\(\int \frac {e^{\frac {20 x^4+36 x^3 \log (2 x+25 x^2)+54 x^2 \log ^2(2 x+25 x^2)+36 x \log ^3(2 x+25 x^2)+9 \log ^4(2 x+25 x^2)}{9 x^4}} (8 x^3+200 x^4+(24 x^2+592 x^3-100 x^4) \log (2 x+25 x^2)+(24 x+576 x^2-300 x^3) \log ^2(2 x+25 x^2)+(8+176 x-300 x^2) \log ^3(2 x+25 x^2)+(-8-100 x) \log ^4(2 x+25 x^2))}{2 x^5+25 x^6} \, dx\) [3493]

Optimal result

Integrand size = 196, antiderivative size = 24 \[ \int \frac {e^{\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}} \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx=e^{\frac {11}{9}+\frac {\left (x+\log \left (2 x+25 x^2\right )\right )^4}{x^4}} \]

[Out]

exp(11/9+(ln(25*x^2+2*x)+x)^4/x^4)

Rubi [F]

\[ \int \frac {e^{\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}} \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx=\int \frac {\exp \left (\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}\right ) \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx \]

[In]

Int[(E^((20*x^4 + 36*x^3*Log[2*x + 25*x^2] + 54*x^2*Log[2*x + 25*x^2]^2 + 36*x*Log[2*x + 25*x^2]^3 + 9*Log[2*x
+ 25*x^2]^4)/(9*x^4))*(8*x^3 + 200*x^4 + (24*x^2 + 592*x^3 - 100*x^4)*Log[2*x + 25*x^2] + (24*x + 576*x^2 - 3
00*x^3)*Log[2*x + 25*x^2]^2 + (8 + 176*x - 300*x^2)*Log[2*x + 25*x^2]^3 + (-8 - 100*x)*Log[2*x + 25*x^2]^4))/(
2*x^5 + 25*x^6),x]

[Out]

200*Defer[Int][E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*
(x*(2 + 25*x))^(-1 + 4/x), x] + 8*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/
x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x))/x, x] - 100*Defer[Int][E^(20/9 + (6*Log[x*(2 + 25*x)
]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)],
x] + 24*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4
/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)])/x^2, x] + 592*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)
/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)])/x,
x] + 24*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/
x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^2)/x^3, x] + 576*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2
)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^2)/
x^2, x] - 300*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*
x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^2)/x, x] + 8*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]
^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^3
)/x^4, x] + 176*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 2
5*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^3)/x^3, x] - 300*Defer[Int][(E^(20/9 + (6*Log[x*(2 +
25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25
*x)]^3)/x^2, x] - 4*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2
+ 25*x)]^4/x^4)*(x*(2 + 25*x))^(4/x)*Log[x*(2 + 25*x)]^4)/x^5, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}\right ) \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{x^5 (2+25 x)} \, dx \\ & = \int \frac {4 \exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} (x+\log (x (2+25 x)))^3 (2+50 x-(2+25 x) \log (x (2+25 x)))}{x^4} \, dx \\ & = 4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} (x+\log (x (2+25 x)))^3 (2+50 x-(2+25 x) \log (x (2+25 x)))}{x^4} \, dx \\ & = 4 \int \left (\frac {2 \exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (1+25 x) (x (2+25 x))^{-1+\frac {4}{x}}}{x}-\frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-6-148 x+25 x^2\right ) \log (x (2+25 x))}{x^2}-\frac {3 \exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-48 x+25 x^2\right ) \log ^2(x (2+25 x))}{x^3}-\frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-44 x+75 x^2\right ) \log ^3(x (2+25 x))}{x^4}-\frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (2+25 x) (x (2+25 x))^{-1+\frac {4}{x}} \log ^4(x (2+25 x))}{x^4}\right ) \, dx \\ & = -\left (4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-6-148 x+25 x^2\right ) \log (x (2+25 x))}{x^2} \, dx\right )-4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-44 x+75 x^2\right ) \log ^3(x (2+25 x))}{x^4} \, dx-4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (2+25 x) (x (2+25 x))^{-1+\frac {4}{x}} \log ^4(x (2+25 x))}{x^4} \, dx+8 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (1+25 x) (x (2+25 x))^{-1+\frac {4}{x}}}{x} \, dx-12 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-48 x+25 x^2\right ) \log ^2(x (2+25 x))}{x^3} \, dx \\ & = -\left (4 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{4/x} \log ^4(x (2+25 x))}{x^5} \, dx\right )-4 \int \left (25 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log (x (2+25 x))-\frac {6 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log (x (2+25 x))}{x^2}-\frac {148 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log (x (2+25 x))}{x}\right ) \, dx-4 \int \left (-\frac {2 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^3(x (2+25 x))}{x^4}-\frac {44 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^3(x (2+25 x))}{x^3}+\frac {75 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^3(x (2+25 x))}{x^2}\right ) \, dx+8 \int \left (25 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}}+\frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}}}{x}\right ) \, dx-12 \int \left (-\frac {2 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^2(x (2+25 x))}{x^3}-\frac {48 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^2(x (2+25 x))}{x^2}+\frac {25 e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^2(x (2+25 x))}{x}\right ) \, dx \\ & = -\left (4 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{4/x} \log ^4(x (2+25 x))}{x^5} \, dx\right )+8 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}}}{x} \, dx+8 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^3(x (2+25 x))}{x^4} \, dx+24 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log (x (2+25 x))}{x^2} \, dx+24 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^2(x (2+25 x))}{x^3} \, dx-100 \int e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log (x (2+25 x)) \, dx+176 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^3(x (2+25 x))}{x^3} \, dx+200 \int e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \, dx-300 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^2(x (2+25 x))}{x} \, dx-300 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^3(x (2+25 x))}{x^2} \, dx+576 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log ^2(x (2+25 x))}{x^2} \, dx+592 \int \frac {e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{-1+\frac {4}{x}} \log (x (2+25 x))}{x} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(24)=48\).

Time = 0.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.67 \[ \int \frac {e^{\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}} \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx=e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{4/x} \]

[In]

Integrate[(E^((20*x^4 + 36*x^3*Log[2*x + 25*x^2] + 54*x^2*Log[2*x + 25*x^2]^2 + 36*x*Log[2*x + 25*x^2]^3 + 9*L
og[2*x + 25*x^2]^4)/(9*x^4))*(8*x^3 + 200*x^4 + (24*x^2 + 592*x^3 - 100*x^4)*Log[2*x + 25*x^2] + (24*x + 576*x
^2 - 300*x^3)*Log[2*x + 25*x^2]^2 + (8 + 176*x - 300*x^2)*Log[2*x + 25*x^2]^3 + (-8 - 100*x)*Log[2*x + 25*x^2]
^4))/(2*x^5 + 25*x^6),x]

[Out]

E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^
(4/x)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(21)=42\).

Time = 2.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.08

method	result	size
parallelrisch	\({\mathrm e}^{\frac {9 \ln \left (25 x^{2}+2 x \right )^{4}+36 x \ln \left (25 x^{2}+2 x \right )^{3}+54 x^{2} \ln \left (25 x^{2}+2 x \right )^{2}+36 x^{3} \ln \left (25 x^{2}+2 x \right )+20 x^{4}}{9 x^{4}}}\)	\(74\)
risch	\(\left (25 x^{2}+2 x \right )^{\frac {4}{x}} {\mathrm e}^{\frac {9 \ln \left (25 x^{2}+2 x \right )^{4}+36 x \ln \left (25 x^{2}+2 x \right )^{3}+54 x^{2} \ln \left (25 x^{2}+2 x \right )^{2}+20 x^{4}}{9 x^{4}}}\)	\(75\)

[In]

int(((-100*x-8)*ln(25*x^2+2*x)^4+(-300*x^2+176*x+8)*ln(25*x^2+2*x)^3+(-300*x^3+576*x^2+24*x)*ln(25*x^2+2*x)^2+
(-100*x^4+592*x^3+24*x^2)*ln(25*x^2+2*x)+200*x^4+8*x^3)*exp(1/9*(9*ln(25*x^2+2*x)^4+36*x*ln(25*x^2+2*x)^3+54*x
^2*ln(25*x^2+2*x)^2+36*x^3*ln(25*x^2+2*x)+20*x^4)/x^4)/(25*x^6+2*x^5),x,method=_RETURNVERBOSE)

[Out]

exp(1/9*(9*ln(25*x^2+2*x)^4+36*x*ln(25*x^2+2*x)^3+54*x^2*ln(25*x^2+2*x)^2+36*x^3*ln(25*x^2+2*x)+20*x^4)/x^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.04 \[ \int \frac {e^{\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}} \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx=e^{\left (\frac {20 \, x^{4} + 36 \, x^{3} \log \left (25 \, x^{2} + 2 \, x\right ) + 54 \, x^{2} \log \left (25 \, x^{2} + 2 \, x\right )^{2} + 36 \, x \log \left (25 \, x^{2} + 2 \, x\right )^{3} + 9 \, \log \left (25 \, x^{2} + 2 \, x\right )^{4}}{9 \, x^{4}}\right )} \]

[In]

integrate(((-100*x-8)*log(25*x^2+2*x)^4+(-300*x^2+176*x+8)*log(25*x^2+2*x)^3+(-300*x^3+576*x^2+24*x)*log(25*x^
2+2*x)^2+(-100*x^4+592*x^3+24*x^2)*log(25*x^2+2*x)+200*x^4+8*x^3)*exp(1/9*(9*log(25*x^2+2*x)^4+36*x*log(25*x^2
+2*x)^3+54*x^2*log(25*x^2+2*x)^2+36*x^3*log(25*x^2+2*x)+20*x^4)/x^4)/(25*x^6+2*x^5),x, algorithm="fricas")

[Out]

e^(1/9*(20*x^4 + 36*x^3*log(25*x^2 + 2*x) + 54*x^2*log(25*x^2 + 2*x)^2 + 36*x*log(25*x^2 + 2*x)^3 + 9*log(25*x
^2 + 2*x)^4)/x^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).

Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.92 \[ \int \frac {e^{\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}} \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx=e^{\frac {\frac {20 x^{4}}{9} + 4 x^{3} \log {\left (25 x^{2} + 2 x \right )} + 6 x^{2} \log {\left (25 x^{2} + 2 x \right )}^{2} + 4 x \log {\left (25 x^{2} + 2 x \right )}^{3} + \log {\left (25 x^{2} + 2 x \right )}^{4}}{x^{4}}} \]

[In]

integrate(((-100*x-8)*ln(25*x**2+2*x)**4+(-300*x**2+176*x+8)*ln(25*x**2+2*x)**3+(-300*x**3+576*x**2+24*x)*ln(2
5*x**2+2*x)**2+(-100*x**4+592*x**3+24*x**2)*ln(25*x**2+2*x)+200*x**4+8*x**3)*exp(1/9*(9*ln(25*x**2+2*x)**4+36*
x*ln(25*x**2+2*x)**3+54*x**2*ln(25*x**2+2*x)**2+36*x**3*ln(25*x**2+2*x)+20*x**4)/x**4)/(25*x**6+2*x**5),x)

[Out]

exp((20*x**4/9 + 4*x**3*log(25*x**2 + 2*x) + 6*x**2*log(25*x**2 + 2*x)**2 + 4*x*log(25*x**2 + 2*x)**3 + log(25
*x**2 + 2*x)**4)/x**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (21) = 42\).

Time = 0.66 (sec) , antiderivative size = 175, normalized size of antiderivative = 7.29 \[ \int \frac {e^{\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}} \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx=e^{\left (\frac {4 \, \log \left (25 \, x + 2\right )}{x} + \frac {6 \, \log \left (25 \, x + 2\right )^{2}}{x^{2}} + \frac {4 \, \log \left (25 \, x + 2\right )^{3}}{x^{3}} + \frac {\log \left (25 \, x + 2\right )^{4}}{x^{4}} + \frac {4 \, \log \left (x\right )}{x} + \frac {12 \, \log \left (25 \, x + 2\right ) \log \left (x\right )}{x^{2}} + \frac {12 \, \log \left (25 \, x + 2\right )^{2} \log \left (x\right )}{x^{3}} + \frac {4 \, \log \left (25 \, x + 2\right )^{3} \log \left (x\right )}{x^{4}} + \frac {6 \, \log \left (x\right )^{2}}{x^{2}} + \frac {12 \, \log \left (25 \, x + 2\right ) \log \left (x\right )^{2}}{x^{3}} + \frac {6 \, \log \left (25 \, x + 2\right )^{2} \log \left (x\right )^{2}}{x^{4}} + \frac {4 \, \log \left (x\right )^{3}}{x^{3}} + \frac {4 \, \log \left (25 \, x + 2\right ) \log \left (x\right )^{3}}{x^{4}} + \frac {\log \left (x\right )^{4}}{x^{4}} + \frac {20}{9}\right )} \]

[In]

integrate(((-100*x-8)*log(25*x^2+2*x)^4+(-300*x^2+176*x+8)*log(25*x^2+2*x)^3+(-300*x^3+576*x^2+24*x)*log(25*x^
2+2*x)^2+(-100*x^4+592*x^3+24*x^2)*log(25*x^2+2*x)+200*x^4+8*x^3)*exp(1/9*(9*log(25*x^2+2*x)^4+36*x*log(25*x^2
+2*x)^3+54*x^2*log(25*x^2+2*x)^2+36*x^3*log(25*x^2+2*x)+20*x^4)/x^4)/(25*x^6+2*x^5),x, algorithm="maxima")

[Out]

e^(4*log(25*x + 2)/x + 6*log(25*x + 2)^2/x^2 + 4*log(25*x + 2)^3/x^3 + log(25*x + 2)^4/x^4 + 4*log(x)/x + 12*l
og(25*x + 2)*log(x)/x^2 + 12*log(25*x + 2)^2*log(x)/x^3 + 4*log(25*x + 2)^3*log(x)/x^4 + 6*log(x)^2/x^2 + 12*l
og(25*x + 2)*log(x)^2/x^3 + 6*log(25*x + 2)^2*log(x)^2/x^4 + 4*log(x)^3/x^3 + 4*log(25*x + 2)*log(x)^3/x^4 + l
og(x)^4/x^4 + 20/9)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (21) = 42\).

Time = 1.37 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.83 \[ \int \frac {e^{\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}} \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx=e^{\left (\frac {4 \, \log \left (25 \, x^{2} + 2 \, x\right )}{x} + \frac {6 \, \log \left (25 \, x^{2} + 2 \, x\right )^{2}}{x^{2}} + \frac {4 \, \log \left (25 \, x^{2} + 2 \, x\right )^{3}}{x^{3}} + \frac {\log \left (25 \, x^{2} + 2 \, x\right )^{4}}{x^{4}} + \frac {20}{9}\right )} \]

[In]

integrate(((-100*x-8)*log(25*x^2+2*x)^4+(-300*x^2+176*x+8)*log(25*x^2+2*x)^3+(-300*x^3+576*x^2+24*x)*log(25*x^
2+2*x)^2+(-100*x^4+592*x^3+24*x^2)*log(25*x^2+2*x)+200*x^4+8*x^3)*exp(1/9*(9*log(25*x^2+2*x)^4+36*x*log(25*x^2
+2*x)^3+54*x^2*log(25*x^2+2*x)^2+36*x^3*log(25*x^2+2*x)+20*x^4)/x^4)/(25*x^6+2*x^5),x, algorithm="giac")

[Out]

e^(4*log(25*x^2 + 2*x)/x + 6*log(25*x^2 + 2*x)^2/x^2 + 4*log(25*x^2 + 2*x)^3/x^3 + log(25*x^2 + 2*x)^4/x^4 + 2
0/9)

Mupad [B] (verification not implemented)

Time = 9.32 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.96 \[ \int \frac {e^{\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}} \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx={\mathrm {e}}^{20/9}\,{\mathrm {e}}^{\frac {{\ln \left (25\,x^2+2\,x\right )}^4}{x^4}}\,{\mathrm {e}}^{\frac {6\,{\ln \left (25\,x^2+2\,x\right )}^2}{x^2}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (25\,x^2+2\,x\right )}^3}{x^3}}\,{\left (25\,x^2+2\,x\right )}^{4/x} \]

[In]

int((exp((4*x*log(2*x + 25*x^2)^3 + 4*x^3*log(2*x + 25*x^2) + log(2*x + 25*x^2)^4 + 6*x^2*log(2*x + 25*x^2)^2
+ (20*x^4)/9)/x^4)*(log(2*x + 25*x^2)^3*(176*x - 300*x^2 + 8) + log(2*x + 25*x^2)*(24*x^2 + 592*x^3 - 100*x^4)
+ log(2*x + 25*x^2)^2*(24*x + 576*x^2 - 300*x^3) - log(2*x + 25*x^2)^4*(100*x + 8) + 8*x^3 + 200*x^4))/(2*x^5
+ 25*x^6),x)

[Out]

exp(20/9)*exp(log(2*x + 25*x^2)^4/x^4)*exp((6*log(2*x + 25*x^2)^2)/x^2)*exp((4*log(2*x + 25*x^2)^3)/x^3)*(2*x
+ 25*x^2)^(4/x)