\(\int \frac {2112 x^2-1664 x^3+320 x^4+(1280 x^2-512 x^3) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx\) [947]

Optimal result

Integrand size = 43, antiderivative size = 22 \[ \int \frac {2112 x^2-1664 x^3+320 x^4+\left (1280 x^2-512 x^3\right ) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx=\frac {16}{625} \left (2+\frac {4}{3} x^3 (3-x+\log (x))^2\right ) \]

[Out]

64/1875*x^3*(3-x+ln(x))^2+32/625

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(22)=44\).

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.68, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {12, 1607, 45, 2371, 2342, 2341} \[ \int \frac {2112 x^2-1664 x^3+320 x^4+\left (1280 x^2-512 x^3\right ) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx=\frac {64 x^5}{1875}-\frac {128 x^4}{625}+\frac {192 x^3}{625}+\frac {64 x^3 \log ^2(x)}{1875}-\frac {128 x^3 \log (x)}{5625}+\frac {128 \left (10 x^3-3 x^4\right ) \log (x)}{5625} \]

[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 45

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 1607

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 2341

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 2342

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 2371

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] && IGtQ[m, 0]

Rubi steps \begin{align*} \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{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).

Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \frac {2112 x^2-1664 x^3+320 x^4+\left (1280 x^2-512 x^3\right ) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx=\frac {192 x^3}{625}-\frac {128 x^4}{625}+\frac {64 x^5}{1875}+\frac {128}{625} x^3 \log (x)-\frac {128 x^4 \log (x)}{1875}+\frac {64 x^3 \log ^2(x)}{1875} \]

[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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(16)=32\).

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.82

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

[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]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (16) = 32\).

Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {2112 x^2-1664 x^3+320 x^4+\left (1280 x^2-512 x^3\right ) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx=\frac {64}{1875} \, x^{5} + \frac {64}{1875} \, x^{3} \log \left (x\right )^{2} - \frac {128}{625} \, x^{4} + \frac {192}{625} \, x^{3} - \frac {128}{1875} \, {\left (x^{4} - 3 \, x^{3}\right )} \log \left (x\right ) \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (17) = 34\).

Time = 0.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {2112 x^2-1664 x^3+320 x^4+\left (1280 x^2-512 x^3\right ) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx=\frac {64 x^{5}}{1875} - \frac {128 x^{4}}{625} + \frac {64 x^{3} \log {\left (x \right )}^{2}}{1875} + \frac {192 x^{3}}{625} + \left (- \frac {128 x^{4}}{1875} + \frac {128 x^{3}}{625}\right ) \log {\left (x \right )} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (16) = 32\).

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {2112 x^2-1664 x^3+320 x^4+\left (1280 x^2-512 x^3\right ) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx=\frac {64}{1875} \, x^{5} + \frac {64}{16875} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} - \frac {128}{625} \, x^{4} + \frac {5056}{16875} \, x^{3} - \frac {128}{5625} \, {\left (3 \, x^{4} - 10 \, x^{3}\right )} \log \left (x\right ) \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (16) = 32\).

Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.77 \[ \int \frac {2112 x^2-1664 x^3+320 x^4+\left (1280 x^2-512 x^3\right ) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx=\frac {64}{1875} \, x^{5} - \frac {128}{1875} \, x^{4} \log \left (x\right ) + \frac {64}{1875} \, x^{3} \log \left (x\right )^{2} - \frac {128}{625} \, x^{4} + \frac {128}{625} \, x^{3} \log \left (x\right ) + \frac {192}{625} \, x^{3} \]

[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

Mupad [B] (verification not implemented)

Time = 8.81 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \frac {2112 x^2-1664 x^3+320 x^4+\left (1280 x^2-512 x^3\right ) \log (x)+192 x^2 \log ^2(x)}{1875} \, dx=\frac {64\,x^3\,{\left (\ln \left (x\right )-x+3\right )}^2}{1875} \]

[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