\(\int \frac {x^2}{\sqrt {a+\frac {(c d^2+a e^2) x^2}{d e}+c x^4}} \, dx\) [710]

Optimal result

Integrand size = 36, antiderivative size = 157 \[ \int \frac {x^2}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=\frac {x \left (a e+c d x^2\right )}{c d \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}}-\frac {a \sqrt {e} \sqrt {\frac {d \left (a e+c d x^2\right )}{a e \left (d+e x^2\right )}} \left (d+e x^2\right ) E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|1-\frac {c d^2}{a e^2}\right )}{c d^{3/2} \sqrt {a+\left (\frac {c d}{e}+\frac {a e}{d}\right ) x^2+c x^4}} \] Output:

x*(c*d*x^2+a*e)/c/d/(a+(c*d/e+a*e/d)*x^2+c*x^4)^(1/2)-a*e^(1/2)*(d*(c*d*x^ 
2+a*e)/a/e/(e*x^2+d))^(1/2)*(e*x^2+d)*EllipticE(e^(1/2)*x/d^(1/2)/(1+e*x^2 
/d)^(1/2),(1-c*d^2/a/e^2)^(1/2))/c/d^(3/2)/(a+(c*d/e+a*e/d)*x^2+c*x^4)^(1/ 
2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.90 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.94 \[ \int \frac {x^2}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=-\frac {i d \sqrt {1+\frac {c d x^2}{a e}} \sqrt {1+\frac {e x^2}{d}} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {c d}{a e}} x\right )|\frac {a e^2}{c d^2}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {c d}{a e}} x\right ),\frac {a e^2}{c d^2}\right )\right )}{\sqrt {\frac {c d}{a e}} e \sqrt {\frac {\left (a e+c d x^2\right ) \left (d+e x^2\right )}{d e}}} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(354\) vs. \(2(157)=314\).

Time = 0.73 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.25, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 1459

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

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

Output:

-((-((x*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2) 
) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + ((c*d)/e + (a*e)/d)*x^2 + c 
*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 
(2 - ((c*d)/e + (a*e)/d)/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + ((c*d)/e 
 + (a*e)/d)*x^2 + c*x^4]))/Sqrt[c]) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqr 
t[(a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellipti 
cF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - ((c*d)/e + (a*e)/d)/(Sqrt[a]*Sqrt[c 
]))/4])/(2*c^(3/4)*Sqrt[a + ((c*d)/e + (a*e)/d)*x^2 + c*x^4])
 

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 = 1.54 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.20

method	result	size
default	\(-\frac {2 a \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )\right )}{\sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}\, \left (\frac {c d}{e}+\frac {a e}{d}+\frac {a \,e^{2}-c \,d^{2}}{d e}\right )}\)	\(188\)
elliptic	\(-\frac {2 a \sqrt {1+\frac {x^{2} d c}{a e}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c d}{a e}}, \sqrt {-1+\frac {\left (\frac {c d}{e}+\frac {a e}{d}\right ) e}{d c}}\right )\right )}{\sqrt {-\frac {c d}{a e}}\, \sqrt {a +\frac {x^{2} e a}{d}+\frac {x^{2} d c}{e}+c \,x^{4}}\, \left (\frac {c d}{e}+\frac {a e}{d}+\frac {a \,e^{2}-c \,d^{2}}{d e}\right )}\)	\(188\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{\sqrt {a+\frac {\left (c d^2+a e^2\right ) x^2}{d e}+c x^4}} \, dx=-\frac {\sqrt {c} d x \sqrt {-\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{x}\right )\,|\,\frac {a e^{2}}{c d^{2}}) - \sqrt {c} d x \sqrt {-\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{x}\right )\,|\,\frac {a e^{2}}{c d^{2}}) - e \sqrt {\frac {c d e x^{4} + a d e + {\left (c d^{2} + a e^{2}\right )} x^{2}}{d e}}}{c e x} \] Input:

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

Output:

-(sqrt(c)*d*x*sqrt(-d/e)*elliptic_e(arcsin(sqrt(-d/e)/x), a*e^2/(c*d^2)) - 
 sqrt(c)*d*x*sqrt(-d/e)*elliptic_f(arcsin(sqrt(-d/e)/x), a*e^2/(c*d^2)) - 
e*sqrt((c*d*e*x^4 + a*d*e + (c*d^2 + a*e^2)*x^2)/(d*e)))/(c*e*x)
 

Sympy [F]

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

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

Output:

Integral(x**2/sqrt(a + a*e*x**2/d + c*d*x**2/e + c*x**4), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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