3.98.60 \(\int \frac {-1020+6124 x-11879 x^2+8384 x^3-2724 x^4+420 x^5-25 x^6+e^{-\frac {12 x}{-2+5 x}} (-64+320 x-400 x^2)+e^{-\frac {9 x}{-2+5 x}} (512-2688 x+3840 x^2-800 x^3)+e^{-\frac {6 x}{-2+5 x}} (-1536+8448 x-13536 x^2+5280 x^3-600 x^4)+e^{-\frac {3 x}{-2+5 x}} (2048-11776 x+20864 x^2-11552 x^3+2560 x^4-200 x^5)+(-1024 x+384 e^{-\frac {12 x}{-2+5 x}} x+5888 x^2-10432 x^3+5776 x^4-1280 x^5+100 x^6+e^{-\frac {9 x}{-2+5 x}} (-2176 x-64 x^2+800 x^3)+e^{-\frac {6 x}{-2+5 x}} (3840 x+1728 x^2-5472 x^3+1200 x^4)+e^{-\frac {3 x}{-2+5 x}} (-1536 x-6144 x^2+12960 x^3-5232 x^4+600 x^5)) \log (x)}{(4 x-20 x^2+25 x^3) \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \frac {-1+\left (-4+2 e^{\frac {3 x}{2-5 x}}+x\right )^4}{\log (x)} \]

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Rubi [F]  time = 23.73, 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 {-1020+6124 x-11879 x^2+8384 x^3-2724 x^4+420 x^5-25 x^6+e^{-\frac {12 x}{-2+5 x}} \left (-64+320 x-400 x^2\right )+e^{-\frac {9 x}{-2+5 x}} \left (512-2688 x+3840 x^2-800 x^3\right )+e^{-\frac {6 x}{-2+5 x}} \left (-1536+8448 x-13536 x^2+5280 x^3-600 x^4\right )+e^{-\frac {3 x}{-2+5 x}} \left (2048-11776 x+20864 x^2-11552 x^3+2560 x^4-200 x^5\right )+\left (-1024 x+384 e^{-\frac {12 x}{-2+5 x}} x+5888 x^2-10432 x^3+5776 x^4-1280 x^5+100 x^6+e^{-\frac {9 x}{-2+5 x}} \left (-2176 x-64 x^2+800 x^3\right )+e^{-\frac {6 x}{-2+5 x}} \left (3840 x+1728 x^2-5472 x^3+1200 x^4\right )+e^{-\frac {3 x}{-2+5 x}} \left (-1536 x-6144 x^2+12960 x^3-5232 x^4+600 x^5\right )\right ) \log (x)}{\left (4 x-20 x^2+25 x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1020 + 6124*x - 11879*x^2 + 8384*x^3 - 2724*x^4 + 420*x^5 - 25*x^6 + (-64 + 320*x - 400*x^2)/E^((12*x)/(
-2 + 5*x)) + (512 - 2688*x + 3840*x^2 - 800*x^3)/E^((9*x)/(-2 + 5*x)) + (-1536 + 8448*x - 13536*x^2 + 5280*x^3
- 600*x^4)/E^((6*x)/(-2 + 5*x)) + (2048 - 11776*x + 20864*x^2 - 11552*x^3 + 2560*x^4 - 200*x^5)/E^((3*x)/(-2
+ 5*x)) + (-1024*x + (384*x)/E^((12*x)/(-2 + 5*x)) + 5888*x^2 - 10432*x^3 + 5776*x^4 - 1280*x^5 + 100*x^6 + (-
2176*x - 64*x^2 + 800*x^3)/E^((9*x)/(-2 + 5*x)) + (3840*x + 1728*x^2 - 5472*x^3 + 1200*x^4)/E^((6*x)/(-2 + 5*x
)) + (-1536*x - 6144*x^2 + 12960*x^3 - 5232*x^4 + 600*x^5)/E^((3*x)/(-2 + 5*x)))*Log[x])/((4*x - 20*x^2 + 25*x
^3)*Log[x]^2),x]

[Out]

192*ExpIntegralEi[2*Log[x]] - 48*ExpIntegralEi[3*Log[x]] + 4*ExpIntegralEi[4*Log[x]] + (32*E^((12*x)/(2 - 5*x)
))/((2 - 5*x)^2*((2 - 5*x)^(-1) + (5*x)/(2 - 5*x)^2)*Log[x]) - 256*LogIntegral[x] - 384*Defer[Int][E^((3*x)/(2
- 5*x))/Log[x]^2, x] + 192*Defer[Int][E^((6*x)/(2 - 5*x))/Log[x]^2, x] - 32*Defer[Int][E^((9*x)/(2 - 5*x))/Lo
g[x]^2, x] + 512*Defer[Int][E^((3*x)/(2 - 5*x))/(x*Log[x]^2), x] - 384*Defer[Int][E^((6*x)/(2 - 5*x))/(x*Log[x
]^2), x] + 128*Defer[Int][E^((9*x)/(2 - 5*x))/(x*Log[x]^2), x] + 96*Defer[Int][(E^((3*x)/(2 - 5*x))*x)/Log[x]^
2, x] - 24*Defer[Int][(E^((6*x)/(2 - 5*x))*x)/Log[x]^2, x] - 8*Defer[Int][(E^((3*x)/(2 - 5*x))*x^2)/Log[x]^2,
x] + 6124*Defer[Int][1/((-2 + 5*x)^2*Log[x]^2), x] - 1020*Defer[Int][1/(x*(-2 + 5*x)^2*Log[x]^2), x] - 11879*D
efer[Int][x/((-2 + 5*x)^2*Log[x]^2), x] + 8384*Defer[Int][x^2/((-2 + 5*x)^2*Log[x]^2), x] - 2724*Defer[Int][x^
3/((-2 + 5*x)^2*Log[x]^2), x] + 420*Defer[Int][x^4/((-2 + 5*x)^2*Log[x]^2), x] - 25*Defer[Int][x^5/((-2 + 5*x)
^2*Log[x]^2), x] + (45312*Defer[Int][E^((3*x)/(2 - 5*x))/Log[x], x])/125 - (4512*Defer[Int][E^((6*x)/(2 - 5*x)
)/Log[x], x])/25 + 32*Defer[Int][E^((9*x)/(2 - 5*x))/Log[x], x] - (4752*Defer[Int][(E^((3*x)/(2 - 5*x))*x)/Log
[x], x])/25 + 48*Defer[Int][(E^((6*x)/(2 - 5*x))*x)/Log[x], x] + 24*Defer[Int][(E^((3*x)/(2 - 5*x))*x^2)/Log[x
], x] - (279936*Defer[Int][E^((3*x)/(2 - 5*x))/((-2 + 5*x)^2*Log[x]), x])/125 + (93312*Defer[Int][E^((6*x)/(2
- 5*x))/((-2 + 5*x)^2*Log[x]), x])/25 - (10368*Defer[Int][E^((9*x)/(2 - 5*x))/((-2 + 5*x)^2*Log[x]), x])/5 + (
46656*Defer[Int][E^((3*x)/(2 - 5*x))/((-2 + 5*x)*Log[x]), x])/125 - (10368*Defer[Int][E^((6*x)/(2 - 5*x))/((-2
+ 5*x)*Log[x]), x])/25 + (576*Defer[Int][E^((9*x)/(2 - 5*x))/((-2 + 5*x)*Log[x]), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1020+6124 x-11879 x^2+8384 x^3-2724 x^4+420 x^5-25 x^6+e^{-\frac {12 x}{-2+5 x}} \left (-64+320 x-400 x^2\right )+e^{-\frac {9 x}{-2+5 x}} \left (512-2688 x+3840 x^2-800 x^3\right )+e^{-\frac {6 x}{-2+5 x}} \left (-1536+8448 x-13536 x^2+5280 x^3-600 x^4\right )+e^{-\frac {3 x}{-2+5 x}} \left (2048-11776 x+20864 x^2-11552 x^3+2560 x^4-200 x^5\right )+\left (-1024 x+384 e^{-\frac {12 x}{-2+5 x}} x+5888 x^2-10432 x^3+5776 x^4-1280 x^5+100 x^6+e^{-\frac {9 x}{-2+5 x}} \left (-2176 x-64 x^2+800 x^3\right )+e^{-\frac {6 x}{-2+5 x}} \left (3840 x+1728 x^2-5472 x^3+1200 x^4\right )+e^{-\frac {3 x}{-2+5 x}} \left (-1536 x-6144 x^2+12960 x^3-5232 x^4+600 x^5\right )\right ) \log (x)}{x \left (4-20 x+25 x^2\right ) \log ^2(x)} \, dx\\ &=\int \frac {-1020+6124 x-11879 x^2+8384 x^3-2724 x^4+420 x^5-25 x^6+e^{-\frac {12 x}{-2+5 x}} \left (-64+320 x-400 x^2\right )+e^{-\frac {9 x}{-2+5 x}} \left (512-2688 x+3840 x^2-800 x^3\right )+e^{-\frac {6 x}{-2+5 x}} \left (-1536+8448 x-13536 x^2+5280 x^3-600 x^4\right )+e^{-\frac {3 x}{-2+5 x}} \left (2048-11776 x+20864 x^2-11552 x^3+2560 x^4-200 x^5\right )+\left (-1024 x+384 e^{-\frac {12 x}{-2+5 x}} x+5888 x^2-10432 x^3+5776 x^4-1280 x^5+100 x^6+e^{-\frac {9 x}{-2+5 x}} \left (-2176 x-64 x^2+800 x^3\right )+e^{-\frac {6 x}{-2+5 x}} \left (3840 x+1728 x^2-5472 x^3+1200 x^4\right )+e^{-\frac {3 x}{-2+5 x}} \left (-1536 x-6144 x^2+12960 x^3-5232 x^4+600 x^5\right )\right ) \log (x)}{x (-2+5 x)^2 \log ^2(x)} \, dx\\ &=\int \frac {-1020-16 e^{\frac {12 x}{2-5 x}} (2-5 x)^2-32 e^{\frac {9 x}{2-5 x}} (2-5 x)^2 (-4+x)-8 e^{\frac {3 x}{2-5 x}} (2-5 x)^2 (-4+x)^3+6124 x-11879 x^2+8384 x^3-2724 x^4+420 x^5-25 x^6-24 e^{\frac {6 x}{2-5 x}} \left (8-22 x+5 x^2\right )^2+4 \left (12 e^{\frac {3 x}{2-5 x}}+(2-5 x)^2\right ) x \left (-4+2 e^{\frac {3 x}{2-5 x}}+x\right )^3 \log (x)}{(2-5 x)^2 x \log ^2(x)} \, dx\\ &=\int \left (\frac {6124}{(-2+5 x)^2 \log ^2(x)}-\frac {1020}{x (-2+5 x)^2 \log ^2(x)}-\frac {11879 x}{(-2+5 x)^2 \log ^2(x)}+\frac {8384 x^2}{(-2+5 x)^2 \log ^2(x)}-\frac {2724 x^3}{(-2+5 x)^2 \log ^2(x)}+\frac {420 x^4}{(-2+5 x)^2 \log ^2(x)}-\frac {25 x^5}{(-2+5 x)^2 \log ^2(x)}-\frac {256}{\log (x)}+\frac {192 x}{\log (x)}-\frac {48 x^2}{\log (x)}+\frac {4 x^3}{\log (x)}-\frac {16 e^{\frac {12 x}{2-5 x}} \left (4-20 x+25 x^2-24 x \log (x)\right )}{x (-2+5 x)^2 \log ^2(x)}+\frac {32 e^{\frac {9 x}{2-5 x}} \left (16-84 x+120 x^2-25 x^3-68 x \log (x)-2 x^2 \log (x)+25 x^3 \log (x)\right )}{x (-2+5 x)^2 \log ^2(x)}+\frac {24 e^{\frac {6 x}{2-5 x}} (-4+x) \left (16-84 x+120 x^2-25 x^3-40 x \log (x)-28 x^2 \log (x)+50 x^3 \log (x)\right )}{x (-2+5 x)^2 \log ^2(x)}+\frac {8 e^{\frac {3 x}{2-5 x}} (-4+x)^2 \left (16-84 x+120 x^2-25 x^3-12 x \log (x)-54 x^2 \log (x)+75 x^3 \log (x)\right )}{x (-2+5 x)^2 \log ^2(x)}\right ) \, dx\\ &=4 \int \frac {x^3}{\log (x)} \, dx+8 \int \frac {e^{\frac {3 x}{2-5 x}} (-4+x)^2 \left (16-84 x+120 x^2-25 x^3-12 x \log (x)-54 x^2 \log (x)+75 x^3 \log (x)\right )}{x (-2+5 x)^2 \log ^2(x)} \, dx-16 \int \frac {e^{\frac {12 x}{2-5 x}} \left (4-20 x+25 x^2-24 x \log (x)\right )}{x (-2+5 x)^2 \log ^2(x)} \, dx+24 \int \frac {e^{\frac {6 x}{2-5 x}} (-4+x) \left (16-84 x+120 x^2-25 x^3-40 x \log (x)-28 x^2 \log (x)+50 x^3 \log (x)\right )}{x (-2+5 x)^2 \log ^2(x)} \, dx-25 \int \frac {x^5}{(-2+5 x)^2 \log ^2(x)} \, dx+32 \int \frac {e^{\frac {9 x}{2-5 x}} \left (16-84 x+120 x^2-25 x^3-68 x \log (x)-2 x^2 \log (x)+25 x^3 \log (x)\right )}{x (-2+5 x)^2 \log ^2(x)} \, dx-48 \int \frac {x^2}{\log (x)} \, dx+192 \int \frac {x}{\log (x)} \, dx-256 \int \frac {1}{\log (x)} \, dx+420 \int \frac {x^4}{(-2+5 x)^2 \log ^2(x)} \, dx-1020 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-2724 \int \frac {x^3}{(-2+5 x)^2 \log ^2(x)} \, dx+6124 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx+8384 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx-11879 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx\\ &=\frac {32 e^{\frac {12 x}{2-5 x}}}{(2-5 x)^2 \left (\frac {1}{2-5 x}+\frac {5 x}{(2-5 x)^2}\right ) \log (x)}-256 \text {li}(x)+4 \operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )+8 \int \left (-\frac {e^{\frac {3 x}{2-5 x}} (-4+x)^3}{x \log ^2(x)}+\frac {3 e^{\frac {3 x}{2-5 x}} (-4+x)^2 \left (-4-18 x+25 x^2\right )}{(-2+5 x)^2 \log (x)}\right ) \, dx+24 \int \left (-\frac {e^{\frac {6 x}{2-5 x}} (-4+x)^2}{x \log ^2(x)}+\frac {2 e^{\frac {6 x}{2-5 x}} \left (80+36 x-114 x^2+25 x^3\right )}{(-2+5 x)^2 \log (x)}\right ) \, dx-25 \int \frac {x^5}{(-2+5 x)^2 \log ^2(x)} \, dx+32 \int \left (\frac {e^{\frac {9 x}{2-5 x}} (4-x)}{x \log ^2(x)}+\frac {e^{\frac {9 x}{2-5 x}} \left (-68-2 x+25 x^2\right )}{(-2+5 x)^2 \log (x)}\right ) \, dx-48 \operatorname {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+192 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+420 \int \frac {x^4}{(-2+5 x)^2 \log ^2(x)} \, dx-1020 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-2724 \int \frac {x^3}{(-2+5 x)^2 \log ^2(x)} \, dx+6124 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx+8384 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx-11879 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx\\ &=192 \text {Ei}(2 \log (x))-48 \text {Ei}(3 \log (x))+4 \text {Ei}(4 \log (x))+\frac {32 e^{\frac {12 x}{2-5 x}}}{(2-5 x)^2 \left (\frac {1}{2-5 x}+\frac {5 x}{(2-5 x)^2}\right ) \log (x)}-256 \text {li}(x)-8 \int \frac {e^{\frac {3 x}{2-5 x}} (-4+x)^3}{x \log ^2(x)} \, dx-24 \int \frac {e^{\frac {6 x}{2-5 x}} (-4+x)^2}{x \log ^2(x)} \, dx+24 \int \frac {e^{\frac {3 x}{2-5 x}} (-4+x)^2 \left (-4-18 x+25 x^2\right )}{(-2+5 x)^2 \log (x)} \, dx-25 \int \frac {x^5}{(-2+5 x)^2 \log ^2(x)} \, dx+32 \int \frac {e^{\frac {9 x}{2-5 x}} (4-x)}{x \log ^2(x)} \, dx+32 \int \frac {e^{\frac {9 x}{2-5 x}} \left (-68-2 x+25 x^2\right )}{(-2+5 x)^2 \log (x)} \, dx+48 \int \frac {e^{\frac {6 x}{2-5 x}} \left (80+36 x-114 x^2+25 x^3\right )}{(-2+5 x)^2 \log (x)} \, dx+420 \int \frac {x^4}{(-2+5 x)^2 \log ^2(x)} \, dx-1020 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-2724 \int \frac {x^3}{(-2+5 x)^2 \log ^2(x)} \, dx+6124 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx+8384 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx-11879 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx\\ &=192 \text {Ei}(2 \log (x))-48 \text {Ei}(3 \log (x))+4 \text {Ei}(4 \log (x))+\frac {32 e^{\frac {12 x}{2-5 x}}}{(2-5 x)^2 \left (\frac {1}{2-5 x}+\frac {5 x}{(2-5 x)^2}\right ) \log (x)}-256 \text {li}(x)-8 \int \left (\frac {48 e^{\frac {3 x}{2-5 x}}}{\log ^2(x)}-\frac {64 e^{\frac {3 x}{2-5 x}}}{x \log ^2(x)}-\frac {12 e^{\frac {3 x}{2-5 x}} x}{\log ^2(x)}+\frac {e^{\frac {3 x}{2-5 x}} x^2}{\log ^2(x)}\right ) \, dx-24 \int \left (-\frac {8 e^{\frac {6 x}{2-5 x}}}{\log ^2(x)}+\frac {16 e^{\frac {6 x}{2-5 x}}}{x \log ^2(x)}+\frac {e^{\frac {6 x}{2-5 x}} x}{\log ^2(x)}\right ) \, dx+24 \int \left (\frac {1888 e^{\frac {3 x}{2-5 x}}}{125 \log (x)}-\frac {198 e^{\frac {3 x}{2-5 x}} x}{25 \log (x)}+\frac {e^{\frac {3 x}{2-5 x}} x^2}{\log (x)}-\frac {11664 e^{\frac {3 x}{2-5 x}}}{125 (-2+5 x)^2 \log (x)}+\frac {1944 e^{\frac {3 x}{2-5 x}}}{125 (-2+5 x) \log (x)}\right ) \, dx-25 \int \frac {x^5}{(-2+5 x)^2 \log ^2(x)} \, dx+32 \int \left (-\frac {e^{\frac {9 x}{2-5 x}}}{\log ^2(x)}+\frac {4 e^{\frac {9 x}{2-5 x}}}{x \log ^2(x)}\right ) \, dx+32 \int \left (\frac {e^{\frac {9 x}{2-5 x}}}{\log (x)}-\frac {324 e^{\frac {9 x}{2-5 x}}}{5 (-2+5 x)^2 \log (x)}+\frac {18 e^{\frac {9 x}{2-5 x}}}{5 (-2+5 x) \log (x)}\right ) \, dx+48 \int \left (-\frac {94 e^{\frac {6 x}{2-5 x}}}{25 \log (x)}+\frac {e^{\frac {6 x}{2-5 x}} x}{\log (x)}+\frac {1944 e^{\frac {6 x}{2-5 x}}}{25 (-2+5 x)^2 \log (x)}-\frac {216 e^{\frac {6 x}{2-5 x}}}{25 (-2+5 x) \log (x)}\right ) \, dx+420 \int \frac {x^4}{(-2+5 x)^2 \log ^2(x)} \, dx-1020 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-2724 \int \frac {x^3}{(-2+5 x)^2 \log ^2(x)} \, dx+6124 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx+8384 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx-11879 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx\\ &=192 \text {Ei}(2 \log (x))-48 \text {Ei}(3 \log (x))+4 \text {Ei}(4 \log (x))+\frac {32 e^{\frac {12 x}{2-5 x}}}{(2-5 x)^2 \left (\frac {1}{2-5 x}+\frac {5 x}{(2-5 x)^2}\right ) \log (x)}-256 \text {li}(x)-8 \int \frac {e^{\frac {3 x}{2-5 x}} x^2}{\log ^2(x)} \, dx-24 \int \frac {e^{\frac {6 x}{2-5 x}} x}{\log ^2(x)} \, dx+24 \int \frac {e^{\frac {3 x}{2-5 x}} x^2}{\log (x)} \, dx-25 \int \frac {x^5}{(-2+5 x)^2 \log ^2(x)} \, dx-32 \int \frac {e^{\frac {9 x}{2-5 x}}}{\log ^2(x)} \, dx+32 \int \frac {e^{\frac {9 x}{2-5 x}}}{\log (x)} \, dx+48 \int \frac {e^{\frac {6 x}{2-5 x}} x}{\log (x)} \, dx+96 \int \frac {e^{\frac {3 x}{2-5 x}} x}{\log ^2(x)} \, dx+\frac {576}{5} \int \frac {e^{\frac {9 x}{2-5 x}}}{(-2+5 x) \log (x)} \, dx+128 \int \frac {e^{\frac {9 x}{2-5 x}}}{x \log ^2(x)} \, dx-\frac {4512}{25} \int \frac {e^{\frac {6 x}{2-5 x}}}{\log (x)} \, dx-\frac {4752}{25} \int \frac {e^{\frac {3 x}{2-5 x}} x}{\log (x)} \, dx+192 \int \frac {e^{\frac {6 x}{2-5 x}}}{\log ^2(x)} \, dx+\frac {45312}{125} \int \frac {e^{\frac {3 x}{2-5 x}}}{\log (x)} \, dx+\frac {46656}{125} \int \frac {e^{\frac {3 x}{2-5 x}}}{(-2+5 x) \log (x)} \, dx-384 \int \frac {e^{\frac {3 x}{2-5 x}}}{\log ^2(x)} \, dx-384 \int \frac {e^{\frac {6 x}{2-5 x}}}{x \log ^2(x)} \, dx-\frac {10368}{25} \int \frac {e^{\frac {6 x}{2-5 x}}}{(-2+5 x) \log (x)} \, dx+420 \int \frac {x^4}{(-2+5 x)^2 \log ^2(x)} \, dx+512 \int \frac {e^{\frac {3 x}{2-5 x}}}{x \log ^2(x)} \, dx-1020 \int \frac {1}{x (-2+5 x)^2 \log ^2(x)} \, dx-\frac {10368}{5} \int \frac {e^{\frac {9 x}{2-5 x}}}{(-2+5 x)^2 \log (x)} \, dx-\frac {279936}{125} \int \frac {e^{\frac {3 x}{2-5 x}}}{(-2+5 x)^2 \log (x)} \, dx-2724 \int \frac {x^3}{(-2+5 x)^2 \log ^2(x)} \, dx+\frac {93312}{25} \int \frac {e^{\frac {6 x}{2-5 x}}}{(-2+5 x)^2 \log (x)} \, dx+6124 \int \frac {1}{(-2+5 x)^2 \log ^2(x)} \, dx+8384 \int \frac {x^2}{(-2+5 x)^2 \log ^2(x)} \, dx-11879 \int \frac {x}{(-2+5 x)^2 \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.42, size = 92, normalized size = 3.54 \begin {gather*} \frac {255+16 e^{\frac {12 x}{2-5 x}}+32 e^{\frac {9 x}{2-5 x}} (-4+x)+24 e^{\frac {6 x}{2-5 x}} (-4+x)^2+8 e^{\frac {3 x}{2-5 x}} (-4+x)^3-256 x+96 x^2-16 x^3+x^4}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1020 + 6124*x - 11879*x^2 + 8384*x^3 - 2724*x^4 + 420*x^5 - 25*x^6 + (-64 + 320*x - 400*x^2)/E^((1
2*x)/(-2 + 5*x)) + (512 - 2688*x + 3840*x^2 - 800*x^3)/E^((9*x)/(-2 + 5*x)) + (-1536 + 8448*x - 13536*x^2 + 52
80*x^3 - 600*x^4)/E^((6*x)/(-2 + 5*x)) + (2048 - 11776*x + 20864*x^2 - 11552*x^3 + 2560*x^4 - 200*x^5)/E^((3*x
)/(-2 + 5*x)) + (-1024*x + (384*x)/E^((12*x)/(-2 + 5*x)) + 5888*x^2 - 10432*x^3 + 5776*x^4 - 1280*x^5 + 100*x^
6 + (-2176*x - 64*x^2 + 800*x^3)/E^((9*x)/(-2 + 5*x)) + (3840*x + 1728*x^2 - 5472*x^3 + 1200*x^4)/E^((6*x)/(-2
+ 5*x)) + (-1536*x - 6144*x^2 + 12960*x^3 - 5232*x^4 + 600*x^5)/E^((3*x)/(-2 + 5*x)))*Log[x])/((4*x - 20*x^2
+ 25*x^3)*Log[x]^2),x]

[Out]

(255 + 16*E^((12*x)/(2 - 5*x)) + 32*E^((9*x)/(2 - 5*x))*(-4 + x) + 24*E^((6*x)/(2 - 5*x))*(-4 + x)^2 + 8*E^((3
*x)/(2 - 5*x))*(-4 + x)^3 - 256*x + 96*x^2 - 16*x^3 + x^4)/Log[x]

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fricas [B]  time = 0.63, size = 99, normalized size = 3.81 \begin {gather*} \frac {x^{4} - 16 \, x^{3} + 96 \, x^{2} + 8 \, {\left (x^{3} - 12 \, x^{2} + 48 \, x - 64\right )} e^{\left (-\frac {3 \, x}{5 \, x - 2}\right )} + 24 \, {\left (x^{2} - 8 \, x + 16\right )} e^{\left (-\frac {6 \, x}{5 \, x - 2}\right )} + 32 \, {\left (x - 4\right )} e^{\left (-\frac {9 \, x}{5 \, x - 2}\right )} - 256 \, x + 16 \, e^{\left (-\frac {12 \, x}{5 \, x - 2}\right )} + 255}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((384*x*exp(-3*x/(5*x-2))^4+(800*x^3-64*x^2-2176*x)*exp(-3*x/(5*x-2))^3+(1200*x^4-5472*x^3+1728*x^2+
3840*x)*exp(-3*x/(5*x-2))^2+(600*x^5-5232*x^4+12960*x^3-6144*x^2-1536*x)*exp(-3*x/(5*x-2))+100*x^6-1280*x^5+57
76*x^4-10432*x^3+5888*x^2-1024*x)*log(x)+(-400*x^2+320*x-64)*exp(-3*x/(5*x-2))^4+(-800*x^3+3840*x^2-2688*x+512
)*exp(-3*x/(5*x-2))^3+(-600*x^4+5280*x^3-13536*x^2+8448*x-1536)*exp(-3*x/(5*x-2))^2+(-200*x^5+2560*x^4-11552*x
^3+20864*x^2-11776*x+2048)*exp(-3*x/(5*x-2))-25*x^6+420*x^5-2724*x^4+8384*x^3-11879*x^2+6124*x-1020)/(25*x^3-2
0*x^2+4*x)/log(x)^2,x, algorithm="fricas")

[Out]

(x^4 - 16*x^3 + 96*x^2 + 8*(x^3 - 12*x^2 + 48*x - 64)*e^(-3*x/(5*x - 2)) + 24*(x^2 - 8*x + 16)*e^(-6*x/(5*x -
2)) + 32*(x - 4)*e^(-9*x/(5*x - 2)) - 256*x + 16*e^(-12*x/(5*x - 2)) + 255)/log(x)

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giac [B]  time = 0.33, size = 165, normalized size = 6.35 \begin {gather*} \frac {x^{4} + 8 \, x^{3} e^{\left (-\frac {3 \, x}{5 \, x - 2}\right )} - 16 \, x^{3} - 96 \, x^{2} e^{\left (-\frac {3 \, x}{5 \, x - 2}\right )} + 24 \, x^{2} e^{\left (-\frac {6 \, x}{5 \, x - 2}\right )} + 96 \, x^{2} + 384 \, x e^{\left (-\frac {3 \, x}{5 \, x - 2}\right )} - 192 \, x e^{\left (-\frac {6 \, x}{5 \, x - 2}\right )} + 32 \, x e^{\left (-\frac {9 \, x}{5 \, x - 2}\right )} - 256 \, x - 512 \, e^{\left (-\frac {3 \, x}{5 \, x - 2}\right )} + 384 \, e^{\left (-\frac {6 \, x}{5 \, x - 2}\right )} - 128 \, e^{\left (-\frac {9 \, x}{5 \, x - 2}\right )} + 16 \, e^{\left (-\frac {12 \, x}{5 \, x - 2}\right )} + 255}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((384*x*exp(-3*x/(5*x-2))^4+(800*x^3-64*x^2-2176*x)*exp(-3*x/(5*x-2))^3+(1200*x^4-5472*x^3+1728*x^2+
3840*x)*exp(-3*x/(5*x-2))^2+(600*x^5-5232*x^4+12960*x^3-6144*x^2-1536*x)*exp(-3*x/(5*x-2))+100*x^6-1280*x^5+57
76*x^4-10432*x^3+5888*x^2-1024*x)*log(x)+(-400*x^2+320*x-64)*exp(-3*x/(5*x-2))^4+(-800*x^3+3840*x^2-2688*x+512
)*exp(-3*x/(5*x-2))^3+(-600*x^4+5280*x^3-13536*x^2+8448*x-1536)*exp(-3*x/(5*x-2))^2+(-200*x^5+2560*x^4-11552*x
^3+20864*x^2-11776*x+2048)*exp(-3*x/(5*x-2))-25*x^6+420*x^5-2724*x^4+8384*x^3-11879*x^2+6124*x-1020)/(25*x^3-2
0*x^2+4*x)/log(x)^2,x, algorithm="giac")

[Out]

(x^4 + 8*x^3*e^(-3*x/(5*x - 2)) - 16*x^3 - 96*x^2*e^(-3*x/(5*x - 2)) + 24*x^2*e^(-6*x/(5*x - 2)) + 96*x^2 + 38
4*x*e^(-3*x/(5*x - 2)) - 192*x*e^(-6*x/(5*x - 2)) + 32*x*e^(-9*x/(5*x - 2)) - 256*x - 512*e^(-3*x/(5*x - 2)) +
384*e^(-6*x/(5*x - 2)) - 128*e^(-9*x/(5*x - 2)) + 16*e^(-12*x/(5*x - 2)) + 255)/log(x)

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maple [B]  time = 0.08, size = 166, normalized size = 6.38

method	result	size
risch	\(\frac {x^{4}+8 \,{\mathrm e}^{-\frac {3 x}{5 x -2}} x^{3}+24 \,{\mathrm e}^{-\frac {6 x}{5 x -2}} x^{2}+32 \,{\mathrm e}^{-\frac {9 x}{5 x -2}} x +16 \,{\mathrm e}^{-\frac {12 x}{5 x -2}}-16 x^{3}-96 \,{\mathrm e}^{-\frac {3 x}{5 x -2}} x^{2}-192 \,{\mathrm e}^{-\frac {6 x}{5 x -2}} x -128 \,{\mathrm e}^{-\frac {9 x}{5 x -2}}+96 x^{2}+384 x \,{\mathrm e}^{-\frac {3 x}{5 x -2}}+384 \,{\mathrm e}^{-\frac {6 x}{5 x -2}}-256 x -512 \,{\mathrm e}^{-\frac {3 x}{5 x -2}}+255}{\ln \relax (x )}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((384*x*exp(-3*x/(5*x-2))^4+(800*x^3-64*x^2-2176*x)*exp(-3*x/(5*x-2))^3+(1200*x^4-5472*x^3+1728*x^2+3840*x
)*exp(-3*x/(5*x-2))^2+(600*x^5-5232*x^4+12960*x^3-6144*x^2-1536*x)*exp(-3*x/(5*x-2))+100*x^6-1280*x^5+5776*x^4
-10432*x^3+5888*x^2-1024*x)*ln(x)+(-400*x^2+320*x-64)*exp(-3*x/(5*x-2))^4+(-800*x^3+3840*x^2-2688*x+512)*exp(-
3*x/(5*x-2))^3+(-600*x^4+5280*x^3-13536*x^2+8448*x-1536)*exp(-3*x/(5*x-2))^2+(-200*x^5+2560*x^4-11552*x^3+2086
4*x^2-11776*x+2048)*exp(-3*x/(5*x-2))-25*x^6+420*x^5-2724*x^4+8384*x^3-11879*x^2+6124*x-1020)/(25*x^3-20*x^2+4
*x)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

(x^4+8*exp(-3*x/(5*x-2))*x^3+24*exp(-6*x/(5*x-2))*x^2+32*exp(-9*x/(5*x-2))*x+16*exp(-12*x/(5*x-2))-16*x^3-96*e
xp(-3*x/(5*x-2))*x^2-192*exp(-6*x/(5*x-2))*x-128*exp(-9*x/(5*x-2))+96*x^2+384*x*exp(-3*x/(5*x-2))+384*exp(-6*x
/(5*x-2))-256*x-512*exp(-3*x/(5*x-2))+255)/ln(x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {{\left (x^{4} e^{\frac {12}{5}} - 16 \, x^{3} e^{\frac {12}{5}} + 96 \, x^{2} e^{\frac {12}{5}} - 256 \, x e^{\frac {12}{5}} + 255 \, e^{\frac {12}{5}} + 16 \, e^{\left (-\frac {24}{5 \, {\left (5 \, x - 2\right )}}\right )}\right )} e^{\left (-\frac {12}{5}\right )}}{\log \relax (x)} + \int -\frac {8 \, {\left (25 \, x^{5} e^{\frac {2}{5}} - 320 \, x^{4} e^{\frac {2}{5}} + 1444 \, x^{3} e^{\frac {2}{5}} - 2608 \, x^{2} e^{\frac {2}{5}} + 1472 \, x e^{\frac {2}{5}} - 3 \, {\left (25 \, x^{5} e^{\frac {2}{5}} - 218 \, x^{4} e^{\frac {2}{5}} + 540 \, x^{3} e^{\frac {2}{5}} - 256 \, x^{2} e^{\frac {2}{5}} - 64 \, x e^{\frac {2}{5}}\right )} \log \relax (x) - 256 \, e^{\frac {2}{5}}\right )} e^{\left (-\frac {6}{5 \, {\left (5 \, x - 2\right )}}\right )}}{{\left (25 \, x^{3} e - 20 \, x^{2} e + 4 \, x e\right )} \log \relax (x)^{2}}\,{d x} + \int -\frac {24 \, {\left (25 \, x^{4} - 220 \, x^{3} + 564 \, x^{2} - 2 \, {\left (25 \, x^{4} - 114 \, x^{3} + 36 \, x^{2} + 80 \, x\right )} \log \relax (x) - 352 \, x + 64\right )} e^{\left (-\frac {12}{5 \, {\left (5 \, x - 2\right )}}\right )}}{{\left (25 \, x^{3} e^{\frac {6}{5}} - 20 \, x^{2} e^{\frac {6}{5}} + 4 \, x e^{\frac {6}{5}}\right )} \log \relax (x)^{2}}\,{d x} + \int -\frac {32 \, {\left (25 \, x^{3} e^{\frac {1}{5}} - 120 \, x^{2} e^{\frac {1}{5}} + 84 \, x e^{\frac {1}{5}} - {\left (25 \, x^{3} e^{\frac {1}{5}} - 2 \, x^{2} e^{\frac {1}{5}} - 68 \, x e^{\frac {1}{5}}\right )} \log \relax (x) - 16 \, e^{\frac {1}{5}}\right )} e^{\left (-\frac {18}{5 \, {\left (5 \, x - 2\right )}}\right )}}{{\left (25 \, x^{3} e^{2} - 20 \, x^{2} e^{2} + 4 \, x e^{2}\right )} \log \relax (x)^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((384*x*exp(-3*x/(5*x-2))^4+(800*x^3-64*x^2-2176*x)*exp(-3*x/(5*x-2))^3+(1200*x^4-5472*x^3+1728*x^2+
3840*x)*exp(-3*x/(5*x-2))^2+(600*x^5-5232*x^4+12960*x^3-6144*x^2-1536*x)*exp(-3*x/(5*x-2))+100*x^6-1280*x^5+57
76*x^4-10432*x^3+5888*x^2-1024*x)*log(x)+(-400*x^2+320*x-64)*exp(-3*x/(5*x-2))^4+(-800*x^3+3840*x^2-2688*x+512
)*exp(-3*x/(5*x-2))^3+(-600*x^4+5280*x^3-13536*x^2+8448*x-1536)*exp(-3*x/(5*x-2))^2+(-200*x^5+2560*x^4-11552*x
^3+20864*x^2-11776*x+2048)*exp(-3*x/(5*x-2))-25*x^6+420*x^5-2724*x^4+8384*x^3-11879*x^2+6124*x-1020)/(25*x^3-2
0*x^2+4*x)/log(x)^2,x, algorithm="maxima")

[Out]

(x^4*e^(12/5) - 16*x^3*e^(12/5) + 96*x^2*e^(12/5) - 256*x*e^(12/5) + 255*e^(12/5) + 16*e^(-24/5/(5*x - 2)))*e^
(-12/5)/log(x) + integrate(-8*(25*x^5*e^(2/5) - 320*x^4*e^(2/5) + 1444*x^3*e^(2/5) - 2608*x^2*e^(2/5) + 1472*x
*e^(2/5) - 3*(25*x^5*e^(2/5) - 218*x^4*e^(2/5) + 540*x^3*e^(2/5) - 256*x^2*e^(2/5) - 64*x*e^(2/5))*log(x) - 25
6*e^(2/5))*e^(-6/5/(5*x - 2))/((25*x^3*e - 20*x^2*e + 4*x*e)*log(x)^2), x) + integrate(-24*(25*x^4 - 220*x^3 +
564*x^2 - 2*(25*x^4 - 114*x^3 + 36*x^2 + 80*x)*log(x) - 352*x + 64)*e^(-12/5/(5*x - 2))/((25*x^3*e^(6/5) - 20
*x^2*e^(6/5) + 4*x*e^(6/5))*log(x)^2), x) + integrate(-32*(25*x^3*e^(1/5) - 120*x^2*e^(1/5) + 84*x*e^(1/5) - (
25*x^3*e^(1/5) - 2*x^2*e^(1/5) - 68*x*e^(1/5))*log(x) - 16*e^(1/5))*e^(-18/5/(5*x - 2))/((25*x^3*e^2 - 20*x^2*
e^2 + 4*x*e^2)*log(x)^2), x)

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mupad [B]  time = 6.71, size = 150, normalized size = 5.77 \begin {gather*} \frac {16\,{\mathrm {e}}^{-\frac {12\,x}{5\,x-2}}}{\ln \relax (x)}-\frac {256\,x-96\,x^2+16\,x^3-x^4+4\,x\,\ln \relax (x)\,{\left (x-4\right )}^3-255}{\ln \relax (x)}-256\,x+192\,x^2-48\,x^3+4\,x^4+\frac {{\mathrm {e}}^{-\frac {3\,x}{5\,x-2}}\,\left (8\,x^3-96\,x^2+384\,x-512\right )}{\ln \relax (x)}+\frac {{\mathrm {e}}^{-\frac {9\,x}{5\,x-2}}\,\left (32\,x-128\right )}{\ln \relax (x)}+\frac {{\mathrm {e}}^{-\frac {6\,x}{5\,x-2}}\,\left (24\,x^2-192\,x+384\right )}{\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(6*x)/(5*x - 2))*(13536*x^2 - 8448*x - 5280*x^3 + 600*x^4 + 1536) - 6124*x + exp(-(3*x)/(5*x - 2))*
(11776*x - 20864*x^2 + 11552*x^3 - 2560*x^4 + 200*x^5 - 2048) + log(x)*(1024*x - exp(-(6*x)/(5*x - 2))*(3840*x
+ 1728*x^2 - 5472*x^3 + 1200*x^4) + exp(-(3*x)/(5*x - 2))*(1536*x + 6144*x^2 - 12960*x^3 + 5232*x^4 - 600*x^5
) - 384*x*exp(-(12*x)/(5*x - 2)) + exp(-(9*x)/(5*x - 2))*(2176*x + 64*x^2 - 800*x^3) - 5888*x^2 + 10432*x^3 -
5776*x^4 + 1280*x^5 - 100*x^6) + exp(-(12*x)/(5*x - 2))*(400*x^2 - 320*x + 64) + exp(-(9*x)/(5*x - 2))*(2688*x
- 3840*x^2 + 800*x^3 - 512) + 11879*x^2 - 8384*x^3 + 2724*x^4 - 420*x^5 + 25*x^6 + 1020)/(log(x)^2*(4*x - 20*
x^2 + 25*x^3)),x)

[Out]

(16*exp(-(12*x)/(5*x - 2)))/log(x) - (256*x - 96*x^2 + 16*x^3 - x^4 + 4*x*log(x)*(x - 4)^3 - 255)/log(x) - 256
*x + 192*x^2 - 48*x^3 + 4*x^4 + (exp(-(3*x)/(5*x - 2))*(384*x - 96*x^2 + 8*x^3 - 512))/log(x) + (exp(-(9*x)/(5
*x - 2))*(32*x - 128))/log(x) + (exp(-(6*x)/(5*x - 2))*(24*x^2 - 192*x + 384))/log(x)

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sympy [B]  time = 0.83, size = 150, normalized size = 5.77 \begin {gather*} \frac {\left (32 x \log {\relax (x )}^{3} - 128 \log {\relax (x )}^{3}\right ) e^{- \frac {9 x}{5 x - 2}} + \left (24 x^{2} \log {\relax (x )}^{3} - 192 x \log {\relax (x )}^{3} + 384 \log {\relax (x )}^{3}\right ) e^{- \frac {6 x}{5 x - 2}} + \left (8 x^{3} \log {\relax (x )}^{3} - 96 x^{2} \log {\relax (x )}^{3} + 384 x \log {\relax (x )}^{3} - 512 \log {\relax (x )}^{3}\right ) e^{- \frac {3 x}{5 x - 2}} + 16 e^{- \frac {12 x}{5 x - 2}} \log {\relax (x )}^{3}}{\log {\relax (x )}^{4}} + \frac {x^{4} - 16 x^{3} + 96 x^{2} - 256 x + 255}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((384*x*exp(-3*x/(5*x-2))**4+(800*x**3-64*x**2-2176*x)*exp(-3*x/(5*x-2))**3+(1200*x**4-5472*x**3+172
8*x**2+3840*x)*exp(-3*x/(5*x-2))**2+(600*x**5-5232*x**4+12960*x**3-6144*x**2-1536*x)*exp(-3*x/(5*x-2))+100*x**
6-1280*x**5+5776*x**4-10432*x**3+5888*x**2-1024*x)*ln(x)+(-400*x**2+320*x-64)*exp(-3*x/(5*x-2))**4+(-800*x**3+
3840*x**2-2688*x+512)*exp(-3*x/(5*x-2))**3+(-600*x**4+5280*x**3-13536*x**2+8448*x-1536)*exp(-3*x/(5*x-2))**2+(
-200*x**5+2560*x**4-11552*x**3+20864*x**2-11776*x+2048)*exp(-3*x/(5*x-2))-25*x**6+420*x**5-2724*x**4+8384*x**3
-11879*x**2+6124*x-1020)/(25*x**3-20*x**2+4*x)/ln(x)**2,x)

[Out]

((32*x*log(x)**3 - 128*log(x)**3)*exp(-9*x/(5*x - 2)) + (24*x**2*log(x)**3 - 192*x*log(x)**3 + 384*log(x)**3)*
exp(-6*x/(5*x - 2)) + (8*x**3*log(x)**3 - 96*x**2*log(x)**3 + 384*x*log(x)**3 - 512*log(x)**3)*exp(-3*x/(5*x -
2)) + 16*exp(-12*x/(5*x - 2))*log(x)**3)/log(x)**4 + (x**4 - 16*x**3 + 96*x**2 - 256*x + 255)/log(x)

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