\(\int \frac {a-c x^4}{(a e+c d x^2) (d+e x^2) \sqrt {a+b x^2+c x^4}} \, dx\) [18]

Optimal result

Integrand size = 46, antiderivative size = 80 \[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {e} \sqrt {c d^2-b d e+a e^2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.19 (sec) , antiderivative size = 383, normalized size of antiderivative = 4.79 \[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\frac {i \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) d}{2 a e},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} d e \sqrt {a+b x^2+c x^4}} \] Input:

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

Output:

(I*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 
+ (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/( 
b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c] 
)] - EllipticPi[((b + Sqrt[b^2 - 4*a*c])*d)/(2*a*e), I*ArcSinh[Sqrt[2]*Sqr 
t[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4 
*a*c])] - EllipticPi[((b + Sqrt[b^2 - 4*a*c])*e)/(2*c*d), I*ArcSinh[Sqrt[2 
]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^ 
2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*Sqrt[a + b*x^2 
 + c*x^4])
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2537, 218}

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.

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

\(\Big \downarrow \) 2537

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 2537

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, 
x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4], d = Coeff[1/u, x, 0], e = Co 
eff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Simp[A   Subst[Int[1/(d - (b*d - a*e 
)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; 
 FreeQ[{A, B}, x] && PolyQ[v, x^2, 2] && PolyQ[1/u, x^2, 2]
 

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.82

method	result	size
default	\(-\frac {\arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\)	\(66\)
elliptic	\(-\frac {\arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\)	\(66\)
pseudoelliptic	\(-\frac {\arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\)	\(66\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 27.61 (sec) , antiderivative size = 472, normalized size of antiderivative = 5.90 \[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\left [-\frac {\sqrt {-c d^{3} e + b d^{2} e^{2} - a d e^{3}} \log \left (-\frac {c^{2} d^{2} e^{2} x^{8} - 2 \, {\left (3 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + 3 \, a c d e^{3}\right )} x^{6} + a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} - 8 \, b c d^{3} e - 8 \, a b d e^{3} + a^{2} e^{4} + 4 \, {\left (2 \, b^{2} + a c\right )} d^{2} e^{2}\right )} x^{4} - 2 \, {\left (3 \, a c d^{3} e - 4 \, a b d^{2} e^{2} + 3 \, a^{2} d e^{3}\right )} x^{2} + 4 \, {\left (c d e x^{5} + a d e x - {\left (c d^{2} - 2 \, b d e + a e^{2}\right )} x^{3}\right )} \sqrt {-c d^{3} e + b d^{2} e^{2} - a d e^{3}} \sqrt {c x^{4} + b x^{2} + a}}{c^{2} d^{2} e^{2} x^{8} + 2 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{6} + a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{4} + 2 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x^{2}}\right )}{4 \, {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )}}, \frac {\arctan \left (\frac {2 \, \sqrt {c d^{3} e - b d^{2} e^{2} + a d e^{3}} \sqrt {c x^{4} + b x^{2} + a} x}{c d e x^{4} + a d e - {\left (c d^{2} - 2 \, b d e + a e^{2}\right )} x^{2}}\right )}{2 \, \sqrt {c d^{3} e - b d^{2} e^{2} + a d e^{3}}}\right ] \] Input:

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

Output:

[-1/4*sqrt(-c*d^3*e + b*d^2*e^2 - a*d*e^3)*log(-(c^2*d^2*e^2*x^8 - 2*(3*c^ 
2*d^3*e - 4*b*c*d^2*e^2 + 3*a*c*d*e^3)*x^6 + a^2*d^2*e^2 + (c^2*d^4 - 8*b* 
c*d^3*e - 8*a*b*d*e^3 + a^2*e^4 + 4*(2*b^2 + a*c)*d^2*e^2)*x^4 - 2*(3*a*c* 
d^3*e - 4*a*b*d^2*e^2 + 3*a^2*d*e^3)*x^2 + 4*(c*d*e*x^5 + a*d*e*x - (c*d^2 
 - 2*b*d*e + a*e^2)*x^3)*sqrt(-c*d^3*e + b*d^2*e^2 - a*d*e^3)*sqrt(c*x^4 + 
 b*x^2 + a))/(c^2*d^2*e^2*x^8 + 2*(c^2*d^3*e + a*c*d*e^3)*x^6 + a^2*d^2*e^ 
2 + (c^2*d^4 + 4*a*c*d^2*e^2 + a^2*e^4)*x^4 + 2*(a*c*d^3*e + a^2*d*e^3)*x^ 
2))/(c*d^3*e - b*d^2*e^2 + a*d*e^3), 1/2*arctan(2*sqrt(c*d^3*e - b*d^2*e^2 
 + a*d*e^3)*sqrt(c*x^4 + b*x^2 + a)*x/(c*d*e*x^4 + a*d*e - (c*d^2 - 2*b*d* 
e + a*e^2)*x^2))/sqrt(c*d^3*e - b*d^2*e^2 + a*d*e^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {a-c x^4}{\left (a e+c d x^2\right ) \left (d+e x^2\right ) \sqrt {a+b x^2+c x^4}} \, dx=\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} d e \,x^{8}+a c \,e^{2} x^{6}+b c d e \,x^{6}+c^{2} d^{2} x^{6}+a b \,e^{2} x^{4}+2 a c d e \,x^{4}+b c \,d^{2} x^{4}+a^{2} e^{2} x^{2}+a b d e \,x^{2}+a c \,d^{2} x^{2}+a^{2} d e}d x \right ) a -\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{4}}{c^{2} d e \,x^{8}+a c \,e^{2} x^{6}+b c d e \,x^{6}+c^{2} d^{2} x^{6}+a b \,e^{2} x^{4}+2 a c d e \,x^{4}+b c \,d^{2} x^{4}+a^{2} e^{2} x^{2}+a b d e \,x^{2}+a c \,d^{2} x^{2}+a^{2} d e}d x \right ) c \] Input:

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

Output:

int(sqrt(a + b*x**2 + c*x**4)/(a**2*d*e + a**2*e**2*x**2 + a*b*d*e*x**2 + 
a*b*e**2*x**4 + a*c*d**2*x**2 + 2*a*c*d*e*x**4 + a*c*e**2*x**6 + b*c*d**2* 
x**4 + b*c*d*e*x**6 + c**2*d**2*x**6 + c**2*d*e*x**8),x)*a - int((sqrt(a + 
 b*x**2 + c*x**4)*x**4)/(a**2*d*e + a**2*e**2*x**2 + a*b*d*e*x**2 + a*b*e* 
*2*x**4 + a*c*d**2*x**2 + 2*a*c*d*e*x**4 + a*c*e**2*x**6 + b*c*d**2*x**4 + 
 b*c*d*e*x**6 + c**2*d**2*x**6 + c**2*d*e*x**8),x)*c