3.10.12 \(\int \frac {b+a x^2}{(b+a x^2+x^4) \sqrt [4]{b x^2+a x^4}} \, dx\) [912]

3.10.12.1 Optimal result

Integrand size = 35, antiderivative size = 69 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {1}{2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+2 \text {$\#$1}^4}\&\right ] \]

output

Unintegrable
 

3.10.12.2 Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.51 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [b-a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}^3+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^3}{-a+2 \text {$\#$1}^4}\&\right ]}{2 \sqrt [4]{x^2 \left (b+a x^2\right )}} \]

input

Integrate[(b + a*x^2)/((b + a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]
 

output

-1/2*(Sqrt[x]*(b + a*x^2)^(1/4)*RootSum[b - a*#1^4 + #1^8 & , (-(Log[Sqrt[ 
x]]*#1^3) + Log[(b + a*x^2)^(1/4) - Sqrt[x]*#1]*#1^3)/(-a + 2*#1^4) & ])/( 
x^2*(b + a*x^2))^(1/4)
 

3.10.12.3 Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(680\) vs. \(2(69)=138\).

Time = 0.94 (sec) , antiderivative size = 680, normalized size of antiderivative = 9.86, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {2467, 1592, 1758, 916, 770, 756, 216, 219, 902, 756, 218, 221}

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.

\(\displaystyle \int \frac {a x^2+b}{\left (a x^2+b+x^4\right ) \sqrt [4]{a x^4+b x^2}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2+b} \int \frac {\left (a x^2+b\right )^{3/4}}{\sqrt {x} \left (x^4+a x^2+b\right )}dx}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 1592

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \int \frac {\left (a x^2+b\right )^{3/4}}{x^4+a x^2+b}d\sqrt {x}}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 1758

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \int \frac {\left (a x^2+b\right )^{3/4}}{2 x^2+a-\sqrt {a^2-4 b}}d\sqrt {x}}{\sqrt {a^2-4 b}}-\frac {2 \int \frac {\left (a x^2+b\right )^{3/4}}{2 x^2+a+\sqrt {a^2-4 b}}d\sqrt {x}}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 916

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} a \int \frac {1}{\sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} a \int \frac {1}{\sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} \left (a \left (\sqrt {a^2-4 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 770

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} a \int \frac {1}{1-a x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} a \int \frac {1}{1-a x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-\frac {1}{2} \left (a \left (\sqrt {a^2-4 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {1}{2} \int \frac {1}{\sqrt {a} x+1}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {1}{2} \int \frac {1}{\sqrt {a} x+1}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-4 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-4 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right ) \int \frac {1}{\left (2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-4 b}+a\right )-2 b\right ) \int \frac {1}{\left (2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 902

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right ) \int \frac {1}{-\left (\left (a \left (a-\sqrt {a^2-4 b}\right )-2 b\right ) x^2\right )+a-\sqrt {a^2-4 b}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-4 b}+a\right )-2 b\right ) \int \frac {1}{-\left (\left (a \left (a+\sqrt {a^2-4 b}\right )-2 b\right ) x^2\right )+a+\sqrt {a^2-4 b}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-\sqrt {a^2-4 b} a-2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {a-\sqrt {a^2-4 b}}}+\frac {\int \frac {1}{\sqrt {a^2-\sqrt {a^2-4 b} a-2 b} x+\sqrt {a-\sqrt {a^2-4 b}}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {a-\sqrt {a^2-4 b}}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-4 b}+a\right )-2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+\sqrt {a^2-4 b} a-2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {\sqrt {a^2-4 b}+a}}+\frac {\int \frac {1}{\sqrt {a^2+\sqrt {a^2-4 b} a-2 b} x+\sqrt {a+\sqrt {a^2-4 b}}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {\sqrt {a^2-4 b}+a}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-\sqrt {a^2-4 b} a-2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {a-\sqrt {a^2-4 b}}}+\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-4 b}+a\right )-2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+\sqrt {a^2-4 b} a-2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {\sqrt {a^2-4 b}+a}}+\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2-2 b\right ) \left (\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2-2 b}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )-\frac {1}{2} \left (a \left (\sqrt {a^2-4 b}+a\right )-2 b\right ) \left (\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2-2 b}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\)

input

Int[(b + a*x^2)/((b + a*x^2 + x^4)*(b*x^2 + a*x^4)^(1/4)),x]
 

output

(2*Sqrt[x]*(b + a*x^2)^(1/4)*((2*((a*(ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2) 
^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/(2*a^(1 
/4))))/2 - ((a^2 - a*Sqrt[a^2 - 4*b] - 2*b)*(ArcTan[((a^2 - a*Sqrt[a^2 - 4 
*b] - 2*b)^(1/4)*Sqrt[x])/((a - Sqrt[a^2 - 4*b])^(1/4)*(b + a*x^2)^(1/4))] 
/(2*(a - Sqrt[a^2 - 4*b])^(3/4)*(a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)) + A 
rcTanh[((a^2 - a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a - Sqrt[a^2 - 4* 
b])^(1/4)*(b + a*x^2)^(1/4))]/(2*(a - Sqrt[a^2 - 4*b])^(3/4)*(a^2 - a*Sqrt 
[a^2 - 4*b] - 2*b)^(1/4))))/2))/Sqrt[a^2 - 4*b] - (2*((a*(ArcTan[(a^(1/4)* 
Sqrt[x])/(b + a*x^2)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a^(1/4)*Sqrt[x])/(b + a 
*x^2)^(1/4)]/(2*a^(1/4))))/2 - ((a*(a + Sqrt[a^2 - 4*b]) - 2*b)*(ArcTan[(( 
a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x])/((a + Sqrt[a^2 - 4*b])^(1/4) 
*(b + a*x^2)^(1/4))]/(2*(a + Sqrt[a^2 - 4*b])^(3/4)*(a^2 + a*Sqrt[a^2 - 4* 
b] - 2*b)^(1/4)) + ArcTanh[((a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4)*Sqrt[x]) 
/((a + Sqrt[a^2 - 4*b])^(1/4)*(b + a*x^2)^(1/4))]/(2*(a + Sqrt[a^2 - 4*b]) 
^(3/4)*(a^2 + a*Sqrt[a^2 - 4*b] - 2*b)^(1/4))))/2))/Sqrt[a^2 - 4*b]))/(b*x 
^2 + a*x^4)^(1/4)
 

3.10.12.3.1 Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ[a 
/b, 0]
 

rule 770

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n)   Subst[In 
t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, 
 b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 
/n]
 

rule 902

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su 
bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b 
, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
 

rule 916

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si 
mp[b/d   Int[(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b* 
x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - 
a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]
 

rule 1592

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c 
_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f   Subst[ 
Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f^2))^q*(a + b*(x^(2*k)/f^k) + c*(x^( 
4*k)/f^4))^p, x], x, (f*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, f, p, q}, x 
] && NeQ[b^2 - 4*a*c, 0] && FractionQ[m] && IntegerQ[p]
 

rule 1758

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ 
)), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/r)   Int[(d + e*x 
^n)^q/(b - r + 2*c*x^n), x], x] - Simp[2*(c/r)   Int[(d + e*x^n)^q/(b + r + 
 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && Ne 
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[q]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

3.10.12.4 Maple [N/A] (verified)

Time = 1.20 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80

method	result	size
pseudoelliptic	\(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-\textit {\_Z}^{4} a +b \right )}{\sum }\left (-\frac {\textit {\_R}^{3} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{-2 \textit {\_R}^{4}+a}\right )\right )}{2}\)	\(55\)

input

int((a*x^2+b)/(x^4+a*x^2+b)/(a*x^4+b*x^2)^(1/4),x,method=_RETURNVERBOSE)
 

output

-1/2*sum(-_R^3*ln((-_R*x+(x^2*(a*x^2+b))^(1/4))/x)/(-2*_R^4+a),_R=RootOf(_ 
Z^8-_Z^4*a+b))
 

3.10.12.5 Fricas [F(-1)]

Timed out. \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]

input

integrate((a*x^2+b)/(x^4+a*x^2+b)/(a*x^4+b*x^2)^(1/4),x, algorithm="fricas 
")
 

output

Timed out
 

3.10.12.6 Sympy [N/A]

Not integrable

Time = 7.55 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.45 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a x^{2} + b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{2} + b + x^{4}\right )}\, dx \]

input

integrate((a*x**2+b)/(x**4+a*x**2+b)/(a*x**4+b*x**2)**(1/4),x)
 

output

Integral((a*x**2 + b)/((x**2*(a*x**2 + b))**(1/4)*(a*x**2 + b + x**4)), x)
 

3.10.12.7 Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}} \,d x } \]

input

integrate((a*x^2+b)/(x^4+a*x^2+b)/(a*x^4+b*x^2)^(1/4),x, algorithm="maxima 
")
 

output

integrate((a*x^2 + b)/((a*x^4 + b*x^2)^(1/4)*(x^4 + a*x^2 + b)), x)
 

3.10.12.8 Giac [N/A]

Not integrable

Time = 3.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + a x^{2} + b\right )}} \,d x } \]

input

integrate((a*x^2+b)/(x^4+a*x^2+b)/(a*x^4+b*x^2)^(1/4),x, algorithm="giac")
 

output

integrate((a*x^2 + b)/((a*x^4 + b*x^2)^(1/4)*(x^4 + a*x^2 + b)), x)
 

3.10.12.9 Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.51 \[ \int \frac {b+a x^2}{\left (b+a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a\,x^2+b}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (x^4+a\,x^2+b\right )} \,d x \]

input

int((b + a*x^2)/((a*x^4 + b*x^2)^(1/4)*(b + a*x^2 + x^4)),x)
 

output

int((b + a*x^2)/((a*x^4 + b*x^2)^(1/4)*(b + a*x^2 + x^4)), x)