∫ 384−44x+5x2+(96−15x) log(4)+(384−40x+5x2+(96−10x) log(4)) log(x)+(80+20 log(4)) log2(x)___________________________ (256−128x+16x2+(128−32x) log(4)+16 log2(4)) log(5)+(256−128x+16x2+(128−32x) log(4)+16 log2(4)) log(5) log(x)+(64−32x+4x2+(32−8x) log(4)+4 log2(4)) log(5) log2(x) dx

3.9.21 \(\int \frac {384-44 x+5 x^2+(96-15 x) \log (4)+(384-40 x+5 x^2+(96-10 x) \log (4)) \log (x)+(80+20 \log (4)) \log ^2(x)}{(256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)) \log (5)+(256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)) \log (5) \log (x)+(64-32 x+4 x^2+(32-8 x) \log (4)+4 \log ^2(4)) \log (5) \log ^2(x)} \, dx\)

Optimal. Leaf size=34 \[ \frac {5+\frac {4-\frac {5 x}{4}}{2+\log (x)}}{-\log (5)+\frac {(4+\log (4)) \log (5)}{x}} \]

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Rubi [F] time = 0.64, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {384-44 x+5 x^2+(96-15 x) \log (4)+\left (384-40 x+5 x^2+(96-10 x) \log (4)\right ) \log (x)+(80+20 \log (4)) \log ^2(x)}{\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5)+\left (256-128 x+16 x^2+(128-32 x) \log (4)+16 \log ^2(4)\right ) \log (5) \log (x)+\left (64-32 x+4 x^2+(32-8 x) \log (4)+4 \log ^2(4)\right ) \log (5) \log ^2(x)} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(384 - 44*x + 5*x^2 + (96 - 15*x)*Log[4] + (384 - 40*x + 5*x^2 + (96 - 10*x)*Log[4])*Log[x] + (80 + 20*Log

[4])*Log[x]^2)/((256 - 128*x + 16*x^2 + (128 - 32*x)*Log[4] + 16*Log[4]^2)*Log[5] + (256 - 128*x + 16*x^2 + (1

28 - 32*x)*Log[4] + 16*Log[4]^2)*Log[5]*Log[x] + (64 - 32*x + 4*x^2 + (32 - 8*x)*Log[4] + 4*Log[4]^2)*Log[5]*L

og[x]^2),x]

[Out]

(5*(4 + Log[4]))/((4 - x + Log[4])*Log[5]) + Defer[Int][(16 - 5*x)/((-4 + x - Log[4])*(2 + Log[x])^2), x]/(4*L

og[5]) + Defer[Int][(5*x^2 + 16*(4 + Log[4]) - 10*x*(4 + Log[4]))/((4 - x + Log[4])^2*(2 + Log[x])), x]/(4*Log

[5])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 x^2+96 (4+\log (4))-x (44+15 \log (4))+\left (5 x^2+96 (4+\log (4))-10 x (4+\log (4))\right ) \log (x)+20 (4+\log (4)) \log ^2(x)}{4 (4-x+\log (4))^2 \log (5) (2+\log (x))^2} \, dx\\ &=\frac {\int \frac {5 x^2+96 (4+\log (4))-x (44+15 \log (4))+\left (5 x^2+96 (4+\log (4))-10 x (4+\log (4))\right ) \log (x)+20 (4+\log (4)) \log ^2(x)}{(4-x+\log (4))^2 (2+\log (x))^2} \, dx}{4 \log (5)}\\ &=\frac {\int \left (\frac {20 (4+\log (4))}{(4-x+\log (4))^2}+\frac {16-5 x}{(-4+x-\log (4)) (2+\log (x))^2}+\frac {5 x^2+16 (4+\log (4))-10 x (4+\log (4))}{(4-x+\log (4))^2 (2+\log (x))}\right ) \, dx}{4 \log (5)}\\ &=\frac {5 (4+\log (4))}{(4-x+\log (4)) \log (5)}+\frac {\int \frac {16-5 x}{(-4+x-\log (4)) (2+\log (x))^2} \, dx}{4 \log (5)}+\frac {\int \frac {5 x^2+16 (4+\log (4))-10 x (4+\log (4))}{(4-x+\log (4))^2 (2+\log (x))} \, dx}{4 \log (5)}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.26, size = 37, normalized size = 1.09 \begin {gather*} \frac {-20 (4+\log (4))+\frac {x (-16+5 x)}{2+\log (x)}}{4 (-4+x-\log (4)) \log (5)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(384 - 44*x + 5*x^2 + (96 - 15*x)*Log[4] + (384 - 40*x + 5*x^2 + (96 - 10*x)*Log[4])*Log[x] + (80 +

20*Log[4])*Log[x]^2)/((256 - 128*x + 16*x^2 + (128 - 32*x)*Log[4] + 16*Log[4]^2)*Log[5] + (256 - 128*x + 16*x^

2 + (128 - 32*x)*Log[4] + 16*Log[4]^2)*Log[5]*Log[x] + (64 - 32*x + 4*x^2 + (32 - 8*x)*Log[4] + 4*Log[4]^2)*Lo

g[5]*Log[x]^2),x]

[Out]

(-20*(4 + Log[4]) + (x*(-16 + 5*x))/(2 + Log[x]))/(4*(-4 + x - Log[4])*Log[5])

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fricas [A] time = 0.61, size = 50, normalized size = 1.47 \begin {gather*} \frac {5 \, x^{2} - 40 \, {\left (\log \relax (2) + 2\right )} \log \relax (x) - 16 \, x - 80 \, \log \relax (2) - 160}{4 \, {\left ({\left (x - 2 \, \log \relax (2) - 4\right )} \log \relax (5) \log \relax (x) + 2 \, {\left (x - 2 \, \log \relax (2) - 4\right )} \log \relax (5)\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*log(2)+80)*log(x)^2+(2*(-10*x+96)*log(2)+5*x^2-40*x+384)*log(x)+2*(-15*x+96)*log(2)+5*x^2-44*x+

384)/((16*log(2)^2+2*(-8*x+32)*log(2)+4*x^2-32*x+64)*log(5)*log(x)^2+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-

128*x+256)*log(5)*log(x)+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-128*x+256)*log(5)),x, algorithm="fricas")

[Out]

1/4*(5*x^2 - 40*(log(2) + 2)*log(x) - 16*x - 80*log(2) - 160)/((x - 2*log(2) - 4)*log(5)*log(x) + 2*(x - 2*log

(2) - 4)*log(5))

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giac [B] time = 0.42, size = 73, normalized size = 2.15 \begin {gather*} \frac {5 \, x^{2} - 16 \, x}{4 \, {\left (x \log \relax (5) \log \relax (x) - 2 \, \log \relax (5) \log \relax (2) \log \relax (x) + 2 \, x \log \relax (5) - 4 \, \log \relax (5) \log \relax (2) - 4 \, \log \relax (5) \log \relax (x) - 8 \, \log \relax (5)\right )}} - \frac {10 \, {\left (\log \relax (2) + 2\right )}}{x \log \relax (5) - 2 \, \log \relax (5) \log \relax (2) - 4 \, \log \relax (5)} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*log(2)+80)*log(x)^2+(2*(-10*x+96)*log(2)+5*x^2-40*x+384)*log(x)+2*(-15*x+96)*log(2)+5*x^2-44*x+

384)/((16*log(2)^2+2*(-8*x+32)*log(2)+4*x^2-32*x+64)*log(5)*log(x)^2+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-

128*x+256)*log(5)*log(x)+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-128*x+256)*log(5)),x, algorithm="giac")

[Out]

1/4*(5*x^2 - 16*x)/(x*log(5)*log(x) - 2*log(5)*log(2)*log(x) + 2*x*log(5) - 4*log(5)*log(2) - 4*log(5)*log(x)

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

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maple [A] time = 0.18, size = 58, normalized size = 1.71


method	result	size

norman	\(\frac {\frac {4 x}{\ln \relax (5)}+\frac {10 \left (\ln \relax (2)+2\right ) \ln \relax (x )}{\ln \relax (5)}-\frac {5 x^{2}}{4 \ln \relax (5)}+\frac {20 \ln \relax (2)+40}{\ln \relax (5)}}{\left (\ln \relax (x )+2\right ) \left (4+2 \ln \relax (2)-x \right )}\)	\(58\)
risch	\(\frac {10 \ln \relax (2)}{\left (4+2 \ln \relax (2)-x \right ) \ln \relax (5)}+\frac {20}{\left (4+2 \ln \relax (2)-x \right ) \ln \relax (5)}-\frac {x \left (5 x -16\right )}{4 \ln \relax (5) \left (4+2 \ln \relax (2)-x \right ) \left (\ln \relax (x )+2\right )}\)	\(67\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((40*ln(2)+80)*ln(x)^2+(2*(-10*x+96)*ln(2)+5*x^2-40*x+384)*ln(x)+2*(-15*x+96)*ln(2)+5*x^2-44*x+384)/((16*l

n(2)^2+2*(-8*x+32)*ln(2)+4*x^2-32*x+64)*ln(5)*ln(x)^2+(64*ln(2)^2+2*(-32*x+128)*ln(2)+16*x^2-128*x+256)*ln(5)*

ln(x)+(64*ln(2)^2+2*(-32*x+128)*ln(2)+16*x^2-128*x+256)*ln(5)),x,method=_RETURNVERBOSE)

[Out]

(4*x/ln(5)+10*(ln(2)+2)/ln(5)*ln(x)-5/4*x^2/ln(5)+20*(ln(2)+2)/ln(5))/(ln(x)+2)/(4+2*ln(2)-x)

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maxima [A] time = 0.62, size = 60, normalized size = 1.76 \begin {gather*} \frac {5 \, x^{2} - 40 \, {\left (\log \relax (2) + 2\right )} \log \relax (x) - 16 \, x - 80 \, \log \relax (2) - 160}{4 \, {\left (2 \, x \log \relax (5) - 4 \, \log \relax (5) \log \relax (2) + {\left (x \log \relax (5) - 2 \, \log \relax (5) \log \relax (2) - 4 \, \log \relax (5)\right )} \log \relax (x) - 8 \, \log \relax (5)\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*log(2)+80)*log(x)^2+(2*(-10*x+96)*log(2)+5*x^2-40*x+384)*log(x)+2*(-15*x+96)*log(2)+5*x^2-44*x+

384)/((16*log(2)^2+2*(-8*x+32)*log(2)+4*x^2-32*x+64)*log(5)*log(x)^2+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-

128*x+256)*log(5)*log(x)+(64*log(2)^2+2*(-32*x+128)*log(2)+16*x^2-128*x+256)*log(5)),x, algorithm="maxima")

[Out]

1/4*(5*x^2 - 40*(log(2) + 2)*log(x) - 16*x - 80*log(2) - 160)/(2*x*log(5) - 4*log(5)*log(2) + (x*log(5) - 2*lo

g(5)*log(2) - 4*log(5))*log(x) - 8*log(5))

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mupad [B] time = 1.79, size = 1650, normalized size = 48.53 result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(44*x + 2*log(2)*(15*x - 96) + log(x)*(40*x + 2*log(2)*(10*x - 96) - 5*x^2 - 384) - log(x)^2*(40*log(2) +

80) - 5*x^2 - 384)/(log(5)*(64*log(2)^2 - 2*log(2)*(32*x - 128) - 128*x + 16*x^2 + 256) + log(5)*log(x)*(64*l

og(2)^2 - 2*log(2)*(32*x - 128) - 128*x + 16*x^2 + 256) + log(5)*log(x)^2*(16*log(2)^2 - 2*log(2)*(8*x - 32) -

32*x + 4*x^2 + 64)),x)

[Out]

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

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

*x*log(2)*log(5) - 4*x*log(2)*log(5)*log(x)) + (11*x^2)/(32*log(5) + 32*log(2)*log(5) - 16*x*log(5) + 8*log(2)

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

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

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

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

symsum(log(576*x - 1984*log(2) - 384*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 21495808

0*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*log(2)^9 +

2097152, z, k)*log(5) + 1056*x*log(2) + 724*x*log(2)^2 + 220*x*log(2)^3 + 25*x*log(2)^4 - 272*log(2)^2 + 648*

log(2)^3 + 340*log(2)^4 + 50*log(2)^5 - 1536*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 +

214958080*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*lo

g(2)^9 + 2097152, z, k)*log(2)*log(5) + 96*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 21

4958080*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*log(

2)^9 + 2097152, z, k)*x*log(5) - 1728*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 2149580

80*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*log(2)^9

+ 2097152, z, k)*log(2)^2*log(5) - 768*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 214958

080*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*log(2)^9

+ 2097152, z, k)*log(2)^3*log(5) - 120*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 21495

8080*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*log(2)^

9 + 2097152, z, k)*log(2)^4*log(5) + 336*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 2149

58080*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*log(2)

^9 + 2097152, z, k)*x*log(2)*log(5) + 264*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 214

958080*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*log(2

)^9 + 2097152, z, k)*x*log(2)^2*log(5) + 60*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^5 + 2

14958080*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 512000*log

(2)^9 + 2097152, z, k)*x*log(2)^3*log(5) - 1536)*root(22020096*log(2) + 287047680*log(2)^4 + 236322816*log(2)^

5 + 214958080*log(2)^3 + 121765888*log(2)^6 + 94371840*log(2)^2 + 38338560*log(2)^7 + 6758400*log(2)^8 + 51200

0*log(2)^9 + 2097152, z, k), k, 1, 3) + (15*x^2*log(2))/(2*(32*log(5) + 32*log(2)*log(5) - 16*x*log(5) + 8*log

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

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

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

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

x)) + (10*x^2*log(x))/(32*log(5) + 32*log(2)*log(5) - 16*x*log(5) + 8*log(2)^2*log(5) + 2*x^2*log(5) + 16*log(

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

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

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

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

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

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

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

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

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

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

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

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sympy [B] time = 0.38, size = 76, normalized size = 2.24 \begin {gather*} \frac {5 x^{2} - 16 x}{8 x \log {\relax (5 )} + \left (4 x \log {\relax (5 )} - 16 \log {\relax (5 )} - 8 \log {\relax (2 )} \log {\relax (5 )}\right ) \log {\relax (x )} - 32 \log {\relax (5 )} - 16 \log {\relax (2 )} \log {\relax (5 )}} - \frac {10 \log {\relax (2 )} + 20}{x \log {\relax (5 )} - 4 \log {\relax (5 )} - 2 \log {\relax (2 )} \log {\relax (5 )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*ln(2)+80)*ln(x)**2+(2*(-10*x+96)*ln(2)+5*x**2-40*x+384)*ln(x)+2*(-15*x+96)*ln(2)+5*x**2-44*x+38

4)/((16*ln(2)**2+2*(-8*x+32)*ln(2)+4*x**2-32*x+64)*ln(5)*ln(x)**2+(64*ln(2)**2+2*(-32*x+128)*ln(2)+16*x**2-128

*x+256)*ln(5)*ln(x)+(64*ln(2)**2+2*(-32*x+128)*ln(2)+16*x**2-128*x+256)*ln(5)),x)

[Out]

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

) - (10*log(2) + 20)/(x*log(5) - 4*log(5) - 2*log(2)*log(5))

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