Integrand size = 165, antiderivative size = 24 \[ \int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (75 x-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+5 x \log ^2(4)+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx=x^{\frac {x}{x+\frac {-1+(-4+x+\log (4))^2}{5 x}}} \] Output:
exp(ln(x)*x/(1/5*((x-4+2*ln(2))^2-1)/x+x))
Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (75 x-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+5 x \log ^2(4)+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx=x^{\frac {5 x^2}{15-8 x+6 x^2-8 \log (4)+2 x \log (4)+\log ^2(4)}} \] Input:
Integrate[(x^((5*x^2)/(15 - 8*x + 6*x^2 + (-8 + 2*x)*Log[4] + Log[4]^2))*( 75*x - 40*x^2 + 30*x^3 + (-40*x + 10*x^2)*Log[4] + 5*x*Log[4]^2 + (150*x - 40*x^2 + (-80*x + 10*x^2)*Log[4] + 10*x*Log[4]^2)*Log[x]))/(225 - 240*x + 244*x^2 - 96*x^3 + 36*x^4 + (-240 + 188*x - 128*x^2 + 24*x^3)*Log[4] + (9 4 - 48*x + 16*x^2)*Log[4]^2 + (-16 + 4*x)*Log[4]^3 + Log[4]^4),x]
Output:
x^((5*x^2)/(15 - 8*x + 6*x^2 - 8*Log[4] + 2*x*Log[4] + Log[4]^2))
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^{\frac {5 x^2}{6 x^2-8 x+(2 x-8) \log (4)+15+\log ^2(4)}} \left (30 x^3-40 x^2+\left (-40 x^2+\left (10 x^2-80 x\right ) \log (4)+150 x+10 x \log ^2(4)\right ) \log (x)+\left (10 x^2-40 x\right ) \log (4)+75 x+5 x \log ^2(4)\right )}{36 x^4-96 x^3+244 x^2+\left (16 x^2-48 x+94\right ) \log ^2(4)+\left (24 x^3-128 x^2+188 x-240\right ) \log (4)-240 x+(4 x-16) \log ^3(4)+225+\log ^4(4)} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^{\frac {5 x^2}{6 x^2-8 x+(2 x-8) \log (4)+15+\log ^2(4)}} \left (30 x^3-40 x^2+\left (-40 x^2+\left (10 x^2-80 x\right ) \log (4)+150 x+10 x \log ^2(4)\right ) \log (x)+\left (10 x^2-40 x\right ) \log (4)+x \left (75+5 \log ^2(4)\right )\right )}{36 x^4-96 x^3+244 x^2+\left (16 x^2-48 x+94\right ) \log ^2(4)+\left (24 x^3-128 x^2+188 x-240\right ) \log (4)-240 x+(4 x-16) \log ^3(4)+225+\log ^4(4)}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {x^{\frac {5 x^2}{6 x^2-8 x+(2 x-8) \log (4)+15+\log ^2(4)}} \left (30 x^3-40 x^2+\left (-40 x^2+\left (10 x^2-80 x\right ) \log (4)+150 x+10 x \log ^2(4)\right ) \log (x)+\left (10 x^2-40 x\right ) \log (4)+x \left (75+5 \log ^2(4)\right )\right )}{\left (6 x^2-8 x+2 x \log (4)+15+\log ^2(4)-8 \log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {x^{\frac {5 x^2}{6 x^2-8 x+(2 x-8) \log (4)+15+\log ^2(4)}} \left (30 x^3-40 x^2+\left (-40 x^2+\left (10 x^2-80 x\right ) \log (4)+150 x+10 x \log ^2(4)\right ) \log (x)+\left (10 x^2-40 x\right ) \log (4)+x \left (75+5 \log ^2(4)\right )\right )}{\left (6 x^2+x (2 \log (4)-8)+15+\log ^2(4)-8 \log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {5 x^{\frac {5 x^2}{6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))}+1} \left (6 x^2-8 x \left (1-\frac {\log (2)}{2}\right ) \log (x)-8 x \left (1-\frac {\log (2)}{2}\right )+30 \left (1+\frac {2}{15} (\log (2)-4) \log (4)\right ) \log (x)+15 \left (1+\frac {2}{15} (\log (2)-4) \log (4)\right )\right )}{\left (6 x^2-2 x (4-\log (4))+(3-\log (4)) (5-\log (4))\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 5 \int \frac {x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1} \left (6 x^2-4 (2-\log (2)) \log (x) x-4 (2-\log (2)) x+2 (15-2 (4-\log (2)) \log (4)) \log (x)+(3-\log (4)) (5-\log (4))\right )}{\left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 5 \int \left (\frac {2 ((3-\log (4)) (5-\log (4))-x (4-\log (4))) \log (x) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1}}{\left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right )^2}+\frac {(3-\log (4)) (5-\log (4)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1}}{\left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right )^2}+\frac {4 (-2+\log (2)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2}}{\left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right )^2}+\frac {6 x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3}}{\left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 5 \left (\frac {\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) \left (4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}\right )^2 x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{20 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-6 x+i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (4)+4\right )}-\frac {\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) \left (4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}\right )^2 x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{20 \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-6 x+i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (4)+4\right )}-\frac {(2-\log (2)) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) \left (4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{5 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-6 x+i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (4)+4\right )}+\frac {(2-\log (2)) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) \left (4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{5 \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-6 x+i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (4)+4\right )}-\frac {3 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) (3-\log (4)) (5-\log (4)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{10 \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-6 x+i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (4)+4\right )}+\frac {3 \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) (3-\log (4)) (5-\log (4)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{10 \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-6 x+i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (4)+4\right )}-\frac {(2-\log (2)) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) \left (8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{5 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-12 x-\log (16)-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}+8\right )}+\frac {(2-\log (2)) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) \left (8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{5 \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-12 x-\log (16)-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}+8\right )}-\frac {\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) \left (2 \sqrt {74-40 \log (4)+5 \log ^2(4)}+i (8-\log (16))\right )^2 x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{40 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-12 x-\log (16)-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}+8\right )}+\frac {\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) \left (2 \sqrt {74-40 \log (4)+5 \log ^2(4)}+i (8-\log (16))\right )^2 x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{40 \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-12 x-\log (16)-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}+8\right )}-\frac {3 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) (3-\log (4)) (5-\log (4)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{5 \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-12 x-\log (16)-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}+8\right )}+\frac {3 \left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right ) (3-\log (4)) (5-\log (4)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}-1}}{5 \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-12 x-\log (16)-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}+8\right )}+\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2,\frac {12 x}{8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)}\right ) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1}}{8 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )}+\frac {3 \operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2,\frac {12 x}{8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)}\right ) (3-\log (4)) (5-\log (4)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1}}{\left (74-40 \log (4)+5 \log ^2(4)\right ) \left (8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)\right )}+\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2,\frac {6 x}{4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}}\right ) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1}}{4 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )}+\frac {3 \operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2,\frac {6 x}{4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}}\right ) (3-\log (4)) (5-\log (4)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1}}{2 \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}\right )}-\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2,\frac {6 x}{4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}}\right ) (2-\log (2)) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )}-\frac {\operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2,\frac {12 x}{8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)}\right ) (2-\log (2)) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )}+\frac {3 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2}}{2 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-6 x+i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (4)+4\right )}-\frac {6 (2-\log (2)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-6 x+i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (4)+4\right )}+\frac {3 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2}}{2 \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-12 x-\log (16)-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}+8\right )}-\frac {12 (2-\log (2)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-12 x-\log (16)-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}+8\right )}+\frac {18 \operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3,\frac {12 x}{8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)}\right ) (3-\log (4)) (5-\log (4)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )^{3/2} \left (2 \sqrt {74-40 \log (4)+5 \log ^2(4)}+i (8-\log (16))\right )}-\frac {9 \operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3,\frac {6 x}{4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}}\right ) (3-\log (4)) (5-\log (4)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )^{3/2} \left (4 i-i \log (4)-\sqrt {74-40 \log (4)+5 \log ^2(4)}\right )}+\frac {9 x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-6 x+i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (4)+4\right )}+\frac {18 x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2\right ) \left (74-40 \log (4)+5 \log ^2(4)\right ) \left (-12 x-\log (16)-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}+8\right )}-\frac {72 \operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+4,\frac {12 x}{8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)}\right ) (2-\log (2)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )^{3/2} \left (2 \sqrt {74-40 \log (4)+5 \log ^2(4)}+i (8-\log (16))\right )}+\frac {36 \operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+4,\frac {6 x}{4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}}\right ) (2-\log (2)) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+3\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )^{3/2} \left (4 i-i \log (4)-\sqrt {74-40 \log (4)+5 \log ^2(4)}\right )}+\frac {108 \operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+4,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+5,\frac {12 x}{8-2 i \sqrt {74-40 \log (4)+5 \log ^2(4)}-\log (16)}\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+4}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+4\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )^{3/2} \left (2 \sqrt {74-40 \log (4)+5 \log ^2(4)}+i (8-\log (16))\right )}-\frac {54 \operatorname {Hypergeometric2F1}\left (1,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+4,\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+5,\frac {6 x}{4-\log (4)+i \sqrt {74-40 \log (4)+5 \log ^2(4)}}\right ) x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+4}}{\left (\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+4\right ) \left (74-40 \log (4)+5 \log ^2(4)\right )^{3/2} \left (4 i-i \log (4)-\sqrt {74-40 \log (4)+5 \log ^2(4)}\right )}+2 (3-\log (4)) (5-\log (4)) \int \frac {x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+1} \log (x)}{\left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right )^2}dx-2 (4-\log (4)) \int \frac {x^{\frac {5 x^2}{6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))}+2} \log (x)}{\left (6 x^2-2 (4-\log (4)) x+(3-\log (4)) (5-\log (4))\right )^2}dx\right )\) |
Input:
Int[(x^((5*x^2)/(15 - 8*x + 6*x^2 + (-8 + 2*x)*Log[4] + Log[4]^2))*(75*x - 40*x^2 + 30*x^3 + (-40*x + 10*x^2)*Log[4] + 5*x*Log[4]^2 + (150*x - 40*x^ 2 + (-80*x + 10*x^2)*Log[4] + 10*x*Log[4]^2)*Log[x]))/(225 - 240*x + 244*x ^2 - 96*x^3 + 36*x^4 + (-240 + 188*x - 128*x^2 + 24*x^3)*Log[4] + (94 - 48 *x + 16*x^2)*Log[4]^2 + (-16 + 4*x)*Log[4]^3 + Log[4]^4),x]
Output:
$Aborted
Time = 9.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46
| method | result | size |
| risch | \(x^{\frac {5 x^{2}}{4 \ln \left (2\right )^{2}+4 x \ln \left (2\right )+6 x^{2}-16 \ln \left (2\right )-8 x +15}}\) | \(35\) |
| parallelrisch | \({\mathrm e}^{\frac {5 x^{2} \ln \left (x \right )}{4 \ln \left (2\right )^{2}+4 x \ln \left (2\right )+6 x^{2}-16 \ln \left (2\right )-8 x +15}}\) | \(36\) |
| norman | \(\frac {\left (4 \ln \left (2\right )^{2}+15-16 \ln \left (2\right )\right ) {\mathrm e}^{\frac {5 x^{2} \ln \left (x \right )}{4 \ln \left (2\right )^{2}+2 \left (2 x -8\right ) \ln \left (2\right )+6 x^{2}-8 x +15}}+\left (4 \ln \left (2\right )-8\right ) x \,{\mathrm e}^{\frac {5 x^{2} \ln \left (x \right )}{4 \ln \left (2\right )^{2}+2 \left (2 x -8\right ) \ln \left (2\right )+6 x^{2}-8 x +15}}+6 x^{2} {\mathrm e}^{\frac {5 x^{2} \ln \left (x \right )}{4 \ln \left (2\right )^{2}+2 \left (2 x -8\right ) \ln \left (2\right )+6 x^{2}-8 x +15}}}{4 \ln \left (2\right )^{2}+4 x \ln \left (2\right )+6 x^{2}-16 \ln \left (2\right )-8 x +15}\) | \(161\) |
Input:
int(((40*x*ln(2)^2+2*(10*x^2-80*x)*ln(2)-40*x^2+150*x)*ln(x)+20*x*ln(2)^2+ 2*(10*x^2-40*x)*ln(2)+30*x^3-40*x^2+75*x)*exp(5*x^2*ln(x)/(4*ln(2)^2+2*(2* x-8)*ln(2)+6*x^2-8*x+15))/(16*ln(2)^4+8*(4*x-16)*ln(2)^3+4*(16*x^2-48*x+94 )*ln(2)^2+2*(24*x^3-128*x^2+188*x-240)*ln(2)+36*x^4-96*x^3+244*x^2-240*x+2 25),x,method=_RETURNVERBOSE)
Output:
x^(5*x^2/(4*ln(2)^2+4*x*ln(2)+6*x^2-16*ln(2)-8*x+15))
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.33 \[ \int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (75 x-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+5 x \log ^2(4)+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx=x^{\frac {5 \, x^{2}}{6 \, x^{2} + 4 \, {\left (x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 15}} \] Input:
integrate(((40*x*log(2)^2+2*(10*x^2-80*x)*log(2)-40*x^2+150*x)*log(x)+20*x *log(2)^2+2*(10*x^2-40*x)*log(2)+30*x^3-40*x^2+75*x)*exp(5*x^2*log(x)/(4*l og(2)^2+2*(2*x-8)*log(2)+6*x^2-8*x+15))/(16*log(2)^4+8*(4*x-16)*log(2)^3+4 *(16*x^2-48*x+94)*log(2)^2+2*(24*x^3-128*x^2+188*x-240)*log(2)+36*x^4-96*x ^3+244*x^2-240*x+225),x, algorithm="fricas")
Output:
x^(5*x^2/(6*x^2 + 4*(x - 4)*log(2) + 4*log(2)^2 - 8*x + 15))
Time = 0.51 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (75 x-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+5 x \log ^2(4)+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx=e^{\frac {5 x^{2} \log {\left (x \right )}}{6 x^{2} - 8 x + \left (4 x - 16\right ) \log {\left (2 \right )} + 4 \log {\left (2 \right )}^{2} + 15}} \] Input:
integrate(((40*x*ln(2)**2+2*(10*x**2-80*x)*ln(2)-40*x**2+150*x)*ln(x)+20*x *ln(2)**2+2*(10*x**2-40*x)*ln(2)+30*x**3-40*x**2+75*x)*exp(5*x**2*ln(x)/(4 *ln(2)**2+2*(2*x-8)*ln(2)+6*x**2-8*x+15))/(16*ln(2)**4+8*(4*x-16)*ln(2)**3 +4*(16*x**2-48*x+94)*ln(2)**2+2*(24*x**3-128*x**2+188*x-240)*ln(2)+36*x**4 -96*x**3+244*x**2-240*x+225),x)
Output:
exp(5*x**2*log(x)/(6*x**2 - 8*x + (4*x - 16)*log(2) + 4*log(2)**2 + 15))
\[ \int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (75 x-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+5 x \log ^2(4)+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx=\int { \frac {5 \, {\left (6 \, x^{3} + 4 \, x \log \left (2\right )^{2} - 8 \, x^{2} + 4 \, {\left (x^{2} - 4 \, x\right )} \log \left (2\right ) + 2 \, {\left (4 \, x \log \left (2\right )^{2} - 4 \, x^{2} + 2 \, {\left (x^{2} - 8 \, x\right )} \log \left (2\right ) + 15 \, x\right )} \log \left (x\right ) + 15 \, x\right )} x^{\frac {5 \, x^{2}}{6 \, x^{2} + 4 \, {\left (x - 4\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x + 15}}}{36 \, x^{4} + 32 \, {\left (x - 4\right )} \log \left (2\right )^{3} + 16 \, \log \left (2\right )^{4} - 96 \, x^{3} + 8 \, {\left (8 \, x^{2} - 24 \, x + 47\right )} \log \left (2\right )^{2} + 244 \, x^{2} + 8 \, {\left (6 \, x^{3} - 32 \, x^{2} + 47 \, x - 60\right )} \log \left (2\right ) - 240 \, x + 225} \,d x } \] Input:
integrate(((40*x*log(2)^2+2*(10*x^2-80*x)*log(2)-40*x^2+150*x)*log(x)+20*x *log(2)^2+2*(10*x^2-40*x)*log(2)+30*x^3-40*x^2+75*x)*exp(5*x^2*log(x)/(4*l og(2)^2+2*(2*x-8)*log(2)+6*x^2-8*x+15))/(16*log(2)^4+8*(4*x-16)*log(2)^3+4 *(16*x^2-48*x+94)*log(2)^2+2*(24*x^3-128*x^2+188*x-240)*log(2)+36*x^4-96*x ^3+244*x^2-240*x+225),x, algorithm="maxima")
Output:
5*integrate((6*x^3 + 4*x*log(2)^2 - 8*x^2 + 4*(x^2 - 4*x)*log(2) + 2*(4*x* log(2)^2 - 4*x^2 + 2*(x^2 - 8*x)*log(2) + 15*x)*log(x) + 15*x)*x^(5*x^2/(6 *x^2 + 4*(x - 4)*log(2) + 4*log(2)^2 - 8*x + 15))/(36*x^4 + 32*(x - 4)*log (2)^3 + 16*log(2)^4 - 96*x^3 + 8*(8*x^2 - 24*x + 47)*log(2)^2 + 244*x^2 + 8*(6*x^3 - 32*x^2 + 47*x - 60)*log(2) - 240*x + 225), x)
Time = 0.67 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.42 \[ \int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (75 x-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+5 x \log ^2(4)+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx=x^{\frac {5 \, x^{2}}{6 \, x^{2} + 4 \, x \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 8 \, x - 16 \, \log \left (2\right ) + 15}} \] Input:
integrate(((40*x*log(2)^2+2*(10*x^2-80*x)*log(2)-40*x^2+150*x)*log(x)+20*x *log(2)^2+2*(10*x^2-40*x)*log(2)+30*x^3-40*x^2+75*x)*exp(5*x^2*log(x)/(4*l og(2)^2+2*(2*x-8)*log(2)+6*x^2-8*x+15))/(16*log(2)^4+8*(4*x-16)*log(2)^3+4 *(16*x^2-48*x+94)*log(2)^2+2*(24*x^3-128*x^2+188*x-240)*log(2)+36*x^4-96*x ^3+244*x^2-240*x+225),x, algorithm="giac")
Output:
x^(5*x^2/(6*x^2 + 4*x*log(2) + 4*log(2)^2 - 8*x - 16*log(2) + 15))
Time = 4.70 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.46 \[ \int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (75 x-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+5 x \log ^2(4)+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx={\mathrm {e}}^{\frac {5\,x^2\,\ln \left (x\right )}{4\,x\,\ln \left (2\right )-16\,\ln \left (2\right )-8\,x+4\,{\ln \left (2\right )}^2+6\,x^2+15}} \] Input:
int((exp((5*x^2*log(x))/(2*log(2)*(2*x - 8) - 8*x + 4*log(2)^2 + 6*x^2 + 1 5))*(75*x - 2*log(2)*(40*x - 10*x^2) + log(x)*(150*x - 2*log(2)*(80*x - 10 *x^2) + 40*x*log(2)^2 - 40*x^2) + 20*x*log(2)^2 - 40*x^2 + 30*x^3))/(8*log (2)^3*(4*x - 16) - 240*x + 2*log(2)*(188*x - 128*x^2 + 24*x^3 - 240) + 4*l og(2)^2*(16*x^2 - 48*x + 94) + 16*log(2)^4 + 244*x^2 - 96*x^3 + 36*x^4 + 2 25),x)
Output:
exp((5*x^2*log(x))/(4*x*log(2) - 16*log(2) - 8*x + 4*log(2)^2 + 6*x^2 + 15 ))
Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50 \[ \int \frac {x^{\frac {5 x^2}{15-8 x+6 x^2+(-8+2 x) \log (4)+\log ^2(4)}} \left (75 x-40 x^2+30 x^3+\left (-40 x+10 x^2\right ) \log (4)+5 x \log ^2(4)+\left (150 x-40 x^2+\left (-80 x+10 x^2\right ) \log (4)+10 x \log ^2(4)\right ) \log (x)\right )}{225-240 x+244 x^2-96 x^3+36 x^4+\left (-240+188 x-128 x^2+24 x^3\right ) \log (4)+\left (94-48 x+16 x^2\right ) \log ^2(4)+(-16+4 x) \log ^3(4)+\log ^4(4)} \, dx=e^{\frac {5 \,\mathrm {log}\left (x \right ) x^{2}}{4 \mathrm {log}\left (2\right )^{2}+4 \,\mathrm {log}\left (2\right ) x -16 \,\mathrm {log}\left (2\right )+6 x^{2}-8 x +15}} \] Input:
int(((40*x*log(2)^2+2*(10*x^2-80*x)*log(2)-40*x^2+150*x)*log(x)+20*x*log(2 )^2+2*(10*x^2-40*x)*log(2)+30*x^3-40*x^2+75*x)*exp(5*x^2*log(x)/(4*log(2)^ 2+2*(2*x-8)*log(2)+6*x^2-8*x+15))/(16*log(2)^4+8*(4*x-16)*log(2)^3+4*(16*x ^2-48*x+94)*log(2)^2+2*(24*x^3-128*x^2+188*x-240)*log(2)+36*x^4-96*x^3+244 *x^2-240*x+225),x)
Output:
e**((5*log(x)*x**2)/(4*log(2)**2 + 4*log(2)*x - 16*log(2) + 6*x**2 - 8*x + 15))