3.99.43 \(\int \frac {24 x+6 x^2-21 x^3+5 x^4+5 x^5+(4+6 x-5 x^2-5 x^3) \log (\frac {-100+100 x}{(16+80 x+140 x^2+100 x^3+25 x^4) \log (2)})}{-4 x^2-6 x^3+5 x^4+5 x^5} \, dx\)

Optimal. Leaf size=29 \[ 5+x+\frac {\log \left (\frac {4 (-1+x)}{\left (\frac {4}{5}+2 x+x^2\right )^2 \log (2)}\right )}{x} \]

________________________________________________________________________________________

Rubi [B]  time = 1.15, antiderivative size = 382, normalized size of antiderivative = 13.17, number of steps used = 34, number of rules used = 8, integrand size = 97, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.082, Rules used = {6741, 6742, 2058, 632, 31, 2074, 2525, 6728} \begin {gather*} \frac {\log \left (-\frac {100 (1-x)}{\left (5 x^2+10 x+4\right )^2 \log (2)}\right )}{x}+x+\frac {3}{19} \left (15+11 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )-\frac {3}{19} \left (1+2 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )-\frac {1}{2} \left (5+\sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )+\frac {1}{19} \left (7-5 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )-\frac {4}{95} \left (15-8 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )+\frac {21}{190} \left (5-9 \sqrt {5}\right ) \log \left (-\sqrt {5} x-\sqrt {5}+1\right )+\frac {21}{190} \left (5+9 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )-\frac {4}{95} \left (15+8 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )+\frac {1}{19} \left (7+5 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )-\frac {1}{2} \left (5-\sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )-\frac {3}{19} \left (1-2 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right )+\frac {3}{19} \left (15-11 \sqrt {5}\right ) \log \left (\sqrt {5} x+\sqrt {5}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(24*x + 6*x^2 - 21*x^3 + 5*x^4 + 5*x^5 + (4 + 6*x - 5*x^2 - 5*x^3)*Log[(-100 + 100*x)/((16 + 80*x + 140*x^
2 + 100*x^3 + 25*x^4)*Log[2])])/(-4*x^2 - 6*x^3 + 5*x^4 + 5*x^5),x]

[Out]

x + (21*(5 - 9*Sqrt[5])*Log[1 - Sqrt[5] - Sqrt[5]*x])/190 - (4*(15 - 8*Sqrt[5])*Log[1 - Sqrt[5] - Sqrt[5]*x])/
95 + ((7 - 5*Sqrt[5])*Log[1 - Sqrt[5] - Sqrt[5]*x])/19 - ((5 + Sqrt[5])*Log[1 - Sqrt[5] - Sqrt[5]*x])/2 - (3*(
1 + 2*Sqrt[5])*Log[1 - Sqrt[5] - Sqrt[5]*x])/19 + (3*(15 + 11*Sqrt[5])*Log[1 - Sqrt[5] - Sqrt[5]*x])/19 + (3*(
15 - 11*Sqrt[5])*Log[1 + Sqrt[5] + Sqrt[5]*x])/19 - (3*(1 - 2*Sqrt[5])*Log[1 + Sqrt[5] + Sqrt[5]*x])/19 - ((5
- Sqrt[5])*Log[1 + Sqrt[5] + Sqrt[5]*x])/2 + ((7 + 5*Sqrt[5])*Log[1 + Sqrt[5] + Sqrt[5]*x])/19 - (4*(15 + 8*Sq
rt[5])*Log[1 + Sqrt[5] + Sqrt[5]*x])/95 + (21*(5 + 9*Sqrt[5])*Log[1 + Sqrt[5] + Sqrt[5]*x])/190 + Log[(-100*(1
- x))/((4 + 10*x + 5*x^2)^2*Log[2])]/x

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), 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 2058

Int[(P_)^(p_), x_Symbol] :> With[{u = Factor[P]}, Int[ExpandIntegrand[u^p, x], x] /;  !SumQ[NonfreeFactors[u,
x]]] /; PolyQ[P, x] && ILtQ[p, 0]

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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

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 {-24 x-6 x^2+21 x^3-5 x^4-5 x^5-\left (4+6 x-5 x^2-5 x^3\right ) \log \left (\frac {-100+100 x}{\left (16+80 x+140 x^2+100 x^3+25 x^4\right ) \log (2)}\right )}{x^2 \left (4+6 x-5 x^2-5 x^3\right )} \, dx\\ &=\int \left (\frac {6}{-4-6 x+5 x^2+5 x^3}+\frac {24}{x \left (-4-6 x+5 x^2+5 x^3\right )}-\frac {21 x}{-4-6 x+5 x^2+5 x^3}+\frac {5 x^2}{-4-6 x+5 x^2+5 x^3}+\frac {5 x^3}{-4-6 x+5 x^2+5 x^3}-\frac {\log \left (\frac {100 (-1+x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x^2}\right ) \, dx\\ &=5 \int \frac {x^2}{-4-6 x+5 x^2+5 x^3} \, dx+5 \int \frac {x^3}{-4-6 x+5 x^2+5 x^3} \, dx+6 \int \frac {1}{-4-6 x+5 x^2+5 x^3} \, dx-21 \int \frac {x}{-4-6 x+5 x^2+5 x^3} \, dx+24 \int \frac {1}{x \left (-4-6 x+5 x^2+5 x^3\right )} \, dx-\int \frac {\log \left (\frac {100 (-1+x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x^2} \, dx\\ &=\frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x}+5 \int \left (\frac {1}{19 (-1+x)}+\frac {2 (2+7 x)}{19 \left (4+10 x+5 x^2\right )}\right ) \, dx+5 \int \left (\frac {1}{5}+\frac {1}{19 (-1+x)}-\frac {8 (7+15 x)}{95 \left (4+10 x+5 x^2\right )}\right ) \, dx+6 \int \left (\frac {1}{19 (-1+x)}-\frac {5 (3+x)}{19 \left (4+10 x+5 x^2\right )}\right ) \, dx-21 \int \left (\frac {1}{19 (-1+x)}+\frac {4-5 x}{19 \left (4+10 x+5 x^2\right )}\right ) \, dx+24 \int \left (\frac {1}{19 (-1+x)}-\frac {1}{4 x}+\frac {5 (26+15 x)}{76 \left (4+10 x+5 x^2\right )}\right ) \, dx-\int \frac {-24-10 x+15 x^2}{(1-x) x \left (4+10 x+5 x^2\right )} \, dx\\ &=x+\log (1-x)-6 \log (x)+\frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x}-\frac {8}{19} \int \frac {7+15 x}{4+10 x+5 x^2} \, dx+\frac {10}{19} \int \frac {2+7 x}{4+10 x+5 x^2} \, dx-\frac {21}{19} \int \frac {4-5 x}{4+10 x+5 x^2} \, dx-\frac {30}{19} \int \frac {3+x}{4+10 x+5 x^2} \, dx+\frac {30}{19} \int \frac {26+15 x}{4+10 x+5 x^2} \, dx-\int \left (\frac {1}{-1+x}-\frac {6}{x}+\frac {5 (6+5 x)}{4+10 x+5 x^2}\right ) \, dx\\ &=x+\frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x}-5 \int \frac {6+5 x}{4+10 x+5 x^2} \, dx+\frac {1}{19} \left (15 \left (15-11 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx+\frac {1}{38} \left (21 \left (5-9 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx-\frac {1}{19} \left (4 \left (15-8 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx+\frac {1}{19} \left (5 \left (7-5 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx-\frac {1}{19} \left (15 \left (1-2 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx-\frac {1}{19} \left (15 \left (1+2 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx+\frac {1}{19} \left (5 \left (7+5 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx-\frac {1}{19} \left (4 \left (15+8 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx+\frac {1}{38} \left (21 \left (5+9 \sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx+\frac {1}{19} \left (15 \left (15+11 \sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx\\ &=x+\frac {21}{190} \left (5-9 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {4}{95} \left (15-8 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {1}{19} \left (7-5 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {3}{19} \left (1+2 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {3}{19} \left (15+11 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {3}{19} \left (15-11 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {3}{19} \left (1-2 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {1}{19} \left (7+5 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {4}{95} \left (15+8 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {21}{190} \left (5+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x}-\frac {1}{2} \left (5 \left (5-\sqrt {5}\right )\right ) \int \frac {1}{5+\sqrt {5}+5 x} \, dx-\frac {1}{2} \left (5 \left (5+\sqrt {5}\right )\right ) \int \frac {1}{5-\sqrt {5}+5 x} \, dx\\ &=x+\frac {21}{190} \left (5-9 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {4}{95} \left (15-8 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {1}{19} \left (7-5 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {1}{2} \left (5+\sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )-\frac {3}{19} \left (1+2 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {3}{19} \left (15+11 \sqrt {5}\right ) \log \left (1-\sqrt {5}-\sqrt {5} x\right )+\frac {3}{19} \left (15-11 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {3}{19} \left (1-2 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {1}{2} \left (5-\sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {1}{19} \left (7+5 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )-\frac {4}{95} \left (15+8 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {21}{190} \left (5+9 \sqrt {5}\right ) \log \left (1+\sqrt {5}+\sqrt {5} x\right )+\frac {\log \left (-\frac {100 (1-x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 28, normalized size = 0.97 \begin {gather*} x+\frac {\log \left (\frac {100 (-1+x)}{\left (4+10 x+5 x^2\right )^2 \log (2)}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*x + 6*x^2 - 21*x^3 + 5*x^4 + 5*x^5 + (4 + 6*x - 5*x^2 - 5*x^3)*Log[(-100 + 100*x)/((16 + 80*x +
140*x^2 + 100*x^3 + 25*x^4)*Log[2])])/(-4*x^2 - 6*x^3 + 5*x^4 + 5*x^5),x]

[Out]

x + Log[(100*(-1 + x))/((4 + 10*x + 5*x^2)^2*Log[2])]/x

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 40, normalized size = 1.38 \begin {gather*} \frac {x^{2} + \log \left (\frac {100 \, {\left (x - 1\right )}}{{\left (25 \, x^{4} + 100 \, x^{3} + 140 \, x^{2} + 80 \, x + 16\right )} \log \relax (2)}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^3-5*x^2+6*x+4)*log((100*x-100)/(25*x^4+100*x^3+140*x^2+80*x+16)/log(2))+5*x^5+5*x^4-21*x^3+6*
x^2+24*x)/(5*x^5+5*x^4-6*x^3-4*x^2),x, algorithm="fricas")

[Out]

(x^2 + log(100*(x - 1)/((25*x^4 + 100*x^3 + 140*x^2 + 80*x + 16)*log(2))))/x

________________________________________________________________________________________

giac [A]  time = 0.25, size = 56, normalized size = 1.93 \begin {gather*} x + \frac {2 \, \log \relax (2)}{x} - \frac {\log \left (25 \, x^{4} \log \relax (2) + 100 \, x^{3} \log \relax (2) + 140 \, x^{2} \log \relax (2) + 80 \, x \log \relax (2) + 16 \, \log \relax (2)\right )}{x} + \frac {\log \left (25 \, x - 25\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^3-5*x^2+6*x+4)*log((100*x-100)/(25*x^4+100*x^3+140*x^2+80*x+16)/log(2))+5*x^5+5*x^4-21*x^3+6*
x^2+24*x)/(5*x^5+5*x^4-6*x^3-4*x^2),x, algorithm="giac")

[Out]

x + 2*log(2)/x - log(25*x^4*log(2) + 100*x^3*log(2) + 140*x^2*log(2) + 80*x*log(2) + 16*log(2))/x + log(25*x -
25)/x

________________________________________________________________________________________

maple [A]  time = 0.20, size = 40, normalized size = 1.38

method	result	size
risch	\(\frac {\ln \left (\frac {100 x -100}{\left (25 x^{4}+100 x^{3}+140 x^{2}+80 x +16\right ) \ln \relax (2)}\right )}{x}+x\)	\(40\)
norman	\(\frac {x^{2}+\ln \left (\frac {100 x -100}{\left (25 x^{4}+100 x^{3}+140 x^{2}+80 x +16\right ) \ln \relax (2)}\right )}{x}\)	\(42\)
default	\(-\frac {\ln \left (\ln \relax (2)\right )}{x}+x +\frac {2 \ln \left (10\right )}{x}+\frac {\ln \left (\frac {x -1}{25 x^{4}+100 x^{3}+140 x^{2}+80 x +16}\right )}{x}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-5*x^3-5*x^2+6*x+4)*ln((100*x-100)/(25*x^4+100*x^3+140*x^2+80*x+16)/ln(2))+5*x^5+5*x^4-21*x^3+6*x^2+24*x
)/(5*x^5+5*x^4-6*x^3-4*x^2),x,method=_RETURNVERBOSE)

[Out]

1/x*ln((100*x-100)/(25*x^4+100*x^3+140*x^2+80*x+16)/ln(2))+x

________________________________________________________________________________________

maxima [B]  time = 0.54, size = 64, normalized size = 2.21 \begin {gather*} x - \frac {{\left (5 \, x + 4\right )} \log \left (5 \, x^{2} + 10 \, x + 4\right ) + 2 \, {\left (x - 1\right )} \log \left (x - 1\right ) - 4 \, \log \relax (5) - 4 \, \log \relax (2) + 2 \, \log \left (\log \relax (2)\right )}{2 \, x} + \frac {5}{2} \, \log \left (5 \, x^{2} + 10 \, x + 4\right ) + \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x^3-5*x^2+6*x+4)*log((100*x-100)/(25*x^4+100*x^3+140*x^2+80*x+16)/log(2))+5*x^5+5*x^4-21*x^3+6*
x^2+24*x)/(5*x^5+5*x^4-6*x^3-4*x^2),x, algorithm="maxima")

[Out]

x - 1/2*((5*x + 4)*log(5*x^2 + 10*x + 4) + 2*(x - 1)*log(x - 1) - 4*log(5) - 4*log(2) + 2*log(log(2)))/x + 5/2
*log(5*x^2 + 10*x + 4) + log(x - 1)

________________________________________________________________________________________

mupad [B]  time = 6.01, size = 38, normalized size = 1.31 \begin {gather*} x+\frac {\ln \left (\frac {100\,\left (x-1\right )}{\ln \relax (2)\,\left (25\,x^4+100\,x^3+140\,x^2+80\,x+16\right )}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(24*x + log((100*x - 100)/(log(2)*(80*x + 140*x^2 + 100*x^3 + 25*x^4 + 16)))*(6*x - 5*x^2 - 5*x^3 + 4) +
6*x^2 - 21*x^3 + 5*x^4 + 5*x^5)/(4*x^2 + 6*x^3 - 5*x^4 - 5*x^5),x)

[Out]

x + log((100*(x - 1))/(log(2)*(80*x + 140*x^2 + 100*x^3 + 25*x^4 + 16)))/x

________________________________________________________________________________________

sympy [A]  time = 0.25, size = 32, normalized size = 1.10 \begin {gather*} x + \frac {\log {\left (\frac {100 x - 100}{\left (25 x^{4} + 100 x^{3} + 140 x^{2} + 80 x + 16\right ) \log {\relax (2 )}} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-5*x**3-5*x**2+6*x+4)*ln((100*x-100)/(25*x**4+100*x**3+140*x**2+80*x+16)/ln(2))+5*x**5+5*x**4-21*x
**3+6*x**2+24*x)/(5*x**5+5*x**4-6*x**3-4*x**2),x)

[Out]

x + log((100*x - 100)/((25*x**4 + 100*x**3 + 140*x**2 + 80*x + 16)*log(2)))/x

________________________________________________________________________________________