3.4.31 \(\int \frac {(1+2 x^2+2 x^4)^{3/2}}{x^6 (3-2 x^2)} \, dx\) [331]

3.4.31.1 Optimal result

Integrand size = 29, antiderivative size = 553 \[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{x^6 \left (3-2 x^2\right )} \, dx=\frac {74 \sqrt {1+2 x^2+2 x^4}}{135 x^3}-\frac {262 \sqrt {1+2 x^2+2 x^4}}{135 x}-\frac {\left (3+40 x^2\right ) \sqrt {1+2 x^2+2 x^4}}{45 x^5}+\frac {262 \sqrt {2} x \sqrt {1+2 x^2+2 x^4}}{135 \left (1+\sqrt {2} x^2\right )}+\frac {17}{27} \sqrt {\frac {17}{3}} \text {arctanh}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )-\frac {262 \sqrt [4]{2} \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{135 \sqrt {1+2 x^2+2 x^4}}+\frac {85\ 2^{3/4} \left (3-\sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{189 \sqrt {1+2 x^2+2 x^4}}+\frac {2^{3/4} \left (37+23 \sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{135 \sqrt {1+2 x^2+2 x^4}}-\frac {289 \left (11-6 \sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{24} \left (12+11 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{1134 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}} \]

output

17/81*arctanh(1/3*x*51^(1/2)/(2*x^4+2*x^2+1)^(1/2))*51^(1/2)+74/135*(2*x^4 
+2*x^2+1)^(1/2)/x^3-262/135*(2*x^4+2*x^2+1)^(1/2)/x-1/45*(40*x^2+3)*(2*x^4 
+2*x^2+1)^(1/2)/x^5+262/135*x*(2*x^4+2*x^2+1)^(1/2)*2^(1/2)/(1+x^2*2^(1/2) 
)-262/135*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*Elli 
pticE(sin(2*arctan(2^(1/4)*x)),1/2*(2-2^(1/2))^(1/2))*(1+x^2*2^(1/2))*((2* 
x^4+2*x^2+1)/(1+x^2*2^(1/2))^2)^(1/2)*2^(1/4)/(2*x^4+2*x^2+1)^(1/2)-289/22 
68*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arctan(2^(1/4)*x))*EllipticPi( 
sin(2*arctan(2^(1/4)*x)),1/2+11/24*2^(1/2),1/2*(2-2^(1/2))^(1/2))*(11-6*2^ 
(1/2))*(1+x^2*2^(1/2))*((2*x^4+2*x^2+1)/(1+x^2*2^(1/2))^2)^(1/2)*2^(3/4)/( 
2*x^4+2*x^2+1)^(1/2)+85/189*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/cos(2*arcta 
n(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),1/2*(2-2^(1/2))^(1/2))*(3 
-2^(1/2))*(1+x^2*2^(1/2))*((2*x^4+2*x^2+1)/(1+x^2*2^(1/2))^2)^(1/2)*2^(3/4 
)/(2*x^4+2*x^2+1)^(1/2)+1/135*2^(3/4)*(cos(2*arctan(2^(1/4)*x))^2)^(1/2)/c 
os(2*arctan(2^(1/4)*x))*EllipticF(sin(2*arctan(2^(1/4)*x)),1/2*(2-2^(1/2)) 
^(1/2))*(37+23*2^(1/2))*(1+x^2*2^(1/2))*((2*x^4+2*x^2+1)/(1+x^2*2^(1/2))^2 
)^(1/2)/(2*x^4+2*x^2+1)^(1/2)
 

3.4.31.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.27 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.41 \[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{x^6 \left (3-2 x^2\right )} \, dx=-\frac {27+192 x^2+1116 x^4+1848 x^6+1572 x^8+786 i \sqrt {1-i} x^5 \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} E\left (\left .i \text {arcsinh}\left (\sqrt {1-i} x\right )\right |i\right )+(543-1329 i) \sqrt {1-i} x^5 \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {1-i} x\right ),i\right )-1445 (1-i)^{3/2} x^5 \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} \operatorname {EllipticPi}\left (-\frac {1}{3}-\frac {i}{3},i \text {arcsinh}\left (\sqrt {1-i} x\right ),i\right )}{405 x^5 \sqrt {1+2 x^2+2 x^4}} \]

input

Integrate[(1 + 2*x^2 + 2*x^4)^(3/2)/(x^6*(3 - 2*x^2)),x]
 

output

-1/405*(27 + 192*x^2 + 1116*x^4 + 1848*x^6 + 1572*x^8 + (786*I)*Sqrt[1 - I 
]*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt 
[1 - I]*x], I] + (543 - 1329*I)*Sqrt[1 - I]*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt 
[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 1445*(1 - I)^(3 
/2)*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[-1/3 - I/3, 
 I*ArcSinh[Sqrt[1 - I]*x], I])/(x^5*Sqrt[1 + 2*x^2 + 2*x^4])
 

3.4.31.3 Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 535, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {1634, 25, 27, 2199, 2199, 1604, 1604, 27, 1604, 25, 1511, 1416, 1509, 2222}

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.

\(\displaystyle \int \frac {\left (2 x^4+2 x^2+1\right )^{3/2}}{x^6 \left (3-2 x^2\right )} \, dx\)

\(\Big \downarrow \) 1634

\(\displaystyle \frac {578}{189} \int -\frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx-\frac {1}{378} \int -\frac {2 \left (2 \left (678-289 \sqrt {2}\right ) x^6+700 x^4+294 x^2+63\right )}{x^6 \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx-\frac {1}{378} \int -\frac {2 \left (2 \left (678-289 \sqrt {2}\right ) x^6+700 x^4+294 x^2+63\right )}{x^6 \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{189} \int \frac {2 \left (678-289 \sqrt {2}\right ) x^6+700 x^4+294 x^2+63}{x^6 \sqrt {2 x^4+2 x^2+1}}dx-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 2199

\(\displaystyle \frac {1}{189} \left (\int \frac {-4 \left (503-289 \sqrt {2}\right ) x^4-3 \left (580-289 \sqrt {2}\right ) x^2+63}{x^6 \sqrt {2 x^4+2 x^2+1}}dx-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 2199

\(\displaystyle \frac {1}{189} \left (\int \frac {\frac {7}{3} \left (404-289 \sqrt {2}\right ) x^2+\frac {17}{3} \left (307-170 \sqrt {2}\right )}{x^6 \sqrt {2 x^4+2 x^2+1}}dx+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {1}{189} \left (-\frac {1}{5} \int \frac {34 \left (307-170 \sqrt {2}\right ) x^2+3 \left (3068-1445 \sqrt {2}\right )}{x^4 \sqrt {2 x^4+2 x^2+1}}dx-\frac {17 \left (307-170 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{15 x^5}+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {1}{189} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {6 \left (\left (3068-1445 \sqrt {2}\right ) x^2+917\right )}{x^2 \sqrt {2 x^4+2 x^2+1}}dx+\frac {\left (3068-1445 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {17 \left (307-170 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{15 x^5}+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{189} \left (\frac {1}{5} \left (2 \int \frac {\left (3068-1445 \sqrt {2}\right ) x^2+917}{x^2 \sqrt {2 x^4+2 x^2+1}}dx+\frac {\left (3068-1445 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {17 \left (307-170 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{15 x^5}+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 1604

\(\displaystyle \frac {1}{189} \left (\frac {1}{5} \left (2 \left (-\int -\frac {1834 x^2-1445 \sqrt {2}+3068}{\sqrt {2 x^4+2 x^2+1}}dx-\frac {917 \sqrt {2 x^4+2 x^2+1}}{x}\right )+\frac {\left (3068-1445 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {17 \left (307-170 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{15 x^5}+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{189} \left (\frac {1}{5} \left (2 \left (\int \frac {1834 x^2-1445 \sqrt {2}+3068}{\sqrt {2 x^4+2 x^2+1}}dx-\frac {917 \sqrt {2 x^4+2 x^2+1}}{x}\right )+\frac {\left (3068-1445 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {17 \left (307-170 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{15 x^5}+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {1}{189} \left (\frac {1}{5} \left (2 \left (4 \left (767-132 \sqrt {2}\right ) \int \frac {1}{\sqrt {2 x^4+2 x^2+1}}dx-917 \sqrt {2} \int \frac {1-\sqrt {2} x^2}{\sqrt {2 x^4+2 x^2+1}}dx-\frac {917 \sqrt {2 x^4+2 x^2+1}}{x}\right )+\frac {\left (3068-1445 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {17 \left (307-170 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{15 x^5}+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {1}{189} \left (\frac {1}{5} \left (2 \left (-917 \sqrt {2} \int \frac {1-\sqrt {2} x^2}{\sqrt {2 x^4+2 x^2+1}}dx+\frac {2^{3/4} \left (767-132 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt {2 x^4+2 x^2+1}}-\frac {917 \sqrt {2 x^4+2 x^2+1}}{x}\right )+\frac {\left (3068-1445 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {17 \left (307-170 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{15 x^5}+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {1}{189} \left (\frac {1}{5} \left (2 \left (\frac {2^{3/4} \left (767-132 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt {2 x^4+2 x^2+1}}-917 \sqrt {2} \left (\frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {x \sqrt {2 x^4+2 x^2+1}}{\sqrt {2} x^2+1}\right )-\frac {917 \sqrt {2 x^4+2 x^2+1}}{x}\right )+\frac {\left (3068-1445 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {17 \left (307-170 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{15 x^5}+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \int \frac {-2 \left (3-\sqrt {2}\right ) x^2-3 \sqrt {2}+2}{\left (3-2 x^2\right ) \sqrt {2 x^4+2 x^2+1}}dx\)

\(\Big \downarrow \) 2222

\(\displaystyle \frac {1}{189} \left (\frac {1}{5} \left (2 \left (\frac {2^{3/4} \left (767-132 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt {2 x^4+2 x^2+1}}-917 \sqrt {2} \left (\frac {\left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \arctan \left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{\sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {x \sqrt {2 x^4+2 x^2+1}}{\sqrt {2} x^2+1}\right )-\frac {917 \sqrt {2 x^4+2 x^2+1}}{x}\right )+\frac {\left (3068-1445 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {17 \left (307-170 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{15 x^5}+\frac {2 \left (503-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{3 x^5}-\frac {\left (678-289 \sqrt {2}\right ) \sqrt {2 x^4+2 x^2+1}}{x^3}\right )-\frac {578}{189} \left (\frac {\left (3-\sqrt {2}\right )^2 \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \operatorname {EllipticPi}\left (\frac {1}{24} \left (12+11 \sqrt {2}\right ),2 \arctan \left (\sqrt [4]{2} x\right ),\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{12 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {7 \text {arctanh}\left (\frac {\sqrt {\frac {17}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{2 \sqrt {51}}\right )\)

input

Int[(1 + 2*x^2 + 2*x^4)^(3/2)/(x^6*(3 - 2*x^2)),x]
 

output

((2*(503 - 289*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4])/(3*x^5) - (17*(307 - 170* 
Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4])/(15*x^5) - ((678 - 289*Sqrt[2])*Sqrt[1 + 
 2*x^2 + 2*x^4])/x^3 + (((3068 - 1445*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4])/x^ 
3 + 2*((-917*Sqrt[1 + 2*x^2 + 2*x^4])/x - 917*Sqrt[2]*(-((x*Sqrt[1 + 2*x^2 
 + 2*x^4])/(1 + Sqrt[2]*x^2)) + ((1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4 
)/(1 + Sqrt[2]*x^2)^2]*EllipticE[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(2 
^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4])) + (2^(3/4)*(767 - 132*Sqrt[2])*(1 + Sqrt[ 
2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*EllipticF[2*ArcTan[2 
^(1/4)*x], (2 - Sqrt[2])/4])/Sqrt[1 + 2*x^2 + 2*x^4]))/5)/189 - (578*((-7* 
ArcTanh[(Sqrt[17/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/(2*Sqrt[51]) + ((3 - Sqrt 
[2])^2*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*Ell 
ipticPi[(12 + 11*Sqrt[2])/24, 2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(12*2 
^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4])))/189
 

3.4.31.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 1604

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[d*(f*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1) 
/(a*f*(m + 1))), x] + Simp[1/(a*f^2*(m + 1))   Int[(f*x)^(m + 2)*(a + b*x^2 
 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x 
], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[ 
m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1634

Int[((x_)^(m_)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_) + (e_.)*(x 
_)^2), x_Symbol] :> Simp[(-(-d/e)^(m/2))*((c*d^2 - b*d*e + a*e^2)^(p + 1/2) 
/(e^(2*p)*(c*d^2 - a*e^2)))   Int[(a*d*Rt[c/a, 2] + a*e + (c*d + a*e*Rt[c/a 
, 2])*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x] + Simp[(-d/e)^(m/2 
)/(e^(2*p)*(c*d^2 - a*e^2))   Int[(x^m/Sqrt[a + b*x^2 + c*x^4])*ExpandToSum 
[((e^(2*p)*(c*d^2 - a*e^2)*(a + b*x^2 + c*x^4)^(p + 1/2))/(-d/e)^(m/2) + (( 
a*d*Rt[c/a, 2] + a*e + (c*d + a*e*Rt[c/a, 2])*x^2)*(c*d^2 - b*d*e + a*e^2)^ 
(p + 1/2))/x^m)/(d + e*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ 
[b^2 - 4*a*c, 0] && IGtQ[p + 1/2, 0] && ILtQ[m/2, 0] && NeQ[c*d^2 - a*e^2, 
0] && PosQ[c/a]
 

rule 2199

Int[(Px_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_S 
ymbol] :> With[{q = Expon[Px, x^2]}, Simp[Coeff[Px, x^2, q]*(d*x)^(m + 2*q 
- 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*d^(2*q - 3)*(m + 4*p + 2*q + 1))), x] 
+ Int[(d*x)^m*(a + b*x^2 + c*x^4)^p*ExpandToSum[Px - Coeff[Px, x^2, q]*x^(2 
*q) - Coeff[Px, x^2, q]*((a*(m + 2*q - 3)*x^(2*(q - 2)) + b*(m + 2*p + 2*q 
- 1)*x^(2*(q - 1)))/(c*(m + 4*p + 2*q + 1))), x], x] /; GtQ[q, 1] && NeQ[m 
+ 4*p + 2*q + 1, 0]] /; FreeQ[{a, b, c, d, m, p}, x] && PolyQ[Px, x^2] && N 
eQ[b^2 - 4*a*c, 0]
 

rule 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

3.4.31.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 5.11 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.48

method	result	size
risch	\(-\frac {524 x^{8}+616 x^{6}+372 x^{4}+64 x^{2}+9}{135 x^{5} \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {362 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\left (-\frac {262}{135}+\frac {262 i}{135}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {578 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{81 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(263\)
elliptic	\(-\frac {\sqrt {2 x^{4}+2 x^{2}+1}}{15 x^{5}}-\frac {46 \sqrt {2 x^{4}+2 x^{2}+1}}{135 x^{3}}-\frac {262 \sqrt {2 x^{4}+2 x^{2}+1}}{135 x}-\frac {208 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{45 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {262 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {262 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {262 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {578 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{81 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(398\)
default	\(-\frac {\sqrt {2 x^{4}+2 x^{2}+1}}{15 x^{5}}-\frac {46 \sqrt {2 x^{4}+2 x^{2}+1}}{135 x^{3}}-\frac {262 \sqrt {2 x^{4}+2 x^{2}+1}}{135 x}+\frac {184 \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{45 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {\left (-\frac {52}{15}+\frac {52 i}{15}\right ) \sqrt {1+\left (1-i\right ) x^{2}}\, \sqrt {1+\left (1+i\right ) x^{2}}\, \left (F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )-E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {236 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{45 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {206 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, F\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}-\frac {206 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {206 i \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, E\left (x \sqrt {-1+i}, \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{135 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {578 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \Pi \left (x \sqrt {-1+i}, -\frac {1}{3}-\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{81 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}\)	\(549\)

input

int((2*x^4+2*x^2+1)^(3/2)/x^6/(-2*x^2+3),x,method=_RETURNVERBOSE)
 

output

-1/135*(524*x^8+616*x^6+372*x^4+64*x^2+9)/x^5/(2*x^4+2*x^2+1)^(1/2)-362/13 
5/(-1+I)^(1/2)*(1+(1-I)*x^2)^(1/2)*(1+(1+I)*x^2)^(1/2)/(2*x^4+2*x^2+1)^(1/ 
2)*EllipticF(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))+(-262/135+262/135*I 
)/(-1+I)^(1/2)*(1+(1-I)*x^2)^(1/2)*(1+(1+I)*x^2)^(1/2)/(2*x^4+2*x^2+1)^(1/ 
2)*(EllipticF(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))-EllipticE(x*(-1+I) 
^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2)))+578/81/(-1+I)^(1/2)*(1-I*x^2+x^2)^(1/2) 
*(1+I*x^2+x^2)^(1/2)/(2*x^4+2*x^2+1)^(1/2)*EllipticPi(x*(-1+I)^(1/2),-1/3- 
1/3*I,(-1-I)^(1/2)/(-1+I)^(1/2))
 

3.4.31.5 Fricas [F]

\[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{x^6 \left (3-2 x^2\right )} \, dx=\int { -\frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (2 \, x^{2} - 3\right )} x^{6}} \,d x } \]

input

integrate((2*x^4+2*x^2+1)^(3/2)/x^6/(-2*x^2+3),x, algorithm="fricas")
 

output

integral(-(2*x^4 + 2*x^2 + 1)^(3/2)/(2*x^8 - 3*x^6), x)
 

3.4.31.6 Sympy [F]

\[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{x^6 \left (3-2 x^2\right )} \, dx=- \int \frac {\sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{8} - 3 x^{6}}\, dx - \int \frac {2 x^{2} \sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{8} - 3 x^{6}}\, dx - \int \frac {2 x^{4} \sqrt {2 x^{4} + 2 x^{2} + 1}}{2 x^{8} - 3 x^{6}}\, dx \]

input

integrate((2*x**4+2*x**2+1)**(3/2)/x**6/(-2*x**2+3),x)
 

output

-Integral(sqrt(2*x**4 + 2*x**2 + 1)/(2*x**8 - 3*x**6), x) - Integral(2*x** 
2*sqrt(2*x**4 + 2*x**2 + 1)/(2*x**8 - 3*x**6), x) - Integral(2*x**4*sqrt(2 
*x**4 + 2*x**2 + 1)/(2*x**8 - 3*x**6), x)
 

3.4.31.7 Maxima [F]

\[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{x^6 \left (3-2 x^2\right )} \, dx=\int { -\frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (2 \, x^{2} - 3\right )} x^{6}} \,d x } \]

input

integrate((2*x^4+2*x^2+1)^(3/2)/x^6/(-2*x^2+3),x, algorithm="maxima")
 

output

-integrate((2*x^4 + 2*x^2 + 1)^(3/2)/((2*x^2 - 3)*x^6), x)
 

3.4.31.8 Giac [F]

\[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{x^6 \left (3-2 x^2\right )} \, dx=\int { -\frac {{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (2 \, x^{2} - 3\right )} x^{6}} \,d x } \]

input

integrate((2*x^4+2*x^2+1)^(3/2)/x^6/(-2*x^2+3),x, algorithm="giac")
 

output

integrate(-(2*x^4 + 2*x^2 + 1)^(3/2)/((2*x^2 - 3)*x^6), x)
 

3.4.31.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+2 x^2+2 x^4\right )^{3/2}}{x^6 \left (3-2 x^2\right )} \, dx=-\int \frac {{\left (2\,x^4+2\,x^2+1\right )}^{3/2}}{x^6\,\left (2\,x^2-3\right )} \,d x \]

input

int(-(2*x^2 + 2*x^4 + 1)^(3/2)/(x^6*(2*x^2 - 3)),x)
 

output

-int((2*x^2 + 2*x^4 + 1)^(3/2)/(x^6*(2*x^2 - 3)), x)