\(\int \frac {x^2}{\sqrt {a+b x^2+\frac {(b d e-a e^2) x^4}{d^2}}} \, dx\) [717]

Optimal result

Integrand size = 33, antiderivative size = 172 \[ \int \frac {x^2}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=\frac {x \left (a d+(b d-a e) x^2\right )}{(b d-a e) \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}}-\frac {a \sqrt {d} \left (d+e x^2\right ) \sqrt {\frac {a d+(b d-a e) x^2}{a \left (d+e x^2\right )}} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|2-\frac {b d}{a e}\right )}{\sqrt {e} (b d-a e) \sqrt {a+b x^2+\frac {e (b d-a e) x^4}{d^2}}} \] Output:

x*(a*d+(-a*e+b*d)*x^2)/(-a*e+b*d)/(a+b*x^2+e*(-a*e+b*d)*x^4/d^2)^(1/2)-a*d 
^(1/2)*(e*x^2+d)*((a*d+(-a*e+b*d)*x^2)/a/(e*x^2+d))^(1/2)*EllipticE(e^(1/2 
)*x/d^(1/2)/(1+e*x^2/d)^(1/2),(2-b*d/a/e)^(1/2))/e^(1/2)/(-a*e+b*d)/(a+b*x 
^2+e*(-a*e+b*d)*x^4/d^2)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=-\frac {i a d \sqrt {1+\frac {b x^2}{a}-\frac {e x^2}{d}} \sqrt {1+\frac {e x^2}{d}} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|-1+\frac {b d}{a e}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),-1+\frac {b d}{a e}\right )\right )}{\sqrt {\frac {e}{d}} (b d-a e) \sqrt {\frac {\left (d+e x^2\right ) \left (b d x^2+a \left (d-e x^2\right )\right )}{d^2}}} \] Input:

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

Output:

((-I)*a*d*Sqrt[1 + (b*x^2)/a - (e*x^2)/d]*Sqrt[1 + (e*x^2)/d]*(EllipticE[I 
*ArcSinh[Sqrt[e/d]*x], -1 + (b*d)/(a*e)] - EllipticF[I*ArcSinh[Sqrt[e/d]*x 
], -1 + (b*d)/(a*e)]))/(Sqrt[e/d]*(b*d - a*e)*Sqrt[((d + e*x^2)*(b*d*x^2 + 
 a*(d - e*x^2)))/d^2])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(510\) vs. \(2(172)=344\).

Time = 0.95 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1459, 27, 1416, 1509}

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.

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

\(\Big \downarrow \) 1459

\(\displaystyle \frac {\sqrt {a} d \int \frac {1}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}-\frac {\sqrt {a} d \int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {a} d \sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a} d \int \frac {1}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}-\frac {\int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\sqrt [4]{a} \sqrt {d} \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 e^{3/4} (b d-a e)^{3/4} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {\int \frac {\sqrt {a} d-\sqrt {e} \sqrt {b d-a e} x^2}{\sqrt {\frac {e (b d-a e) x^4}{d^2}+b x^2+a}}dx}{\sqrt {e} \sqrt {b d-a e}}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\sqrt [4]{a} \sqrt {d} \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right ),\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{2 e^{3/4} (b d-a e)^{3/4} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {\frac {\sqrt [4]{a} \sqrt {d} \left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right ) \sqrt {\frac {d^2 \left (\frac {e x^4 (b d-a e)}{d^2}+a+b x^2\right )}{\left (\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{e} \sqrt [4]{b d-a e} x}{\sqrt [4]{a} \sqrt {d}}\right )|\frac {1}{4} \left (2-\frac {b d}{\sqrt {a} \sqrt {e} \sqrt {b d-a e}}\right )\right )}{\sqrt [4]{e} \sqrt [4]{b d-a e} \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}-\frac {d^2 x \sqrt {\frac {e x^4 (b d-a e)}{d^2}+a+b x^2}}{\sqrt {e} x^2 \sqrt {b d-a e}+\sqrt {a} d}}{\sqrt {e} \sqrt {b d-a e}}\)

Input:

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

Output:

-((-((d^2*x*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2])/(Sqrt[a]*d + Sqrt[e 
]*Sqrt[b*d - a*e]*x^2)) + (a^(1/4)*Sqrt[d]*(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d - 
 a*e]*x^2)*Sqrt[(d^2*(a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2))/(Sqrt[a]*d + S 
qrt[e]*Sqrt[b*d - a*e]*x^2)^2]*EllipticE[2*ArcTan[(e^(1/4)*(b*d - a*e)^(1/ 
4)*x)/(a^(1/4)*Sqrt[d])], (2 - (b*d)/(Sqrt[a]*Sqrt[e]*Sqrt[b*d - a*e]))/4] 
)/(e^(1/4)*(b*d - a*e)^(1/4)*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2]))/( 
Sqrt[e]*Sqrt[b*d - a*e])) + (a^(1/4)*Sqrt[d]*(Sqrt[a]*d + Sqrt[e]*Sqrt[b*d 
 - a*e]*x^2)*Sqrt[(d^2*(a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2))/(Sqrt[a]*d + 
 Sqrt[e]*Sqrt[b*d - a*e]*x^2)^2]*EllipticF[2*ArcTan[(e^(1/4)*(b*d - a*e)^( 
1/4)*x)/(a^(1/4)*Sqrt[d])], (2 - (b*d)/(Sqrt[a]*Sqrt[e]*Sqrt[b*d - a*e]))/ 
4])/(2*e^(3/4)*(b*d - a*e)^(3/4)*Sqrt[a + b*x^2 + (e*(b*d - a*e)*x^4)/d^2] 
)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1459

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 2]}, Simp[1/q   Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[1/q 
 Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c}, x] && 
 NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

Maple [A] (verified)

Time = 1.27 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.17

method	result	size
default	\(-\frac {2 a \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}\, \left (b +\frac {2 a e -b d}{d}\right )}\)	\(202\)
elliptic	\(-\frac {2 a \sqrt {1-\frac {\left (a e -b d \right ) x^{2}}{a d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {a e -b d}{a d}}, \sqrt {-1+\frac {b e}{d \left (-\frac {a \,e^{2}}{d^{2}}+\frac {e b}{d}\right )}}\right )\right )}{\sqrt {\frac {a e -b d}{a d}}\, \sqrt {-\frac {x^{4} a \,e^{2}}{d^{2}}+\frac {e \,x^{4} b}{d}+b \,x^{2}+a}\, \left (b +\frac {2 a e -b d}{d}\right )}\)	\(202\)

Input:

int(x^2/(a+b*x^2+(-a*e^2+b*d*e)*x^4/d^2)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-2*a/((a*e-b*d)/a/d)^(1/2)*(1-(a*e-b*d)/a/d*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/( 
-x^4*a/d^2*e^2+e/d*x^4*b+b*x^2+a)^(1/2)/(b+(2*a*e-b*d)/d)*(EllipticF(x*((a 
*e-b*d)/a/d)^(1/2),(-1+b*e/d/(-a/d^2*e^2+e/d*b))^(1/2))-EllipticE(x*((a*e- 
b*d)/a/d)^(1/2),(-1+b*e/d/(-a/d^2*e^2+e/d*b))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.32 \[ \int \frac {x^2}{\sqrt {a+b x^2+\frac {\left (b d e-a e^2\right ) x^4}{d^2}}} \, dx=-\frac {\sqrt {b d e - a e^{2}} a \sqrt {-\frac {a d}{b d - a e}} d^{2} x E(\arcsin \left (\frac {\sqrt {-\frac {a d}{b d - a e}}}{x}\right )\,|\,\frac {b d - a e}{a e}) - \sqrt {b d e - a e^{2}} a \sqrt {-\frac {a d}{b d - a e}} d^{2} x F(\arcsin \left (\frac {\sqrt {-\frac {a d}{b d - a e}}}{x}\right )\,|\,\frac {b d - a e}{a e}) - {\left (b d^{3} - a d^{2} e\right )} \sqrt {\frac {b d^{2} x^{2} + {\left (b d e - a e^{2}\right )} x^{4} + a d^{2}}{d^{2}}}}{{\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )} x} \] Input:

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

Output:

-(sqrt(b*d*e - a*e^2)*a*sqrt(-a*d/(b*d - a*e))*d^2*x*elliptic_e(arcsin(sqr 
t(-a*d/(b*d - a*e))/x), (b*d - a*e)/(a*e)) - sqrt(b*d*e - a*e^2)*a*sqrt(-a 
*d/(b*d - a*e))*d^2*x*elliptic_f(arcsin(sqrt(-a*d/(b*d - a*e))/x), (b*d - 
a*e)/(a*e)) - (b*d^3 - a*d^2*e)*sqrt((b*d^2*x^2 + (b*d*e - a*e^2)*x^4 + a* 
d^2)/d^2))/((b^2*d^2*e - 2*a*b*d*e^2 + a^2*e^3)*x)
 

Sympy [F]

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

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

Output:

Integral(x**2/sqrt(-(1 + e*x**2/d)*(-a + a*e*x**2/d - b*x**2)), x)
 

Maxima [F]

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

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

Output:

integrate(x^2/sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a), x)
 

Giac [F]

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

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

Output:

integrate(x^2/sqrt(b*x^2 + (b*d*e - a*e^2)*x^4/d^2 + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(d + e*x**2)*sqrt(a*d - a*e*x**2 + b*d*x**2)*x**2)/(a*d**2 - a*e* 
*2*x**4 + b*d**2*x**2 + b*d*e*x**4),x)*d