3.50.66 \(\int \frac {1}{10} (-10+6 x^3-7 x^3 \log (x)+2 x^3 \log ^2(x)) \, dx\)

Optimal. Leaf size=24 \[ -x+\frac {1}{5} x^2 \left (-x+\frac {1}{2} x \log (x)\right )^2 \]

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Rubi [A]  time = 0.03, antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2304, 2305} \begin {gather*} \frac {x^4}{5}+\frac {1}{20} x^4 \log ^2(x)-\frac {1}{5} x^4 \log (x)-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10 + 6*x^3 - 7*x^3*Log[x] + 2*x^3*Log[x]^2)/10,x]

[Out]

-x + x^4/5 - (x^4*Log[x])/5 + (x^4*Log[x]^2)/20

Rule 12

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

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{10} \int \left (-10+6 x^3-7 x^3 \log (x)+2 x^3 \log ^2(x)\right ) \, dx\\ &=-x+\frac {3 x^4}{20}+\frac {1}{5} \int x^3 \log ^2(x) \, dx-\frac {7}{10} \int x^3 \log (x) \, dx\\ &=-x+\frac {31 x^4}{160}-\frac {7}{40} x^4 \log (x)+\frac {1}{20} x^4 \log ^2(x)-\frac {1}{10} \int x^3 \log (x) \, dx\\ &=-x+\frac {x^4}{5}-\frac {1}{5} x^4 \log (x)+\frac {1}{20} x^4 \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 31, normalized size = 1.29 \begin {gather*} -x+\frac {x^4}{5}-\frac {1}{5} x^4 \log (x)+\frac {1}{20} x^4 \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 + 6*x^3 - 7*x^3*Log[x] + 2*x^3*Log[x]^2)/10,x]

[Out]

-x + x^4/5 - (x^4*Log[x])/5 + (x^4*Log[x]^2)/20

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fricas [A]  time = 0.78, size = 25, normalized size = 1.04 \begin {gather*} \frac {1}{20} \, x^{4} \log \relax (x)^{2} - \frac {1}{5} \, x^{4} \log \relax (x) + \frac {1}{5} \, x^{4} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*x^3*log(x)^2-7/10*x^3*log(x)+3/5*x^3-1,x, algorithm="fricas")

[Out]

1/20*x^4*log(x)^2 - 1/5*x^4*log(x) + 1/5*x^4 - x

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giac [A]  time = 0.14, size = 25, normalized size = 1.04 \begin {gather*} \frac {1}{20} \, x^{4} \log \relax (x)^{2} - \frac {1}{5} \, x^{4} \log \relax (x) + \frac {1}{5} \, x^{4} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*x^3*log(x)^2-7/10*x^3*log(x)+3/5*x^3-1,x, algorithm="giac")

[Out]

1/20*x^4*log(x)^2 - 1/5*x^4*log(x) + 1/5*x^4 - x

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maple [A]  time = 0.02, size = 26, normalized size = 1.08

method	result	size
default	\(-x +\frac {x^{4}}{5}-\frac {x^{4} \ln \relax (x )}{5}+\frac {x^{4} \ln \relax (x )^{2}}{20}\)	\(26\)
norman	\(-x +\frac {x^{4}}{5}-\frac {x^{4} \ln \relax (x )}{5}+\frac {x^{4} \ln \relax (x )^{2}}{20}\)	\(26\)
risch	\(-x +\frac {x^{4}}{5}-\frac {x^{4} \ln \relax (x )}{5}+\frac {x^{4} \ln \relax (x )^{2}}{20}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*x^3*ln(x)^2-7/10*x^3*ln(x)+3/5*x^3-1,x,method=_RETURNVERBOSE)

[Out]

-x+1/5*x^4-1/5*x^4*ln(x)+1/20*x^4*ln(x)^2

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maxima [A]  time = 0.50, size = 33, normalized size = 1.38 \begin {gather*} \frac {1}{160} \, {\left (8 \, \log \relax (x)^{2} - 4 \, \log \relax (x) + 1\right )} x^{4} - \frac {7}{40} \, x^{4} \log \relax (x) + \frac {31}{160} \, x^{4} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*x^3*log(x)^2-7/10*x^3*log(x)+3/5*x^3-1,x, algorithm="maxima")

[Out]

1/160*(8*log(x)^2 - 4*log(x) + 1)*x^4 - 7/40*x^4*log(x) + 31/160*x^4 - x

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mupad [B]  time = 3.22, size = 25, normalized size = 1.04 \begin {gather*} \frac {x^4\,{\ln \relax (x)}^2}{20}-\frac {x^4\,\ln \relax (x)}{5}+\frac {x^4}{5}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*log(x)^2)/5 - (7*x^3*log(x))/10 + (3*x^3)/5 - 1,x)

[Out]

(x^4*log(x)^2)/20 - (x^4*log(x))/5 - x + x^4/5

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sympy [A]  time = 0.11, size = 24, normalized size = 1.00 \begin {gather*} \frac {x^{4} \log {\relax (x )}^{2}}{20} - \frac {x^{4} \log {\relax (x )}}{5} + \frac {x^{4}}{5} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*x**3*ln(x)**2-7/10*x**3*ln(x)+3/5*x**3-1,x)

[Out]

x**4*log(x)**2/20 - x**4*log(x)/5 + x**4/5 - x

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