Integrand size = 110, antiderivative size = 31 \[ \int \frac {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{323 x^3-160 x^4+16 x^5} \, dx=\frac {\left (\left (-3+16 \left (5-(5-x)^2\right )\right ) x\right )^{2-5^{\frac {1}{x}}}}{x} \] Output:
exp(ln(x*(77-16*(5-x)^2))*(-exp(ln(5)/x)+2))/x
Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{323 x^3-160 x^4+16 x^5} \, dx=x \left (x \left (-323+160 x-16 x^2\right )\right )^{-5^{\frac {1}{x}}} \left (323-160 x+16 x^2\right )^2 \] Input:
Integrate[((-323*x + 160*x^2 - 16*x^3)^(2 - 5^x^(-1))*(323*x - 480*x^2 + 8 0*x^3 + 5^x^(-1)*(-323*x + 320*x^2 - 48*x^3) + 5^x^(-1)*(323 - 160*x + 16* x^2)*Log[5]*Log[-323*x + 160*x^2 - 16*x^3]))/(323*x^3 - 160*x^4 + 16*x^5), x]
Output:
(x*(323 - 160*x + 16*x^2)^2)/(x*(-323 + 160*x - 16*x^2))^5^x^(-1)
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.
| \(\displaystyle \int \frac {\left (-16 x^3+160 x^2-323 x\right )^{2-5^{\frac {1}{x}}} \left (80 x^3-480 x^2+5^{\frac {1}{x}} \left (-48 x^3+320 x^2-323 x\right )+5^{\frac {1}{x}} \left (16 x^2-160 x+323\right ) \log (5) \log \left (-16 x^3+160 x^2-323 x\right )+323 x\right )}{16 x^5-160 x^4+323 x^3} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (-16 x^3+160 x^2-323 x\right )^{2-5^{\frac {1}{x}}} \left (80 x^3-480 x^2+5^{\frac {1}{x}} \left (-48 x^3+320 x^2-323 x\right )+5^{\frac {1}{x}} \left (16 x^2-160 x+323\right ) \log (5) \log \left (-16 x^3+160 x^2-323 x\right )+323 x\right )}{x^3 \left (16 x^2-160 x+323\right )}dx\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle \int \left (\frac {\left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}} \left (80 x^2-480 x+323\right )}{x^2 \left (16 x^2-160 x+323\right )}-\frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}} \left (48 x^3-320 x^2-16 x^2 \log (5) \log \left (x \left (-16 x^2+160 x-323\right )\right )+160 x \log (5) \log \left (x \left (-16 x^2+160 x-323\right )\right )-323 \log (5) \log \left (x \left (-16 x^2+160 x-323\right )\right )+323 x\right )}{x^3 \left (16 x^2-160 x+323\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {\left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x^2}dx-\int \frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x^2}dx-\frac {320}{323} \int \frac {\left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x}dx+\frac {32}{323} \int \frac {5^{1+\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x}dx+\frac {256}{323} \left (20-\sqrt {77}\right ) \int \frac {\left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{32 x-8 \sqrt {77}-160}dx-\frac {128}{323} \left (20-\sqrt {77}\right ) \int \frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{32 x-8 \sqrt {77}-160}dx+\frac {256}{323} \left (20+\sqrt {77}\right ) \int \frac {\left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{32 x+8 \sqrt {77}-160}dx-\frac {128}{323} \left (20+\sqrt {77}\right ) \int \frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{32 x+8 \sqrt {77}-160}dx+\log (5) \log \left (-x \left (16 x^2-160 x+323\right )\right ) \int \frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x^3}dx-\frac {640 \log (5) \int \frac {\int \frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x^3}dx}{-32 x+8 \sqrt {77}+160}dx}{\sqrt {77}}-\log (5) \int \frac {\int \frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x^3}dx}{x}dx-\frac {32}{77} \left (77+20 \sqrt {77}\right ) \log (5) \int \frac {\int \frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x^3}dx}{32 x-8 \sqrt {77}-160}dx-\frac {32}{77} \left (77-20 \sqrt {77}\right ) \log (5) \int \frac {\int \frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x^3}dx}{32 x+8 \sqrt {77}-160}dx-\frac {640 \log (5) \int \frac {\int \frac {5^{\frac {1}{x}} \left (x \left (-16 x^2+160 x-323\right )\right )^{2-5^{\frac {1}{x}}}}{x^3}dx}{32 x+8 \sqrt {77}-160}dx}{\sqrt {77}}\) |
Input:
Int[((-323*x + 160*x^2 - 16*x^3)^(2 - 5^x^(-1))*(323*x - 480*x^2 + 80*x^3 + 5^x^(-1)*(-323*x + 320*x^2 - 48*x^3) + 5^x^(-1)*(323 - 160*x + 16*x^2)*L og[5]*Log[-323*x + 160*x^2 - 16*x^3]))/(323*x^3 - 160*x^4 + 16*x^5),x]
Output:
$Aborted
Time = 23.15 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06
| method | result | size |
| parallelrisch | \(\frac {{\mathrm e}^{\left (-{\mathrm e}^{\frac {\ln \left (5\right )}{x}}+2\right ) \ln \left (-16 x^{3}+160 x^{2}-323 x \right )}}{x}\) | \(33\) |
| risch | \(\frac {{\mathrm e}^{-\frac {\left (5^{\frac {1}{x}}-2\right ) \left (i \pi {\operatorname {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right )}^{3}+i \pi {\operatorname {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right )}^{2} \operatorname {csgn}\left (i x \right )+i \pi {\operatorname {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right )}^{2} \operatorname {csgn}\left (i \left (x^{2}-10 x +\frac {323}{16}\right )\right )-i \pi \,\operatorname {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (x^{2}-10 x +\frac {323}{16}\right )\right )-2 i \pi {\operatorname {csgn}\left (i x \left (x^{2}-10 x +\frac {323}{16}\right )\right )}^{2}+2 i \pi +2 \ln \left (x \right )+2 \ln \left (x^{2}-10 x +\frac {323}{16}\right )\right )}{2}}}{x}\) | \(162\) |
Input:
int(((16*x^2-160*x+323)*ln(5)*exp(ln(5)/x)*ln(-16*x^3+160*x^2-323*x)+(-48* x^3+320*x^2-323*x)*exp(ln(5)/x)+80*x^3-480*x^2+323*x)*exp((-exp(ln(5)/x)+2 )*ln(-16*x^3+160*x^2-323*x))/(16*x^5-160*x^4+323*x^3),x,method=_RETURNVERB OSE)
Output:
exp((-exp(ln(5)/x)+2)*ln(-16*x^3+160*x^2-323*x))/x
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{323 x^3-160 x^4+16 x^5} \, dx=\frac {{\left (-16 \, x^{3} + 160 \, x^{2} - 323 \, x\right )}^{-5^{\left (\frac {1}{x}\right )} + 2}}{x} \] Input:
integrate(((16*x^2-160*x+323)*log(5)*exp(log(5)/x)*log(-16*x^3+160*x^2-323 *x)+(-48*x^3+320*x^2-323*x)*exp(log(5)/x)+80*x^3-480*x^2+323*x)*exp((-exp( log(5)/x)+2)*log(-16*x^3+160*x^2-323*x))/(16*x^5-160*x^4+323*x^3),x, algor ithm="fricas")
Output:
(-16*x^3 + 160*x^2 - 323*x)^(-5^(1/x) + 2)/x
Time = 0.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{323 x^3-160 x^4+16 x^5} \, dx=\frac {e^{\left (2 - e^{\frac {\log {\left (5 \right )}}{x}}\right ) \log {\left (- 16 x^{3} + 160 x^{2} - 323 x \right )}}}{x} \] Input:
integrate(((16*x**2-160*x+323)*ln(5)*exp(ln(5)/x)*ln(-16*x**3+160*x**2-323 *x)+(-48*x**3+320*x**2-323*x)*exp(ln(5)/x)+80*x**3-480*x**2+323*x)*exp((-e xp(ln(5)/x)+2)*ln(-16*x**3+160*x**2-323*x))/(16*x**5-160*x**4+323*x**3),x)
Output:
exp((2 - exp(log(5)/x))*log(-16*x**3 + 160*x**2 - 323*x))/x
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (26) = 52\).
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{323 x^3-160 x^4+16 x^5} \, dx={\left (256 \, x^{5} - 5120 \, x^{4} + 35936 \, x^{3} - 103360 \, x^{2} + 104329 \, x\right )} e^{\left (-5^{\left (\frac {1}{x}\right )} \log \left (-16 \, x^{2} + 160 \, x - 323\right ) - 5^{\left (\frac {1}{x}\right )} \log \left (x\right )\right )} \] Input:
integrate(((16*x^2-160*x+323)*log(5)*exp(log(5)/x)*log(-16*x^3+160*x^2-323 *x)+(-48*x^3+320*x^2-323*x)*exp(log(5)/x)+80*x^3-480*x^2+323*x)*exp((-exp( log(5)/x)+2)*log(-16*x^3+160*x^2-323*x))/(16*x^5-160*x^4+323*x^3),x, algor ithm="maxima")
Output:
(256*x^5 - 5120*x^4 + 35936*x^3 - 103360*x^2 + 104329*x)*e^(-5^(1/x)*log(- 16*x^2 + 160*x - 323) - 5^(1/x)*log(x))
\[ \int \frac {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{323 x^3-160 x^4+16 x^5} \, dx=\int { \frac {{\left ({\left (16 \, x^{2} - 160 \, x + 323\right )} 5^{\left (\frac {1}{x}\right )} \log \left (5\right ) \log \left (-16 \, x^{3} + 160 \, x^{2} - 323 \, x\right ) + 80 \, x^{3} - {\left (48 \, x^{3} - 320 \, x^{2} + 323 \, x\right )} 5^{\left (\frac {1}{x}\right )} - 480 \, x^{2} + 323 \, x\right )} {\left (-16 \, x^{3} + 160 \, x^{2} - 323 \, x\right )}^{-5^{\left (\frac {1}{x}\right )} + 2}}{16 \, x^{5} - 160 \, x^{4} + 323 \, x^{3}} \,d x } \] Input:
integrate(((16*x^2-160*x+323)*log(5)*exp(log(5)/x)*log(-16*x^3+160*x^2-323 *x)+(-48*x^3+320*x^2-323*x)*exp(log(5)/x)+80*x^3-480*x^2+323*x)*exp((-exp( log(5)/x)+2)*log(-16*x^3+160*x^2-323*x))/(16*x^5-160*x^4+323*x^3),x, algor ithm="giac")
Output:
integrate(((16*x^2 - 160*x + 323)*5^(1/x)*log(5)*log(-16*x^3 + 160*x^2 - 3 23*x) + 80*x^3 - (48*x^3 - 320*x^2 + 323*x)*5^(1/x) - 480*x^2 + 323*x)*(-1 6*x^3 + 160*x^2 - 323*x)^(-5^(1/x) + 2)/(16*x^5 - 160*x^4 + 323*x^3), x)
Time = 3.05 (sec) , antiderivative size = 134, normalized size of antiderivative = 4.32 \[ \int \frac {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{323 x^3-160 x^4+16 x^5} \, dx=\frac {104329\,x}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}}-\frac {103360\,x^2}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}}+\frac {35936\,x^3}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}}-\frac {5120\,x^4}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}}+\frac {256\,x^5}{{\left (-16\,x^3+160\,x^2-323\,x\right )}^{5^{1/x}}} \] Input:
int((exp(-log(160*x^2 - 323*x - 16*x^3)*(exp(log(5)/x) - 2))*(323*x - exp( log(5)/x)*(323*x - 320*x^2 + 48*x^3) - 480*x^2 + 80*x^3 + log(160*x^2 - 32 3*x - 16*x^3)*exp(log(5)/x)*log(5)*(16*x^2 - 160*x + 323)))/(323*x^3 - 160 *x^4 + 16*x^5),x)
Output:
(104329*x)/(160*x^2 - 323*x - 16*x^3)^(5^(1/x)) - (103360*x^2)/(160*x^2 - 323*x - 16*x^3)^(5^(1/x)) + (35936*x^3)/(160*x^2 - 323*x - 16*x^3)^(5^(1/x )) - (5120*x^4)/(160*x^2 - 323*x - 16*x^3)^(5^(1/x)) + (256*x^5)/(160*x^2 - 323*x - 16*x^3)^(5^(1/x))
Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.61 \[ \int \frac {\left (-323 x+160 x^2-16 x^3\right )^{2-5^{\frac {1}{x}}} \left (323 x-480 x^2+80 x^3+5^{\frac {1}{x}} \left (-323 x+320 x^2-48 x^3\right )+5^{\frac {1}{x}} \left (323-160 x+16 x^2\right ) \log (5) \log \left (-323 x+160 x^2-16 x^3\right )\right )}{323 x^3-160 x^4+16 x^5} \, dx=\frac {x \left (256 x^{4}-5120 x^{3}+35936 x^{2}-103360 x +104329\right )}{e^{e^{\frac {\mathrm {log}\left (5\right )}{x}} \mathrm {log}\left (-16 x^{3}+160 x^{2}-323 x \right )}} \] Input:
int(((16*x^2-160*x+323)*log(5)*exp(log(5)/x)*log(-16*x^3+160*x^2-323*x)+(- 48*x^3+320*x^2-323*x)*exp(log(5)/x)+80*x^3-480*x^2+323*x)*exp((-exp(log(5) /x)+2)*log(-16*x^3+160*x^2-323*x))/(16*x^5-160*x^4+323*x^3),x)
Output:
(x*(256*x**4 - 5120*x**3 + 35936*x**2 - 103360*x + 104329))/e**(e**(log(5) /x)*log( - 16*x**3 + 160*x**2 - 323*x))