3.30.8 \(\int \frac {200-75 x-25 x^2-200 x^3-120 x^4+40 x^5+(50-80 x^3-60 x^4) \log (x)}{100+100 x+25 x^2+160 x^3+160 x^4+40 x^5+64 x^6+64 x^7+16 x^8} \, dx\)
Optimal. Leaf size=24 \[ \frac {5 x (3-x+\log (x))}{(2+x) \left (5+4 x^3\right )} \]
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Rubi [F] time = 5.45, 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 {200-75 x-25 x^2-200 x^3-120 x^4+40 x^5+\left (50-80 x^3-60 x^4\right ) \log (x)}{100+100 x+25 x^2+160 x^3+160 x^4+40 x^5+64 x^6+64 x^7+16 x^8} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Int[(200 - 75*x - 25*x^2 - 200*x^3 - 120*x^4 + 40*x^5 + (50 - 80*x^3 - 60*x^4)*Log[x])/(100 + 100*x + 25*x^2 +
160*x^3 + 160*x^4 + 40*x^5 + 64*x^6 + 64*x^7 + 16*x^8),x]
[Out]
50/(27*(2 + x)) - (40*(20 - (28 - 23*x)*x))/(81*(5 + 4*x^3)) + (25*(28 - (23 - 16*x)*x))/(243*(5 + 4*x^3)) + (
5*(115 - 16*(5 - 7*x)*x))/(243*(5 + 4*x^3)) + (40*2^(2/3)*5^(1/3)*(92*2^(2/3) - 55*5^(1/3))*ArcTan[(5^(1/3) -
2*2^(2/3)*x)/(Sqrt[3]*5^(1/3))])/(6561*Sqrt[3]) - (20*2^(2/3)*5^(1/3)*(56*2^(2/3) - 37*5^(1/3))*ArcTan[(5^(1/3
) - 2*2^(2/3)*x)/(Sqrt[3]*5^(1/3))])/(6561*Sqrt[3]) - (2*2^(2/3)*5^(1/3)*(56*2^(2/3) - 23*5^(1/3))*ArcTan[(5^(
1/3) - 2*2^(2/3)*x)/(Sqrt[3]*5^(1/3))])/(81*Sqrt[3]) - (5*10^(1/3)*(1517 - 512*10^(1/3))*ArcTan[(5^(1/3) - 2*2
^(2/3)*x)/(Sqrt[3]*5^(1/3))])/(6561*Sqrt[3]) + (20*10^(1/3)*(74 - 23*10^(1/3))*ArcTan[(5^(1/3) - 2*2^(2/3)*x)/
(Sqrt[3]*5^(1/3))])/(2187*Sqrt[3]) + (1280*10^(1/3)*(19 - 7*10^(1/3))*ArcTan[(5^(1/3) - 2*2^(2/3)*x)/(Sqrt[3]*
5^(1/3))])/(6561*Sqrt[3]) + (5*10^(1/3)*(23 - 4*10^(1/3))*ArcTan[(5^(1/3) - 2*2^(2/3)*x)/(Sqrt[3]*5^(1/3))])/(
243*Sqrt[3]) + (128*10^(2/3)*(19 - 4*10^(2/3))*ArcTan[(5^(1/3) - 2*2^(2/3)*x)/(Sqrt[3]*5^(1/3))])/(2187*Sqrt[3
]) - (4*10^(2/3)*(7 - 2*10^(2/3))*ArcTan[(5^(1/3) - 2*2^(2/3)*x)/(Sqrt[3]*5^(1/3))])/(243*Sqrt[3]) + (8*(-10)^
(1/3)*Log[-((-1)^(2/3)*10^(1/3)) - 2*x])/(9*(1 + (-1)^(1/3))^4) - (5*x*Log[x])/(27*(2 + x)) + (16*10^(1/3)*x*L
og[x])/(81*(10^(1/3) + 2*x)) - (16*(-10)^(1/3)*x*Log[x])/(9*(1 + (-1)^(1/3))^4*((-1)^(2/3)*10^(1/3) + 2*x)) +
(16*10^(1/3)*x*Log[x])/(81*(10^(1/3) + 2*(-1)^(2/3)*x)) + (20*x^3*Log[x])/(27*(5 + 4*x^3)) - (8*10^(1/3)*Log[1
0^(1/3) + 2*x])/81 + (20*(115 + 112*10^(1/3) + 37*10^(2/3))*Log[10^(1/3) + 2*x])/19683 - (40*(160 + 184*10^(1/
3) + 55*10^(2/3))*Log[10^(1/3) + 2*x])/19683 - (160*(185 + 152*10^(1/3) + 56*10^(2/3))*Log[10^(1/3) + 2*x])/19
683 - (5*(275 + 296*10^(1/3) + 92*10^(2/3))*Log[10^(1/3) + 2*x])/6561 + (5*(2432 + 1517*10^(1/3) + 512*10^(2/3
))*Log[10^(1/3) + 2*x])/19683 + (128*(70 + 10^(2/3)*(19 + 4*10^(2/3)))*Log[10^(1/3) + 2*x])/6561 + (8*(-10)^(1
/3)*Log[10^(1/3) + 2*(-1)^(2/3)*x])/81 - (5*10^(1/3)*(23 + 4*10^(1/3))*Log[5^(1/3) + 2^(2/3)*x])/729 + (2*10^(
1/3)*(112 + 23*10^(1/3))*Log[5^(1/3) + 2^(2/3)*x])/243 - (4*10^(2/3)*(7 + 2*10^(2/3))*Log[5^(1/3) + 2^(2/3)*x]
)/729 - (((16*I)/3)*10^(1/3)*Log[x]*Log[1 - (-1/5)^(1/3)*2^(2/3)*x])/(Sqrt[3]*(1 + (-1)^(1/3))^5) - (2*(-2)^(2
/3)*((-5)^(2/3) + 4*2^(2/3)*5^(1/3))*Log[x]*Log[1 - (-1/5)^(1/3)*2^(2/3)*x])/81 + (2*(-10)^(1/3)*(8 - (-10)^(1
/3))*Log[x]*Log[1 + ((-2)^(2/3)*x)/5^(1/3)])/81 + (16*10^(1/3)*Log[x]*Log[1 + (2^(2/3)*x)/5^(1/3)])/81 - (2*10
^(1/3)*(8 + 10^(1/3))*Log[x]*Log[1 + (2^(2/3)*x)/5^(1/3)])/81 - (32*10^(1/3)*Log[x]*Log[1 - ((1 - I*Sqrt[3])*x
)/10^(1/3)])/(81*(1 - I*Sqrt[3])) + (5*(4864 - 1517*10^(1/3) - 512*10^(2/3))*Log[10^(2/3) - 2*10^(1/3)*x + 4*x
^2])/39366 - (20*(320 - 184*10^(1/3) - 55*10^(2/3))*Log[10^(2/3) - 2*10^(1/3)*x + 4*x^2])/19683 - (5*(275 - 14
8*10^(1/3) - 46*10^(2/3))*Log[10^(2/3) - 2*10^(1/3)*x + 4*x^2])/6561 + (10*(230 - 112*10^(1/3) - 37*10^(2/3))*
Log[10^(2/3) - 2*10^(1/3)*x + 4*x^2])/19683 - (160*(185 - 76*10^(1/3) - 28*10^(2/3))*Log[10^(2/3) - 2*10^(1/3)
*x + 4*x^2])/19683 + (64*(140 - 10^(2/3)*(19 + 4*10^(2/3)))*Log[10^(2/3) - 2*10^(1/3)*x + 4*x^2])/6561 + (5*5^
(1/3)*(23 + 4*10^(1/3))*Log[5^(2/3) - 2^(2/3)*5^(1/3)*x + 2*2^(1/3)*x^2])/(2187*2^(2/3)) + (5*10^(1/3)*(23 + 4
*10^(1/3))*Log[5^(2/3) - 2^(2/3)*5^(1/3)*x + 2*2^(1/3)*x^2])/2187 - (10^(1/3)*(112 + 23*10^(1/3))*Log[5^(2/3)
- 2^(2/3)*5^(1/3)*x + 2*2^(1/3)*x^2])/243 + (2*10^(2/3)*(7 + 2*10^(2/3))*Log[5^(2/3) - 2^(2/3)*5^(1/3)*x + 2*2
^(1/3)*x^2])/729 - (5*Log[5 + 4*x^3])/81 - (((16*I)/3)*10^(1/3)*PolyLog[2, (-1/5)^(1/3)*2^(2/3)*x])/(Sqrt[3]*(
1 + (-1)^(1/3))^5) - (2*(-2)^(2/3)*((-5)^(2/3) + 4*2^(2/3)*5^(1/3))*PolyLog[2, (-1/5)^(1/3)*2^(2/3)*x])/81 + (
2*(-10)^(1/3)*(8 - (-10)^(1/3))*PolyLog[2, -(((-2)^(2/3)*x)/5^(1/3))])/81 + (16*10^(1/3)*PolyLog[2, -((2^(2/3)
*x)/5^(1/3))])/81 - (2*10^(1/3)*(8 + 10^(1/3))*PolyLog[2, -((2^(2/3)*x)/5^(1/3))])/81 - (32*10^(1/3)*PolyLog[2
, ((1 - I*Sqrt[3])*x)/10^(1/3)])/(81*(1 - I*Sqrt[3])) - (200*Defer[Int][(x*Log[x])/(5 + 4*x^3)^2, x])/9
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (40-15 x-5 x^2-40 x^3-24 x^4+8 x^5-2 \left (-5+8 x^3+6 x^4\right ) \log (x)\right )}{\left (10+5 x+8 x^3+4 x^4\right )^2} \, dx\\ &=5 \int \frac {40-15 x-5 x^2-40 x^3-24 x^4+8 x^5-2 \left (-5+8 x^3+6 x^4\right ) \log (x)}{\left (10+5 x+8 x^3+4 x^4\right )^2} \, dx\\ &=5 \int \left (\frac {40}{(2+x)^2 \left (5+4 x^3\right )^2}-\frac {15 x}{(2+x)^2 \left (5+4 x^3\right )^2}-\frac {5 x^2}{(2+x)^2 \left (5+4 x^3\right )^2}-\frac {40 x^3}{(2+x)^2 \left (5+4 x^3\right )^2}-\frac {24 x^4}{(2+x)^2 \left (5+4 x^3\right )^2}+\frac {8 x^5}{(2+x)^2 \left (5+4 x^3\right )^2}-\frac {2 \left (-5+8 x^3+6 x^4\right ) \log (x)}{(2+x)^2 \left (5+4 x^3\right )^2}\right ) \, dx\\ &=-\left (10 \int \frac {\left (-5+8 x^3+6 x^4\right ) \log (x)}{(2+x)^2 \left (5+4 x^3\right )^2} \, dx\right )-25 \int \frac {x^2}{(2+x)^2 \left (5+4 x^3\right )^2} \, dx+40 \int \frac {x^5}{(2+x)^2 \left (5+4 x^3\right )^2} \, dx-75 \int \frac {x}{(2+x)^2 \left (5+4 x^3\right )^2} \, dx-120 \int \frac {x^4}{(2+x)^2 \left (5+4 x^3\right )^2} \, dx+200 \int \frac {1}{(2+x)^2 \left (5+4 x^3\right )^2} \, dx-200 \int \frac {x^3}{(2+x)^2 \left (5+4 x^3\right )^2} \, dx\\ &=-\left (10 \int \left (\frac {\log (x)}{27 (2+x)^2}-\frac {10 \left (4-2 x+x^2\right ) \log (x)}{9 \left (5+4 x^3\right )^2}-\frac {4 (-4+x) \log (x)}{27 \left (5+4 x^3\right )}\right ) \, dx\right )-25 \int \left (\frac {4}{729 (2+x)^2}+\frac {92}{6561 (2+x)}+\frac {115-80 x+112 x^2}{243 \left (5+4 x^3\right )^2}-\frac {16 \left (56-37 x+23 x^2\right )}{6561 \left (5+4 x^3\right )}\right ) \, dx+40 \int \left (-\frac {32}{729 (2+x)^2}-\frac {304}{6561 (2+x)}-\frac {5 \left (115-80 x+112 x^2\right )}{972 \left (5+4 x^3\right )^2}+\frac {7585-5120 x+4864 x^2}{26244 \left (5+4 x^3\right )}\right ) \, dx-75 \int \left (-\frac {2}{729 (2+x)^2}-\frac {55}{6561 (2+x)}-\frac {4 \left (20-28 x+23 x^2\right )}{243 \left (5+4 x^3\right )^2}+\frac {4 \left (148-92 x+55 x^2\right )}{6561 \left (5+4 x^3\right )}\right ) \, dx-120 \int \left (\frac {16}{729 (2+x)^2}+\frac {224}{6561 (2+x)}+\frac {5 \left (20-28 x+23 x^2\right )}{243 \left (5+4 x^3\right )^2}-\frac {64 \left (20-19 x+14 x^2\right )}{6561 \left (5+4 x^3\right )}\right ) \, dx+200 \int \left (\frac {1}{729 (2+x)^2}+\frac {32}{6561 (2+x)}+\frac {4 \left (28-23 x+16 x^2\right )}{243 \left (5+4 x^3\right )^2}-\frac {4 \left (92-55 x+32 x^2\right )}{6561 \left (5+4 x^3\right )}\right ) \, dx-200 \int \left (-\frac {8}{729 (2+x)^2}-\frac {148}{6561 (2+x)}-\frac {5 \left (28-23 x+16 x^2\right )}{243 \left (5+4 x^3\right )^2}+\frac {16 \left (76-56 x+37 x^2\right )}{6561 \left (5+4 x^3\right )}\right ) \, dx\\ &=\frac {50}{27 (2+x)}-\frac {5}{27} \log (2+x)+\frac {10 \int \frac {7585-5120 x+4864 x^2}{5+4 x^3} \, dx}{6561}-\frac {100 \int \frac {148-92 x+55 x^2}{5+4 x^3} \, dx}{2187}+\frac {400 \int \frac {56-37 x+23 x^2}{5+4 x^3} \, dx}{6561}-\frac {25}{243} \int \frac {115-80 x+112 x^2}{\left (5+4 x^3\right )^2} \, dx-\frac {800 \int \frac {92-55 x+32 x^2}{5+4 x^3} \, dx}{6561}-\frac {50}{243} \int \frac {115-80 x+112 x^2}{\left (5+4 x^3\right )^2} \, dx-\frac {10}{27} \int \frac {\log (x)}{(2+x)^2} \, dx-\frac {3200 \int \frac {76-56 x+37 x^2}{5+4 x^3} \, dx}{6561}+\frac {2560 \int \frac {20-19 x+14 x^2}{5+4 x^3} \, dx}{2187}+\frac {100}{81} \int \frac {20-28 x+23 x^2}{\left (5+4 x^3\right )^2} \, dx+\frac {40}{27} \int \frac {(-4+x) \log (x)}{5+4 x^3} \, dx-\frac {200}{81} \int \frac {20-28 x+23 x^2}{\left (5+4 x^3\right )^2} \, dx+\frac {800}{243} \int \frac {28-23 x+16 x^2}{\left (5+4 x^3\right )^2} \, dx+\frac {1000}{243} \int \frac {28-23 x+16 x^2}{\left (5+4 x^3\right )^2} \, dx+\frac {100}{9} \int \frac {\left (4-2 x+x^2\right ) \log (x)}{\left (5+4 x^3\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.48, size = 27, normalized size = 1.12 \begin {gather*} \frac {5 x (3-x+\log (x))}{10+5 x+8 x^3+4 x^4} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(200 - 75*x - 25*x^2 - 200*x^3 - 120*x^4 + 40*x^5 + (50 - 80*x^3 - 60*x^4)*Log[x])/(100 + 100*x + 25
*x^2 + 160*x^3 + 160*x^4 + 40*x^5 + 64*x^6 + 64*x^7 + 16*x^8),x]
[Out]
(5*x*(3 - x + Log[x]))/(10 + 5*x + 8*x^3 + 4*x^4)
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fricas [A] time = 1.00, size = 31, normalized size = 1.29 \begin {gather*} -\frac {5 \, {\left (x^{2} - x \log \relax (x) - 3 \, x\right )}}{4 \, x^{4} + 8 \, x^{3} + 5 \, x + 10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-60*x^4-80*x^3+50)*log(x)+40*x^5-120*x^4-200*x^3-25*x^2-75*x+200)/(16*x^8+64*x^7+64*x^6+40*x^5+160
*x^4+160*x^3+25*x^2+100*x+100),x, algorithm="fricas")
[Out]
-5*(x^2 - x*log(x) - 3*x)/(4*x^4 + 8*x^3 + 5*x + 10)
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giac [B] time = 0.53, size = 49, normalized size = 2.04 \begin {gather*} \frac {5 \, x \log \relax (x)}{4 \, x^{4} + 8 \, x^{3} + 5 \, x + 10} - \frac {5 \, {\left (x^{2} - 3 \, x\right )}}{4 \, x^{4} + 8 \, x^{3} + 5 \, x + 10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-60*x^4-80*x^3+50)*log(x)+40*x^5-120*x^4-200*x^3-25*x^2-75*x+200)/(16*x^8+64*x^7+64*x^6+40*x^5+160
*x^4+160*x^3+25*x^2+100*x+100),x, algorithm="giac")
[Out]
5*x*log(x)/(4*x^4 + 8*x^3 + 5*x + 10) - 5*(x^2 - 3*x)/(4*x^4 + 8*x^3 + 5*x + 10)
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maple [A] time = 0.10, size = 33, normalized size = 1.38
|
|
|
| method |
result |
size |
|
|
|
| norman |
\(\frac {15 x -5 x^{2}+5 x \ln \relax (x )}{4 x^{4}+8 x^{3}+5 x +10}\) |
\(33\) |
| risch |
\(\frac {5 x \ln \relax (x )}{4 x^{4}+8 x^{3}+5 x +10}-\frac {5 x \left (x -3\right )}{4 x^{4}+8 x^{3}+5 x +10}\) |
\(47\) |
| default |
\(\frac {-\frac {50}{27} x^{2}+\frac {265}{108} x -\frac {125}{108}}{x^{3}+\frac {5}{4}}+\frac {50}{27 \left (2+x \right )}-\frac {5 \ln \relax (x ) x}{27 \left (2+x \right )}+\frac {20 \ln \relax (x ) x \left (x^{2}-2 x +4\right )}{27 \left (4 x^{3}+5\right )}\) |
\(60\) |
|
|
|
| |
| |
| |
| |
|
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|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((-60*x^4-80*x^3+50)*ln(x)+40*x^5-120*x^4-200*x^3-25*x^2-75*x+200)/(16*x^8+64*x^7+64*x^6+40*x^5+160*x^4+16
0*x^3+25*x^2+100*x+100),x,method=_RETURNVERBOSE)
[Out]
(15*x-5*x^2+5*x*ln(x))/(4*x^4+8*x^3+5*x+10)
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maxima [B] time = 6.63, size = 1153, normalized size = 48.04 result too large to
display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-60*x^4-80*x^3+50)*log(x)+40*x^5-120*x^4-200*x^3-25*x^2-75*x+200)/(16*x^8+64*x^7+64*x^6+40*x^5+160
*x^4+160*x^3+25*x^2+100*x+100),x, algorithm="maxima")
[Out]
5/39366*4^(2/3)*sqrt(3)*(689*5^(2/3)*4^(2/3) - 2848*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*x
- 5^(1/3)*4^(1/3))) - 2/6561*4^(2/3)*sqrt(3)*(241*5^(2/3)*4^(2/3) - 920*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*s
qrt(3)*(2*4^(2/3)*x - 5^(1/3)*4^(1/3))) - 5/39366*4^(2/3)*sqrt(3)*(220*5^(2/3)*4^(2/3) - 1103*5^(1/3))*arctan(
1/60*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*x - 5^(1/3)*4^(1/3))) - 5/39366*4^(2/3)*sqrt(3)*(56*5^(2/3)*4^(2/3) -
241*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*x - 5^(1/3)*4^(1/3))) + 8/19683*4^(2/3)*sqrt(3)*(1
7*5^(2/3)*4^(2/3) + 44*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*x - 5^(1/3)*4^(1/3))) + 4/6561*
4^(2/3)*sqrt(3)*(13*5^(2/3)*4^(2/3) - 95*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*x - 5^(1/3)*4
^(1/3))) + 1/162*4^(2/3)*sqrt(3)*(5^(2/3)*4^(2/3) - 8*5^(1/3))*arctan(1/60*5^(2/3)*4^(2/3)*sqrt(3)*(2*4^(2/3)*
x - 5^(1/3)*4^(1/3))) + 1/78732*5^(1/3)*(9728*5^(2/3) - 4400*5^(1/3)*4^(1/3) - 5515*4^(2/3))*log(4^(2/3)*x^2 -
5^(1/3)*4^(1/3)*x + 5^(2/3)) - 5/19683*5^(1/3)*(1184*5^(2/3) - 689*5^(1/3)*4^(1/3) - 712*4^(2/3))*log(4^(2/3)
*x^2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)) + 4/6561*5^(1/3)*(448*5^(2/3) - 241*5^(1/3)*4^(1/3) - 230*4^(2/3))*log(4^(
2/3)*x^2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)) + 5/78732*5^(1/3)*(368*5^(2/3) - 224*5^(1/3)*4^(1/3) - 241*4^(2/3))*lo
g(4^(2/3)*x^2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)) - 1/6561*5^(1/3)*(275*5^(2/3) - 104*5^(1/3)*4^(1/3) - 190*4^(2/3)
)*log(4^(2/3)*x^2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)) - 16/19683*5^(1/3)*(80*5^(2/3) - 17*5^(1/3)*4^(1/3) + 11*4^(2
/3))*log(4^(2/3)*x^2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)) - 1/81*5^(1/3)*(5^(2/3) - 5^(1/3)*4^(1/3) - 2*4^(2/3))*log
(4^(2/3)*x^2 - 5^(1/3)*4^(1/3)*x + 5^(2/3)) + 1/39366*5^(1/3)*(4864*5^(2/3) + 4400*5^(1/3)*4^(1/3) + 5515*4^(2
/3))*log(1/4*4^(2/3)*(4^(1/3)*x + 5^(1/3))) - 10/19683*5^(1/3)*(592*5^(2/3) + 689*5^(1/3)*4^(1/3) + 712*4^(2/3
))*log(1/4*4^(2/3)*(4^(1/3)*x + 5^(1/3))) - 1/6561*5^(1/3)*(275*5^(2/3) + 208*5^(1/3)*4^(1/3) + 380*4^(2/3))*l
og(1/4*4^(2/3)*(4^(1/3)*x + 5^(1/3))) + 8/6561*5^(1/3)*(224*5^(2/3) + 241*5^(1/3)*4^(1/3) + 230*4^(2/3))*log(1
/4*4^(2/3)*(4^(1/3)*x + 5^(1/3))) + 5/39366*5^(1/3)*(184*5^(2/3) + 224*5^(1/3)*4^(1/3) + 241*4^(2/3))*log(1/4*
4^(2/3)*(4^(1/3)*x + 5^(1/3))) - 32/19683*5^(1/3)*(40*5^(2/3) + 17*5^(1/3)*4^(1/3) - 11*4^(2/3))*log(1/4*4^(2/
3)*(4^(1/3)*x + 5^(1/3))) - 1/81*5^(1/3)*(5^(2/3) + 2*5^(1/3)*4^(1/3) + 4*4^(2/3))*log(1/4*4^(2/3)*(4^(1/3)*x
+ 5^(1/3))) + 5*x*log(x)/(4*x^4 + 8*x^3 + 5*x + 10) + 10/729*(592*x^3 + 45*x^2 - 90*x + 920)/(4*x^4 + 8*x^3 +
5*x + 10) + 10/243*(368*x^3 + 144*x^2 - 45*x + 550)/(4*x^4 + 8*x^3 + 5*x + 10) - 5/243*(152*x^3 + 144*x^2 - 45
*x + 280)/(4*x^4 + 8*x^3 + 5*x + 10) - 40/729*(112*x^3 + 72*x^2 - 144*x + 185)/(4*x^4 + 8*x^3 + 5*x + 10) - 20
0/729*(55*x^3 + 18*x^2 - 36*x + 80)/(4*x^4 + 8*x^3 + 5*x + 10) + 25/729*(32*x^3 + 9*x^2 - 18*x + 76)/(4*x^4 +
8*x^3 + 5*x + 10)
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mupad [B] time = 1.90, size = 27, normalized size = 1.12 \begin {gather*} \frac {5\,x\,\left (\ln \relax (x)-x+3\right )}{4\,x^4+8\,x^3+5\,x+10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(75*x + log(x)*(80*x^3 + 60*x^4 - 50) + 25*x^2 + 200*x^3 + 120*x^4 - 40*x^5 - 200)/(100*x + 25*x^2 + 160*
x^3 + 160*x^4 + 40*x^5 + 64*x^6 + 64*x^7 + 16*x^8 + 100),x)
[Out]
(5*x*(log(x) - x + 3))/(5*x + 8*x^3 + 4*x^4 + 10)
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sympy [B] time = 0.25, size = 44, normalized size = 1.83 \begin {gather*} \frac {5 x \log {\relax (x )}}{4 x^{4} + 8 x^{3} + 5 x + 10} + \frac {- 5 x^{2} + 15 x}{4 x^{4} + 8 x^{3} + 5 x + 10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(((-60*x**4-80*x**3+50)*ln(x)+40*x**5-120*x**4-200*x**3-25*x**2-75*x+200)/(16*x**8+64*x**7+64*x**6+40
*x**5+160*x**4+160*x**3+25*x**2+100*x+100),x)
[Out]
5*x*log(x)/(4*x**4 + 8*x**3 + 5*x + 10) + (-5*x**2 + 15*x)/(4*x**4 + 8*x**3 + 5*x + 10)
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