3.22.23 \(\int \frac {432 x^2-492 x^3+552 x^4+168 x^5+(576 x^2-984 x^3-252 x^4) \log (x^2)+(432 x^2+84 x^3) \log ^2(x^2)}{576+336 x+49 x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac {x^3 \left (-1-\left (-x+\log \left (x^2\right )\right )^2\right )}{-4-\frac {7 x}{6}} \]

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Rubi [B]  time = 0.79, antiderivative size = 127, normalized size of antiderivative = 4.70, number of steps used = 31, number of rules used = 15, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {27, 6741, 12, 6742, 43, 2357, 2295, 2304, 2314, 31, 2317, 2391, 2296, 2305, 2318} \begin {gather*} \frac {6 x^4}{7}-\frac {144 x^3}{49}+\frac {3750 x^2}{343}+\frac {6}{7} x^2 \log ^2\left (x^2\right )+\frac {3456 x \log ^2\left (x^2\right )}{49 (7 x+24)}-\frac {144}{49} x \log ^2\left (x^2\right )+\frac {288}{49} x^2 \log \left (x^2\right )+\frac {165888 x \log \left (x^2\right )}{343 (7 x+24)}-\frac {6912}{343} x \log \left (x^2\right )-\frac {12}{7} x^3 \log \left (x^2\right )-\frac {90000 x}{2401}-\frac {51840000}{16807 (7 x+24)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(432*x^2 - 492*x^3 + 552*x^4 + 168*x^5 + (576*x^2 - 984*x^3 - 252*x^4)*Log[x^2] + (432*x^2 + 84*x^3)*Log[x
^2]^2)/(576 + 336*x + 49*x^2),x]

[Out]

(-90000*x)/2401 + (3750*x^2)/343 - (144*x^3)/49 + (6*x^4)/7 - 51840000/(16807*(24 + 7*x)) - (6912*x*Log[x^2])/
343 + (288*x^2*Log[x^2])/49 - (12*x^3*Log[x^2])/7 + (165888*x*Log[x^2])/(343*(24 + 7*x)) - (144*x*Log[x^2]^2)/
49 + (6*x^2*Log[x^2]^2)/7 + (3456*x*Log[x^2]^2)/(49*(24 + 7*x))

Rule 12

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, 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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*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 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {432 x^2-492 x^3+552 x^4+168 x^5+\left (576 x^2-984 x^3-252 x^4\right ) \log \left (x^2\right )+\left (432 x^2+84 x^3\right ) \log ^2\left (x^2\right )}{(24+7 x)^2} \, dx\\ &=\int \frac {12 x^2 \left (36-41 x+46 x^2+14 x^3+48 \log \left (x^2\right )-82 x \log \left (x^2\right )-21 x^2 \log \left (x^2\right )+36 \log ^2\left (x^2\right )+7 x \log ^2\left (x^2\right )\right )}{(24+7 x)^2} \, dx\\ &=12 \int \frac {x^2 \left (36-41 x+46 x^2+14 x^3+48 \log \left (x^2\right )-82 x \log \left (x^2\right )-21 x^2 \log \left (x^2\right )+36 \log ^2\left (x^2\right )+7 x \log ^2\left (x^2\right )\right )}{(24+7 x)^2} \, dx\\ &=12 \int \left (\frac {36 x^2}{(24+7 x)^2}-\frac {41 x^3}{(24+7 x)^2}+\frac {46 x^4}{(24+7 x)^2}+\frac {14 x^5}{(24+7 x)^2}-\frac {x^2 \left (-48+82 x+21 x^2\right ) \log \left (x^2\right )}{(24+7 x)^2}+\frac {x^2 (36+7 x) \log ^2\left (x^2\right )}{(24+7 x)^2}\right ) \, dx\\ &=-\left (12 \int \frac {x^2 \left (-48+82 x+21 x^2\right ) \log \left (x^2\right )}{(24+7 x)^2} \, dx\right )+12 \int \frac {x^2 (36+7 x) \log ^2\left (x^2\right )}{(24+7 x)^2} \, dx+168 \int \frac {x^5}{(24+7 x)^2} \, dx+432 \int \frac {x^2}{(24+7 x)^2} \, dx-492 \int \frac {x^3}{(24+7 x)^2} \, dx+552 \int \frac {x^4}{(24+7 x)^2} \, dx\\ &=-\left (12 \int \left (\frac {912 \log \left (x^2\right )}{343}-\frac {62}{49} x \log \left (x^2\right )+\frac {3}{7} x^2 \log \left (x^2\right )-\frac {331776 \log \left (x^2\right )}{343 (24+7 x)^2}-\frac {1152 \log \left (x^2\right )}{49 (24+7 x)}\right ) \, dx\right )+12 \int \left (-\frac {12}{49} \log ^2\left (x^2\right )+\frac {1}{7} x \log ^2\left (x^2\right )+\frac {6912 \log ^2\left (x^2\right )}{49 (24+7 x)^2}\right ) \, dx+168 \int \left (-\frac {55296}{16807}+\frac {1728 x}{2401}-\frac {48 x^2}{343}+\frac {x^3}{49}-\frac {7962624}{16807 (24+7 x)^2}+\frac {1658880}{16807 (24+7 x)}\right ) \, dx+432 \int \left (\frac {1}{49}+\frac {576}{49 (24+7 x)^2}-\frac {48}{49 (24+7 x)}\right ) \, dx-492 \int \left (-\frac {48}{343}+\frac {x}{49}-\frac {13824}{343 (24+7 x)^2}+\frac {1728}{343 (24+7 x)}\right ) \, dx+552 \int \left (\frac {1728}{2401}-\frac {48 x}{343}+\frac {x^2}{49}+\frac {331776}{2401 (24+7 x)^2}-\frac {55296}{2401 (24+7 x)}\right ) \, dx\\ &=-\frac {186768 x}{2401}+\frac {5766 x^2}{343}-\frac {200 x^3}{49}+\frac {6 x^4}{7}-\frac {51840000}{16807 (24+7 x)}+\frac {331776 \log (24+7 x)}{2401}+\frac {12}{7} \int x \log ^2\left (x^2\right ) \, dx-\frac {144}{49} \int \log ^2\left (x^2\right ) \, dx-\frac {36}{7} \int x^2 \log \left (x^2\right ) \, dx+\frac {744}{49} \int x \log \left (x^2\right ) \, dx-\frac {10944}{343} \int \log \left (x^2\right ) \, dx+\frac {13824}{49} \int \frac {\log \left (x^2\right )}{24+7 x} \, dx+\frac {82944}{49} \int \frac {\log ^2\left (x^2\right )}{(24+7 x)^2} \, dx+\frac {3981312}{343} \int \frac {\log \left (x^2\right )}{(24+7 x)^2} \, dx\\ &=-\frac {33552 x}{2401}+\frac {3162 x^2}{343}-\frac {144 x^3}{49}+\frac {6 x^4}{7}-\frac {51840000}{16807 (24+7 x)}-\frac {10944}{343} x \log \left (x^2\right )+\frac {372}{49} x^2 \log \left (x^2\right )-\frac {12}{7} x^3 \log \left (x^2\right )+\frac {165888 x \log \left (x^2\right )}{343 (24+7 x)}+\frac {13824}{343} \log \left (1+\frac {7 x}{24}\right ) \log \left (x^2\right )-\frac {144}{49} x \log ^2\left (x^2\right )+\frac {6}{7} x^2 \log ^2\left (x^2\right )+\frac {3456 x \log ^2\left (x^2\right )}{49 (24+7 x)}+\frac {331776 \log (24+7 x)}{2401}-\frac {24}{7} \int x \log \left (x^2\right ) \, dx+\frac {576}{49} \int \log \left (x^2\right ) \, dx-\frac {27648}{343} \int \frac {\log \left (1+\frac {7 x}{24}\right )}{x} \, dx-\frac {13824}{49} \int \frac {\log \left (x^2\right )}{24+7 x} \, dx-\frac {331776}{343} \int \frac {1}{24+7 x} \, dx\\ &=-\frac {90000 x}{2401}+\frac {3750 x^2}{343}-\frac {144 x^3}{49}+\frac {6 x^4}{7}-\frac {51840000}{16807 (24+7 x)}-\frac {6912}{343} x \log \left (x^2\right )+\frac {288}{49} x^2 \log \left (x^2\right )-\frac {12}{7} x^3 \log \left (x^2\right )+\frac {165888 x \log \left (x^2\right )}{343 (24+7 x)}-\frac {144}{49} x \log ^2\left (x^2\right )+\frac {6}{7} x^2 \log ^2\left (x^2\right )+\frac {3456 x \log ^2\left (x^2\right )}{49 (24+7 x)}+\frac {27648}{343} \text {Li}_2\left (-\frac {7 x}{24}\right )+\frac {27648}{343} \int \frac {\log \left (1+\frac {7 x}{24}\right )}{x} \, dx\\ &=-\frac {90000 x}{2401}+\frac {3750 x^2}{343}-\frac {144 x^3}{49}+\frac {6 x^4}{7}-\frac {51840000}{16807 (24+7 x)}-\frac {6912}{343} x \log \left (x^2\right )+\frac {288}{49} x^2 \log \left (x^2\right )-\frac {12}{7} x^3 \log \left (x^2\right )+\frac {165888 x \log \left (x^2\right )}{343 (24+7 x)}-\frac {144}{49} x \log ^2\left (x^2\right )+\frac {6}{7} x^2 \log ^2\left (x^2\right )+\frac {3456 x \log ^2\left (x^2\right )}{49 (24+7 x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.14, size = 71, normalized size = 2.63 \begin {gather*} \frac {6 \left (-8640000-2520000 x+16807 x^3+16807 x^5-9289728 \log (24)-2709504 x \log (24)+387072 (24+7 x) \log (x)-14 \left (331776+96768 x+2401 x^4\right ) \log \left (x^2\right )+16807 x^3 \log ^2\left (x^2\right )\right )}{16807 (24+7 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(432*x^2 - 492*x^3 + 552*x^4 + 168*x^5 + (576*x^2 - 984*x^3 - 252*x^4)*Log[x^2] + (432*x^2 + 84*x^3)
*Log[x^2]^2)/(576 + 336*x + 49*x^2),x]

[Out]

(6*(-8640000 - 2520000*x + 16807*x^3 + 16807*x^5 - 9289728*Log[24] - 2709504*x*Log[24] + 387072*(24 + 7*x)*Log
[x] - 14*(331776 + 96768*x + 2401*x^4)*Log[x^2] + 16807*x^3*Log[x^2]^2))/(16807*(24 + 7*x))

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fricas [A]  time = 1.43, size = 44, normalized size = 1.63 \begin {gather*} \frac {6 \, {\left (16807 \, x^{5} - 33614 \, x^{4} \log \left (x^{2}\right ) + 16807 \, x^{3} \log \left (x^{2}\right )^{2} + 16807 \, x^{3} - 2520000 \, x - 8640000\right )}}{16807 \, {\left (7 \, x + 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((84*x^3+432*x^2)*log(x^2)^2+(-252*x^4-984*x^3+576*x^2)*log(x^2)+168*x^5+552*x^4-492*x^3+432*x^2)/(4
9*x^2+336*x+576),x, algorithm="fricas")

[Out]

6/16807*(16807*x^5 - 33614*x^4*log(x^2) + 16807*x^3*log(x^2)^2 + 16807*x^3 - 2520000*x - 8640000)/(7*x + 24)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {12 \, {\left (14 \, x^{5} + 46 \, x^{4} - 41 \, x^{3} + {\left (7 \, x^{3} + 36 \, x^{2}\right )} \log \left (x^{2}\right )^{2} + 36 \, x^{2} - {\left (21 \, x^{4} + 82 \, x^{3} - 48 \, x^{2}\right )} \log \left (x^{2}\right )\right )}}{49 \, x^{2} + 336 \, x + 576}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((84*x^3+432*x^2)*log(x^2)^2+(-252*x^4-984*x^3+576*x^2)*log(x^2)+168*x^5+552*x^4-492*x^3+432*x^2)/(4
9*x^2+336*x+576),x, algorithm="giac")

[Out]

integrate(12*(14*x^5 + 46*x^4 - 41*x^3 + (7*x^3 + 36*x^2)*log(x^2)^2 + 36*x^2 - (21*x^4 + 82*x^3 - 48*x^2)*log
(x^2))/(49*x^2 + 336*x + 576), x)

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maple [A]  time = 0.45, size = 40, normalized size = 1.48

method	result	size
norman	\(\frac {6 x^{3}+6 x^{5}+6 x^{3} \ln \left (x^{2}\right )^{2}-12 x^{4} \ln \left (x^{2}\right )}{7 x +24}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((84*x^3+432*x^2)*ln(x^2)^2+(-252*x^4-984*x^3+576*x^2)*ln(x^2)+168*x^5+552*x^4-492*x^3+432*x^2)/(49*x^2+33
6*x+576),x,method=_RETURNVERBOSE)

[Out]

(6*x^3+6*x^5+6*x^3*ln(x^2)^2-12*x^4*ln(x^2))/(7*x+24)

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maxima [B]  time = 0.57, size = 83, normalized size = 3.07 \begin {gather*} \frac {6}{7} \, x^{4} - \frac {200}{49} \, x^{3} + \frac {5766}{343} \, x^{2} - \frac {186768}{2401} \, x + \frac {8 \, {\left (7203 \, x^{3} \log \relax (x)^{2} + 2401 \, x^{4} - 4116 \, x^{3} + 42336 \, x^{2} - 3 \, {\left (2401 \, x^{4} + 96768 \, x + 331776\right )} \log \relax (x) + 290304 \, x\right )}}{2401 \, {\left (7 \, x + 24\right )}} - \frac {51840000}{16807 \, {\left (7 \, x + 24\right )}} + \frac {331776}{2401} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((84*x^3+432*x^2)*log(x^2)^2+(-252*x^4-984*x^3+576*x^2)*log(x^2)+168*x^5+552*x^4-492*x^3+432*x^2)/(4
9*x^2+336*x+576),x, algorithm="maxima")

[Out]

6/7*x^4 - 200/49*x^3 + 5766/343*x^2 - 186768/2401*x + 8/2401*(7203*x^3*log(x)^2 + 2401*x^4 - 4116*x^3 + 42336*
x^2 - 3*(2401*x^4 + 96768*x + 331776)*log(x) + 290304*x)/(7*x + 24) - 51840000/16807/(7*x + 24) + 331776/2401*
log(x)

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mupad [B]  time = 1.24, size = 39, normalized size = 1.44 \begin {gather*} \frac {6\,x^5-12\,x^4\,\ln \left (x^2\right )+6\,x^3\,{\ln \left (x^2\right )}^2+6\,x^3}{7\,x+24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^2)^2*(432*x^2 + 84*x^3) - log(x^2)*(984*x^3 - 576*x^2 + 252*x^4) + 432*x^2 - 492*x^3 + 552*x^4 + 16
8*x^5)/(336*x + 49*x^2 + 576),x)

[Out]

(6*x^3 - 12*x^4*log(x^2) + 6*x^5 + 6*x^3*log(x^2)^2)/(7*x + 24)

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sympy [B]  time = 0.33, size = 76, normalized size = 2.81 \begin {gather*} \frac {6 x^{4}}{7} - \frac {144 x^{3}}{49} + \frac {6 x^{3} \log {\left (x^{2} \right )}^{2}}{7 x + 24} + \frac {3750 x^{2}}{343} - \frac {90000 x}{2401} + \frac {331776 \log {\relax (x )}}{2401} - \frac {51840000}{117649 x + 403368} + \frac {\left (- 28812 x^{4} - 1161216 x - 3981312\right ) \log {\left (x^{2} \right )}}{16807 x + 57624} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((84*x**3+432*x**2)*ln(x**2)**2+(-252*x**4-984*x**3+576*x**2)*ln(x**2)+168*x**5+552*x**4-492*x**3+43
2*x**2)/(49*x**2+336*x+576),x)

[Out]

6*x**4/7 - 144*x**3/49 + 6*x**3*log(x**2)**2/(7*x + 24) + 3750*x**2/343 - 90000*x/2401 + 331776*log(x)/2401 -
51840000/(117649*x + 403368) + (-28812*x**4 - 1161216*x - 3981312)*log(x**2)/(16807*x + 57624)

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