3.83.37 \(\int \frac {-686 x-1064 x^2-672 x^3+128 x^4+(-686 x+56 x^2-32 x^3+128 x^4) \log (\frac {2401 x^2+2352 x^3+2144 x^4+768 x^5+256 x^6}{2401-5488 x+4704 x^2-1792 x^3+256 x^4})}{-343+28 x-16 x^2+64 x^3} \, dx\)

Optimal. Leaf size=23 \[ x^2 \log \left (\left (x+\frac {5 x^2}{\left (\frac {7}{4}-x\right )^2}\right )^2\right ) \]

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Rubi [A]  time = 0.66, antiderivative size = 28, normalized size of antiderivative = 1.22, number of steps used = 36, number of rules used = 10, integrand size = 107, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {6688, 12, 6742, 1628, 634, 618, 204, 628, 2074, 2525} \begin {gather*} x^2 \log \left (\frac {x^2 \left (16 x^2+24 x+49\right )^2}{(7-4 x)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-686*x - 1064*x^2 - 672*x^3 + 128*x^4 + (-686*x + 56*x^2 - 32*x^3 + 128*x^4)*Log[(2401*x^2 + 2352*x^3 + 2
144*x^4 + 768*x^5 + 256*x^6)/(2401 - 5488*x + 4704*x^2 - 1792*x^3 + 256*x^4)])/(-343 + 28*x - 16*x^2 + 64*x^3)
,x]

[Out]

x^2*Log[(x^2*(49 + 24*x + 16*x^2)^2)/(7 - 4*x)^4]

Rule 12

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 6688

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

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 (343+532 x+336 x^2-64 x^3-\left (-343+28 x-16 x^2+64 x^3\right ) \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(-7+4 x)^4}\right )\right )}{343-28 x+16 x^2-64 x^3} \, dx\\ &=2 \int \frac {x \left (343+532 x+336 x^2-64 x^3-\left (-343+28 x-16 x^2+64 x^3\right ) \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(-7+4 x)^4}\right )\right )}{343-28 x+16 x^2-64 x^3} \, dx\\ &=2 \int \left (\frac {64 x^4}{(-7+4 x) \left (49+24 x+16 x^2\right )}-\frac {343 x}{-343+28 x-16 x^2+64 x^3}-\frac {532 x^2}{-343+28 x-16 x^2+64 x^3}-\frac {336 x^3}{-343+28 x-16 x^2+64 x^3}+x \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(-7+4 x)^4}\right )\right ) \, dx\\ &=2 \int x \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(-7+4 x)^4}\right ) \, dx+128 \int \frac {x^4}{(-7+4 x) \left (49+24 x+16 x^2\right )} \, dx-672 \int \frac {x^3}{-343+28 x-16 x^2+64 x^3} \, dx-686 \int \frac {x}{-343+28 x-16 x^2+64 x^3} \, dx-1064 \int \frac {x^2}{-343+28 x-16 x^2+64 x^3} \, dx\\ &=x^2 \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(7-4 x)^4}\right )+128 \int \left (\frac {1}{256}+\frac {x}{64}+\frac {343}{5120 (-7+4 x)}+\frac {1421-1852 x}{5120 \left (49+24 x+16 x^2\right )}\right ) \, dx-672 \int \left (\frac {1}{64}+\frac {49}{1280 (-7+4 x)}+\frac {-637-116 x}{1280 \left (49+24 x+16 x^2\right )}\right ) \, dx-686 \int \left (\frac {1}{80 (-7+4 x)}+\frac {7-4 x}{80 \left (49+24 x+16 x^2\right )}\right ) \, dx-1064 \int \left (\frac {7}{320 (-7+4 x)}+\frac {49+52 x}{320 \left (49+24 x+16 x^2\right )}\right ) \, dx-\int \frac {2 x \left (343+532 x+336 x^2-64 x^3\right )}{(7-4 x) \left (49+24 x+16 x^2\right )} \, dx\\ &=-10 x+x^2-\frac {49}{4} \log (7-4 x)+x^2 \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(7-4 x)^4}\right )+\frac {1}{40} \int \frac {1421-1852 x}{49+24 x+16 x^2} \, dx-\frac {21}{40} \int \frac {-637-116 x}{49+24 x+16 x^2} \, dx-2 \int \frac {x \left (343+532 x+336 x^2-64 x^3\right )}{(7-4 x) \left (49+24 x+16 x^2\right )} \, dx-\frac {133}{40} \int \frac {49+52 x}{49+24 x+16 x^2} \, dx-\frac {343}{40} \int \frac {7-4 x}{49+24 x+16 x^2} \, dx\\ &=-10 x+x^2-\frac {49}{4} \log (7-4 x)+x^2 \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(7-4 x)^4}\right )+\frac {343}{320} \int \frac {24+32 x}{49+24 x+16 x^2} \, dx-\frac {463}{320} \int \frac {24+32 x}{49+24 x+16 x^2} \, dx+\frac {609}{320} \int \frac {24+32 x}{49+24 x+16 x^2} \, dx-2 \int \left (-5+x-\frac {49}{2 (-7+4 x)}+\frac {147-124 x}{2 \left (49+24 x+16 x^2\right )}\right ) \, dx-\frac {1729}{320} \int \frac {24+32 x}{49+24 x+16 x^2} \, dx-\frac {133}{4} \int \frac {1}{49+24 x+16 x^2} \, dx+\frac {281}{4} \int \frac {1}{49+24 x+16 x^2} \, dx-\frac {343}{4} \int \frac {1}{49+24 x+16 x^2} \, dx+\frac {1155}{4} \int \frac {1}{49+24 x+16 x^2} \, dx\\ &=-\frac {31}{8} \log \left (49+24 x+16 x^2\right )+x^2 \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(7-4 x)^4}\right )+\frac {133}{2} \operatorname {Subst}\left (\int \frac {1}{-2560-x^2} \, dx,x,24+32 x\right )-\frac {281}{2} \operatorname {Subst}\left (\int \frac {1}{-2560-x^2} \, dx,x,24+32 x\right )+\frac {343}{2} \operatorname {Subst}\left (\int \frac {1}{-2560-x^2} \, dx,x,24+32 x\right )-\frac {1155}{2} \operatorname {Subst}\left (\int \frac {1}{-2560-x^2} \, dx,x,24+32 x\right )-\int \frac {147-124 x}{49+24 x+16 x^2} \, dx\\ &=3 \sqrt {10} \tan ^{-1}\left (\frac {3+4 x}{2 \sqrt {10}}\right )-\frac {31}{8} \log \left (49+24 x+16 x^2\right )+x^2 \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(7-4 x)^4}\right )+\frac {31}{8} \int \frac {24+32 x}{49+24 x+16 x^2} \, dx-240 \int \frac {1}{49+24 x+16 x^2} \, dx\\ &=3 \sqrt {10} \tan ^{-1}\left (\frac {3+4 x}{2 \sqrt {10}}\right )+x^2 \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(7-4 x)^4}\right )+480 \operatorname {Subst}\left (\int \frac {1}{-2560-x^2} \, dx,x,24+32 x\right )\\ &=x^2 \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(7-4 x)^4}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 28, normalized size = 1.22 \begin {gather*} x^2 \log \left (\frac {x^2 \left (49+24 x+16 x^2\right )^2}{(-7+4 x)^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-686*x - 1064*x^2 - 672*x^3 + 128*x^4 + (-686*x + 56*x^2 - 32*x^3 + 128*x^4)*Log[(2401*x^2 + 2352*x
^3 + 2144*x^4 + 768*x^5 + 256*x^6)/(2401 - 5488*x + 4704*x^2 - 1792*x^3 + 256*x^4)])/(-343 + 28*x - 16*x^2 + 6
4*x^3),x]

[Out]

x^2*Log[(x^2*(49 + 24*x + 16*x^2)^2)/(-7 + 4*x)^4]

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fricas [B]  time = 0.50, size = 54, normalized size = 2.35 \begin {gather*} x^{2} \log \left (\frac {256 \, x^{6} + 768 \, x^{5} + 2144 \, x^{4} + 2352 \, x^{3} + 2401 \, x^{2}}{256 \, x^{4} - 1792 \, x^{3} + 4704 \, x^{2} - 5488 \, x + 2401}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^4-32*x^3+56*x^2-686*x)*log((256*x^6+768*x^5+2144*x^4+2352*x^3+2401*x^2)/(256*x^4-1792*x^3+47
04*x^2-5488*x+2401))+128*x^4-672*x^3-1064*x^2-686*x)/(64*x^3-16*x^2+28*x-343),x, algorithm="fricas")

[Out]

x^2*log((256*x^6 + 768*x^5 + 2144*x^4 + 2352*x^3 + 2401*x^2)/(256*x^4 - 1792*x^3 + 4704*x^2 - 5488*x + 2401))

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giac [B]  time = 0.27, size = 54, normalized size = 2.35 \begin {gather*} x^{2} \log \left (\frac {256 \, x^{6} + 768 \, x^{5} + 2144 \, x^{4} + 2352 \, x^{3} + 2401 \, x^{2}}{256 \, x^{4} - 1792 \, x^{3} + 4704 \, x^{2} - 5488 \, x + 2401}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^4-32*x^3+56*x^2-686*x)*log((256*x^6+768*x^5+2144*x^4+2352*x^3+2401*x^2)/(256*x^4-1792*x^3+47
04*x^2-5488*x+2401))+128*x^4-672*x^3-1064*x^2-686*x)/(64*x^3-16*x^2+28*x-343),x, algorithm="giac")

[Out]

x^2*log((256*x^6 + 768*x^5 + 2144*x^4 + 2352*x^3 + 2401*x^2)/(256*x^4 - 1792*x^3 + 4704*x^2 - 5488*x + 2401))

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maple [B]  time = 0.08, size = 52, normalized size = 2.26

method	result	size
default	\(\ln \left (\frac {x^{2} \left (256 x^{4}+768 x^{3}+2144 x^{2}+2352 x +2401\right )}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right ) x^{2}\)	\(52\)
norman	\(x^{2} \ln \left (\frac {256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right )\)	\(55\)
risch	\(x^{2} \ln \left (\frac {256 x^{6}+768 x^{5}+2144 x^{4}+2352 x^{3}+2401 x^{2}}{256 x^{4}-1792 x^{3}+4704 x^{2}-5488 x +2401}\right )\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((128*x^4-32*x^3+56*x^2-686*x)*ln((256*x^6+768*x^5+2144*x^4+2352*x^3+2401*x^2)/(256*x^4-1792*x^3+4704*x^2-
5488*x+2401))+128*x^4-672*x^3-1064*x^2-686*x)/(64*x^3-16*x^2+28*x-343),x,method=_RETURNVERBOSE)

[Out]

ln(x^2*(256*x^4+768*x^3+2144*x^2+2352*x+2401)/(256*x^4-1792*x^3+4704*x^2-5488*x+2401))*x^2

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maxima [B]  time = 1.17, size = 64, normalized size = 2.78 \begin {gather*} 2 \, x^{2} \log \relax (x) + \frac {1}{8} \, {\left (16 \, x^{2} + 31\right )} \log \left (16 \, x^{2} + 24 \, x + 49\right ) - \frac {1}{4} \, {\left (16 \, x^{2} - 49\right )} \log \left (4 \, x - 7\right ) - \frac {31}{8} \, \log \left (16 \, x^{2} + 24 \, x + 49\right ) - \frac {49}{4} \, \log \left (4 \, x - 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x^4-32*x^3+56*x^2-686*x)*log((256*x^6+768*x^5+2144*x^4+2352*x^3+2401*x^2)/(256*x^4-1792*x^3+47
04*x^2-5488*x+2401))+128*x^4-672*x^3-1064*x^2-686*x)/(64*x^3-16*x^2+28*x-343),x, algorithm="maxima")

[Out]

2*x^2*log(x) + 1/8*(16*x^2 + 31)*log(16*x^2 + 24*x + 49) - 1/4*(16*x^2 - 49)*log(4*x - 7) - 31/8*log(16*x^2 +
24*x + 49) - 49/4*log(4*x - 7)

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mupad [B]  time = 5.21, size = 55, normalized size = 2.39 \begin {gather*} x^2\,\left (\ln \left (\frac {1}{256\,x^4-1792\,x^3+4704\,x^2-5488\,x+2401}\right )+\ln \left (256\,x^6+768\,x^5+2144\,x^4+2352\,x^3+2401\,x^2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(686*x + log((2401*x^2 + 2352*x^3 + 2144*x^4 + 768*x^5 + 256*x^6)/(4704*x^2 - 5488*x - 1792*x^3 + 256*x^4
+ 2401))*(686*x - 56*x^2 + 32*x^3 - 128*x^4) + 1064*x^2 + 672*x^3 - 128*x^4)/(28*x - 16*x^2 + 64*x^3 - 343),x
)

[Out]

x^2*(log(1/(4704*x^2 - 5488*x - 1792*x^3 + 256*x^4 + 2401)) + log(2401*x^2 + 2352*x^3 + 2144*x^4 + 768*x^5 + 2
56*x^6))

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sympy [B]  time = 0.26, size = 49, normalized size = 2.13 \begin {gather*} x^{2} \log {\left (\frac {256 x^{6} + 768 x^{5} + 2144 x^{4} + 2352 x^{3} + 2401 x^{2}}{256 x^{4} - 1792 x^{3} + 4704 x^{2} - 5488 x + 2401} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((128*x**4-32*x**3+56*x**2-686*x)*ln((256*x**6+768*x**5+2144*x**4+2352*x**3+2401*x**2)/(256*x**4-179
2*x**3+4704*x**2-5488*x+2401))+128*x**4-672*x**3-1064*x**2-686*x)/(64*x**3-16*x**2+28*x-343),x)

[Out]

x**2*log((256*x**6 + 768*x**5 + 2144*x**4 + 2352*x**3 + 2401*x**2)/(256*x**4 - 1792*x**3 + 4704*x**2 - 5488*x
+ 2401))

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