3.4.55 \(\int \frac {(2000000 x-800000 x^2+80000 x^3+(-800000 x+160000 x^2) \log (x)+80000 x \log ^2(x)) \log (e^5+5 x)+(600000 x-720000 x^2+120000 x^3+e^5 (120000-144000 x+24000 x^2)+(-320000 x+160000 x^2+e^5 (-64000+32000 x)) \log (x)+(8000 e^5+40000 x) \log ^2(x)) \log ^2(e^5+5 x)}{e^5+5 x} \, dx\)

Optimal. Leaf size=20 \[ 8000 x (-5+x+\log (x))^2 \log ^2\left (e^5+5 x\right ) \]

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Rubi [F]  time = 6.08, 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 {\left (2000000 x-800000 x^2+80000 x^3+\left (-800000 x+160000 x^2\right ) \log (x)+80000 x \log ^2(x)\right ) \log \left (e^5+5 x\right )+\left (600000 x-720000 x^2+120000 x^3+e^5 \left (120000-144000 x+24000 x^2\right )+\left (-320000 x+160000 x^2+e^5 (-64000+32000 x)\right ) \log (x)+\left (8000 e^5+40000 x\right ) \log ^2(x)\right ) \log ^2\left (e^5+5 x\right )}{e^5+5 x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((2000000*x - 800000*x^2 + 80000*x^3 + (-800000*x + 160000*x^2)*Log[x] + 80000*x*Log[x]^2)*Log[E^5 + 5*x]
+ (600000*x - 720000*x^2 + 120000*x^3 + E^5*(120000 - 144000*x + 24000*x^2) + (-320000*x + 160000*x^2 + E^5*(-
64000 + 32000*x))*Log[x] + (8000*E^5 + 40000*x)*Log[x]^2)*Log[E^5 + 5*x]^2)/(E^5 + 5*x),x]

[Out]

-192000*x + 17600*E^5*x + (1280*E^10*x)/3 + 44000*x^2 + (1600*E^5*x^2)/3 - (16000*x^3)/9 - 1760*(E^5 + 5*x)^2
- 64*E^5*(E^5 + 5*x)^2 + (128*(E^5 + 5*x)^3)/9 + 32000*E^5*Log[x] - 320*E^10*Log[x] + 96000*x*Log[x] - 3200*E^
5*x*Log[x] - 8000*x^2*Log[x] + 320*(E^5 + 5*x)^2*Log[x] - 16000*x*Log[x]^2 + 6400*E^5*Log[E^5 + 5*x] + 3520*E^
10*Log[E^5 + 5*x] + (128*E^15*Log[E^5 + 5*x])/3 + 32000*x*Log[E^5 + 5*x] - 88000*x^2*Log[E^5 + 5*x] + (16000*x
^3*Log[E^5 + 5*x])/3 + 12800*(E^5 + 5*x)*Log[E^5 + 5*x] - 7040*E^5*(E^5 + 5*x)*Log[E^5 + 5*x] + 3520*(E^5 + 5*
x)^2*Log[E^5 + 5*x] - 640*E^10*Log[x]*Log[E^5 + 5*x] - 32000*x*Log[x]*Log[E^5 + 5*x] + 16000*x^2*Log[x]*Log[E^
5 + 5*x] - 6400*(E^5 + 5*x)*Log[x]*Log[E^5 + 5*x] + 1280*E^5*(E^5 + 5*x)*Log[x]*Log[E^5 + 5*x] - 640*(E^5 + 5*
x)^2*Log[x]*Log[E^5 + 5*x] + 16000*x*Log[x]^2*Log[E^5 + 5*x] - 40000*E^5*Log[E^5 + 5*x]^2 - 3200*E^10*Log[E^5
+ 5*x]^2 + 8000*x^3*Log[E^5 + 5*x]^2 + 36800*(E^5 + 5*x)*Log[E^5 + 5*x]^2 + 6400*E^5*(E^5 + 5*x)*Log[E^5 + 5*x
]^2 - 3200*(E^5 + 5*x)^2*Log[E^5 + 5*x]^2 + 16000*E^5*Log[x]*Log[E^5 + 5*x]^2 + 640*E^10*Log[x]*Log[E^5 + 5*x]
^2 - 12800*(E^5 + 5*x)*Log[x]*Log[E^5 + 5*x]^2 - 1280*E^5*(E^5 + 5*x)*Log[x]*Log[E^5 + 5*x]^2 + 640*(E^5 + 5*x
)^2*Log[x]*Log[E^5 + 5*x]^2 - 3200*E^5*Log[(-5*x)/E^5]*Log[E^5 + 5*x]^2 + 64*E^5*Log[E^5 + 5*x]*(4*E^5*(E^5 +
5*x) - (E^5 + 5*x)^2 - 2*E^10*Log[E^5 + 5*x]) - (64*Log[E^5 + 5*x]*(18*E^10*(E^5 + 5*x) - 9*E^5*(E^5 + 5*x)^2
+ 2*(E^5 + 5*x)^3 - 6*E^15*Log[E^5 + 5*x]))/3 - 6400*E^5*Log[x]*Log[1 + (5*x)/E^5] + 3200*E^5*Log[x]^2*Log[1 +
(5*x)/E^5] - 12800*E^5*PolyLog[2, (-5*x)/E^5] + 6400*E^5*Log[x]*PolyLog[2, (-5*x)/E^5] - 6400*E^5*Log[E^5 + 5
*x]*PolyLog[2, 1 + (5*x)/E^5] - 6400*E^5*PolyLog[3, (-5*x)/E^5] + 6400*E^5*PolyLog[3, 1 + (5*x)/E^5] - 16000*E
^5*Defer[Int][(Log[x]^2*Log[E^5 + 5*x])/(E^5 + 5*x), x] + 8000*Defer[Int][Log[x]^2*Log[E^5 + 5*x]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8000 (5-x-\log (x)) \log \left (e^5+5 x\right ) \left (-10 (-5+x) x-3 (-1+x) \left (e^5+5 x\right ) \log \left (e^5+5 x\right )-\log (x) \left (10 x+\left (e^5+5 x\right ) \log \left (e^5+5 x\right )\right )\right )}{e^5+5 x} \, dx\\ &=8000 \int \frac {(5-x-\log (x)) \log \left (e^5+5 x\right ) \left (-10 (-5+x) x-3 (-1+x) \left (e^5+5 x\right ) \log \left (e^5+5 x\right )-\log (x) \left (10 x+\left (e^5+5 x\right ) \log \left (e^5+5 x\right )\right )\right )}{e^5+5 x} \, dx\\ &=8000 \int \left (\frac {10 x (-5+x+\log (x))^2 \log \left (e^5+5 x\right )}{e^5+5 x}+\left (15-18 x+3 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right ) \log ^2\left (e^5+5 x\right )\right ) \, dx\\ &=8000 \int \left (15-18 x+3 x^2-8 \log (x)+4 x \log (x)+\log ^2(x)\right ) \log ^2\left (e^5+5 x\right ) \, dx+80000 \int \frac {x (-5+x+\log (x))^2 \log \left (e^5+5 x\right )}{e^5+5 x} \, dx\\ &=8000 \int \left (15 \log ^2\left (e^5+5 x\right )-18 x \log ^2\left (e^5+5 x\right )+3 x^2 \log ^2\left (e^5+5 x\right )-8 \log (x) \log ^2\left (e^5+5 x\right )+4 x \log (x) \log ^2\left (e^5+5 x\right )+\log ^2(x) \log ^2\left (e^5+5 x\right )\right ) \, dx+80000 \int \left (\frac {1}{5} (-5+x+\log (x))^2 \log \left (e^5+5 x\right )-\frac {e^5 (-5+x+\log (x))^2 \log \left (e^5+5 x\right )}{5 \left (e^5+5 x\right )}\right ) \, dx\\ &=8000 \int \log ^2(x) \log ^2\left (e^5+5 x\right ) \, dx+16000 \int (-5+x+\log (x))^2 \log \left (e^5+5 x\right ) \, dx+24000 \int x^2 \log ^2\left (e^5+5 x\right ) \, dx+32000 \int x \log (x) \log ^2\left (e^5+5 x\right ) \, dx-64000 \int \log (x) \log ^2\left (e^5+5 x\right ) \, dx+120000 \int \log ^2\left (e^5+5 x\right ) \, dx-144000 \int x \log ^2\left (e^5+5 x\right ) \, dx-\left (16000 e^5\right ) \int \frac {(-5+x+\log (x))^2 \log \left (e^5+5 x\right )}{e^5+5 x} \, dx\\ &=-128000 x \log (x)-12800 e^5 x \log (x)+320 \left (e^5+5 x\right )^2 \log (x)+25600 \left (e^5+5 x\right ) \log (x) \log \left (e^5+5 x\right )+2560 e^5 \left (e^5+5 x\right ) \log (x) \log \left (e^5+5 x\right )-640 \left (e^5+5 x\right )^2 \log (x) \log \left (e^5+5 x\right )+8000 x^3 \log ^2\left (e^5+5 x\right )-12800 \left (e^5+5 x\right ) \log (x) \log ^2\left (e^5+5 x\right )-1280 e^5 \left (e^5+5 x\right ) \log (x) \log ^2\left (e^5+5 x\right )+640 \left (e^5+5 x\right )^2 \log (x) \log ^2\left (e^5+5 x\right )+8000 \int \log ^2(x) \log ^2\left (e^5+5 x\right ) \, dx+16000 \int \left (25 \log \left (e^5+5 x\right )-10 x \log \left (e^5+5 x\right )+x^2 \log \left (e^5+5 x\right )-10 \log (x) \log \left (e^5+5 x\right )+2 x \log (x) \log \left (e^5+5 x\right )+\log ^2(x) \log \left (e^5+5 x\right )\right ) \, dx+24000 \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,e^5+5 x\right )-32000 \int \left (-\frac {2 e^5}{5}+\frac {\left (e^5+5 x\right )^2}{100 x}+\frac {2 e^5 \left (e^5+5 x\right ) \log \left (e^5+5 x\right )}{25 x}-\frac {\left (e^5+5 x\right )^2 \log \left (e^5+5 x\right )}{50 x}-\frac {e^5 \left (e^5+5 x\right ) \log ^2\left (e^5+5 x\right )}{25 x}+\frac {\left (e^5+5 x\right )^2 \log ^2\left (e^5+5 x\right )}{50 x}\right ) \, dx+64000 \int \left (2-\frac {2 \left (e^5+5 x\right ) \log \left (e^5+5 x\right )}{5 x}+\frac {\left (e^5+5 x\right ) \log ^2\left (e^5+5 x\right )}{5 x}\right ) \, dx-80000 \int \frac {x^3 \log \left (e^5+5 x\right )}{e^5+5 x} \, dx-144000 \int \left (-\frac {1}{5} e^5 \log ^2\left (e^5+5 x\right )+\frac {1}{5} \left (e^5+5 x\right ) \log ^2\left (e^5+5 x\right )\right ) \, dx-\left (16000 e^5\right ) \int \left (\frac {25 \log \left (e^5+5 x\right )}{e^5+5 x}-\frac {10 x \log \left (e^5+5 x\right )}{e^5+5 x}+\frac {x^2 \log \left (e^5+5 x\right )}{e^5+5 x}-\frac {10 \log (x) \log \left (e^5+5 x\right )}{e^5+5 x}+\frac {2 x \log (x) \log \left (e^5+5 x\right )}{e^5+5 x}+\frac {\log ^2(x) \log \left (e^5+5 x\right )}{e^5+5 x}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.45, size = 20, normalized size = 1.00 \begin {gather*} 8000 x (-5+x+\log (x))^2 \log ^2\left (e^5+5 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((2000000*x - 800000*x^2 + 80000*x^3 + (-800000*x + 160000*x^2)*Log[x] + 80000*x*Log[x]^2)*Log[E^5 +
5*x] + (600000*x - 720000*x^2 + 120000*x^3 + E^5*(120000 - 144000*x + 24000*x^2) + (-320000*x + 160000*x^2 +
E^5*(-64000 + 32000*x))*Log[x] + (8000*E^5 + 40000*x)*Log[x]^2)*Log[E^5 + 5*x]^2)/(E^5 + 5*x),x]

[Out]

8000*x*(-5 + x + Log[x])^2*Log[E^5 + 5*x]^2

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fricas [B]  time = 0.68, size = 40, normalized size = 2.00 \begin {gather*} 8000 \, {\left (x^{3} + x \log \relax (x)^{2} - 10 \, x^{2} + 2 \, {\left (x^{2} - 5 \, x\right )} \log \relax (x) + 25 \, x\right )} \log \left (5 \, x + e^{5}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8000*exp(5)+40000*x)*log(x)^2+((32000*x-64000)*exp(5)+160000*x^2-320000*x)*log(x)+(24000*x^2-1440
00*x+120000)*exp(5)+120000*x^3-720000*x^2+600000*x)*log(exp(5)+5*x)^2+(80000*x*log(x)^2+(160000*x^2-800000*x)*
log(x)+80000*x^3-800000*x^2+2000000*x)*log(exp(5)+5*x))/(exp(5)+5*x),x, algorithm="fricas")

[Out]

8000*(x^3 + x*log(x)^2 - 10*x^2 + 2*(x^2 - 5*x)*log(x) + 25*x)*log(5*x + e^5)^2

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giac [B]  time = 0.31, size = 87, normalized size = 4.35 \begin {gather*} 8000 \, x^{3} \log \left (5 \, x + e^{5}\right )^{2} + 16000 \, x^{2} \log \left (5 \, x + e^{5}\right )^{2} \log \relax (x) + 8000 \, x \log \left (5 \, x + e^{5}\right )^{2} \log \relax (x)^{2} - 80000 \, x^{2} \log \left (5 \, x + e^{5}\right )^{2} - 80000 \, x \log \left (5 \, x + e^{5}\right )^{2} \log \relax (x) + 200000 \, x \log \left (5 \, x + e^{5}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8000*exp(5)+40000*x)*log(x)^2+((32000*x-64000)*exp(5)+160000*x^2-320000*x)*log(x)+(24000*x^2-1440
00*x+120000)*exp(5)+120000*x^3-720000*x^2+600000*x)*log(exp(5)+5*x)^2+(80000*x*log(x)^2+(160000*x^2-800000*x)*
log(x)+80000*x^3-800000*x^2+2000000*x)*log(exp(5)+5*x))/(exp(5)+5*x),x, algorithm="giac")

[Out]

8000*x^3*log(5*x + e^5)^2 + 16000*x^2*log(5*x + e^5)^2*log(x) + 8000*x*log(5*x + e^5)^2*log(x)^2 - 80000*x^2*l
og(5*x + e^5)^2 - 80000*x*log(5*x + e^5)^2*log(x) + 200000*x*log(5*x + e^5)^2

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maple [B]  time = 0.26, size = 44, normalized size = 2.20

method	result	size
risch	\(\left (8000 x^{3}+16000 x^{2} \ln \relax (x )+8000 x \ln \relax (x )^{2}-80000 x^{2}-80000 x \ln \relax (x )+200000 x \right ) \ln \left ({\mathrm e}^{5}+5 x \right )^{2}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8000*exp(5)+40000*x)*ln(x)^2+((32000*x-64000)*exp(5)+160000*x^2-320000*x)*ln(x)+(24000*x^2-144000*x+120
000)*exp(5)+120000*x^3-720000*x^2+600000*x)*ln(exp(5)+5*x)^2+(80000*x*ln(x)^2+(160000*x^2-800000*x)*ln(x)+8000
0*x^3-800000*x^2+2000000*x)*ln(exp(5)+5*x))/(exp(5)+5*x),x,method=_RETURNVERBOSE)

[Out]

(8000*x^3+16000*x^2*ln(x)+8000*x*ln(x)^2-80000*x^2-80000*x*ln(x)+200000*x)*ln(exp(5)+5*x)^2

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maxima [B]  time = 0.83, size = 312, normalized size = 15.60 \begin {gather*} -\frac {4000}{3} \, x^{2} {\left (e^{5} + 30\right )} + \frac {4000}{3} \, x^{2} e^{5} + 64 \, {\left (125 \, x^{3} + 125 \, x \log \relax (x)^{2} - 1250 \, x^{2} + 250 \, {\left (x^{2} - 5 \, x\right )} \log \relax (x) + 3125 \, x + e^{15} + 50 \, e^{10} + 625 \, e^{5}\right )} \log \left (5 \, x + e^{5}\right )^{2} + 64 \, e^{15} \log \left (5 \, x + e^{5}\right )^{2} + 3200 \, e^{10} \log \left (5 \, x + e^{5}\right )^{2} + 40000 \, e^{5} \log \left (5 \, x + e^{5}\right )^{2} + 40000 \, x^{2} + \frac {320}{3} \, x {\left (11 \, e^{10} + 450 \, e^{5} + 3750\right )} - \frac {3520}{3} \, x e^{10} - 48000 \, x e^{5} - \frac {64}{3} \, {\left (250 \, x^{3} - 75 \, x^{2} {\left (e^{5} + 50\right )} + 30 \, x {\left (e^{10} + 50 \, e^{5} + 625\right )} + 11 \, e^{15} + 450 \, e^{10} + 3750 \, e^{5}\right )} \log \left (5 \, x + e^{5}\right ) + \frac {64}{3} \, {\left (250 \, x^{3} - 75 \, x^{2} e^{5} + 30 \, x e^{10} - 6 \, e^{15} \log \left (5 \, x + e^{5}\right )\right )} \log \left (5 \, x + e^{5}\right ) - 3200 \, {\left (25 \, x^{2} - 10 \, x e^{5} + 2 \, e^{10} \log \left (5 \, x + e^{5}\right )\right )} \log \left (5 \, x + e^{5}\right ) - 80000 \, {\left (e^{5} \log \left (5 \, x + e^{5}\right ) - 5 \, x\right )} \log \left (5 \, x + e^{5}\right ) + \frac {704}{3} \, e^{15} \log \left (5 \, x + e^{5}\right ) + 9600 \, e^{10} \log \left (5 \, x + e^{5}\right ) + 80000 \, e^{5} \log \left (5 \, x + e^{5}\right ) - 400000 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8000*exp(5)+40000*x)*log(x)^2+((32000*x-64000)*exp(5)+160000*x^2-320000*x)*log(x)+(24000*x^2-1440
00*x+120000)*exp(5)+120000*x^3-720000*x^2+600000*x)*log(exp(5)+5*x)^2+(80000*x*log(x)^2+(160000*x^2-800000*x)*
log(x)+80000*x^3-800000*x^2+2000000*x)*log(exp(5)+5*x))/(exp(5)+5*x),x, algorithm="maxima")

[Out]

-4000/3*x^2*(e^5 + 30) + 4000/3*x^2*e^5 + 64*(125*x^3 + 125*x*log(x)^2 - 1250*x^2 + 250*(x^2 - 5*x)*log(x) + 3
125*x + e^15 + 50*e^10 + 625*e^5)*log(5*x + e^5)^2 + 64*e^15*log(5*x + e^5)^2 + 3200*e^10*log(5*x + e^5)^2 + 4
0000*e^5*log(5*x + e^5)^2 + 40000*x^2 + 320/3*x*(11*e^10 + 450*e^5 + 3750) - 3520/3*x*e^10 - 48000*x*e^5 - 64/
3*(250*x^3 - 75*x^2*(e^5 + 50) + 30*x*(e^10 + 50*e^5 + 625) + 11*e^15 + 450*e^10 + 3750*e^5)*log(5*x + e^5) +
64/3*(250*x^3 - 75*x^2*e^5 + 30*x*e^10 - 6*e^15*log(5*x + e^5))*log(5*x + e^5) - 3200*(25*x^2 - 10*x*e^5 + 2*e
^10*log(5*x + e^5))*log(5*x + e^5) - 80000*(e^5*log(5*x + e^5) - 5*x)*log(5*x + e^5) + 704/3*e^15*log(5*x + e^
5) + 9600*e^10*log(5*x + e^5) + 80000*e^5*log(5*x + e^5) - 400000*x

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mupad [B]  time = 0.78, size = 44, normalized size = 2.20 \begin {gather*} {\ln \left (5\,x+{\mathrm {e}}^5\right )}^2\,\left (200000\,x+8000\,x\,{\ln \relax (x)}^2-\ln \relax (x)\,\left (80000\,x-16000\,x^2\right )-80000\,x^2+8000\,x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(5*x + exp(5))*(2000000*x + 80000*x*log(x)^2 - log(x)*(800000*x - 160000*x^2) - 800000*x^2 + 80000*x^3
) + log(5*x + exp(5))^2*(600000*x + exp(5)*(24000*x^2 - 144000*x + 120000) + log(x)^2*(40000*x + 8000*exp(5))
+ log(x)*(160000*x^2 - 320000*x + exp(5)*(32000*x - 64000)) - 720000*x^2 + 120000*x^3))/(5*x + exp(5)),x)

[Out]

log(5*x + exp(5))^2*(200000*x + 8000*x*log(x)^2 - log(x)*(80000*x - 16000*x^2) - 80000*x^2 + 8000*x^3)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: CoercionFailed} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8000*exp(5)+40000*x)*ln(x)**2+((32000*x-64000)*exp(5)+160000*x**2-320000*x)*ln(x)+(24000*x**2-144
000*x+120000)*exp(5)+120000*x**3-720000*x**2+600000*x)*ln(exp(5)+5*x)**2+(80000*x*ln(x)**2+(160000*x**2-800000
*x)*ln(x)+80000*x**3-800000*x**2+2000000*x)*ln(exp(5)+5*x))/(exp(5)+5*x),x)

[Out]

Exception raised: CoercionFailed

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