3.83.17 \(\int \frac {128+148 x-96 x^2-96 x^3+24 x^4+24 x^5-2 x^6-2 x^7+(96+96 x-48 x^2-48 x^3+6 x^4+6 x^5) \log (5)+(24+24 x-6 x^2-6 x^3) \log ^2(5)+(2+2 x) \log ^3(5)}{64-48 x^2+12 x^4-x^6+(48-24 x^2+3 x^4) \log (5)+(12-3 x^2) \log ^2(5)+\log ^3(5)} \, dx\)

Optimal. Leaf size=21 \[ 11+2 x+x^2+\frac {5}{\left (4-x^2+\log (5)\right )^2} \]

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Rubi [A]  time = 0.16, antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 3, number of rules used = 2, integrand size = 144, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {2073, 261} \begin {gather*} x^2+\frac {5}{\left (-x^2+4+\log (5)\right )^2}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(128 + 148*x - 96*x^2 - 96*x^3 + 24*x^4 + 24*x^5 - 2*x^6 - 2*x^7 + (96 + 96*x - 48*x^2 - 48*x^3 + 6*x^4 +
6*x^5)*Log[5] + (24 + 24*x - 6*x^2 - 6*x^3)*Log[5]^2 + (2 + 2*x)*Log[5]^3)/(64 - 48*x^2 + 12*x^4 - x^6 + (48 -
24*x^2 + 3*x^4)*Log[5] + (12 - 3*x^2)*Log[5]^2 + Log[5]^3),x]

[Out]

2*x + x^2 + 5/(4 - x^2 + Log[5])^2

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 2073

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P /. x -> Sqrt[x]]}, Int[ExpandIntegrand[(PP /. x ->
x^2)^p*Q^q, x], x] /;  !SumQ[NonfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x^2] && PolyQ[Q, x] && ILtQ[p,
0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+2 x-\frac {20 x}{\left (-4+x^2-\log (5)\right )^3}\right ) \, dx\\ &=2 x+x^2-20 \int \frac {x}{\left (-4+x^2-\log (5)\right )^3} \, dx\\ &=2 x+x^2+\frac {5}{\left (4-x^2+\log (5)\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 26, normalized size = 1.24 \begin {gather*} 2 \left (x+\frac {x^2}{2}+\frac {5}{2 \left (-4+x^2-\log (5)\right )^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(128 + 148*x - 96*x^2 - 96*x^3 + 24*x^4 + 24*x^5 - 2*x^6 - 2*x^7 + (96 + 96*x - 48*x^2 - 48*x^3 + 6*
x^4 + 6*x^5)*Log[5] + (24 + 24*x - 6*x^2 - 6*x^3)*Log[5]^2 + (2 + 2*x)*Log[5]^3)/(64 - 48*x^2 + 12*x^4 - x^6 +
(48 - 24*x^2 + 3*x^4)*Log[5] + (12 - 3*x^2)*Log[5]^2 + Log[5]^3),x]

[Out]

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

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fricas [B]  time = 0.76, size = 87, normalized size = 4.14 \begin {gather*} \frac {x^{6} + 2 \, x^{5} - 8 \, x^{4} - 16 \, x^{3} + {\left (x^{2} + 2 \, x\right )} \log \relax (5)^{2} + 16 \, x^{2} - 2 \, {\left (x^{4} + 2 \, x^{3} - 4 \, x^{2} - 8 \, x\right )} \log \relax (5) + 32 \, x + 5}{x^{4} - 8 \, x^{2} - 2 \, {\left (x^{2} - 4\right )} \log \relax (5) + \log \relax (5)^{2} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+2)*log(5)^3+(-6*x^3-6*x^2+24*x+24)*log(5)^2+(6*x^5+6*x^4-48*x^3-48*x^2+96*x+96)*log(5)-2*x^7-2
*x^6+24*x^5+24*x^4-96*x^3-96*x^2+148*x+128)/(log(5)^3+(-3*x^2+12)*log(5)^2+(3*x^4-24*x^2+48)*log(5)-x^6+12*x^4
-48*x^2+64),x, algorithm="fricas")

[Out]

(x^6 + 2*x^5 - 8*x^4 - 16*x^3 + (x^2 + 2*x)*log(5)^2 + 16*x^2 - 2*(x^4 + 2*x^3 - 4*x^2 - 8*x)*log(5) + 32*x +
5)/(x^4 - 8*x^2 - 2*(x^2 - 4)*log(5) + log(5)^2 + 16)

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giac [A]  time = 0.16, size = 20, normalized size = 0.95 \begin {gather*} x^{2} + 2 \, x + \frac {5}{{\left (x^{2} - \log \relax (5) - 4\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+2)*log(5)^3+(-6*x^3-6*x^2+24*x+24)*log(5)^2+(6*x^5+6*x^4-48*x^3-48*x^2+96*x+96)*log(5)-2*x^7-2
*x^6+24*x^5+24*x^4-96*x^3-96*x^2+148*x+128)/(log(5)^3+(-3*x^2+12)*log(5)^2+(3*x^4-24*x^2+48)*log(5)-x^6+12*x^4
-48*x^2+64),x, algorithm="giac")

[Out]

x^2 + 2*x + 5/(x^2 - log(5) - 4)^2

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maple [A]  time = 0.10, size = 35, normalized size = 1.67

method	result	size
default	\(x^{2}+2 x +\frac {-20 \ln \relax (5)-80}{\left (-4 \ln \relax (5)-16\right ) \left (-\ln \relax (5)-4+x^{2}\right )^{2}}\)	\(35\)
risch	\(x^{2}+2 x +\frac {5}{x^{4}-2 x^{2} \ln \relax (5)+\ln \relax (5)^{2}-8 x^{2}+8 \ln \relax (5)+16}\)	\(37\)
norman	\(\frac {x^{6}+\left (-4 \ln \relax (5)-16\right ) x^{3}+\left (2 \ln \relax (5)^{2}+16 \ln \relax (5)+32\right ) x +\left (-3 \ln \relax (5)^{2}-24 \ln \relax (5)-48\right ) x^{2}+2 x^{5}+2 \ln \relax (5)^{3}+24 \ln \relax (5)^{2}+96 \ln \relax (5)+133}{\left (\ln \relax (5)+4-x^{2}\right )^{2}}\)	\(79\)
gosper	\(\frac {x^{6}+2 x^{5}-3 x^{2} \ln \relax (5)^{2}-4 x^{3} \ln \relax (5)+2 \ln \relax (5)^{3}+2 x \ln \relax (5)^{2}-24 x^{2} \ln \relax (5)-16 x^{3}+24 \ln \relax (5)^{2}+16 x \ln \relax (5)-48 x^{2}+96 \ln \relax (5)+32 x +133}{x^{4}-2 x^{2} \ln \relax (5)+\ln \relax (5)^{2}-8 x^{2}+8 \ln \relax (5)+16}\)	\(103\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x+2)*ln(5)^3+(-6*x^3-6*x^2+24*x+24)*ln(5)^2+(6*x^5+6*x^4-48*x^3-48*x^2+96*x+96)*ln(5)-2*x^7-2*x^6+24*x
^5+24*x^4-96*x^3-96*x^2+148*x+128)/(ln(5)^3+(-3*x^2+12)*ln(5)^2+(3*x^4-24*x^2+48)*ln(5)-x^6+12*x^4-48*x^2+64),
x,method=_RETURNVERBOSE)

[Out]

x^2+2*x+10*(-2*ln(5)-8)/(-4*ln(5)-16)/(-ln(5)-4+x^2)^2

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maxima [A]  time = 0.37, size = 33, normalized size = 1.57 \begin {gather*} x^{2} + 2 \, x + \frac {5}{x^{4} - 2 \, x^{2} {\left (\log \relax (5) + 4\right )} + \log \relax (5)^{2} + 8 \, \log \relax (5) + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+2)*log(5)^3+(-6*x^3-6*x^2+24*x+24)*log(5)^2+(6*x^5+6*x^4-48*x^3-48*x^2+96*x+96)*log(5)-2*x^7-2
*x^6+24*x^5+24*x^4-96*x^3-96*x^2+148*x+128)/(log(5)^3+(-3*x^2+12)*log(5)^2+(3*x^4-24*x^2+48)*log(5)-x^6+12*x^4
-48*x^2+64),x, algorithm="maxima")

[Out]

x^2 + 2*x + 5/(x^4 - 2*x^2*(log(5) + 4) + log(5)^2 + 8*log(5) + 16)

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mupad [B]  time = 5.59, size = 108, normalized size = 5.14 \begin {gather*} 2\,x+x^2+\left (\sum _{k=1}^3\ln \left (-{\mathrm {root}\left (1,z,k\right )}^2\,\ln \relax (5)\,691200-\mathrm {root}\left (1,z,k\right )\,x^2\,48000-921600\,{\mathrm {root}\left (1,z,k\right )}^2-{\mathrm {root}\left (1,z,k\right )}^2\,{\ln \relax (5)}^2\,172800-{\mathrm {root}\left (1,z,k\right )}^2\,{\ln \relax (5)}^3\,14400+{\mathrm {root}\left (1,z,k\right )}^2\,x^2\,230400+{\mathrm {root}\left (1,z,k\right )}^2\,x^2\,\ln \relax (5)\,115200+{\mathrm {root}\left (1,z,k\right )}^2\,x^2\,{\ln \relax (5)}^2\,14400\right )\,\mathrm {root}\left (1,z,k\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((148*x + log(5)^3*(2*x + 2) + log(5)^2*(24*x - 6*x^2 - 6*x^3 + 24) - 96*x^2 - 96*x^3 + 24*x^4 + 24*x^5 - 2
*x^6 - 2*x^7 + log(5)*(96*x - 48*x^2 - 48*x^3 + 6*x^4 + 6*x^5 + 96) + 128)/(log(5)*(3*x^4 - 24*x^2 + 48) - log
(5)^2*(3*x^2 - 12) + log(5)^3 - 48*x^2 + 12*x^4 - x^6 + 64),x)

[Out]

2*x + x^2 + symsum(log(230400*root(1, z, k)^2*x^2 - 48000*root(1, z, k)*x^2 - 921600*root(1, z, k)^2 - 172800*
root(1, z, k)^2*log(5)^2 - 14400*root(1, z, k)^2*log(5)^3 - 691200*root(1, z, k)^2*log(5) + 115200*root(1, z,
k)^2*x^2*log(5) + 14400*root(1, z, k)^2*x^2*log(5)^2)*root(1, z, k), k, 1, 3)

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sympy [A]  time = 0.42, size = 34, normalized size = 1.62 \begin {gather*} x^{2} + 2 x + \frac {5}{x^{4} + x^{2} \left (-8 - 2 \log {\relax (5 )}\right ) + \log {\relax (5 )}^{2} + 8 \log {\relax (5 )} + 16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x+2)*ln(5)**3+(-6*x**3-6*x**2+24*x+24)*ln(5)**2+(6*x**5+6*x**4-48*x**3-48*x**2+96*x+96)*ln(5)-2*
x**7-2*x**6+24*x**5+24*x**4-96*x**3-96*x**2+148*x+128)/(ln(5)**3+(-3*x**2+12)*ln(5)**2+(3*x**4-24*x**2+48)*ln(
5)-x**6+12*x**4-48*x**2+64),x)

[Out]

x**2 + 2*x + 5/(x**4 + x**2*(-8 - 2*log(5)) + log(5)**2 + 8*log(5) + 16)

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