3.5.17 \(\int \frac {816+512 e+80 e^2+(128+40 e) \log (x)+5 \log ^2(x)+(168+104 e+16 e^2+(26+8 e) \log (x)+\log ^2(x)) \log (80 x)}{9+6 e+e^2} \, dx\)

Optimal. Leaf size=23 \[ x \left (4-\frac {\log (x)}{-3-e}\right )^2 (4+\log (80 x)) \]

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Rubi [B]  time = 0.42, antiderivative size = 207, normalized size of antiderivative = 9.00, number of steps used = 17, number of rules used = 6, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 2295, 2296, 6741, 6742, 2361} \begin {gather*} \frac {16 (17+5 e) x}{3+e}-\frac {8 (16+5 e) x}{(3+e)^2}+\frac {4 (13+4 e) x}{(3+e)^2}-\frac {8 (7+2 e) x}{3+e}+\frac {4 x}{(3+e)^2}+\frac {4 x \log ^2(x)}{(3+e)^2}+\frac {x \log (80 x) \log ^2(x)}{(3+e)^2}+\frac {8 (16+5 e) x \log (x)}{(3+e)^2}-\frac {2 (13+4 e) x \log (x)}{(3+e)^2}-\frac {6 x \log (x)}{(3+e)^2}+\frac {2 (13+4 e) x \log (80 x) \log (x)}{(3+e)^2}-\frac {2 x \log (80 x) \log (x)}{(3+e)^2}-\frac {2 (13+4 e) x \log (80 x)}{(3+e)^2}+\frac {8 (7+2 e) x \log (80 x)}{3+e}+\frac {2 x \log (80 x)}{(3+e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(816 + 512*E + 80*E^2 + (128 + 40*E)*Log[x] + 5*Log[x]^2 + (168 + 104*E + 16*E^2 + (26 + 8*E)*Log[x] + Log
[x]^2)*Log[80*x])/(9 + 6*E + E^2),x]

[Out]

(4*x)/(3 + E)^2 - (8*(7 + 2*E)*x)/(3 + E) + (4*(13 + 4*E)*x)/(3 + E)^2 - (8*(16 + 5*E)*x)/(3 + E)^2 + (16*(17
+ 5*E)*x)/(3 + E) - (6*x*Log[x])/(3 + E)^2 - (2*(13 + 4*E)*x*Log[x])/(3 + E)^2 + (8*(16 + 5*E)*x*Log[x])/(3 +
E)^2 + (4*x*Log[x]^2)/(3 + E)^2 + (2*x*Log[80*x])/(3 + E)^2 + (8*(7 + 2*E)*x*Log[80*x])/(3 + E) - (2*(13 + 4*E
)*x*Log[80*x])/(3 + E)^2 - (2*x*Log[x]*Log[80*x])/(3 + E)^2 + (2*(13 + 4*E)*x*Log[x]*Log[80*x])/(3 + E)^2 + (x
*Log[x]^2*Log[80*x])/(3 + E)^2

Rule 12

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

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 2361

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.)), x_Symbol] :> With[{u =
IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x],
x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, 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} &=\frac {\int \left (816+512 e+80 e^2+(128+40 e) \log (x)+5 \log ^2(x)+\left (168+104 e+16 e^2+(26+8 e) \log (x)+\log ^2(x)\right ) \log (80 x)\right ) \, dx}{(3+e)^2}\\ &=\frac {16 (17+5 e) x}{3+e}+\frac {\int \left (168+104 e+16 e^2+(26+8 e) \log (x)+\log ^2(x)\right ) \log (80 x) \, dx}{(3+e)^2}+\frac {5 \int \log ^2(x) \, dx}{(3+e)^2}+\frac {(8 (16+5 e)) \int \log (x) \, dx}{(3+e)^2}\\ &=-\frac {8 (16+5 e) x}{(3+e)^2}+\frac {16 (17+5 e) x}{3+e}+\frac {8 (16+5 e) x \log (x)}{(3+e)^2}+\frac {5 x \log ^2(x)}{(3+e)^2}+\frac {\int \left (168 \left (1+\frac {1}{21} e (13+2 e)\right )+(26+8 e) \log (x)+\log ^2(x)\right ) \log (80 x) \, dx}{(3+e)^2}-\frac {10 \int \log (x) \, dx}{(3+e)^2}\\ &=\frac {10 x}{(3+e)^2}-\frac {8 (16+5 e) x}{(3+e)^2}+\frac {16 (17+5 e) x}{3+e}-\frac {10 x \log (x)}{(3+e)^2}+\frac {8 (16+5 e) x \log (x)}{(3+e)^2}+\frac {5 x \log ^2(x)}{(3+e)^2}+\frac {\int \left (8 (3+e) (7+2 e) \log (80 x)+2 (13+4 e) \log (x) \log (80 x)+\log ^2(x) \log (80 x)\right ) \, dx}{(3+e)^2}\\ &=\frac {10 x}{(3+e)^2}-\frac {8 (16+5 e) x}{(3+e)^2}+\frac {16 (17+5 e) x}{3+e}-\frac {10 x \log (x)}{(3+e)^2}+\frac {8 (16+5 e) x \log (x)}{(3+e)^2}+\frac {5 x \log ^2(x)}{(3+e)^2}+\frac {\int \log ^2(x) \log (80 x) \, dx}{(3+e)^2}+\frac {(8 (7+2 e)) \int \log (80 x) \, dx}{3+e}+\frac {(2 (13+4 e)) \int \log (x) \log (80 x) \, dx}{(3+e)^2}\\ &=\frac {10 x}{(3+e)^2}-\frac {8 (7+2 e) x}{3+e}-\frac {8 (16+5 e) x}{(3+e)^2}+\frac {16 (17+5 e) x}{3+e}-\frac {10 x \log (x)}{(3+e)^2}+\frac {8 (16+5 e) x \log (x)}{(3+e)^2}+\frac {5 x \log ^2(x)}{(3+e)^2}+\frac {2 x \log (80 x)}{(3+e)^2}+\frac {8 (7+2 e) x \log (80 x)}{3+e}-\frac {2 (13+4 e) x \log (80 x)}{(3+e)^2}-\frac {2 x \log (x) \log (80 x)}{(3+e)^2}+\frac {2 (13+4 e) x \log (x) \log (80 x)}{(3+e)^2}+\frac {x \log ^2(x) \log (80 x)}{(3+e)^2}-\frac {\int \left (2-2 \log (x)+\log ^2(x)\right ) \, dx}{(3+e)^2}-\frac {(2 (13+4 e)) \int (-1+\log (x)) \, dx}{(3+e)^2}\\ &=\frac {8 x}{(3+e)^2}-\frac {8 (7+2 e) x}{3+e}+\frac {2 (13+4 e) x}{(3+e)^2}-\frac {8 (16+5 e) x}{(3+e)^2}+\frac {16 (17+5 e) x}{3+e}-\frac {10 x \log (x)}{(3+e)^2}+\frac {8 (16+5 e) x \log (x)}{(3+e)^2}+\frac {5 x \log ^2(x)}{(3+e)^2}+\frac {2 x \log (80 x)}{(3+e)^2}+\frac {8 (7+2 e) x \log (80 x)}{3+e}-\frac {2 (13+4 e) x \log (80 x)}{(3+e)^2}-\frac {2 x \log (x) \log (80 x)}{(3+e)^2}+\frac {2 (13+4 e) x \log (x) \log (80 x)}{(3+e)^2}+\frac {x \log ^2(x) \log (80 x)}{(3+e)^2}-\frac {\int \log ^2(x) \, dx}{(3+e)^2}+\frac {2 \int \log (x) \, dx}{(3+e)^2}-\frac {(2 (13+4 e)) \int \log (x) \, dx}{(3+e)^2}\\ &=\frac {6 x}{(3+e)^2}-\frac {8 (7+2 e) x}{3+e}+\frac {4 (13+4 e) x}{(3+e)^2}-\frac {8 (16+5 e) x}{(3+e)^2}+\frac {16 (17+5 e) x}{3+e}-\frac {8 x \log (x)}{(3+e)^2}-\frac {2 (13+4 e) x \log (x)}{(3+e)^2}+\frac {8 (16+5 e) x \log (x)}{(3+e)^2}+\frac {4 x \log ^2(x)}{(3+e)^2}+\frac {2 x \log (80 x)}{(3+e)^2}+\frac {8 (7+2 e) x \log (80 x)}{3+e}-\frac {2 (13+4 e) x \log (80 x)}{(3+e)^2}-\frac {2 x \log (x) \log (80 x)}{(3+e)^2}+\frac {2 (13+4 e) x \log (x) \log (80 x)}{(3+e)^2}+\frac {x \log ^2(x) \log (80 x)}{(3+e)^2}+\frac {2 \int \log (x) \, dx}{(3+e)^2}\\ &=\frac {4 x}{(3+e)^2}-\frac {8 (7+2 e) x}{3+e}+\frac {4 (13+4 e) x}{(3+e)^2}-\frac {8 (16+5 e) x}{(3+e)^2}+\frac {16 (17+5 e) x}{3+e}-\frac {6 x \log (x)}{(3+e)^2}-\frac {2 (13+4 e) x \log (x)}{(3+e)^2}+\frac {8 (16+5 e) x \log (x)}{(3+e)^2}+\frac {4 x \log ^2(x)}{(3+e)^2}+\frac {2 x \log (80 x)}{(3+e)^2}+\frac {8 (7+2 e) x \log (80 x)}{3+e}-\frac {2 (13+4 e) x \log (80 x)}{(3+e)^2}-\frac {2 x \log (x) \log (80 x)}{(3+e)^2}+\frac {2 (13+4 e) x \log (x) \log (80 x)}{(3+e)^2}+\frac {x \log ^2(x) \log (80 x)}{(3+e)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 23, normalized size = 1.00 \begin {gather*} \frac {x (4 (3+e)+\log (x))^2 (4+\log (80 x))}{(3+e)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(816 + 512*E + 80*E^2 + (128 + 40*E)*Log[x] + 5*Log[x]^2 + (168 + 104*E + 16*E^2 + (26 + 8*E)*Log[x]
+ Log[x]^2)*Log[80*x])/(9 + 6*E + E^2),x]

[Out]

(x*(4*(3 + E) + Log[x])^2*(4 + Log[80*x]))/(3 + E)^2

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fricas [B]  time = 0.67, size = 95, normalized size = 4.13 \begin {gather*} \frac {x \log \relax (x)^{3} + {\left (8 \, x e + x \log \left (80\right ) + 28 \, x\right )} \log \relax (x)^{2} + 64 \, x e^{2} + 384 \, x e + 16 \, {\left (x e^{2} + 6 \, x e + 9 \, x\right )} \log \left (80\right ) + 8 \, {\left (2 \, x e^{2} + 16 \, x e + {\left (x e + 3 \, x\right )} \log \left (80\right ) + 30 \, x\right )} \log \relax (x) + 576 \, x}{e^{2} + 6 \, e + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)^2+(8*exp(1)+26)*log(x)+16*exp(1)^2+104*exp(1)+168)*log(80*x)+5*log(x)^2+(40*exp(1)+128)*log
(x)+80*exp(1)^2+512*exp(1)+816)/(exp(1)^2+6*exp(1)+9),x, algorithm="fricas")

[Out]

(x*log(x)^3 + (8*x*e + x*log(80) + 28*x)*log(x)^2 + 64*x*e^2 + 384*x*e + 16*(x*e^2 + 6*x*e + 9*x)*log(80) + 8*
(2*x*e^2 + 16*x*e + (x*e + 3*x)*log(80) + 30*x)*log(x) + 576*x)/(e^2 + 6*e + 9)

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giac [B]  time = 0.21, size = 125, normalized size = 5.43 \begin {gather*} \frac {8 \, x e \log \left (80\right ) \log \relax (x) + 8 \, x e \log \relax (x)^{2} + x \log \left (80\right ) \log \relax (x)^{2} + x \log \relax (x)^{3} + 16 \, x e^{2} \log \left (80\right ) + 96 \, x e \log \left (80\right ) + 16 \, x e^{2} \log \relax (x) + 88 \, x e \log \relax (x) + 24 \, x \log \left (80\right ) \log \relax (x) + 28 \, x \log \relax (x)^{2} + 8 \, {\left (x \log \relax (x) - x\right )} {\left (5 \, e + 16\right )} + 64 \, x e^{2} + 424 \, x e + 144 \, x \log \left (80\right ) + 112 \, x \log \relax (x) + 704 \, x}{e^{2} + 6 \, e + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)^2+(8*exp(1)+26)*log(x)+16*exp(1)^2+104*exp(1)+168)*log(80*x)+5*log(x)^2+(40*exp(1)+128)*log
(x)+80*exp(1)^2+512*exp(1)+816)/(exp(1)^2+6*exp(1)+9),x, algorithm="giac")

[Out]

(8*x*e*log(80)*log(x) + 8*x*e*log(x)^2 + x*log(80)*log(x)^2 + x*log(x)^3 + 16*x*e^2*log(80) + 96*x*e*log(80) +
16*x*e^2*log(x) + 88*x*e*log(x) + 24*x*log(80)*log(x) + 28*x*log(x)^2 + 8*(x*log(x) - x)*(5*e + 16) + 64*x*e^
2 + 424*x*e + 144*x*log(80) + 112*x*log(x) + 704*x)/(e^2 + 6*e + 9)

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maple [B]  time = 0.12, size = 72, normalized size = 3.13

method	result	size
norman	\(\frac {\left (64 \,{\mathrm e}+192\right ) x +\left (16 \,{\mathrm e}+48\right ) x \ln \left (80 x \right )+\frac {x \ln \relax (x )^{2} \ln \left (80 x \right )}{3+{\mathrm e}}+32 x \ln \relax (x )+8 x \ln \relax (x ) \ln \left (80 x \right )+\frac {4 x \ln \relax (x )^{2}}{3+{\mathrm e}}}{3+{\mathrm e}}\)	\(72\)
default	\(\frac {576 x +128 x \,{\mathrm e} \ln \relax (x )+384 x \,{\mathrm e}+240 x \ln \relax (x )+16 \,{\mathrm e}^{2} \ln \left (80\right ) x +16 x \,{\mathrm e}^{2} \ln \relax (x )+64 \,{\mathrm e}^{2} x +8 \ln \left (80\right ) {\mathrm e} \ln \relax (x ) x +96 \,{\mathrm e} \ln \left (80\right ) x +8 \,{\mathrm e} \ln \relax (x )^{2} x +\ln \left (80\right ) \ln \relax (x )^{2} x +24 \ln \left (80\right ) \ln \relax (x ) x +144 \ln \left (80\right ) x +x \ln \relax (x )^{3}+28 x \ln \relax (x )^{2}}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}\)	\(118\)
risch	\(\frac {x \ln \relax (x )^{3}}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {\left (56+16 \,{\mathrm e}+8 \ln \relax (2)+2 \ln \relax (5)\right ) x \ln \relax (x )^{2}}{2 \,{\mathrm e}^{2}+12 \,{\mathrm e}+18}+\frac {4 \left (60+4 \,{\mathrm e}^{2}+8 \,{\mathrm e} \ln \relax (2)+2 \,{\mathrm e} \ln \relax (5)+32 \,{\mathrm e}+24 \ln \relax (2)+6 \ln \relax (5)\right ) x \ln \relax (x )}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {64 x \,{\mathrm e}^{2} \ln \relax (2)}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {16 x \,{\mathrm e}^{2} \ln \relax (5)}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {384 x \,{\mathrm e} \ln \relax (2)}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {96 x \,{\mathrm e} \ln \relax (5)}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {64 x \,{\mathrm e}^{2}}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {384 x \,{\mathrm e}}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {576 x \ln \relax (2)}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {144 x \ln \relax (5)}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}+\frac {576 x}{{\mathrm e}^{2}+6 \,{\mathrm e}+9}\)	\(235\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((ln(x)^2+(8*exp(1)+26)*ln(x)+16*exp(1)^2+104*exp(1)+168)*ln(80*x)+5*ln(x)^2+(40*exp(1)+128)*ln(x)+80*exp(
1)^2+512*exp(1)+816)/(exp(1)^2+6*exp(1)+9),x,method=_RETURNVERBOSE)

[Out]

((64*exp(1)+192)*x+(16*exp(1)+48)*x*ln(80*x)+1/(3+exp(1))*x*ln(x)^2*ln(80*x)+32*x*ln(x)+8*x*ln(x)*ln(80*x)+4/(
3+exp(1))*x*ln(x)^2)/(3+exp(1))

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maxima [B]  time = 0.42, size = 133, normalized size = 5.78 \begin {gather*} -\frac {2 \, x {\left (4 \, e + 11\right )} \log \relax (x) + x \log \relax (x)^{2} - 5 \, {\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + 2 \, x {\left (8 \, e^{2} + 44 \, e + 61\right )} - 8 \, {\left (x \log \relax (x) - x\right )} {\left (5 \, e + 16\right )} - 80 \, x e^{2} - 512 \, x e - {\left ({\left (\log \relax (x)^{2} - 2 \, \log \relax (x) + 2\right )} x + 2 \, {\left (x \log \relax (x) - x\right )} {\left (4 \, e + 13\right )} + 16 \, x e^{2} + 104 \, x e + 168 \, x\right )} \log \left (80 \, x\right ) - 816 \, x}{e^{2} + 6 \, e + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((log(x)^2+(8*exp(1)+26)*log(x)+16*exp(1)^2+104*exp(1)+168)*log(80*x)+5*log(x)^2+(40*exp(1)+128)*log
(x)+80*exp(1)^2+512*exp(1)+816)/(exp(1)^2+6*exp(1)+9),x, algorithm="maxima")

[Out]

-(2*x*(4*e + 11)*log(x) + x*log(x)^2 - 5*(log(x)^2 - 2*log(x) + 2)*x + 2*x*(8*e^2 + 44*e + 61) - 8*(x*log(x) -
x)*(5*e + 16) - 80*x*e^2 - 512*x*e - ((log(x)^2 - 2*log(x) + 2)*x + 2*(x*log(x) - x)*(4*e + 13) + 16*x*e^2 +
104*x*e + 168*x)*log(80*x) - 816*x)/(e^2 + 6*e + 9)

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mupad [B]  time = 0.55, size = 76, normalized size = 3.30 \begin {gather*} \frac {x\,\left ({\ln \relax (x)}^3+{\ln \relax (x)}^2\,\left (\ln \left (80\,x\right )+8\,\mathrm {e}-\ln \relax (x)+28\right )+16\,{\left (\mathrm {e}+3\right )}^2\,\left (\ln \left (80\,x\right )-\ln \relax (x)+4\right )+8\,\ln \relax (x)\,\left (\mathrm {e}+3\right )\,\left (\ln \left (80\,x\right )+2\,\mathrm {e}-\ln \relax (x)+10\right )\right )}{6\,\mathrm {e}+{\mathrm {e}}^2+9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((512*exp(1) + 80*exp(2) + log(80*x)*(104*exp(1) + 16*exp(2) + log(x)^2 + log(x)*(8*exp(1) + 26) + 168) + 5
*log(x)^2 + log(x)*(40*exp(1) + 128) + 816)/(6*exp(1) + exp(2) + 9),x)

[Out]

(x*(log(x)^3 + log(x)^2*(log(80*x) + 8*exp(1) - log(x) + 28) + 16*(exp(1) + 3)^2*(log(80*x) - log(x) + 4) + 8*
log(x)*(exp(1) + 3)*(log(80*x) + 2*exp(1) - log(x) + 10)))/(6*exp(1) + exp(2) + 9)

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sympy [B]  time = 0.40, size = 80, normalized size = 3.48 \begin {gather*} \frac {x \log {\relax (x )}^{3}}{e^{2} + 9 + 6 e} + x \left (64 + 16 \log {\left (80 \right )}\right ) + \frac {\left (x \log {\left (80 \right )} + 8 e x + 28 x\right ) \log {\relax (x )}^{2}}{e^{2} + 9 + 6 e} + \frac {\left (8 x \log {\left (80 \right )} + 16 e x + 80 x\right ) \log {\relax (x )}}{e + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((ln(x)**2+(8*exp(1)+26)*ln(x)+16*exp(1)**2+104*exp(1)+168)*ln(80*x)+5*ln(x)**2+(40*exp(1)+128)*ln(x
)+80*exp(1)**2+512*exp(1)+816)/(exp(1)**2+6*exp(1)+9),x)

[Out]

x*log(x)**3/(exp(2) + 9 + 6*E) + x*(64 + 16*log(80)) + (x*log(80) + 8*E*x + 28*x)*log(x)**2/(exp(2) + 9 + 6*E)
+ (8*x*log(80) + 16*E*x + 80*x)*log(x)/(E + 3)

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