3.36.34 \(\int \frac {-125 x-200 x^2-90 x^3-16 x^4-x^5+(-50 x-20 x^2-2 x^3) \log (4)+e^x (-625-1000 x-450 x^2-80 x^3-5 x^4+(-250 x-100 x^2-10 x^3) \log (4))+(e^x (125+100 x+15 x^2) \log (4)+(25 x+20 x^2+3 x^3) \log (4)) \log (\frac {1}{5} (5 e^x+x))}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6)+(e^x (121500-194400 x+29160 x^2+38880 x^3+4860 x^4) \log (4)+(24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5) \log (4)) \log (\frac {1}{5} (5 e^x+x))+(e^x (-24300+19440 x+4860 x^2) \log ^2(4)+(-4860 x+3888 x^2+972 x^3) \log ^2(4)) \log ^2(\frac {1}{5} (5 e^x+x))+(1620 e^x \log ^3(4)+324 x \log ^3(4)) \log ^3(\frac {1}{5} (5 e^x+x))} \, dx\)

Optimal. Leaf size=28 \[ \frac {x}{324 \left (-1+x+\frac {\log (4) \log \left (e^x+\frac {x}{5}\right )}{5+x}\right )^2} \]

________________________________________________________________________________________

Rubi [F]  time = 8.82, 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 {-125 x-200 x^2-90 x^3-16 x^4-x^5+\left (-50 x-20 x^2-2 x^3\right ) \log (4)+e^x \left (-625-1000 x-450 x^2-80 x^3-5 x^4+\left (-250 x-100 x^2-10 x^3\right ) \log (4)\right )+\left (e^x \left (125+100 x+15 x^2\right ) \log (4)+\left (25 x+20 x^2+3 x^3\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )}{-40500 x+97200 x^2-53460 x^3-18144 x^4+10692 x^5+3888 x^6+324 x^7+e^x \left (-202500+486000 x-267300 x^2-90720 x^3+53460 x^4+19440 x^5+1620 x^6\right )+\left (e^x \left (121500-194400 x+29160 x^2+38880 x^3+4860 x^4\right ) \log (4)+\left (24300 x-38880 x^2+5832 x^3+7776 x^4+972 x^5\right ) \log (4)\right ) \log \left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (e^x \left (-24300+19440 x+4860 x^2\right ) \log ^2(4)+\left (-4860 x+3888 x^2+972 x^3\right ) \log ^2(4)\right ) \log ^2\left (\frac {1}{5} \left (5 e^x+x\right )\right )+\left (1620 e^x \log ^3(4)+324 x \log ^3(4)\right ) \log ^3\left (\frac {1}{5} \left (5 e^x+x\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-125*x - 200*x^2 - 90*x^3 - 16*x^4 - x^5 + (-50*x - 20*x^2 - 2*x^3)*Log[4] + E^x*(-625 - 1000*x - 450*x^2
- 80*x^3 - 5*x^4 + (-250*x - 100*x^2 - 10*x^3)*Log[4]) + (E^x*(125 + 100*x + 15*x^2)*Log[4] + (25*x + 20*x^2
+ 3*x^3)*Log[4])*Log[(5*E^x + x)/5])/(-40500*x + 97200*x^2 - 53460*x^3 - 18144*x^4 + 10692*x^5 + 3888*x^6 + 32
4*x^7 + E^x*(-202500 + 486000*x - 267300*x^2 - 90720*x^3 + 53460*x^4 + 19440*x^5 + 1620*x^6) + (E^x*(121500 -
194400*x + 29160*x^2 + 38880*x^3 + 4860*x^4)*Log[4] + (24300*x - 38880*x^2 + 5832*x^3 + 7776*x^4 + 972*x^5)*Lo
g[4])*Log[(5*E^x + x)/5] + (E^x*(-24300 + 19440*x + 4860*x^2)*Log[4]^2 + (-4860*x + 3888*x^2 + 972*x^3)*Log[4]
^2)*Log[(5*E^x + x)/5]^2 + (1620*E^x*Log[4]^3 + 324*x*Log[4]^3)*Log[(5*E^x + x)/5]^3),x]

[Out]

(-25*(8 + Log[16])*Defer[Int][x/(-5 + 4*x + x^2 + Log[4]*Log[E^x + x/5])^3, x])/324 - (5*(18 + Log[16])*Defer[
Int][x^2/(-5 + 4*x + x^2 + Log[4]*Log[E^x + x/5])^3, x])/162 - ((48 + Log[16])*Defer[Int][x^3/(-5 + 4*x + x^2
+ Log[4]*Log[E^x + x/5])^3, x])/324 - Defer[Int][x^4/(-5 + 4*x + x^2 + Log[4]*Log[E^x + x/5])^3, x]/81 - (25*L
og[16]*Defer[Int][x/((5*E^x + x)*(-5 + 4*x + x^2 + Log[4]*Log[E^x + x/5])^3), x])/324 + (5*Log[16]*Defer[Int][
x^2/((5*E^x + x)*(-5 + 4*x + x^2 + Log[4]*Log[E^x + x/5])^3), x])/108 + (Log[16]*Defer[Int][x^3/((5*E^x + x)*(
-5 + 4*x + x^2 + Log[4]*Log[E^x + x/5])^3), x])/36 + (Log[16]*Defer[Int][x^4/((5*E^x + x)*(-5 + 4*x + x^2 + Lo
g[4]*Log[E^x + x/5])^3), x])/324 + (5*Defer[Int][x/(-5 + 4*x + x^2 + Log[4]*Log[E^x + x/5])^2, x])/81 + Defer[
Int][x^2/(-5 + 4*x + x^2 + Log[4]*Log[E^x + x/5])^2, x]/108 + (125*Defer[Subst][Defer[Int][(-5 + 20*x + 25*x^2
+ Log[4]*Log[E^(5*x) + x])^(-2), x], x, x/5])/324

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(5+x) \left ((5+x) \left (x \left (5+6 x+x^2+\log (16)\right )+5 e^x \left (5+x^2+x (6+\log (16))\right )\right )-\left (5 e^x+x\right ) (5+3 x) \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{324 \left (5 e^x+x\right ) \left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\\ &=\frac {1}{324} \int \frac {(5+x) \left ((5+x) \left (x \left (5+6 x+x^2+\log (16)\right )+5 e^x \left (5+x^2+x (6+\log (16))\right )\right )-\left (5 e^x+x\right ) (5+3 x) \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{\left (5 e^x+x\right ) \left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\\ &=\frac {1}{324} \int \left (\frac {(-1+x) x (5+x)^2 \log (16)}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {(5+x) \left (25+x^3+11 x^2 \left (1+\frac {4 \log (2)}{11}\right )+35 x \left (1+\frac {4 \log (2)}{7}\right )-5 \log (4) \log \left (e^x+\frac {x}{5}\right )-3 x \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{\left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}\right ) \, dx\\ &=\frac {1}{324} \int \frac {(5+x) \left (25+x^3+11 x^2 \left (1+\frac {4 \log (2)}{11}\right )+35 x \left (1+\frac {4 \log (2)}{7}\right )-5 \log (4) \log \left (e^x+\frac {x}{5}\right )-3 x \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{\left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{324} \log (16) \int \frac {(-1+x) x (5+x)^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\\ &=\frac {1}{324} \int \frac {(5+x) \left ((5+x) \left (5+x^2+x (6+\log (16))\right )-(5+3 x) \log (4) \log \left (e^x+\frac {x}{5}\right )\right )}{\left (5-4 x-x^2-\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{324} \log (16) \int \left (-\frac {25 x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {15 x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {9 x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}\right ) \, dx\\ &=\frac {1}{324} \int \left (-\frac {x (5+x)^2 (8+4 x+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {25+20 x+3 x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2}\right ) \, dx+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\\ &=-\left (\frac {1}{324} \int \frac {x (5+x)^2 (8+4 x+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\right )+\frac {1}{324} \int \frac {25+20 x+3 x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\\ &=-\left (\frac {1}{324} \int \left (\frac {4 x^4}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {25 x (8+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {10 x^2 (18+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}+\frac {x^3 (48+\log (16))}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3}\right ) \, dx\right )+\frac {1}{324} \int \left (\frac {25}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2}+\frac {20 x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2}+\frac {3 x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2}\right ) \, dx+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\\ &=\frac {1}{108} \int \frac {x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx-\frac {1}{81} \int \frac {x^4}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {5}{81} \int \frac {x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx+\frac {25}{324} \int \frac {1}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 (8+\log (16))) \int \frac {x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{162} (5 (18+\log (16))) \int \frac {x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (48+\log (16)) \int \frac {x^3}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\\ &=\frac {1}{108} \int \frac {x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx-\frac {1}{81} \int \frac {x^4}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {5}{81} \int \frac {x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \, dx+\frac {125}{324} \operatorname {Subst}\left (\int \frac {1}{\left (-5+20 x+25 x^2+\log (4) \log \left (e^{5 x}+x\right )\right )^2} \, dx,x,\frac {x}{5}\right )+\frac {1}{324} \log (16) \int \frac {x^4}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{36} \log (16) \int \frac {x^3}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx+\frac {1}{108} (5 \log (16)) \int \frac {x^2}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 \log (16)) \int \frac {x}{\left (5 e^x+x\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (25 (8+\log (16))) \int \frac {x}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{162} (5 (18+\log (16))) \int \frac {x^2}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx-\frac {1}{324} (48+\log (16)) \int \frac {x^3}{\left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^3} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.55, size = 81, normalized size = 2.89 \begin {gather*} \frac {x (5+x)^2 \left (8 x+4 x^2+\log (16)+5 e^x (8+4 x+\log (16))\right )}{648 \left (4 x+2 x^2+\log (4)+5 e^x (4+2 x+\log (4))\right ) \left (-5+4 x+x^2+\log (4) \log \left (e^x+\frac {x}{5}\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-125*x - 200*x^2 - 90*x^3 - 16*x^4 - x^5 + (-50*x - 20*x^2 - 2*x^3)*Log[4] + E^x*(-625 - 1000*x - 4
50*x^2 - 80*x^3 - 5*x^4 + (-250*x - 100*x^2 - 10*x^3)*Log[4]) + (E^x*(125 + 100*x + 15*x^2)*Log[4] + (25*x + 2
0*x^2 + 3*x^3)*Log[4])*Log[(5*E^x + x)/5])/(-40500*x + 97200*x^2 - 53460*x^3 - 18144*x^4 + 10692*x^5 + 3888*x^
6 + 324*x^7 + E^x*(-202500 + 486000*x - 267300*x^2 - 90720*x^3 + 53460*x^4 + 19440*x^5 + 1620*x^6) + (E^x*(121
500 - 194400*x + 29160*x^2 + 38880*x^3 + 4860*x^4)*Log[4] + (24300*x - 38880*x^2 + 5832*x^3 + 7776*x^4 + 972*x
^5)*Log[4])*Log[(5*E^x + x)/5] + (E^x*(-24300 + 19440*x + 4860*x^2)*Log[4]^2 + (-4860*x + 3888*x^2 + 972*x^3)*
Log[4]^2)*Log[(5*E^x + x)/5]^2 + (1620*E^x*Log[4]^3 + 324*x*Log[4]^3)*Log[(5*E^x + x)/5]^3),x]

[Out]

(x*(5 + x)^2*(8*x + 4*x^2 + Log[16] + 5*E^x*(8 + 4*x + Log[16])))/(648*(4*x + 2*x^2 + Log[4] + 5*E^x*(4 + 2*x
+ Log[4]))*(-5 + 4*x + x^2 + Log[4]*Log[E^x + x/5])^2)

________________________________________________________________________________________

fricas [B]  time = 0.72, size = 68, normalized size = 2.43 \begin {gather*} \frac {x^{3} + 10 \, x^{2} + 25 \, x}{324 \, {\left (x^{4} + 4 \, \log \relax (2)^{2} \log \left (\frac {1}{5} \, x + e^{x}\right )^{2} + 8 \, x^{3} + 4 \, {\left (x^{2} + 4 \, x - 5\right )} \log \relax (2) \log \left (\frac {1}{5} \, x + e^{x}\right ) + 6 \, x^{2} - 40 \, x + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(15*x^2+100*x+125)*log(2)*exp(x)+2*(3*x^3+20*x^2+25*x)*log(2))*log(1/5*x+exp(x))+(2*(-10*x^3-100
*x^2-250*x)*log(2)-5*x^4-80*x^3-450*x^2-1000*x-625)*exp(x)+2*(-2*x^3-20*x^2-50*x)*log(2)-x^5-16*x^4-90*x^3-200
*x^2-125*x)/((12960*log(2)^3*exp(x)+2592*x*log(2)^3)*log(1/5*x+exp(x))^3+(4*(4860*x^2+19440*x-24300)*log(2)^2*
exp(x)+4*(972*x^3+3888*x^2-4860*x)*log(2)^2)*log(1/5*x+exp(x))^2+(2*(4860*x^4+38880*x^3+29160*x^2-194400*x+121
500)*log(2)*exp(x)+2*(972*x^5+7776*x^4+5832*x^3-38880*x^2+24300*x)*log(2))*log(1/5*x+exp(x))+(1620*x^6+19440*x
^5+53460*x^4-90720*x^3-267300*x^2+486000*x-202500)*exp(x)+324*x^7+3888*x^6+10692*x^5-18144*x^4-53460*x^3+97200
*x^2-40500*x),x, algorithm="fricas")

[Out]

1/324*(x^3 + 10*x^2 + 25*x)/(x^4 + 4*log(2)^2*log(1/5*x + e^x)^2 + 8*x^3 + 4*(x^2 + 4*x - 5)*log(2)*log(1/5*x
+ e^x) + 6*x^2 - 40*x + 25)

________________________________________________________________________________________

giac [B]  time = 1.23, size = 133, normalized size = 4.75 \begin {gather*} \frac {x^{3} + 10 \, x^{2} + 25 \, x}{324 \, {\left (x^{4} - 4 \, x^{2} \log \relax (5) \log \relax (2) + 4 \, \log \relax (5)^{2} \log \relax (2)^{2} + 4 \, x^{2} \log \relax (2) \log \left (x + 5 \, e^{x}\right ) - 8 \, \log \relax (5) \log \relax (2)^{2} \log \left (x + 5 \, e^{x}\right ) + 4 \, \log \relax (2)^{2} \log \left (x + 5 \, e^{x}\right )^{2} + 8 \, x^{3} - 16 \, x \log \relax (5) \log \relax (2) + 16 \, x \log \relax (2) \log \left (x + 5 \, e^{x}\right ) + 6 \, x^{2} + 20 \, \log \relax (5) \log \relax (2) - 20 \, \log \relax (2) \log \left (x + 5 \, e^{x}\right ) - 40 \, x + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(15*x^2+100*x+125)*log(2)*exp(x)+2*(3*x^3+20*x^2+25*x)*log(2))*log(1/5*x+exp(x))+(2*(-10*x^3-100
*x^2-250*x)*log(2)-5*x^4-80*x^3-450*x^2-1000*x-625)*exp(x)+2*(-2*x^3-20*x^2-50*x)*log(2)-x^5-16*x^4-90*x^3-200
*x^2-125*x)/((12960*log(2)^3*exp(x)+2592*x*log(2)^3)*log(1/5*x+exp(x))^3+(4*(4860*x^2+19440*x-24300)*log(2)^2*
exp(x)+4*(972*x^3+3888*x^2-4860*x)*log(2)^2)*log(1/5*x+exp(x))^2+(2*(4860*x^4+38880*x^3+29160*x^2-194400*x+121
500)*log(2)*exp(x)+2*(972*x^5+7776*x^4+5832*x^3-38880*x^2+24300*x)*log(2))*log(1/5*x+exp(x))+(1620*x^6+19440*x
^5+53460*x^4-90720*x^3-267300*x^2+486000*x-202500)*exp(x)+324*x^7+3888*x^6+10692*x^5-18144*x^4-53460*x^3+97200
*x^2-40500*x),x, algorithm="giac")

[Out]

1/324*(x^3 + 10*x^2 + 25*x)/(x^4 - 4*x^2*log(5)*log(2) + 4*log(5)^2*log(2)^2 + 4*x^2*log(2)*log(x + 5*e^x) - 8
*log(5)*log(2)^2*log(x + 5*e^x) + 4*log(2)^2*log(x + 5*e^x)^2 + 8*x^3 - 16*x*log(5)*log(2) + 16*x*log(2)*log(x
+ 5*e^x) + 6*x^2 + 20*log(5)*log(2) - 20*log(2)*log(x + 5*e^x) - 40*x + 25)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 33, normalized size = 1.18

method	result	size
risch	\(\frac {x \left (x^{2}+10 x +25\right )}{324 \left (2 \ln \relax (2) \ln \left (\frac {x}{5}+{\mathrm e}^{x}\right )+x^{2}+4 x -5\right )^{2}}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*(15*x^2+100*x+125)*ln(2)*exp(x)+2*(3*x^3+20*x^2+25*x)*ln(2))*ln(1/5*x+exp(x))+(2*(-10*x^3-100*x^2-250*
x)*ln(2)-5*x^4-80*x^3-450*x^2-1000*x-625)*exp(x)+2*(-2*x^3-20*x^2-50*x)*ln(2)-x^5-16*x^4-90*x^3-200*x^2-125*x)
/((12960*ln(2)^3*exp(x)+2592*x*ln(2)^3)*ln(1/5*x+exp(x))^3+(4*(4860*x^2+19440*x-24300)*ln(2)^2*exp(x)+4*(972*x
^3+3888*x^2-4860*x)*ln(2)^2)*ln(1/5*x+exp(x))^2+(2*(4860*x^4+38880*x^3+29160*x^2-194400*x+121500)*ln(2)*exp(x)
+2*(972*x^5+7776*x^4+5832*x^3-38880*x^2+24300*x)*ln(2))*ln(1/5*x+exp(x))+(1620*x^6+19440*x^5+53460*x^4-90720*x
^3-267300*x^2+486000*x-202500)*exp(x)+324*x^7+3888*x^6+10692*x^5-18144*x^4-53460*x^3+97200*x^2-40500*x),x,meth
od=_RETURNVERBOSE)

[Out]

1/324*x*(x^2+10*x+25)/(2*ln(2)*ln(1/5*x+exp(x))+x^2+4*x-5)^2

________________________________________________________________________________________

maxima [B]  time = 1.18, size = 114, normalized size = 4.07 \begin {gather*} \frac {x^{3} + 10 \, x^{2} + 25 \, x}{324 \, {\left (x^{4} + 4 \, \log \relax (5)^{2} \log \relax (2)^{2} + 4 \, \log \relax (2)^{2} \log \left (x + 5 \, e^{x}\right )^{2} - 2 \, {\left (2 \, \log \relax (5) \log \relax (2) - 3\right )} x^{2} + 8 \, x^{3} - 8 \, {\left (2 \, \log \relax (5) \log \relax (2) + 5\right )} x + 20 \, \log \relax (5) \log \relax (2) + 4 \, {\left (x^{2} \log \relax (2) - 2 \, \log \relax (5) \log \relax (2)^{2} + 4 \, x \log \relax (2) - 5 \, \log \relax (2)\right )} \log \left (x + 5 \, e^{x}\right ) + 25\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(15*x^2+100*x+125)*log(2)*exp(x)+2*(3*x^3+20*x^2+25*x)*log(2))*log(1/5*x+exp(x))+(2*(-10*x^3-100
*x^2-250*x)*log(2)-5*x^4-80*x^3-450*x^2-1000*x-625)*exp(x)+2*(-2*x^3-20*x^2-50*x)*log(2)-x^5-16*x^4-90*x^3-200
*x^2-125*x)/((12960*log(2)^3*exp(x)+2592*x*log(2)^3)*log(1/5*x+exp(x))^3+(4*(4860*x^2+19440*x-24300)*log(2)^2*
exp(x)+4*(972*x^3+3888*x^2-4860*x)*log(2)^2)*log(1/5*x+exp(x))^2+(2*(4860*x^4+38880*x^3+29160*x^2-194400*x+121
500)*log(2)*exp(x)+2*(972*x^5+7776*x^4+5832*x^3-38880*x^2+24300*x)*log(2))*log(1/5*x+exp(x))+(1620*x^6+19440*x
^5+53460*x^4-90720*x^3-267300*x^2+486000*x-202500)*exp(x)+324*x^7+3888*x^6+10692*x^5-18144*x^4-53460*x^3+97200
*x^2-40500*x),x, algorithm="maxima")

[Out]

1/324*(x^3 + 10*x^2 + 25*x)/(x^4 + 4*log(5)^2*log(2)^2 + 4*log(2)^2*log(x + 5*e^x)^2 - 2*(2*log(5)*log(2) - 3)
*x^2 + 8*x^3 - 8*(2*log(5)*log(2) + 5)*x + 20*log(5)*log(2) + 4*(x^2*log(2) - 2*log(5)*log(2)^2 + 4*x*log(2) -
5*log(2))*log(x + 5*e^x) + 25)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {125\,x+2\,\ln \relax (2)\,\left (2\,x^3+20\,x^2+50\,x\right )+{\mathrm {e}}^x\,\left (1000\,x+2\,\ln \relax (2)\,\left (10\,x^3+100\,x^2+250\,x\right )+450\,x^2+80\,x^3+5\,x^4+625\right )-\ln \left (\frac {x}{5}+{\mathrm {e}}^x\right )\,\left (2\,\ln \relax (2)\,\left (3\,x^3+20\,x^2+25\,x\right )+2\,{\mathrm {e}}^x\,\ln \relax (2)\,\left (15\,x^2+100\,x+125\right )\right )+200\,x^2+90\,x^3+16\,x^4+x^5}{{\ln \left (\frac {x}{5}+{\mathrm {e}}^x\right )}^3\,\left (12960\,{\mathrm {e}}^x\,{\ln \relax (2)}^3+2592\,x\,{\ln \relax (2)}^3\right )-40500\,x+{\mathrm {e}}^x\,\left (1620\,x^6+19440\,x^5+53460\,x^4-90720\,x^3-267300\,x^2+486000\,x-202500\right )+\ln \left (\frac {x}{5}+{\mathrm {e}}^x\right )\,\left (2\,\ln \relax (2)\,\left (972\,x^5+7776\,x^4+5832\,x^3-38880\,x^2+24300\,x\right )+2\,{\mathrm {e}}^x\,\ln \relax (2)\,\left (4860\,x^4+38880\,x^3+29160\,x^2-194400\,x+121500\right )\right )+97200\,x^2-53460\,x^3-18144\,x^4+10692\,x^5+3888\,x^6+324\,x^7+{\ln \left (\frac {x}{5}+{\mathrm {e}}^x\right )}^2\,\left (4\,{\ln \relax (2)}^2\,\left (972\,x^3+3888\,x^2-4860\,x\right )+4\,{\mathrm {e}}^x\,{\ln \relax (2)}^2\,\left (4860\,x^2+19440\,x-24300\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(125*x + 2*log(2)*(50*x + 20*x^2 + 2*x^3) + exp(x)*(1000*x + 2*log(2)*(250*x + 100*x^2 + 10*x^3) + 450*x^
2 + 80*x^3 + 5*x^4 + 625) - log(x/5 + exp(x))*(2*log(2)*(25*x + 20*x^2 + 3*x^3) + 2*exp(x)*log(2)*(100*x + 15*
x^2 + 125)) + 200*x^2 + 90*x^3 + 16*x^4 + x^5)/(log(x/5 + exp(x))^3*(12960*exp(x)*log(2)^3 + 2592*x*log(2)^3)
- 40500*x + exp(x)*(486000*x - 267300*x^2 - 90720*x^3 + 53460*x^4 + 19440*x^5 + 1620*x^6 - 202500) + log(x/5 +
exp(x))*(2*log(2)*(24300*x - 38880*x^2 + 5832*x^3 + 7776*x^4 + 972*x^5) + 2*exp(x)*log(2)*(29160*x^2 - 194400
*x + 38880*x^3 + 4860*x^4 + 121500)) + 97200*x^2 - 53460*x^3 - 18144*x^4 + 10692*x^5 + 3888*x^6 + 324*x^7 + lo
g(x/5 + exp(x))^2*(4*log(2)^2*(3888*x^2 - 4860*x + 972*x^3) + 4*exp(x)*log(2)^2*(19440*x + 4860*x^2 - 24300)))
,x)

[Out]

int(-(125*x + 2*log(2)*(50*x + 20*x^2 + 2*x^3) + exp(x)*(1000*x + 2*log(2)*(250*x + 100*x^2 + 10*x^3) + 450*x^
2 + 80*x^3 + 5*x^4 + 625) - log(x/5 + exp(x))*(2*log(2)*(25*x + 20*x^2 + 3*x^3) + 2*exp(x)*log(2)*(100*x + 15*
x^2 + 125)) + 200*x^2 + 90*x^3 + 16*x^4 + x^5)/(log(x/5 + exp(x))^3*(12960*exp(x)*log(2)^3 + 2592*x*log(2)^3)
- 40500*x + exp(x)*(486000*x - 267300*x^2 - 90720*x^3 + 53460*x^4 + 19440*x^5 + 1620*x^6 - 202500) + log(x/5 +
exp(x))*(2*log(2)*(24300*x - 38880*x^2 + 5832*x^3 + 7776*x^4 + 972*x^5) + 2*exp(x)*log(2)*(29160*x^2 - 194400
*x + 38880*x^3 + 4860*x^4 + 121500)) + 97200*x^2 - 53460*x^3 - 18144*x^4 + 10692*x^5 + 3888*x^6 + 324*x^7 + lo
g(x/5 + exp(x))^2*(4*log(2)^2*(3888*x^2 - 4860*x + 972*x^3) + 4*exp(x)*log(2)^2*(19440*x + 4860*x^2 - 24300)))
, x)

________________________________________________________________________________________

sympy [A]  time = 0.60, size = 76, normalized size = 2.71 \begin {gather*} \frac {x^{3} + 10 x^{2} + 25 x}{324 x^{4} + 2592 x^{3} + 1944 x^{2} - 12960 x + \left (1296 x^{2} \log {\relax (2 )} + 5184 x \log {\relax (2 )} - 6480 \log {\relax (2 )}\right ) \log {\left (\frac {x}{5} + e^{x} \right )} + 1296 \log {\relax (2 )}^{2} \log {\left (\frac {x}{5} + e^{x} \right )}^{2} + 8100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*(15*x**2+100*x+125)*ln(2)*exp(x)+2*(3*x**3+20*x**2+25*x)*ln(2))*ln(1/5*x+exp(x))+(2*(-10*x**3-10
0*x**2-250*x)*ln(2)-5*x**4-80*x**3-450*x**2-1000*x-625)*exp(x)+2*(-2*x**3-20*x**2-50*x)*ln(2)-x**5-16*x**4-90*
x**3-200*x**2-125*x)/((12960*ln(2)**3*exp(x)+2592*x*ln(2)**3)*ln(1/5*x+exp(x))**3+(4*(4860*x**2+19440*x-24300)
*ln(2)**2*exp(x)+4*(972*x**3+3888*x**2-4860*x)*ln(2)**2)*ln(1/5*x+exp(x))**2+(2*(4860*x**4+38880*x**3+29160*x*
*2-194400*x+121500)*ln(2)*exp(x)+2*(972*x**5+7776*x**4+5832*x**3-38880*x**2+24300*x)*ln(2))*ln(1/5*x+exp(x))+(
1620*x**6+19440*x**5+53460*x**4-90720*x**3-267300*x**2+486000*x-202500)*exp(x)+324*x**7+3888*x**6+10692*x**5-1
8144*x**4-53460*x**3+97200*x**2-40500*x),x)

[Out]

(x**3 + 10*x**2 + 25*x)/(324*x**4 + 2592*x**3 + 1944*x**2 - 12960*x + (1296*x**2*log(2) + 5184*x*log(2) - 6480
*log(2))*log(x/5 + exp(x)) + 1296*log(2)**2*log(x/5 + exp(x))**2 + 8100)

________________________________________________________________________________________