3.54.71 \(\int \frac {-4 x^7+8 x^7 \log (4 x)+(-4 x^4-20 x^5) \log ^2(4 x)+(10 x^4+60 x^5) \log ^3(4 x)+(2 x+30 x^2+100 x^3) \log ^5(4 x)}{\log ^5(4 x)} \, dx\)

Optimal. Leaf size=21 \[ x^2 \left (1+5 x+\frac {x^3}{\log ^2(4 x)}\right )^2 \]

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Rubi [B]  time = 0.68, antiderivative size = 46, normalized size of antiderivative = 2.19, number of steps used = 33, number of rules used = 6, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.076, Rules used = {6742, 14, 2306, 2309, 2178, 2353} \begin {gather*} \frac {x^8}{\log ^4(4 x)}+\frac {10 x^6}{\log ^2(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+25 x^4+10 x^3+x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^7 + 8*x^7*Log[4*x] + (-4*x^4 - 20*x^5)*Log[4*x]^2 + (10*x^4 + 60*x^5)*Log[4*x]^3 + (2*x + 30*x^2 + 1
00*x^3)*Log[4*x]^5)/Log[4*x]^5,x]

[Out]

x^2 + 10*x^3 + 25*x^4 + x^8/Log[4*x]^4 + (2*x^5)/Log[4*x]^2 + (10*x^6)/Log[4*x]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 x \left (1+15 x+50 x^2\right )-\frac {4 x^7}{\log ^5(4 x)}+\frac {8 x^7}{\log ^4(4 x)}-\frac {4 x^4 (1+5 x)}{\log ^3(4 x)}+\frac {10 x^4 (1+6 x)}{\log ^2(4 x)}\right ) \, dx\\ &=2 \int x \left (1+15 x+50 x^2\right ) \, dx-4 \int \frac {x^7}{\log ^5(4 x)} \, dx-4 \int \frac {x^4 (1+5 x)}{\log ^3(4 x)} \, dx+8 \int \frac {x^7}{\log ^4(4 x)} \, dx+10 \int \frac {x^4 (1+6 x)}{\log ^2(4 x)} \, dx\\ &=\frac {x^8}{\log ^4(4 x)}-\frac {8 x^8}{3 \log ^3(4 x)}+2 \int \left (x+15 x^2+50 x^3\right ) \, dx-4 \int \left (\frac {x^4}{\log ^3(4 x)}+\frac {5 x^5}{\log ^3(4 x)}\right ) \, dx-8 \int \frac {x^7}{\log ^4(4 x)} \, dx+10 \int \left (\frac {x^4}{\log ^2(4 x)}+\frac {6 x^5}{\log ^2(4 x)}\right ) \, dx+\frac {64}{3} \int \frac {x^7}{\log ^3(4 x)} \, dx\\ &=x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}-\frac {32 x^8}{3 \log ^2(4 x)}-4 \int \frac {x^4}{\log ^3(4 x)} \, dx+10 \int \frac {x^4}{\log ^2(4 x)} \, dx-20 \int \frac {x^5}{\log ^3(4 x)} \, dx-\frac {64}{3} \int \frac {x^7}{\log ^3(4 x)} \, dx+60 \int \frac {x^5}{\log ^2(4 x)} \, dx+\frac {256}{3} \int \frac {x^7}{\log ^2(4 x)} \, dx\\ &=x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)}-\frac {10 x^5}{\log (4 x)}-\frac {60 x^6}{\log (4 x)}-\frac {256 x^8}{3 \log (4 x)}-10 \int \frac {x^4}{\log ^2(4 x)} \, dx+50 \int \frac {x^4}{\log (4 x)} \, dx-60 \int \frac {x^5}{\log ^2(4 x)} \, dx-\frac {256}{3} \int \frac {x^7}{\log ^2(4 x)} \, dx+360 \int \frac {x^5}{\log (4 x)} \, dx+\frac {2048}{3} \int \frac {x^7}{\log (4 x)} \, dx\\ &=x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)}+\frac {1}{96} \operatorname {Subst}\left (\int \frac {e^{8 x}}{x} \, dx,x,\log (4 x)\right )+\frac {25}{512} \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (4 x)\right )+\frac {45}{512} \operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (4 x)\right )-50 \int \frac {x^4}{\log (4 x)} \, dx-360 \int \frac {x^5}{\log (4 x)} \, dx-\frac {2048}{3} \int \frac {x^7}{\log (4 x)} \, dx\\ &=x^2+10 x^3+25 x^4+\frac {25}{512} \text {Ei}(5 \log (4 x))+\frac {45}{512} \text {Ei}(6 \log (4 x))+\frac {1}{96} \text {Ei}(8 \log (4 x))+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)}-\frac {1}{96} \operatorname {Subst}\left (\int \frac {e^{8 x}}{x} \, dx,x,\log (4 x)\right )-\frac {25}{512} \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (4 x)\right )-\frac {45}{512} \operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (4 x)\right )\\ &=x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.19, size = 46, normalized size = 2.19 \begin {gather*} x^2+10 x^3+25 x^4+\frac {x^8}{\log ^4(4 x)}+\frac {2 x^5}{\log ^2(4 x)}+\frac {10 x^6}{\log ^2(4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^7 + 8*x^7*Log[4*x] + (-4*x^4 - 20*x^5)*Log[4*x]^2 + (10*x^4 + 60*x^5)*Log[4*x]^3 + (2*x + 30*x
^2 + 100*x^3)*Log[4*x]^5)/Log[4*x]^5,x]

[Out]

x^2 + 10*x^3 + 25*x^4 + x^8/Log[4*x]^4 + (2*x^5)/Log[4*x]^2 + (10*x^6)/Log[4*x]^2

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fricas [B]  time = 0.79, size = 49, normalized size = 2.33 \begin {gather*} \frac {x^{8} + {\left (25 \, x^{4} + 10 \, x^{3} + x^{2}\right )} \log \left (4 \, x\right )^{4} + 2 \, {\left (5 \, x^{6} + x^{5}\right )} \log \left (4 \, x\right )^{2}}{\log \left (4 \, x\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^3+30*x^2+2*x)*log(4*x)^5+(60*x^5+10*x^4)*log(4*x)^3+(-20*x^5-4*x^4)*log(4*x)^2+8*x^7*log(4*x
)-4*x^7)/log(4*x)^5,x, algorithm="fricas")

[Out]

(x^8 + (25*x^4 + 10*x^3 + x^2)*log(4*x)^4 + 2*(5*x^6 + x^5)*log(4*x)^2)/log(4*x)^4

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giac [B]  time = 0.22, size = 46, normalized size = 2.19 \begin {gather*} 25 \, x^{4} + \frac {x^{8}}{\log \left (4 \, x\right )^{4}} + \frac {10 \, x^{6}}{\log \left (4 \, x\right )^{2}} + 10 \, x^{3} + \frac {2 \, x^{5}}{\log \left (4 \, x\right )^{2}} + x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^3+30*x^2+2*x)*log(4*x)^5+(60*x^5+10*x^4)*log(4*x)^3+(-20*x^5-4*x^4)*log(4*x)^2+8*x^7*log(4*x
)-4*x^7)/log(4*x)^5,x, algorithm="giac")

[Out]

25*x^4 + x^8/log(4*x)^4 + 10*x^6/log(4*x)^2 + 10*x^3 + 2*x^5/log(4*x)^2 + x^2

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maple [B]  time = 0.06, size = 46, normalized size = 2.19

method	result	size
risch	\(25 x^{4}+10 x^{3}+x^{2}+\frac {x^{5} \left (x^{3}+10 x \ln \left (4 x \right )^{2}+2 \ln \left (4 x \right )^{2}\right )}{\ln \left (4 x \right )^{4}}\)	\(46\)
derivativedivides	\(25 x^{4}+10 x^{3}+\frac {10 x^{6}}{\ln \left (4 x \right )^{2}}+\frac {x^{8}}{\ln \left (4 x \right )^{4}}+x^{2}+\frac {2 x^{5}}{\ln \left (4 x \right )^{2}}\)	\(47\)
default	\(25 x^{4}+10 x^{3}+\frac {10 x^{6}}{\ln \left (4 x \right )^{2}}+\frac {x^{8}}{\ln \left (4 x \right )^{4}}+x^{2}+\frac {2 x^{5}}{\ln \left (4 x \right )^{2}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((100*x^3+30*x^2+2*x)*ln(4*x)^5+(60*x^5+10*x^4)*ln(4*x)^3+(-20*x^5-4*x^4)*ln(4*x)^2+8*x^7*ln(4*x)-4*x^7)/l
n(4*x)^5,x,method=_RETURNVERBOSE)

[Out]

25*x^4+10*x^3+x^2+x^5*(x^3+10*x*ln(4*x)^2+2*ln(4*x)^2)/ln(4*x)^4

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maxima [C]  time = 0.41, size = 74, normalized size = 3.52 \begin {gather*} 25 \, x^{4} + 10 \, x^{3} + x^{2} + \frac {25}{512} \, \Gamma \left (-1, -5 \, \log \left (4 \, x\right )\right ) + \frac {45}{512} \, \Gamma \left (-1, -6 \, \log \left (4 \, x\right )\right ) + \frac {25}{256} \, \Gamma \left (-2, -5 \, \log \left (4 \, x\right )\right ) + \frac {45}{256} \, \Gamma \left (-2, -6 \, \log \left (4 \, x\right )\right ) + \frac {1}{16} \, \Gamma \left (-3, -8 \, \log \left (4 \, x\right )\right ) + \frac {1}{4} \, \Gamma \left (-4, -8 \, \log \left (4 \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x^3+30*x^2+2*x)*log(4*x)^5+(60*x^5+10*x^4)*log(4*x)^3+(-20*x^5-4*x^4)*log(4*x)^2+8*x^7*log(4*x
)-4*x^7)/log(4*x)^5,x, algorithm="maxima")

[Out]

25*x^4 + 10*x^3 + x^2 + 25/512*gamma(-1, -5*log(4*x)) + 45/512*gamma(-1, -6*log(4*x)) + 25/256*gamma(-2, -5*lo
g(4*x)) + 45/256*gamma(-2, -6*log(4*x)) + 1/16*gamma(-3, -8*log(4*x)) + 1/4*gamma(-4, -8*log(4*x))

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mupad [B]  time = 3.58, size = 39, normalized size = 1.86 \begin {gather*} \frac {x^8+2\,x^5\,{\ln \left (4\,x\right )}^2\,\left (5\,x+1\right )}{{\ln \left (4\,x\right )}^4}+x^2\,{\left (5\,x+1\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((8*x^7*log(4*x) + log(4*x)^5*(2*x + 30*x^2 + 100*x^3) - log(4*x)^2*(4*x^4 + 20*x^5) + log(4*x)^3*(10*x^4 +
60*x^5) - 4*x^7)/log(4*x)^5,x)

[Out]

(x^8 + 2*x^5*log(4*x)^2*(5*x + 1))/log(4*x)^4 + x^2*(5*x + 1)^2

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sympy [B]  time = 0.17, size = 39, normalized size = 1.86 \begin {gather*} 25 x^{4} + 10 x^{3} + x^{2} + \frac {x^{8} + \left (10 x^{6} + 2 x^{5}\right ) \log {\left (4 x \right )}^{2}}{\log {\left (4 x \right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((100*x**3+30*x**2+2*x)*ln(4*x)**5+(60*x**5+10*x**4)*ln(4*x)**3+(-20*x**5-4*x**4)*ln(4*x)**2+8*x**7*
ln(4*x)-4*x**7)/ln(4*x)**5,x)

[Out]

25*x**4 + 10*x**3 + x**2 + (x**8 + (10*x**6 + 2*x**5)*log(4*x)**2)/log(4*x)**4

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