3.7.92 \(\int \frac {2 b+a x^2}{\sqrt [4]{b+a x^2} (-2 b-2 a x^2+x^4)} \, dx\) [692]

3.7.92.1 Optimal result

Integrand size = 36, antiderivative size = 54 \[ \int \frac {2 b+a x^2}{\sqrt [4]{b+a x^2} \left (-2 b-2 a x^2+x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{b+a x^2}}{x}\right )}{2^{3/4}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{b+a x^2}}\right )}{2^{3/4}} \]

output

1/2*arctan(2^(1/4)*(a*x^2+b)^(1/4)/x)*2^(1/4)-1/2*arctanh(1/2*x*2^(3/4)/(a 
*x^2+b)^(1/4))*2^(1/4)
 

3.7.92.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.91 \[ \int \frac {2 b+a x^2}{\sqrt [4]{b+a x^2} \left (-2 b-2 a x^2+x^4\right )} \, dx=\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{b+a x^2}}{x}\right )-\text {arctanh}\left (\frac {x}{\sqrt [4]{2} \sqrt [4]{b+a x^2}}\right )}{2^{3/4}} \]

input

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

output

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

3.7.92.3 Rubi [C] (verified)

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

Time = 0.80 (sec) , antiderivative size = 443, normalized size of antiderivative = 8.20, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, 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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

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

input

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

output

(b^(1/4)*(a - Sqrt[a^2 + 2*b])*Sqrt[-((a*x^2)/b)]*EllipticPi[-(Sqrt[b]/Sqr 
t[a^2 + b - a*Sqrt[a^2 + 2*b]]), ArcSin[(b + a*x^2)^(1/4)/b^(1/4)], -1])/( 
2*Sqrt[a^2 + b - a*Sqrt[a^2 + 2*b]]*x) - (b^(1/4)*(a - Sqrt[a^2 + 2*b])*Sq 
rt[-((a*x^2)/b)]*EllipticPi[Sqrt[b]/Sqrt[a^2 + b - a*Sqrt[a^2 + 2*b]], Arc 
Sin[(b + a*x^2)^(1/4)/b^(1/4)], -1])/(2*Sqrt[a^2 + b - a*Sqrt[a^2 + 2*b]]* 
x) + (b^(1/4)*(a + Sqrt[a^2 + 2*b])*Sqrt[-((a*x^2)/b)]*EllipticPi[-(Sqrt[b 
]/Sqrt[a^2 + b + a*Sqrt[a^2 + 2*b]]), ArcSin[(b + a*x^2)^(1/4)/b^(1/4)], - 
1])/(2*Sqrt[a^2 + b + a*Sqrt[a^2 + 2*b]]*x) - (b^(1/4)*(a + Sqrt[a^2 + 2*b 
])*Sqrt[-((a*x^2)/b)]*EllipticPi[Sqrt[b]/Sqrt[a^2 + b + a*Sqrt[a^2 + 2*b]] 
, ArcSin[(b + a*x^2)^(1/4)/b^(1/4)], -1])/(2*Sqrt[a^2 + b + a*Sqrt[a^2 + 2 
*b]]*x)
 

3.7.92.3.1 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]
 

3.7.92.4 Maple [F]

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

input

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

output

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

3.7.92.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.7.92.6 Sympy [F]

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

input

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

output

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

3.7.92.7 Maxima [F]

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

input

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

output

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

3.7.92.8 Giac [F]

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

input

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

output

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

3.7.92.9 Mupad [F(-1)]

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

input

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

output

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