\(\int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} (3 b-2 a x^2+3 x^4)} \, dx\) [2009]

Optimal result

Integrand size = 41, antiderivative size = 142 \[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=-\frac {\arctan \left (\frac {\frac {\sqrt [4]{3} x^2}{\sqrt {2}}-\frac {\sqrt {3 b-2 a x^2}}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt [4]{3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{3} x \sqrt [4]{3 b-2 a x^2}}{\sqrt {3} x^2+\sqrt {3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \] Output:

-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 
)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.80 \[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=-\frac {\arctan \left (\frac {3 x^2-\sqrt {9 b-6 a x^2}}{\sqrt {2} 3^{3/4} x \sqrt [4]{3 b-2 a x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{9 b-6 a x^2}}{\sqrt {3} x^2+\sqrt {3 b-2 a x^2}}\right )}{2 \sqrt {2} \sqrt [4]{3}} \] Input:

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

Output:

-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))
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 1.11 (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

\(\displaystyle \frac {\sqrt [4]{b} \left (\sqrt {a^2-9 b}+a\right ) \sqrt {\frac {a x^2}{b}} \operatorname {EllipticPi}\left (-\frac {3 \sqrt {b}}{\sqrt {-2 a^2-2 \sqrt {a^2-9 b} a+9 b}},\arcsin \left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {-2 a \sqrt {a^2-9 b}-2 a^2+9 b}}-\frac {\sqrt [4]{b} \left (\sqrt {a^2-9 b}+a\right ) \sqrt {\frac {a x^2}{b}} \operatorname {EllipticPi}\left (\frac {3 \sqrt {b}}{\sqrt {-2 a^2-2 \sqrt {a^2-9 b} a+9 b}},\arcsin \left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {-2 a \sqrt {a^2-9 b}-2 a^2+9 b}}+\frac {\sqrt [4]{b} \left (a-\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}} \operatorname {EllipticPi}\left (-\frac {3 \sqrt {b}}{\sqrt {-2 a^2+2 \sqrt {a^2-9 b} a+9 b}},\arcsin \left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {2 a \sqrt {a^2-9 b}-2 a^2+9 b}}-\frac {\sqrt [4]{b} \left (a-\sqrt {a^2-9 b}\right ) \sqrt {\frac {a x^2}{b}} \operatorname {EllipticPi}\left (\frac {3 \sqrt {b}}{\sqrt {-2 a^2+2 \sqrt {a^2-9 b} a+9 b}},\arcsin \left (\frac {\sqrt [4]{3 b-2 a x^2}}{\sqrt [4]{3} \sqrt [4]{b}}\right ),-1\right )}{\sqrt {2} 3^{3/4} x \sqrt {2 a \sqrt {a^2-9 b}-2 a^2+9 b}}\)

Input:

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

Output:

((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)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2256

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]
 

Maple [F]

\[\int \frac {a \,x^{2}-3 b}{\left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}} \left (3 x^{4}-2 a \,x^{2}+3 b \right )}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=\text {Timed out} \] Input:

integrate((a*x^2-3*b)/(-2*a*x^2+3*b)^(1/4)/(3*x^4-2*a*x^2+3*b),x, algorith 
m="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=\int \frac {a x^{2} - 3 b}{\sqrt [4]{- 2 a x^{2} + 3 b} \left (- 2 a x^{2} + 3 b + 3 x^{4}\right )}\, dx \] Input:

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

Output:

Integral((a*x**2 - 3*b)/((-2*a*x**2 + 3*b)**(1/4)*(-2*a*x**2 + 3*b + 3*x** 
4)), x)
 

Maxima [F]

\[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=\int { \frac {a x^{2} - 3 \, b}{{\left (3 \, x^{4} - 2 \, a x^{2} + 3 \, b\right )} {\left (-2 \, a x^{2} + 3 \, b\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((a*x^2-3*b)/(-2*a*x^2+3*b)^(1/4)/(3*x^4-2*a*x^2+3*b),x, algorith 
m="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=\int { \frac {a x^{2} - 3 \, b}{{\left (3 \, x^{4} - 2 \, a x^{2} + 3 \, b\right )} {\left (-2 \, a x^{2} + 3 \, b\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate((a*x^2-3*b)/(-2*a*x^2+3*b)^(1/4)/(3*x^4-2*a*x^2+3*b),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {-3 b+a x^2}{\sqrt [4]{3 b-2 a x^2} \left (3 b-2 a x^2+3 x^4\right )} \, dx=-\left (\int \frac {x^{2}}{2 \left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}} a \,x^{2}-3 \left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}} b -3 \left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}} x^{4}}d x \right ) a +3 \left (\int \frac {1}{2 \left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}} a \,x^{2}-3 \left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}} b -3 \left (-2 a \,x^{2}+3 b \right )^{\frac {1}{4}} x^{4}}d x \right ) b \] Input:

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

Output:

 - int(x**2/(2*( - 2*a*x**2 + 3*b)**(1/4)*a*x**2 - 3*( - 2*a*x**2 + 3*b)** 
(1/4)*b - 3*( - 2*a*x**2 + 3*b)**(1/4)*x**4),x)*a + 3*int(1/(2*( - 2*a*x** 
2 + 3*b)**(1/4)*a*x**2 - 3*( - 2*a*x**2 + 3*b)**(1/4)*b - 3*( - 2*a*x**2 + 
 3*b)**(1/4)*x**4),x)*b