Integrand size = 36, antiderivative size = 74 \[ \int \frac {b+a x^2}{\left (b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {1}{2} \text {RootSum}\left [2 a^2+b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 a+2 \text {$\#$1}^4}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.47 \[ \int \frac {b+a x^2}{\left (b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [2 a^2+b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}^3+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^3}{-3 a+2 \text {$\#$1}^4}\&\right ]}{2 \sqrt [4]{x^2 \left (b+a x^2\right )}} \]
-1/2*(Sqrt[x]*(b + a*x^2)^(1/4)*RootSum[2*a^2 + b - 3*a*#1^4 + #1^8 & , (- (Log[Sqrt[x]]*#1^3) + Log[(b + a*x^2)^(1/4) - Sqrt[x]*#1]*#1^3)/(-3*a + 2* #1^4) & ])/(x^2*(b + a*x^2))^(1/4)
Leaf count is larger than twice the leaf count of optimal. \(681\) vs. \(2(74)=148\).
Time = 0.97 (sec) , antiderivative size = 681, normalized size of antiderivative = 9.20, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.361, Rules used = {2467, 1592, 1758, 25, 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 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \int -\frac {\left (a x^2+b\right )^{3/4}}{-2 x^2+a+\sqrt {a^2-4 b}}d\sqrt {x}}{\sqrt {a^2-4 b}}-\frac {2 \int -\frac {\left (a x^2+b\right )^{3/4}}{-2 x^2+a-\sqrt {a^2-4 b}}d\sqrt {x}}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \int \frac {\left (a x^2+b\right )^{3/4}}{-2 x^2+a-\sqrt {a^2-4 b}}d\sqrt {x}}{\sqrt {a^2-4 b}}-\frac {2 \int \frac {\left (a x^2+b\right )^{3/4}}{-2 x^2+a+\sqrt {a^2-4 b}}d\sqrt {x}}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 916 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \int \frac {1}{\sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} \left (a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \int \frac {1}{\sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \int \frac {1}{1-a x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} \left (a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \int \frac {1}{1-a x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {1}{2} \int \frac {1}{\sqrt {a} x+1}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} \left (a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {1}{2} \int \frac {1}{\sqrt {a} x+1}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} \left (a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a-\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} \left (a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{\left (-2 x^2+a+\sqrt {a^2-4 b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{-\left (\left (a \left (a-\sqrt {a^2-4 b}\right )+2 b\right ) x^2\right )+a-\sqrt {a^2-4 b}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} \left (a \sqrt {a^2-4 b}+a^2+2 b\right ) \int \frac {1}{-\left (\left (a \left (a+\sqrt {a^2-4 b}\right )+2 b\right ) x^2\right )+a+\sqrt {a^2-4 b}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2+2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-\sqrt {a^2-4 b} a+2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {a-\sqrt {a^2-4 b}}}+\frac {\int \frac {1}{\sqrt {a^2-\sqrt {a^2-4 b} a+2 b} x+\sqrt {a-\sqrt {a^2-4 b}}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {a-\sqrt {a^2-4 b}}}\right )-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} \left (a \sqrt {a^2-4 b}+a^2+2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+\sqrt {a^2-4 b} a+2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {\sqrt {a^2-4 b}+a}}+\frac {\int \frac {1}{\sqrt {a^2+\sqrt {a^2-4 b} a+2 b} x+\sqrt {a+\sqrt {a^2-4 b}}}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {\sqrt {a^2-4 b}+a}}\right )-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2+2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2-4 b}}-\sqrt {a^2-\sqrt {a^2-4 b} a+2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {a-\sqrt {a^2-4 b}}}+\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b}}\right )-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} \left (a \sqrt {a^2-4 b}+a^2+2 b\right ) \left (\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2-4 b}}-\sqrt {a^2+\sqrt {a^2-4 b} a+2 b} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {\sqrt {a^2-4 b}+a}}+\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b}}\right )-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \left (\frac {1}{2} \left (-a \sqrt {a^2-4 b}+a^2+2 b\right ) \left (\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{a-\sqrt {a^2-4 b}} \sqrt [4]{a x^2+b}}\right )}{2 \left (a-\sqrt {a^2-4 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2-4 b}+a^2+2 b}}\right )-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}-\frac {2 \left (\frac {1}{2} \left (a \sqrt {a^2-4 b}+a^2+2 b\right ) \left (\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b}}{\sqrt [4]{\sqrt {a^2-4 b}+a} \sqrt [4]{a x^2+b}}\right )}{2 \left (\sqrt {a^2-4 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2-4 b}+a^2+2 b}}\right )-\frac {1}{2} a \left (\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )\right )}{\sqrt {a^2-4 b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
(2*Sqrt[x]*(b + a*x^2)^(1/4)*((2*(-1/2*(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)))) + ((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)) + 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*S qrt[a^2 - 4*b] + 2*b)^(1/4))))/2))/Sqrt[a^2 - 4*b] - (2*(-1/2*(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)))) + ((a^2 + a*Sqrt[a^2 - 4*b] + 2*b)*(Arc Tan[((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)*Sq rt[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)
3.10.78.3.1 Defintions of rubi rules used
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])
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]
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])
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]
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]
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]
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]
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]
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]
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]
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]
Time = 1.22 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 \textit {\_Z}^{4} a +2 a^{2}+b \right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{2 \textit {\_R}^{4}-3 a}\right )}{2}\) | \(61\) |
-1/2*sum(_R^3*ln((-_R*x+(x^2*(a*x^2+b))^(1/4))/x)/(2*_R^4-3*a),_R=RootOf(_ Z^8-3*_Z^4*a+2*a^2+b))
Timed out. \[ \int \frac {b+a x^2}{\left (b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]
Not integrable
Time = 7.85 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.42 \[ \int \frac {b+a x^2}{\left (b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a x^{2} + b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (- a x^{2} + b + x^{4}\right )}\, dx \]
Not integrable
Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.49 \[ \int \frac {b+a x^2}{\left (b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} + b\right )}} \,d x } \]
Not integrable
Time = 6.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.49 \[ \int \frac {b+a x^2}{\left (b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - a x^{2} + b\right )}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.49 \[ \int \frac {b+a x^2}{\left (b-a x^2+x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a\,x^2+b}{{\left (a\,x^4+b\,x^2\right )}^{1/4}\,\left (x^4-a\,x^2+b\right )} \,d x \]