\(\int \frac {1}{(8+24 x+8 x^2-15 x^3+8 x^4)^{3/2}} \, dx\) [90]

Optimal result

Integrand size = 24, antiderivative size = 432 \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx=-\frac {\left (172-7 \left (3+\frac {4}{x}\right )^2\right ) x^2}{208 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (50896-2455 \left (3+\frac {4}{x}\right )^2\right ) \left (3+\frac {4}{x}\right ) x^2}{322608 \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {2455 \left (517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4\right ) \left (3+\frac {4}{x}\right ) x^2}{322608 \left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right ) \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}-\frac {2455 \left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right ) \sqrt {\frac {517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4}{\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right )^2}} x^2 E\left (2 \arctan \left (\frac {3+\frac {4}{x}}{\sqrt [4]{517}}\right )|\frac {517+19 \sqrt {517}}{1034}\right )}{624\ 517^{3/4} \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}}+\frac {\left (4910-203 \sqrt {517}\right ) \left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right ) \sqrt {\frac {517-38 \left (3+\frac {4}{x}\right )^2+\left (3+\frac {4}{x}\right )^4}{\left (\sqrt {517}+\left (3+\frac {4}{x}\right )^2\right )^2}} x^2 \operatorname {EllipticF}\left (2 \arctan \left (\frac {3+\frac {4}{x}}{\sqrt [4]{517}}\right ),\frac {517+19 \sqrt {517}}{1034}\right )}{2496\ 517^{3/4} \sqrt {8+24 x+8 x^2-15 x^3+8 x^4}} \] Output:

-1/208*(172-7*(3+4/x)^2)*x^2/(8*x^4-15*x^3+8*x^2+24*x+8)^(1/2)+1/322608*(5 
0896-2455*(3+4/x)^2)*(3+4/x)*x^2/(8*x^4-15*x^3+8*x^2+24*x+8)^(1/2)+2455/32 
2608*(517-38*(3+4/x)^2+(3+4/x)^4)*(3+4/x)*x^2/(517^(1/2)+(3+4/x)^2)/(8*x^4 
-15*x^3+8*x^2+24*x+8)^(1/2)-2455/322608*(517^(1/2)+(3+4/x)^2)*((517-38*(3+ 
4/x)^2+(3+4/x)^4)/(517^(1/2)+(3+4/x)^2)^2)^(1/2)*x^2*EllipticE(sin(2*arcta 
n(1/517*(3+4/x)*517^(3/4))),1/1034*(534578+19646*517^(1/2))^(1/2))*517^(1/ 
4)/(8*x^4-15*x^3+8*x^2+24*x+8)^(1/2)+1/1290432*(4910-203*517^(1/2))*(517^( 
1/2)+(3+4/x)^2)*((517-38*(3+4/x)^2+(3+4/x)^4)/(517^(1/2)+(3+4/x)^2)^2)^(1/ 
2)*x^2*InverseJacobiAM(2*arctan(1/517*(3+4/x)*517^(3/4)),1/1034*(534578+19 
646*517^(1/2))^(1/2))*517^(1/4)/(8*x^4-15*x^3+8*x^2+24*x+8)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 16.07 (sec) , antiderivative size = 6019, normalized size of antiderivative = 13.93 \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-3/2),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.22, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {2504, 27, 7270, 2202, 1576, 27, 1158, 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.

\(\displaystyle \int \frac {1}{\left (8 x^4-15 x^3+8 x^2+24 x+8\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2504

\(\displaystyle -1024 \int \frac {1}{1024 \sqrt {2} \left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^2 \left (\frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}{\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^4}\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {1}{\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^2 \left (\frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}{\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^4}\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {2}}\)

\(\Big \downarrow \) 7270

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \int \frac {\left (3-4 \left (\frac {3}{4}+\frac {1}{x}\right )\right )^4}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 2202

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\int \frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4+864 \left (\frac {3}{4}+\frac {1}{x}\right )^2+81}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\int \frac {\left (-768 \left (\frac {3}{4}+\frac {1}{x}\right )^2-432\right ) \left (\frac {3}{4}+\frac {1}{x}\right )}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 1576

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {1}{2} \int -\frac {48 \left (16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+9\right )}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )^2+\int \frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4+864 \left (\frac {3}{4}+\frac {1}{x}\right )^2+81}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\int \frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4+864 \left (\frac {3}{4}+\frac {1}{x}\right )^2+81}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )-24 \int \frac {16 \left (\frac {3}{4}+\frac {1}{x}\right )^2+9}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )^2\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 1158

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\int \frac {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4+864 \left (\frac {3}{4}+\frac {1}{x}\right )^2+81}{\left (256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517\right )^{3/2}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {2 \left (43-28 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right )}{13 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\int \frac {4096 \left (104951-78560 \left (\frac {3}{4}+\frac {1}{x}\right )^2\right )}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{82587648}+\frac {2 \left (43-28 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right )}{13 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {32 \left (3181-2455 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\int \frac {104951-78560 \left (\frac {3}{4}+\frac {1}{x}\right )^2}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{20163}+\frac {2 \left (43-28 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right )}{13 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {32 \left (3181-2455 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 1511

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\left (104951-4910 \sqrt {517}\right ) \int \frac {1}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+4910 \sqrt {517} \int \frac {\sqrt {517}-16 \left (\frac {3}{4}+\frac {1}{x}\right )^2}{\sqrt {517} \sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{20163}+\frac {2 \left (43-28 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right )}{13 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {32 \left (3181-2455 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {\left (104951-4910 \sqrt {517}\right ) \int \frac {1}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+4910 \int \frac {\sqrt {517}-16 \left (\frac {3}{4}+\frac {1}{x}\right )^2}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )}{20163}+\frac {2 \left (43-28 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right )}{13 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {32 \left (3181-2455 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517} \left (\frac {4910 \int \frac {\sqrt {517}-16 \left (\frac {3}{4}+\frac {1}{x}\right )^2}{\sqrt {256 \left (\frac {3}{4}+\frac {1}{x}\right )^4-608 \left (\frac {3}{4}+\frac {1}{x}\right )^2+517}}d\left (\frac {3}{4}+\frac {1}{x}\right )+\frac {\left (104951-4910 \sqrt {517}\right ) \left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right ) \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (16 \left (\frac {1}{x}+\frac {3}{4}\right )^2+\sqrt {517}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 \left (\frac {3}{4}+\frac {1}{x}\right )}{\sqrt [4]{517}}\right ),\frac {517+19 \sqrt {517}}{1034}\right )}{8 \sqrt [4]{517} \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}}{20163}+\frac {2 \left (43-28 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right )}{13 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {32 \left (3181-2455 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {3}{4}\right )}{20163 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}\right )}{\sqrt {2} \left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}{\left (3-4 \left (\frac {1}{x}+\frac {3}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 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 {\left (104951-4910 \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}}+4910 \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 {2 \left (43-28 \left (\frac {1}{x}+\frac {3}{4}\right )^2\right )}{13 \sqrt {256 \left (\frac {1}{x}+\frac {3}{4}\right )^4-608 \left (\frac {1}{x}+\frac {3}{4}\right )^2+517}}-\frac {32 \left (3181-2455 \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}}\right )}{\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}}}\)

Input:

Int[(8 + 24*x + 8*x^2 - 15*x^3 + 8*x^4)^(-3/2),x]
 

Output:

-((Sqrt[517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4]*((2*(43 - 28*(3 
/4 + x^(-1))^2))/(13*Sqrt[517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^ 
4]) - (32*(3181 - 2455*(3/4 + x^(-1))^2)*(3/4 + x^(-1)))/(20163*Sqrt[517 - 
 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4]) + (4910*((-517*Sqrt[517 - 6 
08*(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)*S 
qrt[(517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4)/(Sqrt[517] + 16*(3 
/4 + x^(-1))^2)^2]*EllipticE[2*ArcTan[(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])) + ((104951 - 4910*Sqrt[517])*(Sqrt[517] + 16*(3/4 + x^(-1))^2)*S 
qrt[(517 - 608*(3/4 + x^(-1))^2 + 256*(3/4 + x^(-1))^4)/(Sqrt[517] + 16*(3 
/4 + x^(-1))^2)^2]*EllipticF[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]))/20163))/(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]))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1158

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]
 

rule 1416

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]
 

rule 1509

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]
 

rule 1511

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]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 2202

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]
 

rule 2206

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]
 

rule 2504

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 
]
 

rule 7270

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(5420\) vs. \(2(387)=774\).

Time = 3.77 (sec) , antiderivative size = 5421, normalized size of antiderivative = 12.55

method	result	size
default	\(\text {Expression too large to display}\)	\(5421\)
risch	\(\text {Expression too large to display}\)	\(5421\)
elliptic	\(\text {Expression too large to display}\)	\(5421\)

Input:

int(1/(8*x^4-15*x^3+8*x^2+24*x+8)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [F]

\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)/(64*x^8 - 240*x^7 + 353*x 
^6 + 144*x^5 - 528*x^4 + 144*x^3 + 704*x^2 + 384*x + 64), x)
 

Sympy [F]

\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (8 x^{4} - 15 x^{3} + 8 x^{2} + 24 x + 8\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(8*x**4-15*x**3+8*x**2+24*x+8)**(3/2),x)
 

Output:

Integral((8*x**4 - 15*x**3 + 8*x**2 + 24*x + 8)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8)^(3/2),x, algorithm="maxima")
 

Output:

integrate((8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(8*x^4-15*x^3+8*x^2+24*x+8)^(3/2),x, algorithm="giac")
 

Output:

integrate((8*x^4 - 15*x^3 + 8*x^2 + 24*x + 8)^(-3/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (8\,x^4-15\,x^3+8\,x^2+24\,x+8\right )}^{3/2}} \,d x \] Input:

int(1/(24*x + 8*x^2 - 15*x^3 + 8*x^4 + 8)^(3/2),x)
 

Output:

int(1/(24*x + 8*x^2 - 15*x^3 + 8*x^4 + 8)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (8+24 x+8 x^2-15 x^3+8 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {8 x^{4}-15 x^{3}+8 x^{2}+24 x +8}}{64 x^{8}-240 x^{7}+353 x^{6}+144 x^{5}-528 x^{4}+144 x^{3}+704 x^{2}+384 x +64}d x \] Input:

int(1/(8*x^4-15*x^3+8*x^2+24*x+8)^(3/2),x)
 

Output:

int(sqrt(8*x**4 - 15*x**3 + 8*x**2 + 24*x + 8)/(64*x**8 - 240*x**7 + 353*x 
**6 + 144*x**5 - 528*x**4 + 144*x**3 + 704*x**2 + 384*x + 64),x)