Integrand size = 41, antiderivative size = 142 \[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=-\frac {\arctan \left (\frac {\frac {\sqrt [4]{3} x^2}{\sqrt {2}}-\frac {\sqrt {3 b-2 a x^2}}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt [4]{3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{3} x \sqrt [4]{3 b-2 a x^2}}{\sqrt {3} x^2+\sqrt {3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \]
-1/12*arctan((1/2*3^(1/4)*x^2*2^(1/2)-1/6*(-2*a*x^2+3*b)^(1/2)*2^(1/2)*3^( 3/4))/x/(-2*a*x^2+3*b)^(1/4))*2^(1/2)*3^(3/4)-1/12*arctanh(2^(1/2)*3^(1/4) *x*(-2*a*x^2+3*b)^(1/4)/(3^(1/2)*x^2+(-2*a*x^2+3*b)^(1/2)))*2^(1/2)*3^(3/4 )
Time = 0.37 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.80 \[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=-\frac {\arctan \left (\frac {3 x^2-\sqrt {9 b-6 a x^2}}{\sqrt {2} 3^{3/4} x \sqrt [4]{3 b-2 a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{9 b-6 a x^2}}{\sqrt {3} x^2+\sqrt {3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \]
-1/2*(ArcTan[(3*x^2 - Sqrt[9*b - 6*a*x^2])/(Sqrt[2]*3^(3/4)*x*(3*b - 2*a*x ^2)^(1/4))] + ArcTanh[(Sqrt[2]*x*(9*b - 6*a*x^2)^(1/4))/(Sqrt[3]*x^2 + Sqr t[3*b - 2*a*x^2])])/(Sqrt[2]*3^(1/4))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.10 (sec) , antiderivative size = 539, normalized size of antiderivative = 3.80, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {2256, 2009}
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.
\(\displaystyle \int \frac {a x^2-3 b}{\sqrt [4]{3 b-2 a x^2} \left (-2 a x^2+3 b+3 x^4\right )} \, dx\) |
\(\Big \downarrow \) 2256 |
\(\displaystyle \int \left (\frac {a-\sqrt {a^2-9 b}}{\left (2 \sqrt {a^2-9 b}-2 a+6 x^2\right ) \sqrt [4]{3 b-2 a x^2}}+\frac {\sqrt {a^2-9 b}+a}{\left (-2 \sqrt {a^2-9 b}-2 a+6 x^2\right ) \sqrt [4]{3 b-2 a x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt [4]{b} \left (\sqrt {a^2-9 b}+a\right ) \sqrt {\frac {a x^2}{b}} \operatorname {EllipticPi}\left (-\frac {3 \sqrt {b}}{\sqrt {-2 a^2-2 \sqrt {a^2-9 b} a+9 b}},\arcsin \left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {-2 a \sqrt {a^2-9 b}-2 a^2+9 b}}-\frac {\sqrt [4]{b} \left (\sqrt {a^2-9 b}+a\right ) \sqrt {\frac {a x^2}{b}} \operatorname {EllipticPi}\left (\frac {3 \sqrt {b}}{\sqrt {-2 a^2-2 \sqrt {a^2-9 b} a+9 b}},\arcsin \left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {-2 a \sqrt {a^2-9 b}-2 a^2+9 b}}+\frac {\sqrt [4]{b} \left (a-\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}} \operatorname {EllipticPi}\left (-\frac {3 \sqrt {b}}{\sqrt {-2 a^2+2 \sqrt {a^2-9 b} a+9 b}},\arcsin \left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {2 a \sqrt {a^2-9 b}-2 a^2+9 b}}-\frac {\sqrt [4]{b} \left (a-\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}} \operatorname {EllipticPi}\left (\frac {3 \sqrt {b}}{\sqrt {-2 a^2+2 \sqrt {a^2-9 b} a+9 b}},\arcsin \left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {2 a \sqrt {a^2-9 b}-2 a^2+9 b}}\) |
((a + Sqrt[a^2 - 9*b])*b^(1/4)*Sqrt[(a*x^2)/b]*EllipticPi[(-3*Sqrt[b])/Sqr t[-2*a^2 - 2*a*Sqrt[a^2 - 9*b] + 9*b], ArcSin[(3*b - 2*a*x^2)^(1/4)/(3^(1/ 4)*b^(1/4))], -1])/(Sqrt[2]*3^(3/4)*Sqrt[-2*a^2 - 2*a*Sqrt[a^2 - 9*b] + 9* b]*x) - ((a + Sqrt[a^2 - 9*b])*b^(1/4)*Sqrt[(a*x^2)/b]*EllipticPi[(3*Sqrt[ b])/Sqrt[-2*a^2 - 2*a*Sqrt[a^2 - 9*b] + 9*b], ArcSin[(3*b - 2*a*x^2)^(1/4) /(3^(1/4)*b^(1/4))], -1])/(Sqrt[2]*3^(3/4)*Sqrt[-2*a^2 - 2*a*Sqrt[a^2 - 9* b] + 9*b]*x) + ((a - Sqrt[a^2 - 9*b])*b^(1/4)*Sqrt[(a*x^2)/b]*EllipticPi[( -3*Sqrt[b])/Sqrt[-2*a^2 + 2*a*Sqrt[a^2 - 9*b] + 9*b], ArcSin[(3*b - 2*a*x^ 2)^(1/4)/(3^(1/4)*b^(1/4))], -1])/(Sqrt[2]*3^(3/4)*Sqrt[-2*a^2 + 2*a*Sqrt[ a^2 - 9*b] + 9*b]*x) - ((a - Sqrt[a^2 - 9*b])*b^(1/4)*Sqrt[(a*x^2)/b]*Elli pticPi[(3*Sqrt[b])/Sqrt[-2*a^2 + 2*a*Sqrt[a^2 - 9*b] + 9*b], ArcSin[(3*b - 2*a*x^2)^(1/4)/(3^(1/4)*b^(1/4))], -1])/(Sqrt[2]*3^(3/4)*Sqrt[-2*a^2 + 2* a*Sqrt[a^2 - 9*b] + 9*b]*x)
3.21.9.3.1 Defintions of rubi rules used
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4 )^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
\[\int \frac {a \,x^{2}-3 b}{\left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}} \left (3 x^{4}-2 a \,x^{2}+3 b \right )}d x\]
Timed out. \[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=\int \frac {a x^{2} - 3 b}{\sqrt [4]{- 2 a x^{2} + 3 b} \left (- 2 a x^{2} + 3 b + 3 x^{4}\right )}\, dx \]
\[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=\int { \frac {a x^{2} - 3 \, b}{{\left (3 \, x^{4} - 2 \, a x^{2} + 3 \, b\right )} {\left (-2 \, a x^{2} + 3 \, b\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=\int { \frac {a x^{2} - 3 \, b}{{\left (3 \, x^{4} - 2 \, a x^{2} + 3 \, b\right )} {\left (-2 \, a x^{2} + 3 \, b\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=\int -\frac {3\,b-a\,x^2}{{\left (3\,b-2\,a\,x^2\right )}^{1/4}\,\left (3\,x^4-2\,a\,x^2+3\,b\right )} \,d x \]