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

Optimal. Leaf size=18 \[ -6+\frac {7 x}{9}-5 x^2 (x+\log (x))^2 \]

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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {12, 1593, 43, 2334, 2305, 2304} \begin {gather*} -5 x^4-5 x^2 \log ^2(x)+5 x^2 \log (x)-5 \left (2 x^3+x^2\right ) \log (x)+\frac {7 x}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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

Rule 12

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

Rule 43

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])

Rule 1593

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]

Rule 2304

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
]

Rule 2305

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[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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}{9} \int \left (7-90 x^2-180 x^3+\left (-90 x-270 x^2\right ) \log (x)-90 x \log ^2(x)\right ) \, dx\\ &=\frac {7 x}{9}-\frac {10 x^3}{3}-5 x^4+\frac {1}{9} \int \left (-90 x-270 x^2\right ) \log (x) \, dx-10 \int x \log ^2(x) \, dx\\ &=\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) \, dx+10 \int x \log (x) \, dx\\ &=\frac {7 x}{9}-\frac {5 x^2}{2}-\frac {10 x^3}{3}-5 x^4+5 x^2 \log (x)-5 \left (x^2+2 x^3\right ) \log (x)-5 x^2 \log ^2(x)-\frac {1}{9} \int 45 (-1-2 x) x \, dx\\ &=\frac {7 x}{9}-\frac {5 x^2}{2}-\frac {10 x^3}{3}-5 x^4+5 x^2 \log (x)-5 \left (x^2+2 x^3\right ) \log (x)-5 x^2 \log ^2(x)-5 \int (-1-2 x) x \, dx\\ &=\frac {7 x}{9}-\frac {5 x^2}{2}-\frac {10 x^3}{3}-5 x^4+5 x^2 \log (x)-5 \left (x^2+2 x^3\right ) \log (x)-5 x^2 \log ^2(x)-5 \int \left (-x-2 x^2\right ) \, dx\\ &=\frac {7 x}{9}-5 x^4+5 x^2 \log (x)-5 \left (x^2+2 x^3\right ) \log (x)-5 x^2 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.50 \begin {gather*} \frac {7 x}{9}-5 x^4-10 x^3 \log (x)-5 x^2 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*x)/9 - 5*x^4 - 10*x^3*Log[x] - 5*x^2*Log[x]^2

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fricas [A]  time = 0.57, size = 25, normalized size = 1.39 \begin {gather*} -5 \, x^{4} - 10 \, x^{3} \log \relax (x) - 5 \, x^{2} \log \relax (x)^{2} + \frac {7}{9} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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

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giac [A]  time = 0.14, size = 25, normalized size = 1.39 \begin {gather*} -5 \, x^{4} - 10 \, x^{3} \log \relax (x) - 5 \, x^{2} \log \relax (x)^{2} + \frac {7}{9} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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

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maple [A]  time = 0.02, size = 26, normalized size = 1.44

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

7/9*x-5*x^4-5*x^2*ln(x)^2-10*x^3*ln(x)

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maxima [B]  time = 0.36, size = 44, normalized size = 2.44 \begin {gather*} -5 \, x^{4} - \frac {5}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} + \frac {5}{2} \, x^{2} - 5 \, {\left (2 \, x^{3} + x^{2}\right )} \log \relax (x) + \frac {7}{9} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

-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 {gather*} -5\,x^4-10\,x^3\,\ln \relax (x)-5\,x^2\,{\ln \relax (x)}^2+\frac {7\,x}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.12, size = 27, normalized size = 1.50 \begin {gather*} - 5 x^{4} - 10 x^{3} \log {\relax (x )} - 5 x^{2} \log {\relax (x )}^{2} + \frac {7 x}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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