3.10.47 \(\int \frac {2112 x^2-1664 x^3+320 x^4+(1280 x^2-512 x^3) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx\)

Optimal. Leaf size=22 \[ \frac {16}{625} \left (2+\frac {4}{3} x^3 (3-x+\log (x))^2\right ) \]

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Rubi [B]  time = 0.07, antiderivative size = 59, normalized size of antiderivative = 2.68, number of steps used = 9, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {12, 1593, 43, 2334, 2305, 2304} \begin {gather*} \frac {64 x^5}{1875}-\frac {128 x^4}{625}+\frac {192 x^3}{625}+\frac {64 x^3 \log ^2(x)}{1875}-\frac {128 x^3 \log (x)}{5625}+\frac {128 \left (10 x^3-3 x^4\right ) \log (x)}{5625} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2112*x^2 - 1664*x^3 + 320*x^4 + (1280*x^2 - 512*x^3)*Log[x] + 192*x^2*Log[x]^2)/1875,x]

[Out]

(192*x^3)/625 - (128*x^4)/625 + (64*x^5)/1875 - (128*x^3*Log[x])/5625 + (128*(10*x^3 - 3*x^4)*Log[x])/5625 + (
64*x^3*Log[x]^2)/1875

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 {\int \left (2112 x^2-1664 x^3+320 x^4+\left (1280 x^2-512 x^3\right ) \log (x)+192 x^2 \log ^2(x)\right ) \, dx}{1875}\\ &=\frac {704 x^3}{1875}-\frac {416 x^4}{1875}+\frac {64 x^5}{1875}+\frac {\int \left (1280 x^2-512 x^3\right ) \log (x) \, dx}{1875}+\frac {64}{625} \int x^2 \log ^2(x) \, dx\\ &=\frac {704 x^3}{1875}-\frac {416 x^4}{1875}+\frac {64 x^5}{1875}+\frac {64 x^3 \log ^2(x)}{1875}+\frac {\int (1280-512 x) x^2 \log (x) \, dx}{1875}-\frac {128 \int x^2 \log (x) \, dx}{1875}\\ &=\frac {6464 x^3}{16875}-\frac {416 x^4}{1875}+\frac {64 x^5}{1875}-\frac {128 x^3 \log (x)}{5625}+\frac {128 \left (10 x^3-3 x^4\right ) \log (x)}{5625}+\frac {64 x^3 \log ^2(x)}{1875}-\frac {\int \frac {128}{3} (10-3 x) x^2 \, dx}{1875}\\ &=\frac {6464 x^3}{16875}-\frac {416 x^4}{1875}+\frac {64 x^5}{1875}-\frac {128 x^3 \log (x)}{5625}+\frac {128 \left (10 x^3-3 x^4\right ) \log (x)}{5625}+\frac {64 x^3 \log ^2(x)}{1875}-\frac {128 \int (10-3 x) x^2 \, dx}{5625}\\ &=\frac {6464 x^3}{16875}-\frac {416 x^4}{1875}+\frac {64 x^5}{1875}-\frac {128 x^3 \log (x)}{5625}+\frac {128 \left (10 x^3-3 x^4\right ) \log (x)}{5625}+\frac {64 x^3 \log ^2(x)}{1875}-\frac {128 \int \left (10 x^2-3 x^3\right ) \, dx}{5625}\\ &=\frac {192 x^3}{625}-\frac {128 x^4}{625}+\frac {64 x^5}{1875}-\frac {128 x^3 \log (x)}{5625}+\frac {128 \left (10 x^3-3 x^4\right ) \log (x)}{5625}+\frac {64 x^3 \log ^2(x)}{1875}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 51, normalized size = 2.32 \begin {gather*} \frac {192 x^3}{625}-\frac {128 x^4}{625}+\frac {64 x^5}{1875}+\frac {128}{625} x^3 \log (x)-\frac {128 x^4 \log (x)}{1875}+\frac {64 x^3 \log ^2(x)}{1875} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2112*x^2 - 1664*x^3 + 320*x^4 + (1280*x^2 - 512*x^3)*Log[x] + 192*x^2*Log[x]^2)/1875,x]

[Out]

(192*x^3)/625 - (128*x^4)/625 + (64*x^5)/1875 + (128*x^3*Log[x])/625 - (128*x^4*Log[x])/1875 + (64*x^3*Log[x]^
2)/1875

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fricas [B]  time = 1.26, size = 38, normalized size = 1.73 \begin {gather*} \frac {64}{1875} \, x^{5} + \frac {64}{1875} \, x^{3} \log \relax (x)^{2} - \frac {128}{625} \, x^{4} + \frac {192}{625} \, x^{3} - \frac {128}{1875} \, {\left (x^{4} - 3 \, x^{3}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(64/625*x^2*log(x)^2+1/1875*(-512*x^3+1280*x^2)*log(x)+64/375*x^4-1664/1875*x^3+704/625*x^2,x, algori
thm="fricas")

[Out]

64/1875*x^5 + 64/1875*x^3*log(x)^2 - 128/625*x^4 + 192/625*x^3 - 128/1875*(x^4 - 3*x^3)*log(x)

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giac [B]  time = 0.42, size = 39, normalized size = 1.77 \begin {gather*} \frac {64}{1875} \, x^{5} - \frac {128}{1875} \, x^{4} \log \relax (x) + \frac {64}{1875} \, x^{3} \log \relax (x)^{2} - \frac {128}{625} \, x^{4} + \frac {128}{625} \, x^{3} \log \relax (x) + \frac {192}{625} \, x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(64/625*x^2*log(x)^2+1/1875*(-512*x^3+1280*x^2)*log(x)+64/375*x^4-1664/1875*x^3+704/625*x^2,x, algori
thm="giac")

[Out]

64/1875*x^5 - 128/1875*x^4*log(x) + 64/1875*x^3*log(x)^2 - 128/625*x^4 + 128/625*x^3*log(x) + 192/625*x^3

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maple [B]  time = 0.02, size = 40, normalized size = 1.82

method	result	size
default	\(\frac {192 x^{3}}{625}-\frac {128 x^{4}}{625}+\frac {64 x^{5}}{1875}+\frac {64 x^{3} \ln \relax (x )^{2}}{1875}+\frac {128 x^{3} \ln \relax (x )}{625}-\frac {128 x^{4} \ln \relax (x )}{1875}\)	\(40\)
norman	\(\frac {192 x^{3}}{625}-\frac {128 x^{4}}{625}+\frac {64 x^{5}}{1875}+\frac {64 x^{3} \ln \relax (x )^{2}}{1875}+\frac {128 x^{3} \ln \relax (x )}{625}-\frac {128 x^{4} \ln \relax (x )}{1875}\)	\(40\)
risch	\(\frac {192 x^{3}}{625}-\frac {128 x^{4}}{625}+\frac {64 x^{5}}{1875}+\frac {64 x^{3} \ln \relax (x )^{2}}{1875}+\frac {128 x^{3} \ln \relax (x )}{625}-\frac {128 x^{4} \ln \relax (x )}{1875}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(64/625*x^2*ln(x)^2+1/1875*(-512*x^3+1280*x^2)*ln(x)+64/375*x^4-1664/1875*x^3+704/625*x^2,x,method=_RETURNV
ERBOSE)

[Out]

192/625*x^3-128/625*x^4+64/1875*x^5+64/1875*x^3*ln(x)^2+128/625*x^3*ln(x)-128/1875*x^4*ln(x)

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maxima [B]  time = 0.38, size = 48, normalized size = 2.18 \begin {gather*} \frac {64}{1875} \, x^{5} + \frac {64}{16875} \, {\left (9 \, \log \relax (x)^{2} - 6 \, \log \relax (x) + 2\right )} x^{3} - \frac {128}{625} \, x^{4} + \frac {5056}{16875} \, x^{3} - \frac {128}{5625} \, {\left (3 \, x^{4} - 10 \, x^{3}\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(64/625*x^2*log(x)^2+1/1875*(-512*x^3+1280*x^2)*log(x)+64/375*x^4-1664/1875*x^3+704/625*x^2,x, algori
thm="maxima")

[Out]

64/1875*x^5 + 64/16875*(9*log(x)^2 - 6*log(x) + 2)*x^3 - 128/625*x^4 + 5056/16875*x^3 - 128/5625*(3*x^4 - 10*x
^3)*log(x)

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mupad [B]  time = 0.76, size = 14, normalized size = 0.64 \begin {gather*} \frac {64\,x^3\,{\left (\ln \relax (x)-x+3\right )}^2}{1875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(1280*x^2 - 512*x^3))/1875 + (64*x^2*log(x)^2)/625 + (704*x^2)/625 - (1664*x^3)/1875 + (64*x^4)/37
5,x)

[Out]

(64*x^3*(log(x) - x + 3)^2)/1875

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sympy [B]  time = 0.14, size = 48, normalized size = 2.18 \begin {gather*} \frac {64 x^{5}}{1875} - \frac {128 x^{4}}{625} + \frac {64 x^{3} \log {\relax (x )}^{2}}{1875} + \frac {192 x^{3}}{625} + \left (- \frac {128 x^{4}}{1875} + \frac {128 x^{3}}{625}\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(64/625*x**2*ln(x)**2+1/1875*(-512*x**3+1280*x**2)*ln(x)+64/375*x**4-1664/1875*x**3+704/625*x**2,x)

[Out]

64*x**5/1875 - 128*x**4/625 + 64*x**3*log(x)**2/1875 + 192*x**3/625 + (-128*x**4/1875 + 128*x**3/625)*log(x)

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