3.68.56 \(\int \frac {20-3 x^2+8 x^3+(20-32 x+3 x^2-4 x^3) \log (\frac {-20+32 x-3 x^2+4 x^3}{4 x})}{-20 x^2+32 x^3-3 x^4+4 x^5} \, dx\)

Optimal. Leaf size=24 \[ 5+\frac {\log \left (8-\frac {5}{x}+\frac {5 x}{4}+(-2+x) x\right )}{x} \]

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Rubi [A]  time = 5.74, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 26, number of rules used = 11, integrand size = 75, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {6741, 6742, 2100, 2081, 2079, 800, 634, 618, 204, 628, 2525} \begin {gather*} \frac {\log \left (x^2-\frac {3 x}{4}-\frac {5}{x}+8\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(20 - 3*x^2 + 8*x^3 + (20 - 32*x + 3*x^2 - 4*x^3)*Log[(-20 + 32*x - 3*x^2 + 4*x^3)/(4*x)])/(-20*x^2 + 32*x
^3 - 3*x^4 + 4*x^5),x]

[Out]

Log[8 - 5/x - (3*x)/4 + x^2]/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 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 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 2100

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Log[Qn])/(n*Coe
ff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, 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 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} &=\int \frac {-20+3 x^2-8 x^3-\left (20-32 x+3 x^2-4 x^3\right ) \log \left (\frac {-20+32 x-3 x^2+4 x^3}{4 x}\right )}{x^2 \left (20-32 x+3 x^2-4 x^3\right )} \, dx\\ &=\int \left (\frac {20-3 x^2+8 x^3}{x^2 \left (-20+32 x-3 x^2+4 x^3\right )}-\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x^2}\right ) \, dx\\ &=\int \frac {20-3 x^2+8 x^3}{x^2 \left (-20+32 x-3 x^2+4 x^3\right )} \, dx-\int \frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x^2} \, dx\\ &=\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\int \frac {\frac {3}{4}-\frac {5}{x^2}-2 x}{5-8 x+\frac {3 x^2}{4}-x^3} \, dx+\int \left (-\frac {1}{x^2}-\frac {8}{5 x}+\frac {2 \left (113+18 x+16 x^2\right )}{5 \left (-20+32 x-3 x^2+4 x^3\right )}\right ) \, dx\\ &=\frac {1}{x}-\frac {8 \log (x)}{5}+\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}+\frac {2}{5} \int \frac {113+18 x+16 x^2}{-20+32 x-3 x^2+4 x^3} \, dx-\int \left (-\frac {1}{x^2}-\frac {8}{5 x}+\frac {2 \left (113+18 x+16 x^2\right )}{5 \left (-20+32 x-3 x^2+4 x^3\right )}\right ) \, dx\\ &=\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}+\frac {8}{15} \log \left (-20+32 x-3 x^2+4 x^3\right )+\frac {1}{30} \int \frac {844+312 x}{-20+32 x-3 x^2+4 x^3} \, dx-\frac {2}{5} \int \frac {113+18 x+16 x^2}{-20+32 x-3 x^2+4 x^3} \, dx\\ &=\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\frac {1}{30} \int \frac {844+312 x}{-20+32 x-3 x^2+4 x^3} \, dx+\frac {1}{30} \operatorname {Subst}\left (\int \frac {922+312 x}{-\frac {97}{8}+\frac {125 x}{4}+4 x^3} \, dx,x,-\frac {1}{4}+x\right )\\ &=\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\frac {1}{30} \operatorname {Subst}\left (\int \frac {922+312 x}{-\frac {97}{8}+\frac {125 x}{4}+4 x^3} \, dx,x,-\frac {1}{4}+x\right )+\frac {8}{15} \operatorname {Subst}\left (\int \frac {922+312 x}{\left (\frac {1}{3} \left (\frac {125\ 3^{2/3}}{\sqrt [3]{873+8 \sqrt {103461}}}-\sqrt [3]{3 \left (873+8 \sqrt {103461}\right )}\right )+4 x\right ) \left (\frac {1}{9} \left (375+\frac {46875 \sqrt [3]{3}}{\left (873+8 \sqrt {103461}\right )^{2/3}}+\left (3 \left (873+8 \sqrt {103461}\right )\right )^{2/3}\right )-\frac {4 \left (125 \sqrt [3]{\frac {3}{873+8 \sqrt {103461}}}-\sqrt [3]{873+8 \sqrt {103461}}\right ) x}{3^{2/3}}+16 x^2\right )} \, dx,x,-\frac {1}{4}+x\right )\\ &=\frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x}-\frac {8}{15} \operatorname {Subst}\left (\int \frac {922+312 x}{\left (\frac {1}{3} \left (\frac {125\ 3^{2/3}}{\sqrt [3]{873+8 \sqrt {103461}}}-\sqrt [3]{3 \left (873+8 \sqrt {103461}\right )}\right )+4 x\right ) \left (\frac {1}{9} \left (375+\frac {46875 \sqrt [3]{3}}{\left (873+8 \sqrt {103461}\right )^{2/3}}+\left (3 \left (873+8 \sqrt {103461}\right )\right )^{2/3}\right )-\frac {4 \left (125 \sqrt [3]{\frac {3}{873+8 \sqrt {103461}}}-\sqrt [3]{873+8 \sqrt {103461}}\right ) x}{3^{2/3}}+16 x^2\right )} \, dx,x,-\frac {1}{4}+x\right )+\frac {8}{15} \operatorname {Subst}\left (\int \left (\frac {6 \left (873+8 \sqrt {103461}\right )^{2/3} \left (-1625 3^{2/3}+461 \sqrt [3]{873+8 \sqrt {103461}}+13 \sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}\right )}{\left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right ) \left (125\ 3^{2/3}-\sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}+12 \sqrt [3]{873+8 \sqrt {103461}} x\right )}+\frac {2 \left (873+8 \sqrt {103461}\right )^{2/3} \left (\left (126599\ 3^{2/3}+312 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+4875 \left (873+8 \sqrt {103461}\right )^{2/3}-\sqrt [3]{3} \left (195531+7376 \sqrt {103461}\right )+12 \sqrt [3]{873+8 \sqrt {103461}} \left (1625\ 3^{2/3}-461 \sqrt [3]{873+8 \sqrt {103461}}-13 \sqrt [3]{3} \left (873+8 \sqrt {103461}\right )^{2/3}\right ) x\right )}{\left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}-125 \left (873+8 \sqrt {103461}\right )^{2/3}\right ) \left (15625 \sqrt [3]{3}+\left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}\right ) \sqrt [3]{873+8 \sqrt {103461}}+125 \left (873+8 \sqrt {103461}\right )^{2/3}+4\ 3^{2/3} \left (291\ 3^{2/3}+8 \sqrt [6]{3} \sqrt {34487}-125 \sqrt [3]{873+8 \sqrt {103461}}\right ) x+48 \left (873+8 \sqrt {103461}\right )^{2/3} x^2\right )}\right ) \, dx,x,-\frac {1}{4}+x\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 20, normalized size = 0.83 \begin {gather*} \frac {\log \left (8-\frac {5}{x}-\frac {3 x}{4}+x^2\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20 - 3*x^2 + 8*x^3 + (20 - 32*x + 3*x^2 - 4*x^3)*Log[(-20 + 32*x - 3*x^2 + 4*x^3)/(4*x)])/(-20*x^2
+ 32*x^3 - 3*x^4 + 4*x^5),x]

[Out]

Log[8 - 5/x - (3*x)/4 + x^2]/x

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fricas [A]  time = 0.46, size = 25, normalized size = 1.04 \begin {gather*} \frac {\log \left (\frac {4 \, x^{3} - 3 \, x^{2} + 32 \, x - 20}{4 \, x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+3*x^2-32*x+20)*log(1/4*(4*x^3-3*x^2+32*x-20)/x)+8*x^3-3*x^2+20)/(4*x^5-3*x^4+32*x^3-20*x^2)
,x, algorithm="fricas")

[Out]

log(1/4*(4*x^3 - 3*x^2 + 32*x - 20)/x)/x

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giac [A]  time = 0.19, size = 25, normalized size = 1.04 \begin {gather*} \frac {\log \left (\frac {4 \, x^{3} - 3 \, x^{2} + 32 \, x - 20}{4 \, x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+3*x^2-32*x+20)*log(1/4*(4*x^3-3*x^2+32*x-20)/x)+8*x^3-3*x^2+20)/(4*x^5-3*x^4+32*x^3-20*x^2)
,x, algorithm="giac")

[Out]

log(1/4*(4*x^3 - 3*x^2 + 32*x - 20)/x)/x

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maple [A]  time = 0.06, size = 26, normalized size = 1.08

method	result	size
norman	\(\frac {\ln \left (\frac {4 x^{3}-3 x^{2}+32 x -20}{4 x}\right )}{x}\)	\(26\)
risch	\(\frac {\ln \left (\frac {4 x^{3}-3 x^{2}+32 x -20}{4 x}\right )}{x}\)	\(26\)
default	\(\frac {\ln \left (\frac {4 x^{3}-3 x^{2}+32 x -20}{x}\right )}{x}-\frac {2 \ln \relax (2)}{x}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3+3*x^2-32*x+20)*ln(1/4*(4*x^3-3*x^2+32*x-20)/x)+8*x^3-3*x^2+20)/(4*x^5-3*x^4+32*x^3-20*x^2),x,meth
od=_RETURNVERBOSE)

[Out]

ln(1/4*(4*x^3-3*x^2+32*x-20)/x)/x

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maxima [A]  time = 0.46, size = 30, normalized size = 1.25 \begin {gather*} -\frac {2 \, \log \relax (2) - \log \left (4 \, x^{3} - 3 \, x^{2} + 32 \, x - 20\right ) + \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3+3*x^2-32*x+20)*log(1/4*(4*x^3-3*x^2+32*x-20)/x)+8*x^3-3*x^2+20)/(4*x^5-3*x^4+32*x^3-20*x^2)
,x, algorithm="maxima")

[Out]

-(2*log(2) - log(4*x^3 - 3*x^2 + 32*x - 20) + log(x))/x

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mupad [B]  time = 4.16, size = 22, normalized size = 0.92 \begin {gather*} \frac {\ln \left (\frac {x^3-\frac {3\,x^2}{4}+8\,x-5}{x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x^2 - 8*x^3 + log((8*x - (3*x^2)/4 + x^3 - 5)/x)*(32*x - 3*x^2 + 4*x^3 - 20) - 20)/(20*x^2 - 32*x^3 + 3
*x^4 - 4*x^5),x)

[Out]

log((8*x - (3*x^2)/4 + x^3 - 5)/x)/x

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sympy [A]  time = 0.19, size = 19, normalized size = 0.79 \begin {gather*} \frac {\log {\left (\frac {x^{3} - \frac {3 x^{2}}{4} + 8 x - 5}{x} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3+3*x**2-32*x+20)*ln(1/4*(4*x**3-3*x**2+32*x-20)/x)+8*x**3-3*x**2+20)/(4*x**5-3*x**4+32*x**3
-20*x**2),x)

[Out]

log((x**3 - 3*x**2/4 + 8*x - 5)/x)/x

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