3.32.24 \(\int \frac {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+(200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5) \log (5)+(-200 x-50 x^2-50 x^4) \log ^2(5)+(-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+(400 x+100 x^2+100 x^4) \log (5)) \log (\frac {x}{4+x+x^3})+(-200 x-50 x^2-50 x^4) \log ^2(\frac {x}{4+x+x^3})}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+(-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8) \log (5)+(48 x^3+36 x^4+6 x^5+12 x^6+6 x^7) \log ^2(5)+(-4 x^3-x^4-x^6) \log ^3(5)+(192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+(-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7) \log (5)+(12 x^3+3 x^4+3 x^6) \log ^2(5)) \log (\frac {x}{4+x+x^3})+(48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+(-12 x^3-3 x^4-3 x^6) \log (5)) \log ^2(\frac {x}{4+x+x^3})+(4 x^3+x^4+x^6) \log ^3(\frac {x}{4+x+x^3})} \, dx\)

Optimal. Leaf size=31 \[ \left (5-\frac {5}{x \left (-4-2 x+\log (5)-\log \left (\frac {x}{4+x+x^3}\right )\right )}\right )^2 \]

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Rubi [F]  time = 12.52, 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 {-1000-5000 x-6200 x^2-2900 x^3-1000 x^4-1000 x^5-400 x^6+\left (200+1850 x+1600 x^2+350 x^3+300 x^4+300 x^5\right ) \log (5)+\left (-200 x-50 x^2-50 x^4\right ) \log ^2(5)+\left (-200-1850 x-1600 x^2-350 x^3-300 x^4-300 x^5+\left (400 x+100 x^2+100 x^4\right ) \log (5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (-200 x-50 x^2-50 x^4\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{256 x^3+448 x^4+288 x^5+144 x^6+104 x^7+48 x^8+8 x^9+\left (-192 x^3-240 x^4-96 x^5-60 x^6-48 x^7-12 x^8\right ) \log (5)+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7\right ) \log ^2(5)+\left (-4 x^3-x^4-x^6\right ) \log ^3(5)+\left (192 x^3+240 x^4+96 x^5+60 x^6+48 x^7+12 x^8+\left (-96 x^3-72 x^4-12 x^5-24 x^6-12 x^7\right ) \log (5)+\left (12 x^3+3 x^4+3 x^6\right ) \log ^2(5)\right ) \log \left (\frac {x}{4+x+x^3}\right )+\left (48 x^3+36 x^4+6 x^5+12 x^6+6 x^7+\left (-12 x^3-3 x^4-3 x^6\right ) \log (5)\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )+\left (4 x^3+x^4+x^6\right ) \log ^3\left (\frac {x}{4+x+x^3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1000 - 5000*x - 6200*x^2 - 2900*x^3 - 1000*x^4 - 1000*x^5 - 400*x^6 + (200 + 1850*x + 1600*x^2 + 350*x^3
+ 300*x^4 + 300*x^5)*Log[5] + (-200*x - 50*x^2 - 50*x^4)*Log[5]^2 + (-200 - 1850*x - 1600*x^2 - 350*x^3 - 300
*x^4 - 300*x^5 + (400*x + 100*x^2 + 100*x^4)*Log[5])*Log[x/(4 + x + x^3)] + (-200*x - 50*x^2 - 50*x^4)*Log[x/(
4 + x + x^3)]^2)/(256*x^3 + 448*x^4 + 288*x^5 + 144*x^6 + 104*x^7 + 48*x^8 + 8*x^9 + (-192*x^3 - 240*x^4 - 96*
x^5 - 60*x^6 - 48*x^7 - 12*x^8)*Log[5] + (48*x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7)*Log[5]^2 + (-4*x^3 - x^4 -
x^6)*Log[5]^3 + (192*x^3 + 240*x^4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8 + (-96*x^3 - 72*x^4 - 12*x^5 - 24*x^6
- 12*x^7)*Log[5] + (12*x^3 + 3*x^4 + 3*x^6)*Log[5]^2)*Log[x/(4 + x + x^3)] + (48*x^3 + 36*x^4 + 6*x^5 + 12*x^6
+ 6*x^7 + (-12*x^3 - 3*x^4 - 3*x^6)*Log[5])*Log[x/(4 + x + x^3)]^2 + (4*x^3 + x^4 + x^6)*Log[x/(4 + x + x^3)]
^3),x]

[Out]

-400*Defer[Int][(4 + 2*x + Log[x/(5*(4 + x + x^3))])^(-3), x] - 50*(5 - Log[5])*Defer[Int][1/(x^3*(4 + 2*x + L
og[x/(5*(4 + x + x^3))])^3), x] + (25*(5 - Log[5])*Defer[Int][1/(x^2*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3),
x])/2 - (25*(100 - 37*Log[5] + 4*Log[5]^2)*Defer[Int][1/(x^2*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/2 -
(25*(5 - Log[5])*Defer[Int][1/(x*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/8 - (25*(124 - 32*Log[5] + Log[5
]^2)*Defer[Int][1/(x*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/2 + (25*(100 - 37*Log[5] + 4*Log[5]^2)*Defer
[Int][1/(x*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/8 + 1600*Defer[Int][1/((4 + x + x^3)*(4 + 2*x + Log[x/
(5*(4 + x + x^3))])^3), x] - 50*(58 - 7*Log[5])*Defer[Int][1/((4 + x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))
])^3), x] + (425*(5 - Log[5])*Defer[Int][1/((4 + x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/8 + (25
*(124 - 32*Log[5] + Log[5]^2)*Defer[Int][1/((4 + x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/2 - (25
*(100 - 37*Log[5] + 4*Log[5]^2)*Defer[Int][1/((4 + x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/8 + 4
00*Defer[Int][x/((4 + x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x] - (25*(5 - Log[5])*Defer[Int][x/((4
+ x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/2 - 50*(20 - 6*Log[5] + Log[5]^2)*Defer[Int][x/((4 +
x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x] + (25*(100 - 37*Log[5] + 4*Log[5]^2)*Defer[Int][x/((4 + x
+ x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/2 + (25*(5 - Log[5])*Defer[Int][x^2/((4 + x + x^3)*(4 + 2
*x + Log[x/(5*(4 + x + x^3))])^3), x])/8 + (25*(124 - 32*Log[5] + Log[5]^2)*Defer[Int][x^2/((4 + x + x^3)*(4 +
2*x + Log[x/(5*(4 + x + x^3))])^3), x])/2 - (25*(100 - 37*Log[5] + 4*Log[5]^2)*Defer[Int][x^2/((4 + x + x^3)*
(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/8 - 100*(10 - Log[125])*Defer[Int][x^2/((4 + x + x^3)*(4 + 2*x +
Log[x/(5*(4 + x + x^3))])^3), x] - 50*Defer[Int][Log[x/(4 + x + x^3)]/(x^3*(4 + 2*x + Log[x/(5*(4 + x + x^3))]
)^3), x] - 50*(9 - Log[25])*Defer[Int][Log[x/(4 + x + x^3)]/(x^2*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x] -
(575*Defer[Int][Log[x/(4 + x + x^3)]/(x*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/2 - (25*Defer[Int][Log[x
/(4 + x + x^3)]/((4 + x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/2 + 150*Defer[Int][(x*Log[x/(4 + x
+ x^3)])/((4 + x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x] - (25*Defer[Int][(x^2*Log[x/(4 + x + x^3)
])/((4 + x + x^3)*(4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x])/2 - 50*Defer[Int][Log[x/(4 + x + x^3)]^2/(x^2*(
4 + 2*x + Log[x/(5*(4 + x + x^3))])^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {50 \left (-8 x^6-x^3 (58-7 \log (5))-x^5 (20-6 \log (5))+4 (-5+\log (5))-x^2 \left (124-32 \log (5)+\log ^2(5)\right )-x^4 \left (20-6 \log (5)+\log ^2(5)\right )-x \left (100-37 \log (5)+4 \log ^2(5)\right )-\left (4+7 x^3+6 x^5+x (37-8 \log (5))-2 x^2 (-16+\log (5))-2 x^4 (-3+\log (5))\right ) \log \left (\frac {x}{4+x+x^3}\right )-x \left (4+x+x^3\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )\right )}{x^3 \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx\\ &=50 \int \frac {-8 x^6-x^3 (58-7 \log (5))-x^5 (20-6 \log (5))+4 (-5+\log (5))-x^2 \left (124-32 \log (5)+\log ^2(5)\right )-x^4 \left (20-6 \log (5)+\log ^2(5)\right )-x \left (100-37 \log (5)+4 \log ^2(5)\right )-\left (4+7 x^3+6 x^5+x (37-8 \log (5))-2 x^2 (-16+\log (5))-2 x^4 (-3+\log (5))\right ) \log \left (\frac {x}{4+x+x^3}\right )-x \left (4+x+x^3\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )}{x^3 \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx\\ &=50 \int \left (-\frac {8 x^3}{\left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3}+\frac {4 (-5+\log (5))}{x^3 \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3}+\frac {-58+7 \log (5)}{\left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3}+\frac {-100+37 \log (5)-4 \log ^2(5)}{x^2 \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3}+\frac {-124+32 \log (5)-\log ^2(5)}{x \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3}-\frac {x \left (20-6 \log (5)+\log ^2(5)\right )}{\left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3}+\frac {2 x^2 (-10+\log (125))}{\left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3}+\frac {\left (-4-7 x^3-6 x^5-6 x^4 \left (1-\frac {\log (5)}{3}\right )-37 x \left (1-\frac {8 \log (5)}{37}\right )-32 x^2 \left (1-\frac {\log (5)}{16}\right )\right ) \log \left (\frac {x}{4+x+x^3}\right )}{x^3 \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3}-\frac {\log ^2\left (\frac {x}{4+x+x^3}\right )}{x^2 \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3}\right ) \, dx\\ &=50 \int \frac {\left (-4-7 x^3-6 x^5-6 x^4 \left (1-\frac {\log (5)}{3}\right )-37 x \left (1-\frac {8 \log (5)}{37}\right )-32 x^2 \left (1-\frac {\log (5)}{16}\right )\right ) \log \left (\frac {x}{4+x+x^3}\right )}{x^3 \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx-50 \int \frac {\log ^2\left (\frac {x}{4+x+x^3}\right )}{x^2 \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx-400 \int \frac {x^3}{\left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx-(50 (58-7 \log (5))) \int \frac {1}{\left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx-(200 (5-\log (5))) \int \frac {1}{x^3 \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx-\left (50 \left (124-32 \log (5)+\log ^2(5)\right )\right ) \int \frac {1}{x \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx-\left (50 \left (20-6 \log (5)+\log ^2(5)\right )\right ) \int \frac {x}{\left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx-\left (50 \left (100-37 \log (5)+4 \log ^2(5)\right )\right ) \int \frac {1}{x^2 \left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx-(100 (10-\log (125))) \int \frac {x^2}{\left (4+x+x^3\right ) \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.83, size = 1066, normalized size = 34.39 \begin {gather*} -\frac {25 \left (x \left (32+136 x+96 x^2-48 x^3+66 x^4+45 x^5-18 x^6+20 x^7+6 x^8-x^9+2 x^{10}\right ) \log ^3\left (\frac {x}{5 \left (4+x+x^3\right )}\right )+\left (4+x+x^3\right ) \log ^2\left (\frac {x}{5 \left (4+x+x^3\right )}\right ) \left (-16+8 x^9+x^7 (13-2 \log (5))+2 x (-1+8 \log (5))+8 x^2 (23+8 \log (5))+x^6 (68+8 \log (5))-4 x^4 (11+9 \log (5))+x^5 (-2+26 \log (5))+x^3 (227+32 \log (5))+x^8 (-2+\log (625))-2 x \left (8+32 x+16 x^2-18 x^3+13 x^4+4 x^5-x^6+2 x^7\right ) \log \left (\frac {x}{4+x+x^3}\right )\right )+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right ) \left (8 x^{13}+2 x^{12} (6+5 \log (5))+x^4 \left (1356-31 \log (5)-48 \log ^2(5)\right )+x^7 \left (754+50 \log (5)-18 \log ^2(5)\right )+x^{10} \left (160+21 \log (5)-\log ^2(5)\right )+2 x^9 \left (71+73 \log (5)+3 \log ^2(5)\right )+8 x \left (-63+9 \log (5)+4 \log ^2(5)\right )+x^8 \left (-92+63 \log (5)+20 \log ^2(5)\right )+4 x^3 \left (655+385 \log (5)+24 \log ^2(5)\right )+2 x^5 \left (94+142 \log (5)+33 \log ^2(5)\right )+x^6 \left (1072+727 \log (5)+45 \log ^2(5)\right )+2 x^2 \left (404+603 \log (5)+68 \log ^2(5)\right )-32 (7+\log (25))+2 x^{11} \left (-14+\log ^2(5)+\log (625)\right )-\left (4+x+x^3\right ) \left (-16+10 x^9+x^7 (11-2 \log (5))+4 x^8 (2+\log (5))+2 x (11+8 \log (5))+8 x^2 (37+8 \log (5))+x^6 (98+8 \log (5))-4 x^4 (23+9 \log (5))+x^5 (20+26 \log (5))+x^3 (315+32 \log (5))\right ) \log \left (\frac {x}{4+x+x^3}\right )+x \left (32+136 x+96 x^2-48 x^3+66 x^4+45 x^5-18 x^6+20 x^7+6 x^8-x^9+2 x^{10}\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )\right )-2 \left (8 x^{14}-8 x^{13} (1+\log (5))+16 (1+7 \log (5))-4 x^{10} (1+25 \log (5))+x^6 \left (230-536 \log (5)-115 \log ^2(5)\right )+x \left (224+220 \log (5)-48 \log ^2(5)\right )+x^8 \left (656-38 \log (5)-30 \log ^2(5)\right )+x^{12} \left (2+6 \log (5)-\log ^2(5)\right )+x^7 \left (730-549 \log (5)+13 \log ^2(5)\right )+2 x^4 \left (1250-539 \log (5)+20 \log ^2(5)\right )-x^9 \left (14+23 \log (5)+24 \log ^2(5)\right )-2 x^5 \left (-544+83 \log (5)+38 \log ^2(5)\right )-4 x^2 \left (-262+149 \log (5)+59 \log ^2(5)\right )-2 x^3 \left (-1156+871 \log (5)+116 \log ^2(5)\right )-5 x^{11} \left (-26+\log ^2(5)+\log (25)\right )+\left (-112+100 x^{10}+8 x^{13}+x^4 (1078-80 \log (5))+x^7 (549-26 \log (5))+10 x^{11} (1+\log (5))+4 x (-55+24 \log (5))+x^9 (23+48 \log (5))+x^8 (38+60 \log (5))+2 x^5 (83+76 \log (5))+x^6 (536+230 \log (5))+2 x^3 (871+232 \log (5))+x^2 (596+472 \log (5))+x^{12} (-6+\log (25))\right ) \log \left (\frac {x}{4+x+x^3}\right )-x \left (48+236 x+232 x^2-40 x^3+76 x^4+115 x^5-13 x^6+30 x^7+24 x^8+5 x^{10}+x^{11}\right ) \log ^2\left (\frac {x}{4+x+x^3}\right )\right )\right )}{4 x^2 \left (2+4 x+x^2-x^3+x^4\right )^3 \left (4+2 x+\log \left (\frac {x}{5 \left (4+x+x^3\right )}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1000 - 5000*x - 6200*x^2 - 2900*x^3 - 1000*x^4 - 1000*x^5 - 400*x^6 + (200 + 1850*x + 1600*x^2 + 3
50*x^3 + 300*x^4 + 300*x^5)*Log[5] + (-200*x - 50*x^2 - 50*x^4)*Log[5]^2 + (-200 - 1850*x - 1600*x^2 - 350*x^3
- 300*x^4 - 300*x^5 + (400*x + 100*x^2 + 100*x^4)*Log[5])*Log[x/(4 + x + x^3)] + (-200*x - 50*x^2 - 50*x^4)*L
og[x/(4 + x + x^3)]^2)/(256*x^3 + 448*x^4 + 288*x^5 + 144*x^6 + 104*x^7 + 48*x^8 + 8*x^9 + (-192*x^3 - 240*x^4
- 96*x^5 - 60*x^6 - 48*x^7 - 12*x^8)*Log[5] + (48*x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7)*Log[5]^2 + (-4*x^3 -
x^4 - x^6)*Log[5]^3 + (192*x^3 + 240*x^4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8 + (-96*x^3 - 72*x^4 - 12*x^5 - 2
4*x^6 - 12*x^7)*Log[5] + (12*x^3 + 3*x^4 + 3*x^6)*Log[5]^2)*Log[x/(4 + x + x^3)] + (48*x^3 + 36*x^4 + 6*x^5 +
12*x^6 + 6*x^7 + (-12*x^3 - 3*x^4 - 3*x^6)*Log[5])*Log[x/(4 + x + x^3)]^2 + (4*x^3 + x^4 + x^6)*Log[x/(4 + x +
x^3)]^3),x]

[Out]

(-25*(x*(32 + 136*x + 96*x^2 - 48*x^3 + 66*x^4 + 45*x^5 - 18*x^6 + 20*x^7 + 6*x^8 - x^9 + 2*x^10)*Log[x/(5*(4
+ x + x^3))]^3 + (4 + x + x^3)*Log[x/(5*(4 + x + x^3))]^2*(-16 + 8*x^9 + x^7*(13 - 2*Log[5]) + 2*x*(-1 + 8*Log
[5]) + 8*x^2*(23 + 8*Log[5]) + x^6*(68 + 8*Log[5]) - 4*x^4*(11 + 9*Log[5]) + x^5*(-2 + 26*Log[5]) + x^3*(227 +
32*Log[5]) + x^8*(-2 + Log[625]) - 2*x*(8 + 32*x + 16*x^2 - 18*x^3 + 13*x^4 + 4*x^5 - x^6 + 2*x^7)*Log[x/(4 +
x + x^3)]) + Log[x/(5*(4 + x + x^3))]*(8*x^13 + 2*x^12*(6 + 5*Log[5]) + x^4*(1356 - 31*Log[5] - 48*Log[5]^2)
+ x^7*(754 + 50*Log[5] - 18*Log[5]^2) + x^10*(160 + 21*Log[5] - Log[5]^2) + 2*x^9*(71 + 73*Log[5] + 3*Log[5]^2
) + 8*x*(-63 + 9*Log[5] + 4*Log[5]^2) + x^8*(-92 + 63*Log[5] + 20*Log[5]^2) + 4*x^3*(655 + 385*Log[5] + 24*Log
[5]^2) + 2*x^5*(94 + 142*Log[5] + 33*Log[5]^2) + x^6*(1072 + 727*Log[5] + 45*Log[5]^2) + 2*x^2*(404 + 603*Log[
5] + 68*Log[5]^2) - 32*(7 + Log[25]) + 2*x^11*(-14 + Log[5]^2 + Log[625]) - (4 + x + x^3)*(-16 + 10*x^9 + x^7*
(11 - 2*Log[5]) + 4*x^8*(2 + Log[5]) + 2*x*(11 + 8*Log[5]) + 8*x^2*(37 + 8*Log[5]) + x^6*(98 + 8*Log[5]) - 4*x
^4*(23 + 9*Log[5]) + x^5*(20 + 26*Log[5]) + x^3*(315 + 32*Log[5]))*Log[x/(4 + x + x^3)] + x*(32 + 136*x + 96*x
^2 - 48*x^3 + 66*x^4 + 45*x^5 - 18*x^6 + 20*x^7 + 6*x^8 - x^9 + 2*x^10)*Log[x/(4 + x + x^3)]^2) - 2*(8*x^14 -
8*x^13*(1 + Log[5]) + 16*(1 + 7*Log[5]) - 4*x^10*(1 + 25*Log[5]) + x^6*(230 - 536*Log[5] - 115*Log[5]^2) + x*(
224 + 220*Log[5] - 48*Log[5]^2) + x^8*(656 - 38*Log[5] - 30*Log[5]^2) + x^12*(2 + 6*Log[5] - Log[5]^2) + x^7*(
730 - 549*Log[5] + 13*Log[5]^2) + 2*x^4*(1250 - 539*Log[5] + 20*Log[5]^2) - x^9*(14 + 23*Log[5] + 24*Log[5]^2)
- 2*x^5*(-544 + 83*Log[5] + 38*Log[5]^2) - 4*x^2*(-262 + 149*Log[5] + 59*Log[5]^2) - 2*x^3*(-1156 + 871*Log[5
] + 116*Log[5]^2) - 5*x^11*(-26 + Log[5]^2 + Log[25]) + (-112 + 100*x^10 + 8*x^13 + x^4*(1078 - 80*Log[5]) + x
^7*(549 - 26*Log[5]) + 10*x^11*(1 + Log[5]) + 4*x*(-55 + 24*Log[5]) + x^9*(23 + 48*Log[5]) + x^8*(38 + 60*Log[
5]) + 2*x^5*(83 + 76*Log[5]) + x^6*(536 + 230*Log[5]) + 2*x^3*(871 + 232*Log[5]) + x^2*(596 + 472*Log[5]) + x^
12*(-6 + Log[25]))*Log[x/(4 + x + x^3)] - x*(48 + 236*x + 232*x^2 - 40*x^3 + 76*x^4 + 115*x^5 - 13*x^6 + 30*x^
7 + 24*x^8 + 5*x^10 + x^11)*Log[x/(4 + x + x^3)]^2)))/(4*x^2*(2 + 4*x + x^2 - x^3 + x^4)^3*(4 + 2*x + Log[x/(5
*(4 + x + x^3))])^2)

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fricas [B]  time = 0.68, size = 118, normalized size = 3.81 \begin {gather*} \frac {25 \, {\left (4 \, x^{2} - 2 \, x \log \relax (5) + 2 \, x \log \left (\frac {x}{x^{3} + x + 4}\right ) + 8 \, x + 1\right )}}{4 \, x^{4} + x^{2} \log \relax (5)^{2} + x^{2} \log \left (\frac {x}{x^{3} + x + 4}\right )^{2} + 16 \, x^{3} + 16 \, x^{2} - 4 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \relax (5) + 2 \, {\left (2 \, x^{3} - x^{2} \log \relax (5) + 4 \, x^{2}\right )} \log \left (\frac {x}{x^{3} + x + 4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x^4-50*x^2-200*x)*log(x/(x^3+x+4))^2+((100*x^4+100*x^2+400*x)*log(5)-300*x^5-300*x^4-350*x^3-1
600*x^2-1850*x-200)*log(x/(x^3+x+4))+(-50*x^4-50*x^2-200*x)*log(5)^2+(300*x^5+300*x^4+350*x^3+1600*x^2+1850*x+
200)*log(5)-400*x^6-1000*x^5-1000*x^4-2900*x^3-6200*x^2-5000*x-1000)/((x^6+x^4+4*x^3)*log(x/(x^3+x+4))^3+((-3*
x^6-3*x^4-12*x^3)*log(5)+6*x^7+12*x^6+6*x^5+36*x^4+48*x^3)*log(x/(x^3+x+4))^2+((3*x^6+3*x^4+12*x^3)*log(5)^2+(
-12*x^7-24*x^6-12*x^5-72*x^4-96*x^3)*log(5)+12*x^8+48*x^7+60*x^6+96*x^5+240*x^4+192*x^3)*log(x/(x^3+x+4))+(-x^
6-x^4-4*x^3)*log(5)^3+(6*x^7+12*x^6+6*x^5+36*x^4+48*x^3)*log(5)^2+(-12*x^8-48*x^7-60*x^6-96*x^5-240*x^4-192*x^
3)*log(5)+8*x^9+48*x^8+104*x^7+144*x^6+288*x^5+448*x^4+256*x^3),x, algorithm="fricas")

[Out]

25*(4*x^2 - 2*x*log(5) + 2*x*log(x/(x^3 + x + 4)) + 8*x + 1)/(4*x^4 + x^2*log(5)^2 + x^2*log(x/(x^3 + x + 4))^
2 + 16*x^3 + 16*x^2 - 4*(x^3 + 2*x^2)*log(5) + 2*(2*x^3 - x^2*log(5) + 4*x^2)*log(x/(x^3 + x + 4)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x^4-50*x^2-200*x)*log(x/(x^3+x+4))^2+((100*x^4+100*x^2+400*x)*log(5)-300*x^5-300*x^4-350*x^3-1
600*x^2-1850*x-200)*log(x/(x^3+x+4))+(-50*x^4-50*x^2-200*x)*log(5)^2+(300*x^5+300*x^4+350*x^3+1600*x^2+1850*x+
200)*log(5)-400*x^6-1000*x^5-1000*x^4-2900*x^3-6200*x^2-5000*x-1000)/((x^6+x^4+4*x^3)*log(x/(x^3+x+4))^3+((-3*
x^6-3*x^4-12*x^3)*log(5)+6*x^7+12*x^6+6*x^5+36*x^4+48*x^3)*log(x/(x^3+x+4))^2+((3*x^6+3*x^4+12*x^3)*log(5)^2+(
-12*x^7-24*x^6-12*x^5-72*x^4-96*x^3)*log(5)+12*x^8+48*x^7+60*x^6+96*x^5+240*x^4+192*x^3)*log(x/(x^3+x+4))+(-x^
6-x^4-4*x^3)*log(5)^3+(6*x^7+12*x^6+6*x^5+36*x^4+48*x^3)*log(5)^2+(-12*x^8-48*x^7-60*x^6-96*x^5-240*x^4-192*x^
3)*log(5)+8*x^9+48*x^8+104*x^7+144*x^6+288*x^5+448*x^4+256*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.07, size = 57, normalized size = 1.84

method	result	size
risch	\(-\frac {25 \left (2 x \ln \relax (5)-4 x^{2}-2 x \ln \left (\frac {x}{x^{3}+x +4}\right )-8 x -1\right )}{x^{2} \left (\ln \relax (5)-\ln \left (\frac {x}{x^{3}+x +4}\right )-4-2 x \right )^{2}}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-50*x^4-50*x^2-200*x)*ln(x/(x^3+x+4))^2+((100*x^4+100*x^2+400*x)*ln(5)-300*x^5-300*x^4-350*x^3-1600*x^2-
1850*x-200)*ln(x/(x^3+x+4))+(-50*x^4-50*x^2-200*x)*ln(5)^2+(300*x^5+300*x^4+350*x^3+1600*x^2+1850*x+200)*ln(5)
-400*x^6-1000*x^5-1000*x^4-2900*x^3-6200*x^2-5000*x-1000)/((x^6+x^4+4*x^3)*ln(x/(x^3+x+4))^3+((-3*x^6-3*x^4-12
*x^3)*ln(5)+6*x^7+12*x^6+6*x^5+36*x^4+48*x^3)*ln(x/(x^3+x+4))^2+((3*x^6+3*x^4+12*x^3)*ln(5)^2+(-12*x^7-24*x^6-
12*x^5-72*x^4-96*x^3)*ln(5)+12*x^8+48*x^7+60*x^6+96*x^5+240*x^4+192*x^3)*ln(x/(x^3+x+4))+(-x^6-x^4-4*x^3)*ln(5
)^3+(6*x^7+12*x^6+6*x^5+36*x^4+48*x^3)*ln(5)^2+(-12*x^8-48*x^7-60*x^6-96*x^5-240*x^4-192*x^3)*ln(5)+8*x^9+48*x
^8+104*x^7+144*x^6+288*x^5+448*x^4+256*x^3),x,method=_RETURNVERBOSE)

[Out]

-25*(2*x*ln(5)-4*x^2-2*x*ln(x/(x^3+x+4))-8*x-1)/x^2/(ln(5)-ln(x/(x^3+x+4))-4-2*x)^2

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maxima [B]  time = 0.86, size = 132, normalized size = 4.26 \begin {gather*} \frac {25 \, {\left (4 \, x^{2} - 2 \, x {\left (\log \relax (5) - 4\right )} - 2 \, x \log \left (x^{3} + x + 4\right ) + 2 \, x \log \relax (x) + 1\right )}}{4 \, x^{4} - 4 \, x^{3} {\left (\log \relax (5) - 4\right )} + x^{2} \log \left (x^{3} + x + 4\right )^{2} + x^{2} \log \relax (x)^{2} + {\left (\log \relax (5)^{2} - 8 \, \log \relax (5) + 16\right )} x^{2} - 2 \, {\left (2 \, x^{3} - x^{2} {\left (\log \relax (5) - 4\right )} + x^{2} \log \relax (x)\right )} \log \left (x^{3} + x + 4\right ) + 2 \, {\left (2 \, x^{3} - x^{2} {\left (\log \relax (5) - 4\right )}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x^4-50*x^2-200*x)*log(x/(x^3+x+4))^2+((100*x^4+100*x^2+400*x)*log(5)-300*x^5-300*x^4-350*x^3-1
600*x^2-1850*x-200)*log(x/(x^3+x+4))+(-50*x^4-50*x^2-200*x)*log(5)^2+(300*x^5+300*x^4+350*x^3+1600*x^2+1850*x+
200)*log(5)-400*x^6-1000*x^5-1000*x^4-2900*x^3-6200*x^2-5000*x-1000)/((x^6+x^4+4*x^3)*log(x/(x^3+x+4))^3+((-3*
x^6-3*x^4-12*x^3)*log(5)+6*x^7+12*x^6+6*x^5+36*x^4+48*x^3)*log(x/(x^3+x+4))^2+((3*x^6+3*x^4+12*x^3)*log(5)^2+(
-12*x^7-24*x^6-12*x^5-72*x^4-96*x^3)*log(5)+12*x^8+48*x^7+60*x^6+96*x^5+240*x^4+192*x^3)*log(x/(x^3+x+4))+(-x^
6-x^4-4*x^3)*log(5)^3+(6*x^7+12*x^6+6*x^5+36*x^4+48*x^3)*log(5)^2+(-12*x^8-48*x^7-60*x^6-96*x^5-240*x^4-192*x^
3)*log(5)+8*x^9+48*x^8+104*x^7+144*x^6+288*x^5+448*x^4+256*x^3),x, algorithm="maxima")

[Out]

25*(4*x^2 - 2*x*(log(5) - 4) - 2*x*log(x^3 + x + 4) + 2*x*log(x) + 1)/(4*x^4 - 4*x^3*(log(5) - 4) + x^2*log(x^
3 + x + 4)^2 + x^2*log(x)^2 + (log(5)^2 - 8*log(5) + 16)*x^2 - 2*(2*x^3 - x^2*(log(5) - 4) + x^2*log(x))*log(x
^3 + x + 4) + 2*(2*x^3 - x^2*(log(5) - 4))*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {5000\,x+\ln \left (\frac {x}{x^3+x+4}\right )\,\left (1850\,x-\ln \relax (5)\,\left (100\,x^4+100\,x^2+400\,x\right )+1600\,x^2+350\,x^3+300\,x^4+300\,x^5+200\right )+{\ln \left (\frac {x}{x^3+x+4}\right )}^2\,\left (50\,x^4+50\,x^2+200\,x\right )+{\ln \relax (5)}^2\,\left (50\,x^4+50\,x^2+200\,x\right )+6200\,x^2+2900\,x^3+1000\,x^4+1000\,x^5+400\,x^6-\ln \relax (5)\,\left (300\,x^5+300\,x^4+350\,x^3+1600\,x^2+1850\,x+200\right )+1000}{{\ln \left (\frac {x}{x^3+x+4}\right )}^2\,\left (48\,x^3-\ln \relax (5)\,\left (3\,x^6+3\,x^4+12\,x^3\right )+36\,x^4+6\,x^5+12\,x^6+6\,x^7\right )+{\ln \left (\frac {x}{x^3+x+4}\right )}^3\,\left (x^6+x^4+4\,x^3\right )-{\ln \relax (5)}^3\,\left (x^6+x^4+4\,x^3\right )-\ln \relax (5)\,\left (12\,x^8+48\,x^7+60\,x^6+96\,x^5+240\,x^4+192\,x^3\right )+{\ln \relax (5)}^2\,\left (6\,x^7+12\,x^6+6\,x^5+36\,x^4+48\,x^3\right )+\ln \left (\frac {x}{x^3+x+4}\right )\,\left ({\ln \relax (5)}^2\,\left (3\,x^6+3\,x^4+12\,x^3\right )-\ln \relax (5)\,\left (12\,x^7+24\,x^6+12\,x^5+72\,x^4+96\,x^3\right )+192\,x^3+240\,x^4+96\,x^5+60\,x^6+48\,x^7+12\,x^8\right )+256\,x^3+448\,x^4+288\,x^5+144\,x^6+104\,x^7+48\,x^8+8\,x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5000*x + log(x/(x + x^3 + 4))*(1850*x - log(5)*(400*x + 100*x^2 + 100*x^4) + 1600*x^2 + 350*x^3 + 300*x^
4 + 300*x^5 + 200) + log(x/(x + x^3 + 4))^2*(200*x + 50*x^2 + 50*x^4) + log(5)^2*(200*x + 50*x^2 + 50*x^4) + 6
200*x^2 + 2900*x^3 + 1000*x^4 + 1000*x^5 + 400*x^6 - log(5)*(1850*x + 1600*x^2 + 350*x^3 + 300*x^4 + 300*x^5 +
200) + 1000)/(log(x/(x + x^3 + 4))^2*(48*x^3 - log(5)*(12*x^3 + 3*x^4 + 3*x^6) + 36*x^4 + 6*x^5 + 12*x^6 + 6*
x^7) + log(x/(x + x^3 + 4))^3*(4*x^3 + x^4 + x^6) - log(5)^3*(4*x^3 + x^4 + x^6) - log(5)*(192*x^3 + 240*x^4 +
96*x^5 + 60*x^6 + 48*x^7 + 12*x^8) + log(5)^2*(48*x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7) + log(x/(x + x^3 + 4
))*(log(5)^2*(12*x^3 + 3*x^4 + 3*x^6) - log(5)*(96*x^3 + 72*x^4 + 12*x^5 + 24*x^6 + 12*x^7) + 192*x^3 + 240*x^
4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8) + 256*x^3 + 448*x^4 + 288*x^5 + 144*x^6 + 104*x^7 + 48*x^8 + 8*x^9),x)

[Out]

int(-(5000*x + log(x/(x + x^3 + 4))*(1850*x - log(5)*(400*x + 100*x^2 + 100*x^4) + 1600*x^2 + 350*x^3 + 300*x^
4 + 300*x^5 + 200) + log(x/(x + x^3 + 4))^2*(200*x + 50*x^2 + 50*x^4) + log(5)^2*(200*x + 50*x^2 + 50*x^4) + 6
200*x^2 + 2900*x^3 + 1000*x^4 + 1000*x^5 + 400*x^6 - log(5)*(1850*x + 1600*x^2 + 350*x^3 + 300*x^4 + 300*x^5 +
200) + 1000)/(log(x/(x + x^3 + 4))^2*(48*x^3 - log(5)*(12*x^3 + 3*x^4 + 3*x^6) + 36*x^4 + 6*x^5 + 12*x^6 + 6*
x^7) + log(x/(x + x^3 + 4))^3*(4*x^3 + x^4 + x^6) - log(5)^3*(4*x^3 + x^4 + x^6) - log(5)*(192*x^3 + 240*x^4 +
96*x^5 + 60*x^6 + 48*x^7 + 12*x^8) + log(5)^2*(48*x^3 + 36*x^4 + 6*x^5 + 12*x^6 + 6*x^7) + log(x/(x + x^3 + 4
))*(log(5)^2*(12*x^3 + 3*x^4 + 3*x^6) - log(5)*(96*x^3 + 72*x^4 + 12*x^5 + 24*x^6 + 12*x^7) + 192*x^3 + 240*x^
4 + 96*x^5 + 60*x^6 + 48*x^7 + 12*x^8) + 256*x^3 + 448*x^4 + 288*x^5 + 144*x^6 + 104*x^7 + 48*x^8 + 8*x^9), x)

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sympy [B]  time = 0.56, size = 114, normalized size = 3.68 \begin {gather*} \frac {100 x^{2} + 50 x \log {\left (\frac {x}{x^{3} + x + 4} \right )} - 50 x \log {\relax (5 )} + 200 x + 25}{4 x^{4} - 4 x^{3} \log {\relax (5 )} + 16 x^{3} + x^{2} \log {\left (\frac {x}{x^{3} + x + 4} \right )}^{2} - 8 x^{2} \log {\relax (5 )} + x^{2} \log {\relax (5 )}^{2} + 16 x^{2} + \left (4 x^{3} - 2 x^{2} \log {\relax (5 )} + 8 x^{2}\right ) \log {\left (\frac {x}{x^{3} + x + 4} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-50*x**4-50*x**2-200*x)*ln(x/(x**3+x+4))**2+((100*x**4+100*x**2+400*x)*ln(5)-300*x**5-300*x**4-350
*x**3-1600*x**2-1850*x-200)*ln(x/(x**3+x+4))+(-50*x**4-50*x**2-200*x)*ln(5)**2+(300*x**5+300*x**4+350*x**3+160
0*x**2+1850*x+200)*ln(5)-400*x**6-1000*x**5-1000*x**4-2900*x**3-6200*x**2-5000*x-1000)/((x**6+x**4+4*x**3)*ln(
x/(x**3+x+4))**3+((-3*x**6-3*x**4-12*x**3)*ln(5)+6*x**7+12*x**6+6*x**5+36*x**4+48*x**3)*ln(x/(x**3+x+4))**2+((
3*x**6+3*x**4+12*x**3)*ln(5)**2+(-12*x**7-24*x**6-12*x**5-72*x**4-96*x**3)*ln(5)+12*x**8+48*x**7+60*x**6+96*x*
*5+240*x**4+192*x**3)*ln(x/(x**3+x+4))+(-x**6-x**4-4*x**3)*ln(5)**3+(6*x**7+12*x**6+6*x**5+36*x**4+48*x**3)*ln
(5)**2+(-12*x**8-48*x**7-60*x**6-96*x**5-240*x**4-192*x**3)*ln(5)+8*x**9+48*x**8+104*x**7+144*x**6+288*x**5+44
8*x**4+256*x**3),x)

[Out]

(100*x**2 + 50*x*log(x/(x**3 + x + 4)) - 50*x*log(5) + 200*x + 25)/(4*x**4 - 4*x**3*log(5) + 16*x**3 + x**2*lo
g(x/(x**3 + x + 4))**2 - 8*x**2*log(5) + x**2*log(5)**2 + 16*x**2 + (4*x**3 - 2*x**2*log(5) + 8*x**2)*log(x/(x
**3 + x + 4)))

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