3.67.67 \(\int \frac {-144 x^2-240 x^3-100 x^4+(192 x^2+460 x^3+250 x^4) \log (x)+(72 x^2+80 x^3) \log ^2(x)+(12 x+6 x^2) \log ^3(x)}{\log ^3(x)} \, dx\)

Optimal. Leaf size=29 \[ 2 x^2 \left (3+x \left (1+\frac {2 x}{\log (x)}+\frac {3 (2+x)}{\log (x)}\right )^2\right ) \]

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Rubi [A]  time = 0.86, antiderivative size = 56, normalized size of antiderivative = 1.93, number of steps used = 38, number of rules used = 10, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6741, 12, 6688, 6742, 43, 2353, 2306, 2309, 2178, 2356} \begin {gather*} \frac {50 x^5}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {20 x^4}{\log (x)}+2 x^3+\frac {72 x^3}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+6 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-144*x^2 - 240*x^3 - 100*x^4 + (192*x^2 + 460*x^3 + 250*x^4)*Log[x] + (72*x^2 + 80*x^3)*Log[x]^2 + (12*x
+ 6*x^2)*Log[x]^3)/Log[x]^3,x]

[Out]

6*x^2 + 2*x^3 + (72*x^3)/Log[x]^2 + (120*x^4)/Log[x]^2 + (50*x^5)/Log[x]^2 + (24*x^3)/Log[x] + (20*x^4)/Log[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 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 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 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] && LtQ[p, -1]

Rule 2309

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Polyx*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && PolynomialQ[Polyx, x]

Rule 6688

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

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 {2 x \left (-72 x-120 x^2-50 x^3+96 x \log (x)+230 x^2 \log (x)+125 x^3 \log (x)+36 x \log ^2(x)+40 x^2 \log ^2(x)+6 \log ^3(x)+3 x \log ^3(x)\right )}{\log ^3(x)} \, dx\\ &=2 \int \frac {x \left (-72 x-120 x^2-50 x^3+96 x \log (x)+230 x^2 \log (x)+125 x^3 \log (x)+36 x \log ^2(x)+40 x^2 \log ^2(x)+6 \log ^3(x)+3 x \log ^3(x)\right )}{\log ^3(x)} \, dx\\ &=2 \int \frac {x \left (-2 x (6+5 x)^2+x \left (96+230 x+125 x^2\right ) \log (x)+4 x (9+10 x) \log ^2(x)+3 (2+x) \log ^3(x)\right )}{\log ^3(x)} \, dx\\ &=2 \int \left (3 x (2+x)-\frac {2 x^2 (6+5 x)^2}{\log ^3(x)}+\frac {x^2 (6+5 x) (16+25 x)}{\log ^2(x)}+\frac {4 x^2 (9+10 x)}{\log (x)}\right ) \, dx\\ &=2 \int \frac {x^2 (6+5 x) (16+25 x)}{\log ^2(x)} \, dx-4 \int \frac {x^2 (6+5 x)^2}{\log ^3(x)} \, dx+6 \int x (2+x) \, dx+8 \int \frac {x^2 (9+10 x)}{\log (x)} \, dx\\ &=2 \int \left (\frac {96 x^2}{\log ^2(x)}+\frac {230 x^3}{\log ^2(x)}+\frac {125 x^4}{\log ^2(x)}\right ) \, dx-4 \int \left (\frac {36 x^2}{\log ^3(x)}+\frac {60 x^3}{\log ^3(x)}+\frac {25 x^4}{\log ^3(x)}\right ) \, dx+6 \int \left (2 x+x^2\right ) \, dx+8 \int \left (\frac {9 x^2}{\log (x)}+\frac {10 x^3}{\log (x)}\right ) \, dx\\ &=6 x^2+2 x^3+72 \int \frac {x^2}{\log (x)} \, dx+80 \int \frac {x^3}{\log (x)} \, dx-100 \int \frac {x^4}{\log ^3(x)} \, dx-144 \int \frac {x^2}{\log ^3(x)} \, dx+192 \int \frac {x^2}{\log ^2(x)} \, dx-240 \int \frac {x^3}{\log ^3(x)} \, dx+250 \int \frac {x^4}{\log ^2(x)} \, dx+460 \int \frac {x^3}{\log ^2(x)} \, dx\\ &=6 x^2+2 x^3+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}-\frac {192 x^3}{\log (x)}-\frac {460 x^4}{\log (x)}-\frac {250 x^5}{\log (x)}+72 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+80 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )-216 \int \frac {x^2}{\log ^2(x)} \, dx-250 \int \frac {x^4}{\log ^2(x)} \, dx-480 \int \frac {x^3}{\log ^2(x)} \, dx+576 \int \frac {x^2}{\log (x)} \, dx+1250 \int \frac {x^4}{\log (x)} \, dx+1840 \int \frac {x^3}{\log (x)} \, dx\\ &=6 x^2+2 x^3+72 \text {Ei}(3 \log (x))+80 \text {Ei}(4 \log (x))+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+\frac {20 x^4}{\log (x)}+576 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-648 \int \frac {x^2}{\log (x)} \, dx-1250 \int \frac {x^4}{\log (x)} \, dx+1250 \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )+1840 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )-1920 \int \frac {x^3}{\log (x)} \, dx\\ &=6 x^2+2 x^3+648 \text {Ei}(3 \log (x))+1920 \text {Ei}(4 \log (x))+1250 \text {Ei}(5 \log (x))+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+\frac {20 x^4}{\log (x)}-648 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-1250 \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )-1920 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\\ &=6 x^2+2 x^3+\frac {72 x^3}{\log ^2(x)}+\frac {120 x^4}{\log ^2(x)}+\frac {50 x^5}{\log ^2(x)}+\frac {24 x^3}{\log (x)}+\frac {20 x^4}{\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.22, size = 37, normalized size = 1.28 \begin {gather*} \frac {2 x^2 \left (x (6+5 x)^2+2 x (6+5 x) \log (x)+(3+x) \log ^2(x)\right )}{\log ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-144*x^2 - 240*x^3 - 100*x^4 + (192*x^2 + 460*x^3 + 250*x^4)*Log[x] + (72*x^2 + 80*x^3)*Log[x]^2 +
(12*x + 6*x^2)*Log[x]^3)/Log[x]^3,x]

[Out]

(2*x^2*(x*(6 + 5*x)^2 + 2*x*(6 + 5*x)*Log[x] + (3 + x)*Log[x]^2))/Log[x]^2

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fricas [A]  time = 0.51, size = 51, normalized size = 1.76 \begin {gather*} \frac {2 \, {\left (25 \, x^{5} + 60 \, x^{4} + 36 \, x^{3} + {\left (x^{3} + 3 \, x^{2}\right )} \log \relax (x)^{2} + 2 \, {\left (5 \, x^{4} + 6 \, x^{3}\right )} \log \relax (x)\right )}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+12*x)*log(x)^3+(80*x^3+72*x^2)*log(x)^2+(250*x^4+460*x^3+192*x^2)*log(x)-100*x^4-240*x^3-144
*x^2)/log(x)^3,x, algorithm="fricas")

[Out]

2*(25*x^5 + 60*x^4 + 36*x^3 + (x^3 + 3*x^2)*log(x)^2 + 2*(5*x^4 + 6*x^3)*log(x))/log(x)^2

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giac [A]  time = 0.20, size = 56, normalized size = 1.93 \begin {gather*} 2 \, x^{3} + \frac {50 \, x^{5}}{\log \relax (x)^{2}} + \frac {20 \, x^{4}}{\log \relax (x)} + 6 \, x^{2} + \frac {120 \, x^{4}}{\log \relax (x)^{2}} + \frac {24 \, x^{3}}{\log \relax (x)} + \frac {72 \, x^{3}}{\log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+12*x)*log(x)^3+(80*x^3+72*x^2)*log(x)^2+(250*x^4+460*x^3+192*x^2)*log(x)-100*x^4-240*x^3-144
*x^2)/log(x)^3,x, algorithm="giac")

[Out]

2*x^3 + 50*x^5/log(x)^2 + 20*x^4/log(x) + 6*x^2 + 120*x^4/log(x)^2 + 24*x^3/log(x) + 72*x^3/log(x)^2

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maple [A]  time = 0.02, size = 40, normalized size = 1.38

method	result	size
risch	\(2 x^{3}+6 x^{2}+\frac {2 x^{3} \left (25 x^{2}+10 x \ln \relax (x )+60 x +12 \ln \relax (x )+36\right )}{\ln \relax (x )^{2}}\)	\(40\)
norman	\(\frac {72 x^{3}+120 x^{4}+50 x^{5}+6 x^{2} \ln \relax (x )^{2}+24 x^{3} \ln \relax (x )+2 x^{3} \ln \relax (x )^{2}+20 x^{4} \ln \relax (x )}{\ln \relax (x )^{2}}\)	\(54\)
default	\(2 x^{3}+6 x^{2}+\frac {20 x^{4}}{\ln \relax (x )}+\frac {50 x^{5}}{\ln \relax (x )^{2}}+\frac {24 x^{3}}{\ln \relax (x )}+\frac {120 x^{4}}{\ln \relax (x )^{2}}+\frac {72 x^{3}}{\ln \relax (x )^{2}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((6*x^2+12*x)*ln(x)^3+(80*x^3+72*x^2)*ln(x)^2+(250*x^4+460*x^3+192*x^2)*ln(x)-100*x^4-240*x^3-144*x^2)/ln(
x)^3,x,method=_RETURNVERBOSE)

[Out]

2*x^3+6*x^2+2*x^3*(25*x^2+10*x*ln(x)+60*x+12*ln(x)+36)/ln(x)^2

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maxima [C]  time = 0.43, size = 73, normalized size = 2.52 \begin {gather*} 2 \, x^{3} + 6 \, x^{2} + 80 \, {\rm Ei}\left (4 \, \log \relax (x)\right ) + 72 \, {\rm Ei}\left (3 \, \log \relax (x)\right ) + 576 \, \Gamma \left (-1, -3 \, \log \relax (x)\right ) + 1840 \, \Gamma \left (-1, -4 \, \log \relax (x)\right ) + 1250 \, \Gamma \left (-1, -5 \, \log \relax (x)\right ) + 1296 \, \Gamma \left (-2, -3 \, \log \relax (x)\right ) + 3840 \, \Gamma \left (-2, -4 \, \log \relax (x)\right ) + 2500 \, \Gamma \left (-2, -5 \, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x^2+12*x)*log(x)^3+(80*x^3+72*x^2)*log(x)^2+(250*x^4+460*x^3+192*x^2)*log(x)-100*x^4-240*x^3-144
*x^2)/log(x)^3,x, algorithm="maxima")

[Out]

2*x^3 + 6*x^2 + 80*Ei(4*log(x)) + 72*Ei(3*log(x)) + 576*gamma(-1, -3*log(x)) + 1840*gamma(-1, -4*log(x)) + 125
0*gamma(-1, -5*log(x)) + 1296*gamma(-2, -3*log(x)) + 3840*gamma(-2, -4*log(x)) + 2500*gamma(-2, -5*log(x))

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mupad [B]  time = 4.25, size = 50, normalized size = 1.72 \begin {gather*} 2\,x^2\,\left (x+3\right )+\frac {2\,x^2\,\left (25\,x^3+60\,x^2+36\,x\right )+2\,x^2\,\ln \relax (x)\,\left (10\,x^2+12\,x\right )}{{\ln \relax (x)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^3*(12*x + 6*x^2) + log(x)*(192*x^2 + 460*x^3 + 250*x^4) + log(x)^2*(72*x^2 + 80*x^3) - 144*x^2 - 2
40*x^3 - 100*x^4)/log(x)^3,x)

[Out]

2*x^2*(x + 3) + (2*x^2*(36*x + 60*x^2 + 25*x^3) + 2*x^2*log(x)*(12*x + 10*x^2))/log(x)^2

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sympy [A]  time = 0.12, size = 42, normalized size = 1.45 \begin {gather*} 2 x^{3} + 6 x^{2} + \frac {50 x^{5} + 120 x^{4} + 72 x^{3} + \left (20 x^{4} + 24 x^{3}\right ) \log {\relax (x )}}{\log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((6*x**2+12*x)*ln(x)**3+(80*x**3+72*x**2)*ln(x)**2+(250*x**4+460*x**3+192*x**2)*ln(x)-100*x**4-240*x
**3-144*x**2)/ln(x)**3,x)

[Out]

2*x**3 + 6*x**2 + (50*x**5 + 120*x**4 + 72*x**3 + (20*x**4 + 24*x**3)*log(x))/log(x)**2

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