3.47.67 \(\int \frac {(-157464-96228 e-9396 e^2-332 e^3-4 e^4) \log (2)+(-87480-50220 e-3360 e^2-60 e^3) \log (2) \log (x)+(-16200-8700 e-300 e^2) \log (2) \log ^2(x)+(-1000-500 e) \log (2) \log ^3(x)}{3125 x+3125 x \log (x)+1250 x \log ^2(x)+250 x \log ^3(x)+25 x \log ^4(x)+x \log ^5(x)} \, dx\)

Optimal. Leaf size=17 \[ \log (2) \left (5+\frac {2+e}{5+\log (x)}\right )^4 \]

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Rubi [B]  time = 0.21, antiderivative size = 58, normalized size of antiderivative = 3.41, number of steps used = 10, number of rules used = 3, integrand size = 112, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6, 12, 1850} \begin {gather*} \frac {500 (2+e) \log (2)}{\log (x)+5}+\frac {150 (2+e)^2 \log (2)}{(\log (x)+5)^2}+\frac {20 (2+e)^3 \log (2)}{(\log (x)+5)^3}+\frac {(2+e)^4 \log (2)}{(\log (x)+5)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-157464 - 96228*E - 9396*E^2 - 332*E^3 - 4*E^4)*Log[2] + (-87480 - 50220*E - 3360*E^2 - 60*E^3)*Log[2]*L
og[x] + (-16200 - 8700*E - 300*E^2)*Log[2]*Log[x]^2 + (-1000 - 500*E)*Log[2]*Log[x]^3)/(3125*x + 3125*x*Log[x]
+ 1250*x*Log[x]^2 + 250*x*Log[x]^3 + 25*x*Log[x]^4 + x*Log[x]^5),x]

[Out]

((2 + E)^4*Log[2])/(5 + Log[x])^4 + (20*(2 + E)^3*Log[2])/(5 + Log[x])^3 + (150*(2 + E)^2*Log[2])/(5 + Log[x])
^2 + (500*(2 + E)*Log[2])/(5 + Log[x])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, 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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int -\frac {4 \left (39366 \log (2)+24057 e \log (2)+2349 e^2 \log (2)+83 e^3 \log (2)+e^4 \log (2)+21870 x \log (2)+12555 e x \log (2)+840 e^2 x \log (2)+15 e^3 x \log (2)+4050 x^2 \log (2)+2175 e x^2 \log (2)+75 e^2 x^2 \log (2)+250 x^3 \log (2)+125 e x^3 \log (2)\right )}{(5+x)^5} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int -\frac {4 \left (39366 \log (2)+24057 e \log (2)+2349 e^2 \log (2)+83 e^3 \log (2)+e^4 \log (2)+840 e^2 x \log (2)+15 e^3 x \log (2)+(21870+12555 e) x \log (2)+4050 x^2 \log (2)+2175 e x^2 \log (2)+75 e^2 x^2 \log (2)+250 x^3 \log (2)+125 e x^3 \log (2)\right )}{(5+x)^5} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int -\frac {4 \left (39366 \log (2)+24057 e \log (2)+2349 e^2 \log (2)+83 e^3 \log (2)+e^4 \log (2)+(21870+12555 e) x \log (2)+\left (840 e^2+15 e^3\right ) x \log (2)+4050 x^2 \log (2)+2175 e x^2 \log (2)+75 e^2 x^2 \log (2)+250 x^3 \log (2)+125 e x^3 \log (2)\right )}{(5+x)^5} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int -\frac {4 \left (39366 \log (2)+24057 e \log (2)+2349 e^2 \log (2)+83 e^3 \log (2)+e^4 \log (2)+\left (21870+12555 e+840 e^2+15 e^3\right ) x \log (2)+4050 x^2 \log (2)+2175 e x^2 \log (2)+75 e^2 x^2 \log (2)+250 x^3 \log (2)+125 e x^3 \log (2)\right )}{(5+x)^5} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int -\frac {4 \left (39366 \log (2)+24057 e \log (2)+2349 e^2 \log (2)+83 e^3 \log (2)+e^4 \log (2)+\left (21870+12555 e+840 e^2+15 e^3\right ) x \log (2)+75 e^2 x^2 \log (2)+(4050+2175 e) x^2 \log (2)+250 x^3 \log (2)+125 e x^3 \log (2)\right )}{(5+x)^5} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int -\frac {4 \left (39366 \log (2)+24057 e \log (2)+2349 e^2 \log (2)+83 e^3 \log (2)+e^4 \log (2)+\left (21870+12555 e+840 e^2+15 e^3\right ) x \log (2)+\left (4050+2175 e+75 e^2\right ) x^2 \log (2)+250 x^3 \log (2)+125 e x^3 \log (2)\right )}{(5+x)^5} \, dx,x,\log (x)\right )\\ &=\operatorname {Subst}\left (\int -\frac {4 \left (39366 \log (2)+24057 e \log (2)+2349 e^2 \log (2)+83 e^3 \log (2)+e^4 \log (2)+\left (21870+12555 e+840 e^2+15 e^3\right ) x \log (2)+\left (4050+2175 e+75 e^2\right ) x^2 \log (2)+(250+125 e) x^3 \log (2)\right )}{(5+x)^5} \, dx,x,\log (x)\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {39366 \log (2)+24057 e \log (2)+2349 e^2 \log (2)+83 e^3 \log (2)+e^4 \log (2)+\left (21870+12555 e+840 e^2+15 e^3\right ) x \log (2)+\left (4050+2175 e+75 e^2\right ) x^2 \log (2)+(250+125 e) x^3 \log (2)}{(5+x)^5} \, dx,x,\log (x)\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \left (\frac {(2+e)^4 \log (2)}{(5+x)^5}+\frac {15 (2+e)^3 \log (2)}{(5+x)^4}+\frac {75 (2+e)^2 \log (2)}{(5+x)^3}+\frac {125 (2+e) \log (2)}{(5+x)^2}\right ) \, dx,x,\log (x)\right )\right )\\ &=\frac {(2+e)^4 \log (2)}{(5+\log (x))^4}+\frac {20 (2+e)^3 \log (2)}{(5+\log (x))^3}+\frac {150 (2+e)^2 \log (2)}{(5+\log (x))^2}+\frac {500 (2+e) \log (2)}{5+\log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.12, size = 48, normalized size = 2.82 \begin {gather*} \frac {(2+e) \log (2) \left ((2+e)^3+20 (2+e)^2 (5+\log (x))+150 (2+e) (5+\log (x))^2+500 (5+\log (x))^3\right )}{(5+\log (x))^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-157464 - 96228*E - 9396*E^2 - 332*E^3 - 4*E^4)*Log[2] + (-87480 - 50220*E - 3360*E^2 - 60*E^3)*Lo
g[2]*Log[x] + (-16200 - 8700*E - 300*E^2)*Log[2]*Log[x]^2 + (-1000 - 500*E)*Log[2]*Log[x]^3)/(3125*x + 3125*x*
Log[x] + 1250*x*Log[x]^2 + 250*x*Log[x]^3 + 25*x*Log[x]^4 + x*Log[x]^5),x]

[Out]

((2 + E)*Log[2]*((2 + E)^3 + 20*(2 + E)^2*(5 + Log[x]) + 150*(2 + E)*(5 + Log[x])^2 + 500*(5 + Log[x])^3))/(5
+ Log[x])^4

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fricas [B]  time = 0.84, size = 91, normalized size = 5.35 \begin {gather*} \frac {500 \, {\left (e + 2\right )} \log \relax (2) \log \relax (x)^{3} + 150 \, {\left (e^{2} + 54 \, e + 104\right )} \log \relax (2) \log \relax (x)^{2} + 20 \, {\left (e^{3} + 81 \, e^{2} + 2187 \, e + 4058\right )} \log \relax (2) \log \relax (x) + {\left (e^{4} + 108 \, e^{3} + 4374 \, e^{2} + 78732 \, e + 140816\right )} \log \relax (2)}{\log \relax (x)^{4} + 20 \, \log \relax (x)^{3} + 150 \, \log \relax (x)^{2} + 500 \, \log \relax (x) + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-500*exp(1)-1000)*log(2)*log(x)^3+(-300*exp(1)^2-8700*exp(1)-16200)*log(2)*log(x)^2+(-60*exp(1)^3-
3360*exp(1)^2-50220*exp(1)-87480)*log(2)*log(x)+(-4*exp(1)^4-332*exp(1)^3-9396*exp(1)^2-96228*exp(1)-157464)*l
og(2))/(x*log(x)^5+25*x*log(x)^4+250*x*log(x)^3+1250*x*log(x)^2+3125*x*log(x)+3125*x),x, algorithm="fricas")

[Out]

(500*(e + 2)*log(2)*log(x)^3 + 150*(e^2 + 54*e + 104)*log(2)*log(x)^2 + 20*(e^3 + 81*e^2 + 2187*e + 4058)*log(
2)*log(x) + (e^4 + 108*e^3 + 4374*e^2 + 78732*e + 140816)*log(2))/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500
*log(x) + 625)

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giac [B]  time = 0.17, size = 111, normalized size = 6.53 \begin {gather*} \frac {500 \, e \log \relax (2) \log \relax (x)^{3} + 150 \, e^{2} \log \relax (2) \log \relax (x)^{2} + 8100 \, e \log \relax (2) \log \relax (x)^{2} + 1000 \, \log \relax (2) \log \relax (x)^{3} + 20 \, e^{3} \log \relax (2) \log \relax (x) + 1620 \, e^{2} \log \relax (2) \log \relax (x) + 43740 \, e \log \relax (2) \log \relax (x) + 15600 \, \log \relax (2) \log \relax (x)^{2} + e^{4} \log \relax (2) + 108 \, e^{3} \log \relax (2) + 4374 \, e^{2} \log \relax (2) + 78732 \, e \log \relax (2) + 81160 \, \log \relax (2) \log \relax (x) + 140816 \, \log \relax (2)}{{\left (\log \relax (x) + 5\right )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-500*exp(1)-1000)*log(2)*log(x)^3+(-300*exp(1)^2-8700*exp(1)-16200)*log(2)*log(x)^2+(-60*exp(1)^3-
3360*exp(1)^2-50220*exp(1)-87480)*log(2)*log(x)+(-4*exp(1)^4-332*exp(1)^3-9396*exp(1)^2-96228*exp(1)-157464)*l
og(2))/(x*log(x)^5+25*x*log(x)^4+250*x*log(x)^3+1250*x*log(x)^2+3125*x*log(x)+3125*x),x, algorithm="giac")

[Out]

(500*e*log(2)*log(x)^3 + 150*e^2*log(2)*log(x)^2 + 8100*e*log(2)*log(x)^2 + 1000*log(2)*log(x)^3 + 20*e^3*log(
2)*log(x) + 1620*e^2*log(2)*log(x) + 43740*e*log(2)*log(x) + 15600*log(2)*log(x)^2 + e^4*log(2) + 108*e^3*log(
2) + 4374*e^2*log(2) + 78732*e*log(2) + 81160*log(2)*log(x) + 140816*log(2))/(log(x) + 5)^4

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maple [B]  time = 0.12, size = 70, normalized size = 4.12

method	result	size
default	\(-4 \ln \relax (2) \left ({\mathrm e}+2\right ) \left (-\frac {15 \,{\mathrm e}^{2}+60 \,{\mathrm e}+60}{3 \left (5+\ln \relax (x )\right )^{3}}-\frac {{\mathrm e}^{3}+6 \,{\mathrm e}^{2}+12 \,{\mathrm e}+8}{4 \left (5+\ln \relax (x )\right )^{4}}-\frac {125}{5+\ln \relax (x )}-\frac {75 \,{\mathrm e}+150}{2 \left (5+\ln \relax (x )\right )^{2}}\right )\)	\(70\)
risch	\(\frac {\ln \relax (2) \left ({\mathrm e}^{4}+20 \ln \relax (x ) {\mathrm e}^{3}+150 \,{\mathrm e}^{2} \ln \relax (x )^{2}+500 \ln \relax (x )^{3} {\mathrm e}+108 \,{\mathrm e}^{3}+1620 \,{\mathrm e}^{2} \ln \relax (x )+8100 \,{\mathrm e} \ln \relax (x )^{2}+1000 \ln \relax (x )^{3}+4374 \,{\mathrm e}^{2}+43740 \,{\mathrm e} \ln \relax (x )+15600 \ln \relax (x )^{2}+78732 \,{\mathrm e}+81160 \ln \relax (x )+140816\right )}{\left (5+\ln \relax (x )\right )^{4}}\)	\(84\)
norman	\(\frac {\left (500 \,{\mathrm e} \ln \relax (2)+1000 \ln \relax (2)\right ) \ln \relax (x )^{3}+\left (150 \,{\mathrm e}^{2} \ln \relax (2)+8100 \,{\mathrm e} \ln \relax (2)+15600 \ln \relax (2)\right ) \ln \relax (x )^{2}+\left (20 \,{\mathrm e}^{3} \ln \relax (2)+1620 \,{\mathrm e}^{2} \ln \relax (2)+43740 \,{\mathrm e} \ln \relax (2)+81160 \ln \relax (2)\right ) \ln \relax (x )+78732 \,{\mathrm e} \ln \relax (2)+140816 \ln \relax (2)+4374 \,{\mathrm e}^{2} \ln \relax (2)+108 \,{\mathrm e}^{3} \ln \relax (2)+{\mathrm e}^{4} \ln \relax (2)}{\left (5+\ln \relax (x )\right )^{4}}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-500*exp(1)-1000)*ln(2)*ln(x)^3+(-300*exp(1)^2-8700*exp(1)-16200)*ln(2)*ln(x)^2+(-60*exp(1)^3-3360*exp(1
)^2-50220*exp(1)-87480)*ln(2)*ln(x)+(-4*exp(1)^4-332*exp(1)^3-9396*exp(1)^2-96228*exp(1)-157464)*ln(2))/(x*ln(
x)^5+25*x*ln(x)^4+250*x*ln(x)^3+1250*x*ln(x)^2+3125*x*ln(x)+3125*x),x,method=_RETURNVERBOSE)

[Out]

-4*ln(2)*(exp(1)+2)*(-1/3*(15*exp(2)+60*exp(1)+60)/(5+ln(x))^3-1/4*(exp(3)+6*exp(2)+12*exp(1)+8)/(5+ln(x))^4-1
25/(5+ln(x))-1/2*(75*exp(1)+150)/(5+ln(x))^2)

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maxima [B]  time = 0.40, size = 692, normalized size = 40.71 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-500*exp(1)-1000)*log(2)*log(x)^3+(-300*exp(1)^2-8700*exp(1)-16200)*log(2)*log(x)^2+(-60*exp(1)^3-
3360*exp(1)^2-50220*exp(1)-87480)*log(2)*log(x)+(-4*exp(1)^4-332*exp(1)^3-9396*exp(1)^2-96228*exp(1)-157464)*l
og(2))/(x*log(x)^5+25*x*log(x)^4+250*x*log(x)^3+1250*x*log(x)^2+3125*x*log(x)+3125*x),x, algorithm="maxima")

[Out]

125*e*log(2)*log(x)^3/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 75*e^2*log(2)*log(x)^2/(log
(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 2175*e*log(2)*log(x)^2/(log(x)^4 + 20*log(x)^3 + 150*
log(x)^2 + 500*log(x) + 625) + 250*log(2)*log(x)^3/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625)
+ 25*(3*log(x) + 5)*e^2*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 125*(3*log(x)^2 + 15*log(x) + 25)*
e*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 725*(3*log(x) + 5)*e*log(2)/(log(x)^3 + 15*log(x)^2 + 75
*log(x) + 125) + 15*e^3*log(2)*log(x)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 840*e^2*log
(2)*log(x)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 12555*e*log(2)*log(x)/(log(x)^4 + 20*l
og(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 4050*log(2)*log(x)^2/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500
*log(x) + 625) + 250*(3*log(x)^2 + 15*log(x) + 25)*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 1350*(3
*log(x) + 5)*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + e^4*log(2)/(log(x)^4 + 20*log(x)^3 + 150*log(
x)^2 + 500*log(x) + 625) + 83*e^3*log(2)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 5*e^3*lo
g(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 2349*e^2*log(2)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500
*log(x) + 625) + 280*e^2*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 24057*e*log(2)/(log(x)^4 + 20*log
(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 4185*e*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125) + 21870*lo
g(2)*log(x)/(log(x)^4 + 20*log(x)^3 + 150*log(x)^2 + 500*log(x) + 625) + 39366*log(2)/(log(x)^4 + 20*log(x)^3
+ 150*log(x)^2 + 500*log(x) + 625) + 7290*log(2)/(log(x)^3 + 15*log(x)^2 + 75*log(x) + 125)

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mupad [B]  time = 3.40, size = 62, normalized size = 3.65 \begin {gather*} \frac {500\,\ln \relax (2)\,\left (\mathrm {e}+2\right )}{\ln \relax (x)+5}+\frac {150\,\ln \relax (2)\,{\left (\mathrm {e}+2\right )}^2}{{\left (\ln \relax (x)+5\right )}^2}+\frac {20\,\ln \relax (2)\,{\left (\mathrm {e}+2\right )}^3}{{\left (\ln \relax (x)+5\right )}^3}+\frac {\ln \relax (2)\,{\left (\mathrm {e}+2\right )}^4}{{\left (\ln \relax (x)+5\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2)*(96228*exp(1) + 9396*exp(2) + 332*exp(3) + 4*exp(4) + 157464) + log(2)*log(x)^3*(500*exp(1) + 100
0) + log(2)*log(x)*(50220*exp(1) + 3360*exp(2) + 60*exp(3) + 87480) + log(2)*log(x)^2*(8700*exp(1) + 300*exp(2
) + 16200))/(3125*x + 1250*x*log(x)^2 + 250*x*log(x)^3 + 25*x*log(x)^4 + x*log(x)^5 + 3125*x*log(x)),x)

[Out]

(500*log(2)*(exp(1) + 2))/(log(x) + 5) + (150*log(2)*(exp(1) + 2)^2)/(log(x) + 5)^2 + (20*log(2)*(exp(1) + 2)^
3)/(log(x) + 5)^3 + (log(2)*(exp(1) + 2)^4)/(log(x) + 5)^4

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sympy [B]  time = 0.22, size = 141, normalized size = 8.29 \begin {gather*} \frac {\left (1000 \log {\relax (2 )} + 500 e \log {\relax (2 )}\right ) \log {\relax (x )}^{3} + \left (150 e^{2} \log {\relax (2 )} + 15600 \log {\relax (2 )} + 8100 e \log {\relax (2 )}\right ) \log {\relax (x )}^{2} + \left (20 e^{3} \log {\relax (2 )} + 1620 e^{2} \log {\relax (2 )} + 81160 \log {\relax (2 )} + 43740 e \log {\relax (2 )}\right ) \log {\relax (x )} + e^{4} \log {\relax (2 )} + 108 e^{3} \log {\relax (2 )} + 4374 e^{2} \log {\relax (2 )} + 140816 \log {\relax (2 )} + 78732 e \log {\relax (2 )}}{\log {\relax (x )}^{4} + 20 \log {\relax (x )}^{3} + 150 \log {\relax (x )}^{2} + 500 \log {\relax (x )} + 625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-500*exp(1)-1000)*ln(2)*ln(x)**3+(-300*exp(1)**2-8700*exp(1)-16200)*ln(2)*ln(x)**2+(-60*exp(1)**3-
3360*exp(1)**2-50220*exp(1)-87480)*ln(2)*ln(x)+(-4*exp(1)**4-332*exp(1)**3-9396*exp(1)**2-96228*exp(1)-157464)
*ln(2))/(x*ln(x)**5+25*x*ln(x)**4+250*x*ln(x)**3+1250*x*ln(x)**2+3125*x*ln(x)+3125*x),x)

[Out]

((1000*log(2) + 500*E*log(2))*log(x)**3 + (150*exp(2)*log(2) + 15600*log(2) + 8100*E*log(2))*log(x)**2 + (20*e
xp(3)*log(2) + 1620*exp(2)*log(2) + 81160*log(2) + 43740*E*log(2))*log(x) + exp(4)*log(2) + 108*exp(3)*log(2)
+ 4374*exp(2)*log(2) + 140816*log(2) + 78732*E*log(2))/(log(x)**4 + 20*log(x)**3 + 150*log(x)**2 + 500*log(x)
+ 625)

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