\(\int \frac {2 a+b x^2}{\sqrt [4]{a+b x^2} (a+b x^2+c x^4)} \, dx\) [13]

Optimal result

Integrand size = 35, antiderivative size = 119 \[ \int \frac {2 a+b x^2}{\sqrt [4]{a+b x^2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {\arctan \left (\frac {-\sqrt {c} x^2+\sqrt {a+b x^2}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a+b x^2}}\right )}{\sqrt {2} \sqrt [4]{c}}+\frac {\text {arctanh}\left (\frac {\sqrt {c} x^2+\sqrt {a+b x^2}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a+b x^2}}\right )}{\sqrt {2} \sqrt [4]{c}} \] Output:

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

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.92 \[ \int \frac {2 a+b x^2}{\sqrt [4]{a+b x^2} \left (a+b x^2+c x^4\right )} \, dx=\frac {-\arctan \left (\frac {-\sqrt {c} x^2+\sqrt {a+b x^2}}{\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a+b x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{c} x \sqrt [4]{a+b x^2}}{\sqrt {c} x^2+\sqrt {a+b x^2}}\right )}{\sqrt {2} \sqrt [4]{c}} \] Input:

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

Output:

(-ArcTan[(-(Sqrt[c]*x^2) + Sqrt[a + b*x^2])/(Sqrt[2]*c^(1/4)*x*(a + b*x^2) 
^(1/4))] + ArcTanh[(Sqrt[2]*c^(1/4)*x*(a + b*x^2)^(1/4))/(Sqrt[c]*x^2 + Sq 
rt[a + b*x^2])])/(Sqrt[2]*c^(1/4))
 

Rubi [C] (verified)

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

Time = 1.08 (sec) , antiderivative size = 565, normalized size of antiderivative = 4.75, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, 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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(a^(1/4)*(b + Sqrt[b^2 - 4*a*c])*Sqrt[-((b*x^2)/a)]*EllipticPi[-((Sqrt[2]* 
Sqrt[a]*Sqrt[c])/Sqrt[-b^2 + 2*a*c - b*Sqrt[b^2 - 4*a*c]]), ArcSin[(a + b* 
x^2)^(1/4)/a^(1/4)], -1])/(Sqrt[2]*Sqrt[c]*Sqrt[-b^2 + 2*a*c - b*Sqrt[b^2 
- 4*a*c]]*x) - (a^(1/4)*(b + Sqrt[b^2 - 4*a*c])*Sqrt[-((b*x^2)/a)]*Ellipti 
cPi[(Sqrt[2]*Sqrt[a]*Sqrt[c])/Sqrt[-b^2 + 2*a*c - b*Sqrt[b^2 - 4*a*c]], Ar 
cSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(Sqrt[2]*Sqrt[c]*Sqrt[-b^2 + 2*a*c - 
 b*Sqrt[b^2 - 4*a*c]]*x) + (a^(1/4)*(b - Sqrt[b^2 - 4*a*c])*Sqrt[-((b*x^2) 
/a)]*EllipticPi[-((Sqrt[2]*Sqrt[a]*Sqrt[c])/Sqrt[-b^2 + 2*a*c + b*Sqrt[b^2 
 - 4*a*c]]), ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(Sqrt[2]*Sqrt[c]*Sqrt 
[-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c]]*x) - (a^(1/4)*(b - Sqrt[b^2 - 4*a*c]) 
*Sqrt[-((b*x^2)/a)]*EllipticPi[(Sqrt[2]*Sqrt[a]*Sqrt[c])/Sqrt[-b^2 + 2*a*c 
 + b*Sqrt[b^2 - 4*a*c]], ArcSin[(a + b*x^2)^(1/4)/a^(1/4)], -1])/(Sqrt[2]* 
Sqrt[c]*Sqrt[-b^2 + 2*a*c + b*Sqrt[b^2 - 4*a*c]]*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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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