3.8.97 \(\int \frac {1}{(8+8 x-x^3+8 x^4)^{3/2}} \, dx\) [797]

3.8.97.1 Optimal result

Integrand size = 19, antiderivative size = 431 \[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^{3/2}} \, dx=-\frac {\left (66-\left (1+\frac {4}{x}\right )^2\right ) x^2}{1008 \sqrt {8+8 x-x^3+8 x^4}}+\frac {\left (216-7 \left (1+\frac {4}{x}\right )^2\right ) \left (1+\frac {4}{x}\right ) x^2}{12528 \sqrt {8+8 x-x^3+8 x^4}}+\frac {7 \left (261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4\right ) \left (1+\frac {4}{x}\right ) x^2}{432 \sqrt {29} \sqrt {8+8 x-x^3+8 x^4} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )}-\frac {7 x^2 \sqrt {\frac {261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4}{\left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )^2}} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right ) E\left (2 \arctan \left (\frac {4+x}{\sqrt {3} \sqrt [4]{29} x}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{144 \sqrt {3} 29^{3/4} \sqrt {8+8 x-x^3+8 x^4}}+\frac {\left (14-5 \sqrt {29}\right ) x^2 \sqrt {\frac {261-6 \left (1+\frac {4}{x}\right )^2+\left (1+\frac {4}{x}\right )^4}{\left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right )^2}} \left (87+\frac {\sqrt {29} (4+x)^2}{x^2}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {4+x}{\sqrt {3} \sqrt [4]{29} x}\right ),\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{576 \sqrt {3} 29^{3/4} \sqrt {8+8 x-x^3+8 x^4}} \]

output

-1/1008*(66-(1+4/x)^2)*x^2/(8*x^4-x^3+8*x+8)^(1/2)+1/12528*(216-7*(1+4/x)^ 
2)*(1+4/x)*x^2/(8*x^4-x^3+8*x+8)^(1/2)+7/12528*(261-6*(1+4/x)^2+(1+4/x)^4) 
*(1+4/x)*x^2*29^(1/2)/(87+(4+x)^2*29^(1/2)/x^2)/(8*x^4-x^3+8*x+8)^(1/2)-7/ 
12528*x^2*(cos(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))^2)^(1/2)/cos(2*arc 
tan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))*EllipticE(sin(2*arctan(1/87*(4+x)*29^( 
3/4)/x*3^(1/2))),1/58*(1682+58*29^(1/2))^(1/2))*(87+(4+x)^2*29^(1/2)/x^2)* 
((261-6*(1+4/x)^2+(1+4/x)^4)/(87+(4+x)^2*29^(1/2)/x^2)^2)^(1/2)*29^(1/4)*3 
^(1/2)/(8*x^4-x^3+8*x+8)^(1/2)+1/50112*x^2*(cos(2*arctan(1/87*(4+x)*29^(3/ 
4)/x*3^(1/2)))^2)^(1/2)/cos(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2)))*Ellip 
ticF(sin(2*arctan(1/87*(4+x)*29^(3/4)/x*3^(1/2))),1/58*(1682+58*29^(1/2))^ 
(1/2))*(14-5*29^(1/2))*(87+(4+x)^2*29^(1/2)/x^2)*((261-6*(1+4/x)^2+(1+4/x) 
^4)/(87+(4+x)^2*29^(1/2)/x^2)^2)^(1/2)*29^(1/4)*3^(1/2)/(8*x^4-x^3+8*x+8)^ 
(1/2)
 

3.8.97.2 Mathematica [C] (warning: unable to verify)

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

Time = 16.15 (sec) , antiderivative size = 4865, normalized size of antiderivative = 11.29 \[ \int \frac {1}{\left (8+8 x-x^3+8 x^4\right )^{3/2}} \, dx=\text {Result too large to show} \]

input

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

output

(544 + 1539*x - 1146*x^2 + 784*x^3)/(21924*Sqrt[8 + 8*x - x^3 + 8*x^4]) + 
((28*(x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])^2*(-(EllipticF[ArcSin[S 
qrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 
 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((x - Root[8 
 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0 
] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))]], -(((Root[8 + 8*#1 - #1^3 
+ 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8* 
#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/( 
(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*#1^4 
& , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^3 
 + 8*#1^4 & , 4, 0])))]*Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0]) + Ellipti 
cPi[(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] + Root[8 + 8*#1 - #1^3 + 8*# 
1^4 & , 4, 0])/(-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] + Root[8 + 8*#1 - 
 #1^3 + 8*#1^4 & , 4, 0]), ArcSin[Sqrt[((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 
 & , 1, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1^ 
3 + 8*#1^4 & , 4, 0]))/((x - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0])*(Roo 
t[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 4 
, 0]))]], -(((Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 2, 0] - Root[8 + 8*#1 - #1 
^3 + 8*#1^4 & , 3, 0])*(Root[8 + 8*#1 - #1^3 + 8*#1^4 & , 1, 0] - Root[8 + 
 8*#1 - #1^3 + 8*#1^4 & , 4, 0]))/((-Root[8 + 8*#1 - #1^3 + 8*#1^4 & , ...
 

3.8.97.3 Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.28, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, 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.

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

\(\Big \downarrow \) 2504

\(\displaystyle -1024 \int \frac {1}{1024 \sqrt {2} \left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^2 \left (\frac {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}{\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^4}\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {1}{\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^2 \left (\frac {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}{\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^4}\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {2}}\)

\(\Big \downarrow \) 7270

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \int \frac {\left (1-4 \left (\frac {1}{4}+\frac {1}{x}\right )\right )^4}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 2202

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\int \frac {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4+96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\int \frac {\left (-256 \left (\frac {1}{4}+\frac {1}{x}\right )^2-16\right ) \left (\frac {1}{4}+\frac {1}{x}\right )}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 1576

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\frac {1}{2} \int -\frac {16 \left (16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1\right )}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )^2+\int \frac {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4+96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\int \frac {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4+96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )-8 \int \frac {16 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )^2\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 1158

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\int \frac {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4+96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+1}{\left (256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261\right )^{3/2}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {33-8 \left (\frac {1}{x}+\frac {1}{4}\right )^2}{63 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\frac {\int \frac {86016 \left (435-224 \left (\frac {1}{4}+\frac {1}{x}\right )^2\right )}{\sqrt {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{67350528}+\frac {33-8 \left (\frac {1}{x}+\frac {1}{4}\right )^2}{63 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}-\frac {16 \left (27-14 \left (\frac {1}{x}+\frac {1}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{783 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\frac {1}{783} \int \frac {435-224 \left (\frac {1}{4}+\frac {1}{x}\right )^2}{\sqrt {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {33-8 \left (\frac {1}{x}+\frac {1}{4}\right )^2}{63 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}-\frac {16 \left (27-14 \left (\frac {1}{x}+\frac {1}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{783 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 1511

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\frac {1}{783} \left (3 \left (145-14 \sqrt {29}\right ) \int \frac {1}{\sqrt {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}}d\left (\frac {1}{4}+\frac {1}{x}\right )+42 \sqrt {29} \int \frac {87-16 \sqrt {29} \left (\frac {1}{4}+\frac {1}{x}\right )^2}{87 \sqrt {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}}d\left (\frac {1}{4}+\frac {1}{x}\right )\right )+\frac {33-8 \left (\frac {1}{x}+\frac {1}{4}\right )^2}{63 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}-\frac {16 \left (27-14 \left (\frac {1}{x}+\frac {1}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{783 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\frac {1}{783} \left (3 \left (145-14 \sqrt {29}\right ) \int \frac {1}{\sqrt {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}}d\left (\frac {1}{4}+\frac {1}{x}\right )+\frac {14 \int \frac {87-16 \sqrt {29} \left (\frac {1}{4}+\frac {1}{x}\right )^2}{\sqrt {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {29}}\right )+\frac {33-8 \left (\frac {1}{x}+\frac {1}{4}\right )^2}{63 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}-\frac {16 \left (27-14 \left (\frac {1}{x}+\frac {1}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{783 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\frac {1}{783} \left (\frac {14 \int \frac {87-16 \sqrt {29} \left (\frac {1}{4}+\frac {1}{x}\right )^2}{\sqrt {256 \left (\frac {1}{4}+\frac {1}{x}\right )^4-96 \left (\frac {1}{4}+\frac {1}{x}\right )^2+261}}d\left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {29}}+\frac {\sqrt {3} \left (145-14 \sqrt {29}\right ) \left (16 \sqrt {29} \left (\frac {1}{x}+\frac {1}{4}\right )^2+87\right ) \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (16 \sqrt {29} \left (\frac {1}{x}+\frac {1}{4}\right )^2+87\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 \left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {3} \sqrt [4]{29}}\right ),\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{8 \sqrt [4]{29} \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}\right )+\frac {33-8 \left (\frac {1}{x}+\frac {1}{4}\right )^2}{63 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}-\frac {16 \left (27-14 \left (\frac {1}{x}+\frac {1}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{783 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

\(\Big \downarrow \) 1509

\(\displaystyle -\frac {\sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\frac {1}{783} \left (\frac {\sqrt {3} \left (145-14 \sqrt {29}\right ) \left (16 \sqrt {29} \left (\frac {1}{x}+\frac {1}{4}\right )^2+87\right ) \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (16 \sqrt {29} \left (\frac {1}{x}+\frac {1}{4}\right )^2+87\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {4 \left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {3} \sqrt [4]{29}}\right ),\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{8 \sqrt [4]{29} \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}+\frac {14 \left (\frac {\sqrt {3} 29^{3/4} \left (16 \sqrt {29} \left (\frac {1}{x}+\frac {1}{4}\right )^2+87\right ) \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (16 \sqrt {29} \left (\frac {1}{x}+\frac {1}{4}\right )^2+87\right )^2}} E\left (2 \arctan \left (\frac {4 \left (\frac {1}{4}+\frac {1}{x}\right )}{\sqrt {3} \sqrt [4]{29}}\right )|\frac {1}{58} \left (29+\sqrt {29}\right )\right )}{4 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}-\frac {29 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261} \left (\frac {1}{x}+\frac {1}{4}\right )}{16 \sqrt {29} \left (\frac {1}{x}+\frac {1}{4}\right )^2+87}\right )}{\sqrt {29}}\right )+\frac {33-8 \left (\frac {1}{x}+\frac {1}{4}\right )^2}{63 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}-\frac {16 \left (27-14 \left (\frac {1}{x}+\frac {1}{4}\right )^2\right ) \left (\frac {1}{x}+\frac {1}{4}\right )}{783 \sqrt {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}}\right )}{\sqrt {2} \left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^2 \sqrt {\frac {256 \left (\frac {1}{x}+\frac {1}{4}\right )^4-96 \left (\frac {1}{x}+\frac {1}{4}\right )^2+261}{\left (1-4 \left (\frac {1}{x}+\frac {1}{4}\right )\right )^4}}}\)

input

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

output

-((Sqrt[261 - 96*(1/4 + x^(-1))^2 + 256*(1/4 + x^(-1))^4]*((33 - 8*(1/4 + 
x^(-1))^2)/(63*Sqrt[261 - 96*(1/4 + x^(-1))^2 + 256*(1/4 + x^(-1))^4]) - ( 
16*(27 - 14*(1/4 + x^(-1))^2)*(1/4 + x^(-1)))/(783*Sqrt[261 - 96*(1/4 + x^ 
(-1))^2 + 256*(1/4 + x^(-1))^4]) + ((14*((-29*Sqrt[261 - 96*(1/4 + x^(-1)) 
^2 + 256*(1/4 + x^(-1))^4]*(1/4 + x^(-1)))/(87 + 16*Sqrt[29]*(1/4 + x^(-1) 
)^2) + (Sqrt[3]*29^(3/4)*(87 + 16*Sqrt[29]*(1/4 + x^(-1))^2)*Sqrt[(261 - 9 
6*(1/4 + x^(-1))^2 + 256*(1/4 + x^(-1))^4)/(87 + 16*Sqrt[29]*(1/4 + x^(-1) 
)^2)^2]*EllipticE[2*ArcTan[(4*(1/4 + x^(-1)))/(Sqrt[3]*29^(1/4))], (29 + S 
qrt[29])/58])/(4*Sqrt[261 - 96*(1/4 + x^(-1))^2 + 256*(1/4 + x^(-1))^4]))) 
/Sqrt[29] + (Sqrt[3]*(145 - 14*Sqrt[29])*(87 + 16*Sqrt[29]*(1/4 + x^(-1))^ 
2)*Sqrt[(261 - 96*(1/4 + x^(-1))^2 + 256*(1/4 + x^(-1))^4)/(87 + 16*Sqrt[2 
9]*(1/4 + x^(-1))^2)^2]*EllipticF[2*ArcTan[(4*(1/4 + x^(-1)))/(Sqrt[3]*29^ 
(1/4))], (29 + Sqrt[29])/58])/(8*29^(1/4)*Sqrt[261 - 96*(1/4 + x^(-1))^2 + 
 256*(1/4 + x^(-1))^4]))/783))/(Sqrt[2]*(1 - 4*(1/4 + x^(-1)))^2*Sqrt[(261 
 - 96*(1/4 + x^(-1))^2 + 256*(1/4 + x^(-1))^4)/(1 - 4*(1/4 + x^(-1)))^4]))
 

3.8.97.3.1 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]
 

3.8.97.4 Maple [C] (verified)

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

Time = 2.70 (sec) , antiderivative size = 4426, normalized size of antiderivative = 10.27

method	result	size
default	\(\text {Expression too large to display}\)	\(4426\)
risch	\(\text {Expression too large to display}\)	\(4426\)
elliptic	\(\text {Expression too large to display}\)	\(4426\)

input

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

output

-16*(-17/10962-57/12992*x+191/58464*x^2-7/3132*x^3)/(8*x^4-x^3+8*x+8)^(1/2 
)+421/12528*(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1)-RootOf(8*_Z^4-_Z^3+8*_Z+8, 
index=4))*((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,i 
ndex=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(RootOf(8*_Z^4-_Z^3+8*_Z+8 
,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8 
,index=2)))^(1/2)*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))^2*((RootOf(8*_Z^4 
-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4 
-_Z^3+8*_Z+8,index=3))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3)-RootOf(8*_Z^4-_ 
Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2)*((RootO 
f(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootO 
f(8*_Z^4-_Z^3+8*_Z+8,index=4))/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf( 
8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2) 
/(RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2))/( 
RootOf(8*_Z^4-_Z^3+8*_Z+8,index=2)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*2^( 
1/2)/((x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8, 
index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=3))*(x-RootOf(8*_Z^4-_Z^3+8*_ 
Z+8,index=4)))^(1/2)*EllipticF(((RootOf(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf 
(8*_Z^4-_Z^3+8*_Z+8,index=2))*(x-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(Root 
Of(8*_Z^4-_Z^3+8*_Z+8,index=4)-RootOf(8*_Z^4-_Z^3+8*_Z+8,index=1))/(x-Root 
Of(8*_Z^4-_Z^3+8*_Z+8,index=2)))^(1/2),((RootOf(8*_Z^4-_Z^3+8*_Z+8,inde...
 

3.8.97.5 Fricas [F]

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

input

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

output

integral(sqrt(8*x^4 - x^3 + 8*x + 8)/(64*x^8 - 16*x^7 + x^6 + 128*x^5 + 11 
2*x^4 - 16*x^3 + 64*x^2 + 128*x + 64), x)
 

3.8.97.6 Sympy [F]

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

input

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

output

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

3.8.97.7 Maxima [F]

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

input

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

output

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

3.8.97.8 Giac [F]

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

input

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

output

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

3.8.97.9 Mupad [F(-1)]

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

input

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

output

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