Integrand size = 24, antiderivative size = 577 \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{5/2}} \, dx=-\frac {\left (124415-6308 \left (3+\frac {4}{x}\right )^2\right ) x^2}{97344 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}-\frac {\left (64489-1399 \left (3+\frac {4}{x}\right )^2\right ) x^2}{624 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (18932921731-1086525994 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{78056941248 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (11921698-359497 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{483912 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {543262997 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \left (3+\frac {4}{x}\right ) x^2}{39028470624 \left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right ) \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}-\frac {543262997 \left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right ) \sqrt {\frac {517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4}{\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right )^2}} x^2 E\left (2 \arctan \left (\frac {4+3 x}{\sqrt [4]{517} x}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{75490272\ 517^{3/4} \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (4346103976-175318963 \sqrt {517}\right ) \left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right ) \sqrt {\frac {517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4}{\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right )^2}} x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {4+3 x}{\sqrt [4]{517} x}\right ),\frac {517+19 \sqrt {517}}{1034}\right )}{1207844352\ 517^{3/4} \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \]
-1/97344*(124415-6308*(3+4/x)^2)*x^2/(8*x^4-15*x^3+8*x^2+24*x+8)^(1/2)-1/6 24*(64489-1399*(3+4/x)^2)*x^2/(517-38*(3+4/x)^2+(3+4/x)^4)/(8*x^4-15*x^3+8 *x^2+24*x+8)^(1/2)+1/78056941248*(18932921731-1086525994*(3+4/x)^2)*(3+4/x )*x^2/(8*x^4-15*x^3+8*x^2+24*x+8)^(1/2)+1/483912*(11921698-359497*(3+4/x)^ 2)*(3+4/x)*x^2/(517-38*(3+4/x)^2+(3+4/x)^4)/(8*x^4-15*x^3+8*x^2+24*x+8)^(1 /2)+543262997/39028470624*(517-38*(3+4/x)^2+(3+4/x)^4)*(3+4/x)*x^2/((3+4/x )^2+517^(1/2))/(8*x^4-15*x^3+8*x^2+24*x+8)^(1/2)-543262997/39028470624*x^2 *(cos(2*arctan(1/517*(4+3*x)*517^(3/4)/x))^2)^(1/2)/cos(2*arctan(1/517*(4+ 3*x)*517^(3/4)/x))*EllipticE(sin(2*arctan(1/517*(4+3*x)*517^(3/4)/x)),1/10 34*(534578+19646*517^(1/2))^(1/2))*((3+4/x)^2+517^(1/2))*((517-38*(3+4/x)^ 2+(3+4/x)^4)/((3+4/x)^2+517^(1/2))^2)^(1/2)*517^(1/4)/(8*x^4-15*x^3+8*x^2+ 24*x+8)^(1/2)+1/624455529984*x^2*(cos(2*arctan(1/517*(4+3*x)*517^(3/4)/x)) ^2)^(1/2)/cos(2*arctan(1/517*(4+3*x)*517^(3/4)/x))*EllipticF(sin(2*arctan( 1/517*(4+3*x)*517^(3/4)/x)),1/1034*(534578+19646*517^(1/2))^(1/2))*(434610 3976-175318963*517^(1/2))*((3+4/x)^2+517^(1/2))*((517-38*(3+4/x)^2+(3+4/x) ^4)/((3+4/x)^2+517^(1/2))^2)^(1/2)*517^(1/4)/(8*x^4-15*x^3+8*x^2+24*x+8)^( 1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.
Time = 16.10 (sec) , antiderivative size = 6084, normalized size of antiderivative = 10.54 \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{5/2}} \, dx=\text {Result too large to show} \]
Time = 1.02 (sec) , antiderivative size = 636, normalized size of antiderivative = 1.10, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {2504, 27, 7270, 2202, 2194, 27, 2191, 27, 1158, 2206, 27, 2206, 27, 1511, 27, 1416, 1509}
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.
| \(\displaystyle \int \frac {1}{\left (8 x^4-15 x^3+8 x^2+24 x+8\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2504 |
| \(\displaystyle -1024 \int \frac {1}{8192 \sqrt {2} \left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^2 \left (\frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}{\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^4}\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {1}{\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^2 \left (\frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}{\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^4}\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{8 \sqrt {2}}\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \int \frac {\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^8}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\int \frac {65536 \left (\frac {3}{4}+\frac {1}{x}\right )^8+1032192 \left (\frac {3}{4}+\frac {1}{x}\right )^6+1451520 \left (\frac {3}{4}+\frac {1}{x}\right )^4+326592 \left (\frac {3}{4}+\frac {1}{x}\right )^2+6561}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\int \frac {\left (-393216 \left (\frac {3}{4}+\frac {1}{x}\right )^6-1548288 \left (\frac {3}{4}+\frac {1}{x}\right )^4-870912 \left (\frac {3}{4}+\frac {1}{x}\right )^2-69984\right ) \left (\frac {3}{4}+\frac {1}{x}\right )}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {1}{2} \int -\frac {96 \left (4096 \left (\frac {3}{4}+\frac {1}{x}\right )^6+16128 \left (\frac {3}{4}+\frac {1}{x}\right )^4+9072 \left (\frac {3}{4}+\frac {1}{x}\right )^2+729\right )}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )^2+\int \frac {65536 \left (\frac {3}{4}+\frac {1}{x}\right )^8+1032192 \left (\frac {3}{4}+\frac {1}{x}\right )^6+1451520 \left (\frac {3}{4}+\frac {1}{x}\right )^4+326592 \left (\frac {3}{4}+\frac {1}{x}\right )^2+6561}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\int \frac {65536 \left (\frac {3}{4}+\frac {1}{x}\right )^8+1032192 \left (\frac {3}{4}+\frac {1}{x}\right )^6+1451520 \left (\frac {3}{4}+\frac {1}{x}\right )^4+326592 \left (\frac {3}{4}+\frac {1}{x}\right )^2+6561}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )-48 \int \frac {4096 \left (\frac {3}{4}+\frac {1}{x}\right )^6+16128 \left (\frac {3}{4}+\frac {1}{x}\right )^4+9072 \left (\frac {3}{4}+\frac {1}{x}\right )^2+729}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )^2\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\int \frac {65536 \left (\frac {3}{4}+\frac {1}{x}\right )^8+1032192 \left (\frac {3}{4}+\frac {1}{x}\right )^6+1451520 \left (\frac {3}{4}+\frac {1}{x}\right )^4+326592 \left (\frac {3}{4}+\frac {1}{x}\right )^2+6561}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )-48 \left (\frac {\int \frac {2048 \left (1872 \left (\frac {3}{4}+\frac {1}{x}\right )^2+23009\right )}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )^2}{239616}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\int \frac {65536 \left (\frac {3}{4}+\frac {1}{x}\right )^8+1032192 \left (\frac {3}{4}+\frac {1}{x}\right )^6+1451520 \left (\frac {3}{4}+\frac {1}{x}\right )^4+326592 \left (\frac {3}{4}+\frac {1}{x}\right )^2+6561}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )-48 \left (\frac {1}{117} \int \frac {1872 \left (\frac {3}{4}+\frac {1}{x}\right )^2+23009}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )^2-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 1158 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\int \frac {65536 \left (\frac {3}{4}+\frac {1}{x}\right )^8+1032192 \left (\frac {3}{4}+\frac {1}{x}\right )^6+1451520 \left (\frac {3}{4}+\frac {1}{x}\right )^4+326592 \left (\frac {3}{4}+\frac {1}{x}\right )^2+6561}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{5/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )-48 \left (-\frac {124415-100928 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{73008 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\int \frac {4096 \left (15485184 \left (\frac {3}{4}+\frac {1}{x}\right )^4+832856352 \left (\frac {3}{4}+\frac {1}{x}\right )^2+382261973\right )}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{247762944}-48 \left (-\frac {124415-100928 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{73008 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )-\frac {64 \left (5960849-2875976 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{60489 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\int \frac {15485184 \left (\frac {3}{4}+\frac {1}{x}\right )^4+832856352 \left (\frac {3}{4}+\frac {1}{x}\right )^2+382261973}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{60489}-48 \left (-\frac {124415-100928 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{73008 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )-\frac {64 \left (5960849-2875976 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{60489 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\frac {\int \frac {4096 \left (90639903871-69537663616 \left (\frac {3}{4}+\frac {1}{x}\right )^2\right )}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{82587648}-\frac {4 \left (18932921731-17384415904 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}}{60489}-48 \left (-\frac {124415-100928 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{73008 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )-\frac {64 \left (5960849-2875976 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{60489 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\frac {\int \frac {90639903871-69537663616 \left (\frac {3}{4}+\frac {1}{x}\right )^2}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{20163}-\frac {4 \left (18932921731-17384415904 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}}{60489}-48 \left (-\frac {124415-100928 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{73008 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )-\frac {64 \left (5960849-2875976 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{60489 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\frac {\left (90639903871-4346103976 \sqrt {517}\right ) \int \frac {1}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+4346103976 \sqrt {517} \int \frac {\sqrt {517}-16 \left (\frac {3}{4}+\frac {1}{x}\right )^2}{\sqrt {517} \sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{20163}-\frac {4 \left (18932921731-17384415904 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}}{60489}-48 \left (-\frac {124415-100928 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{73008 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )-\frac {64 \left (5960849-2875976 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{60489 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\frac {\left (90639903871-4346103976 \sqrt {517}\right ) \int \frac {1}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+4346103976 \int \frac {\sqrt {517}-16 \left (\frac {3}{4}+\frac {1}{x}\right )^2}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{20163}-\frac {4 \left (18932921731-17384415904 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}}{60489}-48 \left (-\frac {124415-100928 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{73008 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )-\frac {64 \left (5960849-2875976 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{60489 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\frac {4346103976 \int \frac {\sqrt {517}-16 \left (\frac {3}{4}+\frac {1}{x}\right )^2}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {\left (90639903871-4346103976 \sqrt {517}\right ) \left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right ) \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 \left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt [4]{517}}\right ),\frac {517+19 \sqrt {517}}{1034}\right )}{8 \sqrt [4]{517} \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}}{20163}-\frac {4 \left (18932921731-17384415904 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}}{60489}-48 \left (-\frac {124415-100928 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{73008 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )-\frac {64 \left (5960849-2875976 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{60489 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\frac {\frac {\left (90639903871-4346103976 \sqrt {517}\right ) \left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right ) \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 \left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt [4]{517}}\right ),\frac {517+19 \sqrt {517}}{1034}\right )}{8 \sqrt [4]{517} \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}+4346103976 \left (\frac {\sqrt [4]{517} \left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right ) \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right )^2}} E\left (2 \arctan \left (\frac {4 \left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt [4]{517}}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{4 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {517 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {1}{x}+\frac {3}{4}\right )}{8272 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517 \sqrt {517}}\right )}{20163}-\frac {4 \left (18932921731-17384415904 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}}{60489}-48 \left (-\frac {124415-100928 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{73008 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {64489-22384 \left (\frac {1}{x}+\frac {3}{4}\right )^2}{468 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )-\frac {64 \left (5960849-2875976 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{60489 \left (256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517\right )^{3/2}}\right )}{8 \sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\) |
-1/8*(Sqrt[517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4]*(-48*(-1/468 *(64489 - 22384*(3/4 + x^(-1))^2)/(517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4)^(3/2) - (124415 - 100928*(3/4 + x^(-1))^2)/(73008*Sqrt[517 - 6 08*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4])) - (64*(5960849 - 2875976*(3/ 4 + x^(-1))^2)*(3/4 + x^(-1)))/(60489*(517 - 608*(3/4 + x^(-1))^2 + 256*(3 /4 + x^(-1))^4)^(3/2)) + ((-4*(18932921731 - 17384415904*(3/4 + x^(-1))^2) *(3/4 + x^(-1)))/(20163*Sqrt[517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1 ))^4]) + (4346103976*((-517*Sqrt[517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x ^(-1))^4]*(3/4 + x^(-1)))/(517*Sqrt[517] + 8272*(3/4 + x^(-1))^2) + (517^( 1/4)*(Sqrt[517] + 16*(3/4 + x^(-1))^2)*Sqrt[(517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4)/(Sqrt[517] + 16*(3/4 + x^(-1))^2)^2]*EllipticE[2*Arc Tan[(4*(3/4 + x^(-1)))/517^(1/4)], (517 + 19*Sqrt[517])/1034])/(4*Sqrt[517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4])) + ((90639903871 - 434610 3976*Sqrt[517])*(Sqrt[517] + 16*(3/4 + x^(-1))^2)*Sqrt[(517 - 608*(3/4 + x ^(-1))^2 + 256*(3/4 + x^(-1))^4)/(Sqrt[517] + 16*(3/4 + x^(-1))^2)^2]*Elli pticF[2*ArcTan[(4*(3/4 + x^(-1)))/517^(1/4)], (517 + 19*Sqrt[517])/1034])/ (8*517^(1/4)*Sqrt[517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4]))/201 63)/60489))/(Sqrt[2]*(3 - 4*(3/4 + x^(-1)))^2*Sqrt[(517 - 608*(3/4 + x^(-1 ))^2 + 256*(3/4 + x^(-1))^4)/(3 - 4*(3/4 + x^(-1)))^4])
3.9.2.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[c/a]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Int[(P4_)^(p_), x_Symbol] :> With[{a = Coeff[P4, x, 0], b = Coeff[P4, x, 1] , c = Coeff[P4, x, 2], d = Coeff[P4, x, 3], e = Coeff[P4, x, 4]}, Simp[-16* a^2 Subst[Int[(1/(b - 4*a*x)^2)*(a*((-3*b^4 + 16*a*b^2*c - 64*a^2*b*d + 2 56*a^3*e - 32*a^2*(3*b^2 - 8*a*c)*x^2 + 256*a^4*x^4)/(b - 4*a*x)^4))^p, x], x, b/(4*a) + 1/x], x] /; NeQ[a, 0] && NeQ[b, 0] && EqQ[b^3 - 4*a*b*c + 8*a ^2*d, 0]] /; FreeQ[p, x] && PolyQ[P4, x, 4] && IntegerQ[2*p] && !IGtQ[p, 0 ]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.84 (sec) , antiderivative size = 5441, normalized size of antiderivative = 9.43
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(5441\) |
| default | \(\text {Expression too large to display}\) | \(5477\) |
| elliptic | \(\text {Expression too large to display}\) | \(5477\) |
\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {5}{2}}} \,d x } \]
integral(sqrt(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)/(512*x^12 - 2880*x^11 + 6 936*x^10 - 4527*x^9 - 8808*x^8 + 16776*x^7 + 5528*x^6 - 17856*x^5 - 384*x^ 4 + 20160*x^3 + 15360*x^2 + 4608*x + 512), x)
\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (8 x^{4} - 15 x^{3} + 8 x^{2} + 24 x + 8\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (8\,x^4-15\,x^3+8\,x^2+24\,x+8\right )}^{5/2}} \,d x \]