3.62.74 \(\int \frac {20+5 x+12 x^3+3 x^4+e^x (12 x^2+3 x^3)+(-20+24 x^3+9 x^4+e^x (12 x^2+18 x^3+3 x^4)) \log (x)+(-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x (-480 x^2-288 x^5)) \log ^2(x)}{x^2+(-80 x+20 x^2-48 x^4+12 x^5+e^x (-48 x^3+12 x^4)) \log (x)+(1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} (576 x^4-288 x^5+36 x^6)+e^x (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7)) \log ^2(x)} \, dx\)

Optimal. Leaf size=34 \[ 1+\frac {4+x}{2 (-4+x)+\frac {x}{\left (5+3 x^2 \left (e^x+x\right )\right ) \log (x)}} \]

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Rubi [F]  time = 137.49, 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 {20+5 x+12 x^3+3 x^4+e^x \left (12 x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+e^x \left (12 x^2+18 x^3+3 x^4\right )\right ) \log (x)+\left (-400-480 x^3-144 e^{2 x} x^4-144 x^6+e^x \left (-480 x^2-288 x^5\right )\right ) \log ^2(x)}{x^2+\left (-80 x+20 x^2-48 x^4+12 x^5+e^x \left (-48 x^3+12 x^4\right )\right ) \log (x)+\left (1600-800 x+100 x^2+1920 x^3-960 x^4+120 x^5+576 x^6-288 x^7+36 x^8+e^{2 x} \left (576 x^4-288 x^5+36 x^6\right )+e^x \left (1920 x^2-960 x^3+120 x^4+1152 x^5-576 x^6+72 x^7\right )\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(20 + 5*x + 12*x^3 + 3*x^4 + E^x*(12*x^2 + 3*x^3) + (-20 + 24*x^3 + 9*x^4 + E^x*(12*x^2 + 18*x^3 + 3*x^4))
*Log[x] + (-400 - 480*x^3 - 144*E^(2*x)*x^4 - 144*x^6 + E^x*(-480*x^2 - 288*x^5))*Log[x]^2)/(x^2 + (-80*x + 20
*x^2 - 48*x^4 + 12*x^5 + E^x*(-48*x^3 + 12*x^4))*Log[x] + (1600 - 800*x + 100*x^2 + 1920*x^3 - 960*x^4 + 120*x
^5 + 576*x^6 - 288*x^7 + 36*x^8 + E^(2*x)*(576*x^4 - 288*x^5 + 36*x^6) + E^x*(1920*x^2 - 960*x^3 + 120*x^4 + 1
152*x^5 - 576*x^6 + 72*x^7))*Log[x]^2),x]

[Out]

-4/(4 - x) - 26*Defer[Int][(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^(-2), x] - 64*Defer[Int][1/((4 - x)
^2*(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^2), x] - 120*Defer[Int][1/((-4 + x)*(x + 2*(-4 + x)*(5 + 3*
E^x*x^2 + 3*x^3)*Log[x])^2), x] - 5*Defer[Int][x/(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^2, x] - Defer
[Int][x^2/(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^2, x]/2 - 4*Defer[Int][1/(Log[x]*(x + 2*(-4 + x)*(5
+ 3*E^x*x^2 + 3*x^3)*Log[x])^2), x] - 16*Defer[Int][1/((-4 + x)*Log[x]*(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)
*Log[x])^2), x] - Defer[Int][x/(Log[x]*(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^2), x]/2 - 40*Defer[Int
][Log[x]/(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^2, x] - 30*Defer[Int][(x*Log[x])/(x + 2*(-4 + x)*(5 +
3*E^x*x^2 + 3*x^3)*Log[x])^2, x] - 5*Defer[Int][(x^2*Log[x])/(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^
2, x] + 12*Defer[Int][(x^3*Log[x])/(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^2, x] - 9*Defer[Int][(x^4*L
og[x])/(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^2, x] - 3*Defer[Int][(x^5*Log[x])/(x + 2*(-4 + x)*(5 +
3*E^x*x^2 + 3*x^3)*Log[x])^2, x] + 5*Defer[Int][(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])^(-1), x] + 32*
Defer[Int][1/((4 - x)^2*(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])), x] + 30*Defer[Int][1/((-4 + x)*(x +
2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])), x] + Defer[Int][x/(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])
, x]/2 + Defer[Int][1/(Log[x]*(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])), x]/2 + 4*Defer[Int][1/((-4 + x
)*Log[x]*(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(4+x) \left (5+3 e^x x^2+3 x^3\right )+\left (-20+24 x^3+9 x^4+3 e^x x^2 \left (4+6 x+x^2\right )\right ) \log (x)-16 \left (5+3 e^x x^2+3 x^3\right )^2 \log ^2(x)}{\left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )^2} \, dx\\ &=\int \left (-\frac {4}{(-4+x)^2}+\frac {-16+x^2-16 \log (x)-4 x \log (x)+2 x^2 \log (x)+x^3 \log (x)}{2 (-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )}-\frac {(4+x) \left (-4 x+x^2-4 x \log (x)-2 x^2 \log (x)+x^3 \log (x)+320 \log ^2(x)-60 x^2 \log ^2(x)-86 x^3 \log ^2(x)+144 x^4 \log ^2(x)-54 x^5 \log ^2(x)+6 x^6 \log ^2(x)\right )}{2 (-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )^2}\right ) \, dx\\ &=-\frac {4}{4-x}+\frac {1}{2} \int \frac {-16+x^2-16 \log (x)-4 x \log (x)+2 x^2 \log (x)+x^3 \log (x)}{(-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )} \, dx-\frac {1}{2} \int \frac {(4+x) \left (-4 x+x^2-4 x \log (x)-2 x^2 \log (x)+x^3 \log (x)+320 \log ^2(x)-60 x^2 \log ^2(x)-86 x^3 \log ^2(x)+144 x^4 \log ^2(x)-54 x^5 \log ^2(x)+6 x^6 \log ^2(x)\right )}{(-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )^2} \, dx\\ &=-\frac {4}{4-x}+\frac {1}{2} \int \frac {-16+x^2+\left (-16-4 x+2 x^2+x^3\right ) \log (x)}{(4-x)^2 \log (x) \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )} \, dx-\frac {1}{2} \int \frac {(4+x) \left ((-4+x) x+x \left (-4-2 x+x^2\right ) \log (x)+2 (-4+x)^2 \left (10+5 x-3 x^3+3 x^4\right ) \log ^2(x)\right )}{(4-x)^2 \log (x) \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )^2} \, dx\\ &=-\frac {4}{4-x}+\frac {1}{2} \int \left (-\frac {16}{(-4+x)^2 \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )}-\frac {4 x}{(-4+x)^2 \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )}+\frac {2 x^2}{(-4+x)^2 \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )}+\frac {x^3}{(-4+x)^2 \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )}-\frac {16}{(-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )}+\frac {x^2}{(-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )}\right ) \, dx-\frac {1}{2} \int \left (\frac {8 \left (-4 x+x^2-4 x \log (x)-2 x^2 \log (x)+x^3 \log (x)+320 \log ^2(x)-60 x^2 \log ^2(x)-86 x^3 \log ^2(x)+144 x^4 \log ^2(x)-54 x^5 \log ^2(x)+6 x^6 \log ^2(x)\right )}{(-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )^2}+\frac {-4 x+x^2-4 x \log (x)-2 x^2 \log (x)+x^3 \log (x)+320 \log ^2(x)-60 x^2 \log ^2(x)-86 x^3 \log ^2(x)+144 x^4 \log ^2(x)-54 x^5 \log ^2(x)+6 x^6 \log ^2(x)}{(-4+x) \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )^2}\right ) \, dx\\ &=-\frac {4}{4-x}+\frac {1}{2} \int \frac {x^3}{(-4+x)^2 \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )} \, dx+\frac {1}{2} \int \frac {x^2}{(-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )} \, dx-\frac {1}{2} \int \frac {-4 x+x^2-4 x \log (x)-2 x^2 \log (x)+x^3 \log (x)+320 \log ^2(x)-60 x^2 \log ^2(x)-86 x^3 \log ^2(x)+144 x^4 \log ^2(x)-54 x^5 \log ^2(x)+6 x^6 \log ^2(x)}{(-4+x) \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )^2} \, dx-2 \int \frac {x}{(-4+x)^2 \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )} \, dx-4 \int \frac {-4 x+x^2-4 x \log (x)-2 x^2 \log (x)+x^3 \log (x)+320 \log ^2(x)-60 x^2 \log ^2(x)-86 x^3 \log ^2(x)+144 x^4 \log ^2(x)-54 x^5 \log ^2(x)+6 x^6 \log ^2(x)}{(-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )^2} \, dx-8 \int \frac {1}{(-4+x)^2 \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )} \, dx-8 \int \frac {1}{(-4+x)^2 \log (x) \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )} \, dx+\int \frac {x^2}{(-4+x)^2 \left (x-40 \log (x)+10 x \log (x)-24 e^x x^2 \log (x)-24 x^3 \log (x)+6 e^x x^3 \log (x)+6 x^4 \log (x)\right )} \, dx\\ &=-\frac {4}{4-x}+\frac {1}{2} \int \frac {x^3}{(4-x)^2 \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )} \, dx+\frac {1}{2} \int \frac {x^2}{(4-x)^2 \log (x) \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )} \, dx-\frac {1}{2} \int \frac {-((-4+x) x)-x \left (-4-2 x+x^2\right ) \log (x)-2 (-4+x)^2 \left (10+5 x-3 x^3+3 x^4\right ) \log ^2(x)}{(4-x) \log (x) \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )^2} \, dx-2 \int \frac {x}{(4-x)^2 \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )} \, dx-4 \int \frac {(-4+x) x+x \left (-4-2 x+x^2\right ) \log (x)+2 (-4+x)^2 \left (10+5 x-3 x^3+3 x^4\right ) \log ^2(x)}{(4-x)^2 \log (x) \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )^2} \, dx-8 \int \frac {1}{(4-x)^2 \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )} \, dx-8 \int \frac {1}{(4-x)^2 \log (x) \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )} \, dx+\int \frac {x^2}{(4-x)^2 \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 51, normalized size = 1.50 \begin {gather*} -\frac {x-16 \left (5+3 e^x x^2+3 x^3\right ) \log (x)}{2 \left (x+2 (-4+x) \left (5+3 e^x x^2+3 x^3\right ) \log (x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20 + 5*x + 12*x^3 + 3*x^4 + E^x*(12*x^2 + 3*x^3) + (-20 + 24*x^3 + 9*x^4 + E^x*(12*x^2 + 18*x^3 + 3
*x^4))*Log[x] + (-400 - 480*x^3 - 144*E^(2*x)*x^4 - 144*x^6 + E^x*(-480*x^2 - 288*x^5))*Log[x]^2)/(x^2 + (-80*
x + 20*x^2 - 48*x^4 + 12*x^5 + E^x*(-48*x^3 + 12*x^4))*Log[x] + (1600 - 800*x + 100*x^2 + 1920*x^3 - 960*x^4 +
120*x^5 + 576*x^6 - 288*x^7 + 36*x^8 + E^(2*x)*(576*x^4 - 288*x^5 + 36*x^6) + E^x*(1920*x^2 - 960*x^3 + 120*x
^4 + 1152*x^5 - 576*x^6 + 72*x^7))*Log[x]^2),x]

[Out]

-1/2*(x - 16*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])/(x + 2*(-4 + x)*(5 + 3*E^x*x^2 + 3*x^3)*Log[x])

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fricas [A]  time = 0.78, size = 60, normalized size = 1.76 \begin {gather*} \frac {16 \, {\left (3 \, x^{3} + 3 \, x^{2} e^{x} + 5\right )} \log \relax (x) - x}{2 \, {\left (2 \, {\left (3 \, x^{4} - 12 \, x^{3} + 3 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{x} + 5 \, x - 20\right )} \log \relax (x) + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*exp(x)^2*x^4+(-288*x^5-480*x^2)*exp(x)-144*x^6-480*x^3-400)*log(x)^2+((3*x^4+18*x^3+12*x^2)*e
xp(x)+9*x^4+24*x^3-20)*log(x)+(3*x^3+12*x^2)*exp(x)+3*x^4+12*x^3+5*x+20)/(((36*x^6-288*x^5+576*x^4)*exp(x)^2+(
72*x^7-576*x^6+1152*x^5+120*x^4-960*x^3+1920*x^2)*exp(x)+36*x^8-288*x^7+576*x^6+120*x^5-960*x^4+1920*x^3+100*x
^2-800*x+1600)*log(x)^2+((12*x^4-48*x^3)*exp(x)+12*x^5-48*x^4+20*x^2-80*x)*log(x)+x^2),x, algorithm="fricas")

[Out]

1/2*(16*(3*x^3 + 3*x^2*e^x + 5)*log(x) - x)/(2*(3*x^4 - 12*x^3 + 3*(x^3 - 4*x^2)*e^x + 5*x - 20)*log(x) + x)

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giac [B]  time = 1.14, size = 71, normalized size = 2.09 \begin {gather*} \frac {48 \, x^{3} \log \relax (x) + 48 \, x^{2} e^{x} \log \relax (x) - x + 80 \, \log \relax (x)}{2 \, {\left (6 \, x^{4} \log \relax (x) + 6 \, x^{3} e^{x} \log \relax (x) - 24 \, x^{3} \log \relax (x) - 24 \, x^{2} e^{x} \log \relax (x) + 10 \, x \log \relax (x) + x - 40 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*exp(x)^2*x^4+(-288*x^5-480*x^2)*exp(x)-144*x^6-480*x^3-400)*log(x)^2+((3*x^4+18*x^3+12*x^2)*e
xp(x)+9*x^4+24*x^3-20)*log(x)+(3*x^3+12*x^2)*exp(x)+3*x^4+12*x^3+5*x+20)/(((36*x^6-288*x^5+576*x^4)*exp(x)^2+(
72*x^7-576*x^6+1152*x^5+120*x^4-960*x^3+1920*x^2)*exp(x)+36*x^8-288*x^7+576*x^6+120*x^5-960*x^4+1920*x^3+100*x
^2-800*x+1600)*log(x)^2+((12*x^4-48*x^3)*exp(x)+12*x^5-48*x^4+20*x^2-80*x)*log(x)+x^2),x, algorithm="giac")

[Out]

1/2*(48*x^3*log(x) + 48*x^2*e^x*log(x) - x + 80*log(x))/(6*x^4*log(x) + 6*x^3*e^x*log(x) - 24*x^3*log(x) - 24*
x^2*e^x*log(x) + 10*x*log(x) + x - 40*log(x))

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maple [A]  time = 0.08, size = 65, normalized size = 1.91

method	result	size
risch	\(\frac {4}{x -4}-\frac {\left (4+x \right ) x}{2 \left (x -4\right ) \left (6 x^{3} {\mathrm e}^{x} \ln \relax (x )+6 x^{4} \ln \relax (x )-24 x^{2} {\mathrm e}^{x} \ln \relax (x )-24 x^{3} \ln \relax (x )+10 x \ln \relax (x )-40 \ln \relax (x )+x \right )}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-144*exp(x)^2*x^4+(-288*x^5-480*x^2)*exp(x)-144*x^6-480*x^3-400)*ln(x)^2+((3*x^4+18*x^3+12*x^2)*exp(x)+9
*x^4+24*x^3-20)*ln(x)+(3*x^3+12*x^2)*exp(x)+3*x^4+12*x^3+5*x+20)/(((36*x^6-288*x^5+576*x^4)*exp(x)^2+(72*x^7-5
76*x^6+1152*x^5+120*x^4-960*x^3+1920*x^2)*exp(x)+36*x^8-288*x^7+576*x^6+120*x^5-960*x^4+1920*x^3+100*x^2-800*x
+1600)*ln(x)^2+((12*x^4-48*x^3)*exp(x)+12*x^5-48*x^4+20*x^2-80*x)*ln(x)+x^2),x,method=_RETURNVERBOSE)

[Out]

4/(x-4)-1/2*(4+x)*x/(x-4)/(6*x^3*exp(x)*ln(x)+6*x^4*ln(x)-24*x^2*exp(x)*ln(x)-24*x^3*ln(x)+10*x*ln(x)-40*ln(x)
+x)

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maxima [A]  time = 0.48, size = 64, normalized size = 1.88 \begin {gather*} \frac {48 \, x^{2} e^{x} \log \relax (x) + 16 \, {\left (3 \, x^{3} + 5\right )} \log \relax (x) - x}{2 \, {\left (6 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{x} \log \relax (x) + 2 \, {\left (3 \, x^{4} - 12 \, x^{3} + 5 \, x - 20\right )} \log \relax (x) + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*exp(x)^2*x^4+(-288*x^5-480*x^2)*exp(x)-144*x^6-480*x^3-400)*log(x)^2+((3*x^4+18*x^3+12*x^2)*e
xp(x)+9*x^4+24*x^3-20)*log(x)+(3*x^3+12*x^2)*exp(x)+3*x^4+12*x^3+5*x+20)/(((36*x^6-288*x^5+576*x^4)*exp(x)^2+(
72*x^7-576*x^6+1152*x^5+120*x^4-960*x^3+1920*x^2)*exp(x)+36*x^8-288*x^7+576*x^6+120*x^5-960*x^4+1920*x^3+100*x
^2-800*x+1600)*log(x)^2+((12*x^4-48*x^3)*exp(x)+12*x^5-48*x^4+20*x^2-80*x)*log(x)+x^2),x, algorithm="maxima")

[Out]

1/2*(48*x^2*e^x*log(x) + 16*(3*x^3 + 5)*log(x) - x)/(6*(x^3 - 4*x^2)*e^x*log(x) + 2*(3*x^4 - 12*x^3 + 5*x - 20
)*log(x) + x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {5\,x+{\mathrm {e}}^x\,\left (3\,x^3+12\,x^2\right )+\ln \relax (x)\,\left ({\mathrm {e}}^x\,\left (3\,x^4+18\,x^3+12\,x^2\right )+24\,x^3+9\,x^4-20\right )+12\,x^3+3\,x^4-{\ln \relax (x)}^2\,\left ({\mathrm {e}}^x\,\left (288\,x^5+480\,x^2\right )+144\,x^4\,{\mathrm {e}}^{2\,x}+480\,x^3+144\,x^6+400\right )+20}{x^2-\ln \relax (x)\,\left (80\,x+{\mathrm {e}}^x\,\left (48\,x^3-12\,x^4\right )-20\,x^2+48\,x^4-12\,x^5\right )+{\ln \relax (x)}^2\,\left ({\mathrm {e}}^{2\,x}\,\left (36\,x^6-288\,x^5+576\,x^4\right )-800\,x+100\,x^2+1920\,x^3-960\,x^4+120\,x^5+576\,x^6-288\,x^7+36\,x^8+{\mathrm {e}}^x\,\left (72\,x^7-576\,x^6+1152\,x^5+120\,x^4-960\,x^3+1920\,x^2\right )+1600\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + exp(x)*(12*x^2 + 3*x^3) + log(x)*(exp(x)*(12*x^2 + 18*x^3 + 3*x^4) + 24*x^3 + 9*x^4 - 20) + 12*x^3
+ 3*x^4 - log(x)^2*(exp(x)*(480*x^2 + 288*x^5) + 144*x^4*exp(2*x) + 480*x^3 + 144*x^6 + 400) + 20)/(x^2 - log(
x)*(80*x + exp(x)*(48*x^3 - 12*x^4) - 20*x^2 + 48*x^4 - 12*x^5) + log(x)^2*(exp(2*x)*(576*x^4 - 288*x^5 + 36*x
^6) - 800*x + 100*x^2 + 1920*x^3 - 960*x^4 + 120*x^5 + 576*x^6 - 288*x^7 + 36*x^8 + exp(x)*(1920*x^2 - 960*x^3
+ 120*x^4 + 1152*x^5 - 576*x^6 + 72*x^7) + 1600)),x)

[Out]

int((5*x + exp(x)*(12*x^2 + 3*x^3) + log(x)*(exp(x)*(12*x^2 + 18*x^3 + 3*x^4) + 24*x^3 + 9*x^4 - 20) + 12*x^3
+ 3*x^4 - log(x)^2*(exp(x)*(480*x^2 + 288*x^5) + 144*x^4*exp(2*x) + 480*x^3 + 144*x^6 + 400) + 20)/(x^2 - log(
x)*(80*x + exp(x)*(48*x^3 - 12*x^4) - 20*x^2 + 48*x^4 - 12*x^5) + log(x)^2*(exp(2*x)*(576*x^4 - 288*x^5 + 36*x
^6) - 800*x + 100*x^2 + 1920*x^3 - 960*x^4 + 120*x^5 + 576*x^6 - 288*x^7 + 36*x^8 + exp(x)*(1920*x^2 - 960*x^3
+ 120*x^4 + 1152*x^5 - 576*x^6 + 72*x^7) + 1600)), x)

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sympy [B]  time = 1.10, size = 95, normalized size = 2.79 \begin {gather*} \frac {- x^{2} - 4 x}{12 x^{5} \log {\relax (x )} - 96 x^{4} \log {\relax (x )} + 192 x^{3} \log {\relax (x )} + 20 x^{2} \log {\relax (x )} + 2 x^{2} - 160 x \log {\relax (x )} - 8 x + \left (12 x^{4} \log {\relax (x )} - 96 x^{3} \log {\relax (x )} + 192 x^{2} \log {\relax (x )}\right ) e^{x} + 320 \log {\relax (x )}} + \frac {4}{x - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-144*exp(x)**2*x**4+(-288*x**5-480*x**2)*exp(x)-144*x**6-480*x**3-400)*ln(x)**2+((3*x**4+18*x**3+1
2*x**2)*exp(x)+9*x**4+24*x**3-20)*ln(x)+(3*x**3+12*x**2)*exp(x)+3*x**4+12*x**3+5*x+20)/(((36*x**6-288*x**5+576
*x**4)*exp(x)**2+(72*x**7-576*x**6+1152*x**5+120*x**4-960*x**3+1920*x**2)*exp(x)+36*x**8-288*x**7+576*x**6+120
*x**5-960*x**4+1920*x**3+100*x**2-800*x+1600)*ln(x)**2+((12*x**4-48*x**3)*exp(x)+12*x**5-48*x**4+20*x**2-80*x)
*ln(x)+x**2),x)

[Out]

(-x**2 - 4*x)/(12*x**5*log(x) - 96*x**4*log(x) + 192*x**3*log(x) + 20*x**2*log(x) + 2*x**2 - 160*x*log(x) - 8*
x + (12*x**4*log(x) - 96*x**3*log(x) + 192*x**2*log(x))*exp(x) + 320*log(x)) + 4/(x - 4)

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