∫ −25x4+60x5−21x6+2x7+(120x2−264x3+48x4) log(3)+(−144+288x) log2(3)+(16x4 log(3)−384x log2(3)) log(5)_______________________________________________________________________________ (25x4−10x5+x6+(−120x2+24x3) log(3)+144 log2(3)) log(5) dx

3.96.63 \(\int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+(120 x^2-264 x^3+48 x^4) \log (3)+(-144+288 x) \log ^2(3)+(16 x^4 \log (3)-384 x \log ^2(3)) \log (5)}{(25 x^4-10 x^5+x^6+(-120 x^2+24 x^3) \log (3)+144 \log ^2(3)) \log (5)} \, dx\)

Optimal. Leaf size=38 \[ \frac {4}{-\frac {3}{x^2}+\frac {5-x}{4 \log (3)}}-\frac {-5+x-x^2}{\log (5)} \]

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Rubi [C] time = 79.58, antiderivative size = 4395, normalized size of antiderivative = 115.66, number of steps used = 21, number of rules used = 11, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {12, 2074, 2081, 2079, 800, 634, 618, 206, 628, 2101, 822}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(-25*x^4 + 60*x^5 - 21*x^6 + 2*x^7 + (120*x^2 - 264*x^3 + 48*x^4)*Log[3] + (-144 + 288*x)*Log[3]^2 + (16*x

^4*Log[3] - 384*x*Log[3]^2)*Log[5])/((25*x^4 - 10*x^5 + x^6 + (-120*x^2 + 24*x^3)*Log[3] + 144*Log[3]^2)*Log[5

]),x]

[Out]

(400*Log[3])/(3*(5*x^2 - x^3 - 12*Log[3])) - (48*Log[3]*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log

[3]])^(1/3))/(25 - (5 - 3*x)*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + (-125 + 162*L

og[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) + (48*ArcTanh[(125 - 162*Log[3] + (18*I)*Sqrt[(125 - 81*

Log[3])*Log[3]] - 25*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) - 2*(5 - 3*x)*(-125 + 1

62*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3))/Sqrt[3*((5832*I)*Sqrt[125 - 81*Log[3]]*Log[3]^(3/2)

- 52488*Log[3]^2 + 8100*Log[3]*(10 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)) - 25*(

5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))*((36*I)*Sqrt[(125 - 81*Log[3])*Log[3]]

+ 25*(5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))))]]*Log[3]*(506250*Log[3] - (750*

I)*Sqrt[(125 - 81*Log[3])*Log[3]]*(35 - 2*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)) -

15625*(5 - (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)) + (1944*I)*Sqrt[125 - 81*Log[3]

]*Log[3]^(3/2)*(20 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)) - 17496*Log[3]^2*(20 +

(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))))/((-125 + 162*Log[3] - (18*I)*Sqrt[(125 -

81*Log[3])*Log[3]])^(1/3)*(625 + 25*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) + (-125

+ 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(4/3))*Sqrt[((5832*I)*Sqrt[125 - 81*Log[3]]*Log[3]^(3/2

) - 52488*Log[3]^2 + 8100*Log[3]*(10 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)) - 25

*(5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))*((36*I)*Sqrt[(125 - 81*Log[3])*Log[3]

] + 25*(5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))))/3]) - (144*Log[3]*(-125 + 162

*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(4/3)*(12*(125 - 81*Log[3])*Log[3] - ((5 - 3*x)*((125*I)*Sqrt

[(125 - 81*Log[3])*Log[3]]*(5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)) - 750*Log[3

]*(10 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)) - (54*I)*Sqrt[125 - 81*Log[3]]*Log[

3]^(3/2)*(10 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)) + 486*Log[3]^2*(10 + (-125 +

162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))))/(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3

])*Log[3]])^(2/3)))/((2250*Log[3] + (162*I)*Sqrt[125 - 81*Log[3]]*Log[3]^(3/2) - 1458*Log[3]^2 - (125*I)*Sqrt[

(125 - 81*Log[3])*Log[3]])*(5 - 3*x - 25/(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) - (

-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))*(25 - (-125 + 162*Log[3] - (18*I)*Sqrt[(125

- 81*Log[3])*Log[3]])^(2/3))*(25 - (5 - 3*x)^2 - 625/(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]

])^(2/3) - (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) - ((5 - 3*x)*(25 + (-125 + 162*Lo

g[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)))/(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3

]])^(1/3))) - (62208*ArcTanh[(125 - 162*Log[3] + (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]] - 25*(-125 + 162*Log[3]

- (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) - 2*(5 - 3*x)*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3

])*Log[3]])^(2/3))/Sqrt[3*((5832*I)*Sqrt[125 - 81*Log[3]]*Log[3]^(3/2) - 52488*Log[3]^2 + 8100*Log[3]*(10 + (-

125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)) - 25*(5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(1

25 - 81*Log[3])*Log[3]])^(1/3))*((36*I)*Sqrt[(125 - 81*Log[3])*Log[3]] + 25*(5 + (-125 + 162*Log[3] - (18*I)*S

qrt[(125 - 81*Log[3])*Log[3]])^(1/3))))]]*Log[3]^2*((48201307559424*I)*Log[3]^6 + (286978140000*I)*Log[3]^4*(6

75 - 47*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - (1487694677760*I)*Log[3]^5*(125 -

2*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) + (1992903750000*I)*Log[3]^3*(25 + 11*(-

125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) + (32958984375*I)*Log[3]*(2025 + 113*(-125 +

162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - (128144531250*I)*Log[3]^2*(1325 + 117*(-125 + 162

*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - 1220703125*((125*I)*(25 + (-125 + 162*Log[3] - (18*I

)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - 6*Sqrt[(125 - 81*Log[3])*Log[3]]*(275 + 13*(-125 + 162*Log[3] - (18

*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3))) - 486*Sqrt[125 - 81*Log[3]]*Log[3]^(3/2)*(9765625*(2275 + 157*(-12

5 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - 1296*Log[3]*(8503056*Log[3]^3 + 658125*Log[3]

*(25 - 3*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - 524880*Log[3]^2*(50 - (-125 + 16

2*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) + 625000*(25 + 4*(-125 + 162*Log[3] - (18*I)*Sqrt[(12

5 - 81*Log[3])*Log[3]])^(2/3))))))/((125*I - (162*I)*Log[3] - 18*Sqrt[(125 - 81*Log[3])*Log[3]])*(25 - (-125 +

162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3))^2*(625 + 25*(-125 + 162*Log[3] - (18*I)*Sqrt[(125

- 81*Log[3])*Log[3]])^(2/3) + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(4/3))^3*Sqrt[((5832

*I)*Sqrt[125 - 81*Log[3]]*Log[3]^(3/2) - 52488*Log[3]^2 + 8100*Log[3]*(10 + (-125 + 162*Log[3] - (18*I)*Sqrt[(

125 - 81*Log[3])*Log[3]])^(1/3)) - 25*(5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))*

((36*I)*Sqrt[(125 - 81*Log[3])*Log[3]] + 25*(5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(

1/3))))/3]) - x/Log[5] + x^2/Log[5] - (16*Log[3]*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(

1/3)*(25 - 20*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + (-125 + 162*Log[3] - (18*I)*

Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3))*Log[25 - (5 - 3*x)*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*L

og[3]])^(1/3) + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)])/(625 + 25*(-125 + 162*Log[

3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]]

)^(4/3)) + (20736*Log[3]^2*(1377495072*Log[3]^4 - 21257640*Log[3]^3*(250 - 5*(-125 + 162*Log[3] - (18*I)*Sqrt[

(125 - 81*Log[3])*Log[3]])^(1/3) + 4*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) + 8201

250*Log[3]^2*(825 - 40*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 32*(-125 + 162*Log[

3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - 6328125*Log[3]*(475 - 45*(-125 + 162*Log[3] - (18*I)*Sqrt

[(125 - 81*Log[3])*Log[3]])^(1/3) + 36*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) + 78

125*(125*(25 - 5*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 4*(-125 + 162*Log[3] - (1

8*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) + (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]]*(75 - 10*(-125 + 162*Log[3

] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 8*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]

])^(2/3))) - (236196*I)*Sqrt[125 - 81*Log[3]]*Log[3]^(3/2)*(648*Log[3]^2 - 10*Log[3]*(200 - 5*(-125 + 162*Log[

3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 4*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3

]])^(2/3)) + (625*(475 - 30*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 24*(-125 + 162

*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)))/162))*Log[25 - (5 - 3*x)*(-125 + 162*Log[3] - (18*I)*

Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)])/((

25 - (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3))^2*(625 + 25*(-125 + 162*Log[3] - (18*I

)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(4/3))^3

) + (8*Log[3]*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3)*(25 - 20*(-125 + 162*Log[3] -

(18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/

3))*Log[125*x - 162*x*Log[3] + (18*I)*x*Sqrt[(125 - 81*Log[3])*Log[3]] - 25*x*(-125 + 162*Log[3] - (18*I)*Sqrt

[(125 - 81*Log[3])*Log[3]])^(1/3) - 10*x*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) + 3

*x^2*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) + 6*(9*Log[3] - I*Sqrt[(125 - 81*Log[3]

)*Log[3]])*(5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))])/(625 + 25*(-125 + 162*Log

[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]

])^(4/3)) + (10368*Log[3]^2*((446308403328*I)*Log[3]^5 + (26572050000*I)*Log[3]^3*(130 - 5*(-125 + 162*Log[3]

- (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 4*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])

^(2/3)) - (6887475360*I)*Log[3]^4*(300 - 5*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) +

4*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - (3075468750*I)*Log[3]^2*(800 - 55*(-12

5 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 44*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*

Log[3])*Log[3]])^(2/3)) + (791015625*I)*Log[3]*(825 - 95*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Lo

g[3]])^(1/3) + 76*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - 9765625*((125*I)*(25 -

5*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 4*(-125 + 162*Log[3] - (18*I)*Sqrt[(125

- 81*Log[3])*Log[3]])^(2/3)) - 18*Sqrt[(125 - 81*Log[3])*Log[3]]*(100 - 15*(-125 + 162*Log[3] - (18*I)*Sqrt[(1

25 - 81*Log[3])*Log[3]])^(1/3) + 12*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3))) + 1458

*Sqrt[125 - 81*Log[3]]*Log[3]^(3/2)*(34012224*Log[3]^3 - 524880*Log[3]^2*(250 - 5*(-125 + 162*Log[3] - (18*I)*

Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 4*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) +

405000*Log[3]*(425 - 20*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3) + 16*(-125 + 162*Lo

g[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3)) - 78125*(1100 - 95*(-125 + 162*Log[3] - (18*I)*Sqrt[(125

- 81*Log[3])*Log[3]])^(1/3) + 76*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3))))*Log[125*

x - 162*x*Log[3] + (18*I)*x*Sqrt[(125 - 81*Log[3])*Log[3]] - 25*x*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*L

og[3])*Log[3]])^(1/3) - 10*x*(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) + 3*x^2*(-125 +

162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) + 6*(9*Log[3] - I*Sqrt[(125 - 81*Log[3])*Log[3]])*(

5 + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(1/3))])/((125*I - (162*I)*Log[3] - 18*Sqrt[(1

25 - 81*Log[3])*Log[3]])*(25 - (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3))^2*(625 + 25*

(-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81*Log[3])*Log[3]])^(2/3) + (-125 + 162*Log[3] - (18*I)*Sqrt[(125 - 81

*Log[3])*Log[3]])^(4/3))^3)

Rule 12

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]

, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +

c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +

b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &

& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp

andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -

4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp

[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x

)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*

c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2

*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -

b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,

c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||

IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /; !SumQ[N

onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rule 2079

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S

qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[(18^(1/3)*b*d)/(3*r) - r/18^(1/

3) + d*x, x]^p*Simp[(b*d)/3 + (12^(1/3)*b^2*d^2)/(3*r^2) + r^2/(3*12^(1/3)) - d*((2^(1/3)*b*d)/(3^(1/3)*r) - r

/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0

]

Rule 2081

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3, x, 0], b = Coeff[P3, x, 1], c = C

oeff[P3, x, 2], d = Coeff[P3, x, 3]}, Subst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2

)/(27*d^2) - ((c^2 - 3*b*d)*x)/(3*d) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c, 0]] /; FreeQ[{e, f, m, p}, x

] && PolyQ[P3, x, 3]

Rule 2101

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Qn^(p + 1)

)/(n*(p + 1)*Coeff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[P

m, x, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm, x] && PolyQ[Qn, x] && NeQ[p,

-1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-25 x^4+60 x^5-21 x^6+2 x^7+\left (120 x^2-264 x^3+48 x^4\right ) \log (3)+(-144+288 x) \log ^2(3)+\left (16 x^4 \log (3)-384 x \log ^2(3)\right ) \log (5)}{25 x^4-10 x^5+x^6+\left (-120 x^2+24 x^3\right ) \log (3)+144 \log ^2(3)} \, dx}{\log (5)}\\ &=\frac {\int \left (-1+2 x+\frac {16 (5+x) \log (3) \log (5)}{-5 x^2+x^3+12 \log (3)}-\frac {16 \log (3) \left (-25 x^2+60 \log (3)+36 x \log (3)\right ) \log (5)}{\left (-5 x^2+x^3+12 \log (3)\right )^2}\right ) \, dx}{\log (5)}\\ &=-\frac {x}{\log (5)}+\frac {x^2}{\log (5)}+(16 \log (3)) \int \frac {5+x}{-5 x^2+x^3+12 \log (3)} \, dx-(16 \log (3)) \int \frac {-25 x^2+60 \log (3)+36 x \log (3)}{\left (-5 x^2+x^3+12 \log (3)\right )^2} \, dx\\ &=\frac {400 \log (3)}{3 \left (5 x^2-x^3-12 \log (3)\right )}-\frac {x}{\log (5)}+\frac {x^2}{\log (5)}-\frac {1}{3} (16 \log (3)) \int \frac {-2 x (125-54 \log (3))+180 \log (3)}{\left (-5 x^2+x^3+12 \log (3)\right )^2} \, dx+(16 \log (3)) \operatorname {Subst}\left (\int \frac {\frac {20}{3}+x}{-\frac {25 x}{3}+x^3-\frac {2}{27} (125-162 \log (3))} \, dx,x,-\frac {5}{3}+x\right )\\ &=\frac {400 \log (3)}{3 \left (5 x^2-x^3-12 \log (3)\right )}-\frac {x}{\log (5)}+\frac {x^2}{\log (5)}-\frac {1}{3} (16 \log (3)) \operatorname {Subst}\left (\int \frac {-2 x (125-54 \log (3))+\frac {1}{3} (-10 (125-54 \log (3))+540 \log (3))}{\left (-\frac {25 x}{3}+x^3-\frac {2}{27} (125-162 \log (3))\right )^2} \, dx,x,-\frac {5}{3}+x\right )+(16 \log (3)) \operatorname {Subst}\left (\int \frac {\frac {20}{3}+x}{\left (x+\frac {25+\left (-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}\right )^{2/3}}{3 \sqrt [3]{-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}}}\right ) \left (x^2-\frac {x \left (25+\left (-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}\right )^{2/3}\right )}{3 \sqrt [3]{-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}}}+\frac {1}{9} \left (-25+\frac {625}{\left (-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}\right )^{2/3}}+\left (-125+162 \log (3)-18 i \sqrt {(125-81 \log (3)) \log (3)}\right )^{2/3}\right )\right )} \, dx,x,-\frac {5}{3}+x\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B] time = 0.06, size = 90, normalized size = 2.37 \begin {gather*} \frac {x \left (x^3 (750-486 \log (3))+12 (125-81 \log (3)) \log (3)+5 x^2 (-125+81 \log (3))+x^4 (-125+81 \log (3))-4 x \log (3) (375-500 \log (5)+81 \log (3) (-3+\log (625)))\right )}{\left (-5 x^2+x^3+12 \log (3)\right ) (-125+81 \log (3)) \log (5)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-25*x^4 + 60*x^5 - 21*x^6 + 2*x^7 + (120*x^2 - 264*x^3 + 48*x^4)*Log[3] + (-144 + 288*x)*Log[3]^2 +

(16*x^4*Log[3] - 384*x*Log[3]^2)*Log[5])/((25*x^4 - 10*x^5 + x^6 + (-120*x^2 + 24*x^3)*Log[3] + 144*Log[3]^2)

*Log[5]),x]

[Out]

(x*(x^3*(750 - 486*Log[3]) + 12*(125 - 81*Log[3])*Log[3] + 5*x^2*(-125 + 81*Log[3]) + x^4*(-125 + 81*Log[3]) -

4*x*Log[3]*(375 - 500*Log[5] + 81*Log[3]*(-3 + Log[625]))))/((-5*x^2 + x^3 + 12*Log[3])*(-125 + 81*Log[3])*Lo

g[5])

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fricas [A] time = 0.60, size = 54, normalized size = 1.42 \begin {gather*} \frac {x^{5} - 6 \, x^{4} - 16 \, x^{2} \log \relax (5) \log \relax (3) + 5 \, x^{3} + 12 \, {\left (x^{2} - x\right )} \log \relax (3)}{{\left (x^{3} - 5 \, x^{2} + 12 \, \log \relax (3)\right )} \log \relax (5)} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-384*x*log(3)^2+16*x^4*log(3))*log(5)+(288*x-144)*log(3)^2+(48*x^4-264*x^3+120*x^2)*log(3)+2*x^7-2

1*x^6+60*x^5-25*x^4)/(144*log(3)^2+(24*x^3-120*x^2)*log(3)+x^6-10*x^5+25*x^4)/log(5),x, algorithm="fricas")

[Out]

(x^5 - 6*x^4 - 16*x^2*log(5)*log(3) + 5*x^3 + 12*(x^2 - x)*log(3))/((x^3 - 5*x^2 + 12*log(3))*log(5))

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giac [A] time = 0.13, size = 37, normalized size = 0.97 \begin {gather*} -\frac {\frac {16 \, x^{2} \log \relax (5) \log \relax (3)}{x^{3} - 5 \, x^{2} + 12 \, \log \relax (3)} - x^{2} + x}{\log \relax (5)} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-384*x*log(3)^2+16*x^4*log(3))*log(5)+(288*x-144)*log(3)^2+(48*x^4-264*x^3+120*x^2)*log(3)+2*x^7-2

1*x^6+60*x^5-25*x^4)/(144*log(3)^2+(24*x^3-120*x^2)*log(3)+x^6-10*x^5+25*x^4)/log(5),x, algorithm="giac")

[Out]

-(16*x^2*log(5)*log(3)/(x^3 - 5*x^2 + 12*log(3)) - x^2 + x)/log(5)

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maple [A] time = 0.12, size = 37, normalized size = 0.97


method	result	size

default	\(\frac {x^{2}-x -\frac {4 \ln \relax (3) \ln \relax (5) x^{2}}{3 \left (\frac {x^{3}}{12}-\frac {5 x^{2}}{12}+\ln \relax (3)\right )}}{\ln \relax (5)}\)	\(37\)
risch	\(\frac {x^{2}}{\ln \relax (5)}-\frac {x}{\ln \relax (5)}-\frac {4 \ln \relax (3) x^{2}}{3 \left (\frac {x^{3}}{12}-\frac {5 x^{2}}{12}+\ln \relax (3)\right )}\)	\(39\)
gosper	\(-\frac {-x^{5}+16 x^{2} \ln \relax (3) \ln \relax (5)+6 x^{4}-12 x^{2} \ln \relax (3)+12 x \ln \relax (3)-25 x^{2}+60 \ln \relax (3)}{\ln \relax (5) \left (x^{3}-5 x^{2}+12 \ln \relax (3)\right )}\)	\(63\)
norman	\(\frac {-\frac {\left (16 \ln \relax (3) \ln \relax (5)-12 \ln \relax (3)-25\right ) x^{2}}{\ln \relax (5)}+\frac {x^{5}}{\ln \relax (5)}-\frac {6 x^{4}}{\ln \relax (5)}-\frac {12 \ln \relax (3) x}{\ln \relax (5)}-\frac {60 \ln \relax (3)}{\ln \relax (5)}}{x^{3}-5 x^{2}+12 \ln \relax (3)}\)	\(73\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((-384*x*ln(3)^2+16*x^4*ln(3))*ln(5)+(288*x-144)*ln(3)^2+(48*x^4-264*x^3+120*x^2)*ln(3)+2*x^7-21*x^6+60*x^

5-25*x^4)/(144*ln(3)^2+(24*x^3-120*x^2)*ln(3)+x^6-10*x^5+25*x^4)/ln(5),x,method=_RETURNVERBOSE)

[Out]

1/ln(5)*(x^2-x-4/3*ln(3)*ln(5)*x^2/(1/12*x^3-5/12*x^2+ln(3)))

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maxima [A] time = 0.37, size = 37, normalized size = 0.97 \begin {gather*} -\frac {\frac {16 \, x^{2} \log \relax (5) \log \relax (3)}{x^{3} - 5 \, x^{2} + 12 \, \log \relax (3)} - x^{2} + x}{\log \relax (5)} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-384*x*log(3)^2+16*x^4*log(3))*log(5)+(288*x-144)*log(3)^2+(48*x^4-264*x^3+120*x^2)*log(3)+2*x^7-2

1*x^6+60*x^5-25*x^4)/(144*log(3)^2+(24*x^3-120*x^2)*log(3)+x^6-10*x^5+25*x^4)/log(5),x, algorithm="maxima")

[Out]

-(16*x^2*log(5)*log(3)/(x^3 - 5*x^2 + 12*log(3)) - x^2 + x)/log(5)

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mupad [B] time = 0.38, size = 38, normalized size = 1.00 \begin {gather*} \frac {x^2}{\ln \relax (5)}-\frac {x}{\ln \relax (5)}-\frac {16\,x^2\,\ln \relax (3)}{x^3-5\,x^2+12\,\ln \relax (3)} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((log(3)^2*(288*x - 144) - log(5)*(384*x*log(3)^2 - 16*x^4*log(3)) + log(3)*(120*x^2 - 264*x^3 + 48*x^4) -

25*x^4 + 60*x^5 - 21*x^6 + 2*x^7)/(log(5)*(144*log(3)^2 - log(3)*(120*x^2 - 24*x^3) + 25*x^4 - 10*x^5 + x^6)),

[Out]

x^2/log(5) - x/log(5) - (16*x^2*log(3))/(12*log(3) - 5*x^2 + x^3)

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sympy [A] time = 0.72, size = 32, normalized size = 0.84 \begin {gather*} \frac {x^{2}}{\log {\relax (5 )}} - \frac {16 x^{2} \log {\relax (3 )}}{x^{3} - 5 x^{2} + 12 \log {\relax (3 )}} - \frac {x}{\log {\relax (5 )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-384*x*ln(3)**2+16*x**4*ln(3))*ln(5)+(288*x-144)*ln(3)**2+(48*x**4-264*x**3+120*x**2)*ln(3)+2*x**7

-21*x**6+60*x**5-25*x**4)/(144*ln(3)**2+(24*x**3-120*x**2)*ln(3)+x**6-10*x**5+25*x**4)/ln(5),x)

[Out]

x**2/log(5) - 16*x**2*log(3)/(x**3 - 5*x**2 + 12*log(3)) - x/log(5)

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