac{10 x^{3}}{3} - 5 x^{4} - 5 x^{2} \log^{2} (x) + \frac{1}{9} \int (- 90 - 270 x) x \log (x) d x + 10 \int x \log (x) d x \\ = \frac{7 x}{9} - \frac{5 x^{2}}{2} - \frac{10 x^{3}}{3} - 5 x^{4} + 5 x^{2} \log (x) - 5 (x^{2} + 2 x^{3}) \log (x) - 5 x^{2} \log^{2} (x) - \frac{1}{9} \int 45 (- 1 - 2 x) x d x \\ = \frac{7 x}{9} - \frac{5 x^{2}}{2} - \frac{10 x^{3}}{3} - 5 x^{4} + 5 x^{2} \log (x) - 5 (x^{2} + 2 x^{3}) \log (x) - 5 x^{2} \log^{2} (x) - 5 \int (- 1 - 2 x) x d x \\ = \frac{7 x}{9} - \frac{5 x^{2}}{2} - \frac{10 x^{3}}{3} - 5 x^{4} + 5 x^{2} \log (x) - 5 (x^{2} + 2 x^{3}) \log (x) - 5 x^{2} \log^{2} (x) - 5 \int (- x - 2 x^{2}) d x \\ = \frac{7 x}{9} - 5 x^{4} + 5 x^{2} \log (x) - 5 (x^{2} + 2 x^{3}) \log (x) - 5 x^{2} \log^{2} (x) \end{aligned} \end{matrix}

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Mathematica [A] time = 0.01, size = 27, normalized size = 1.50

\begin{matrix} \frac{7 x}{9} - 5 x^{4} - 10 x^{3} \log (x) - 5 x^{2} \log^{2} (x) \end{matrix}

Integrate[(7 - 90*x^2 - 180*x^3 + (-90*x - 270*x^2)*Log[x] - 90*x*Log[x]^2)/9,x]

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fricas [A] time = 0.57, size = 25, normalized size = 1.39

\begin{matrix} - 5 x^{4} - 10 x^{3} \log (x) - 5 x^{2} \log (x)^{2} + \frac{7}{9} x \end{matrix}

integrate(-10*x*log(x)^2+1/9*(-270*x^2-90*x)*log(x)-20*x^3-10*x^2+7/9,x, algorithm="fricas")

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giac [A] time = 0.14, size = 25, normalized size = 1.39

\begin{matrix} - 5 x^{4} - 10 x^{3} \log (x) - 5 x^{2} \log (x)^{2} + \frac{7}{9} x \end{matrix}

integrate(-10*x*log(x)^2+1/9*(-270*x^2-90*x)*log(x)-20*x^3-10*x^2+7/9,x, algorithm="giac")

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method	result	size

default	$\frac{7 x}{9} - 5 x^{4} - 5 x^{2} \ln (x)^{2} - 10 x^{3} \ln (x)$	$26$
norman	$\frac{7 x}{9} - 5 x^{4} - 5 x^{2} \ln (x)^{2} - 10 x^{3} \ln (x)$	$26$
risch	$\frac{7 x}{9} - 5 x^{4} - 5 x^{2} \ln (x)^{2} - 10 x^{3} \ln (x)$	$26$

int(-10*x*ln(x)^2+1/9*(-270*x^2-90*x)*ln(x)-20*x^3-10*x^2+7/9,x,method=_RETURNVERBOSE)

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maxima [B] time = 0.36, size = 44, normalized size = 2.44

\begin{matrix} - 5 x^{4} - \frac{5}{2} (2 \log (x)^{2} - 2 \log (x) + 1) x^{2} + \frac{5}{2} x^{2} - 5 (2 x^{3} + x^{2}) \log (x) + \frac{7}{9} x \end{matrix}

integrate(-10*x*log(x)^2+1/9*(-270*x^2-90*x)*log(x)-20*x^3-10*x^2+7/9,x, algorithm="maxima")

-5*x^4 - 5/2*(2*log(x)^2 - 2*log(x) + 1)*x^2 + 5/2*x^2 - 5*(2*x^3 + x^2)*log(x) + 7/9*x

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mupad [B] time = 5.14, size = 25, normalized size = 1.39

\begin{matrix} - 5 x^{4} - 10 x^{3} \ln (x) - 5 x^{2} {\ln (x)}^{2} + \frac{7 x}{9} \end{matrix}

int(7/9 - (log(x)*(90*x + 270*x^2))/9 - 10*x^2 - 20*x^3 - 10*x*log(x)^2,x)

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sympy [A] time = 0.12, size = 27, normalized size = 1.50

\begin{matrix} - 5 x^{4} - 10 x^{3} \log (x) - 5 x^{2} \log {(x)}^{2} + \frac{7 x}{9} \end{matrix}

integrate(-10*x*ln(x)**2+1/9*(-270*x**2-90*x)*ln(x)-20*x**3-10*x**2+7/9,x)

-5*x**4 - 10*x**3*log(x) - 5*x**2*log(x)**2 + 7*x/9

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3.89.37 ∫19(7−90x2−180x3+(−90x−270x2)log⁡(x)−90xlog2⁡(x))dx

3.89.37 $\int \frac{1}{9} (7 - 90 x^{2} - 180 x^{3} + (- 90 x - 270 x^{2}) \log (x) - 90 x \log^{2} (x)) d x$