\(\int \frac {(a-c x^4) \sqrt {a+b x^2+c x^4}}{(a d e+(c d^2+a e^2) x^2+c d e x^4)^2} \, dx\) [11]

Optimal result

Integrand size = 54, antiderivative size = 140 \[ \int \frac {\left (a-c x^4\right ) \sqrt {a+b x^2+c x^4}}{\left (a d e+\left (c d^2+a e^2\right ) x^2+c d e x^4\right )^2} \, dx=\frac {x \sqrt {a+b x^2+c x^4}}{2 d e \left (a d e+\left (c d^2+a e^2\right ) x^2+c d e x^4\right )}+\frac {\arctan \left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 d^{3/2} e^{3/2} \sqrt {c d^2-b d e+a e^2}} \] Output:

1/2*x*(c*x^4+b*x^2+a)^(1/2)/d/e/(a*d*e+(a*e^2+c*d^2)*x^2+c*d*e*x^4)+1/2*ar 
ctan((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^ 
(3/2)/e^(3/2)/(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 = 13.85 (sec) , antiderivative size = 1265, normalized size of antiderivative = 9.04 \[ \int \frac {\left (a-c x^4\right ) \sqrt {a+b x^2+c x^4}}{\left (a d e+\left (c d^2+a e^2\right ) x^2+c d e x^4\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 3.58 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {7293, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\arctan \left (\frac {x \sqrt {a e^2-b d e+c d^2}}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 d^{3/2} e^{3/2} \sqrt {a e^2-b d e+c d^2}}+\frac {c x \sqrt {a+b x^2+c x^4}}{2 e \left (c d^2-a e^2\right ) \left (a e+c d x^2\right )}-\frac {x \sqrt {a+b x^2+c x^4}}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 4.05 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.86

method	result	size
default	\(\frac {\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x}{c d e \,x^{4}+a \,e^{2} x^{2}+c \,d^{2} x^{2}+a d e}-\frac {\arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}}{2 d e}\)	\(121\)
pseudoelliptic	\(\frac {\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x}{c d e \,x^{4}+a \,e^{2} x^{2}+c \,d^{2} x^{2}+a d e}-\frac {\arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{\sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}}{2 d e}\)	\(121\)
elliptic	\(\frac {\left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {2}}{4 d e x \left (\frac {d e \left (c \,x^{4}+b \,x^{2}+a \right )}{2 x^{2}}+\frac {a \,e^{2}}{2}-\frac {b d e}{2}+\frac {c \,d^{2}}{2}\right )}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, d e}{x \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right )}{2 d e \sqrt {\left (a \,e^{2}-b d e +c \,d^{2}\right ) d e}}\right ) \sqrt {2}}{2}\)	\(148\)

Input:

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

Output:

1/2/d/e*((c*x^4+b*x^2+a)^(1/2)*x/(c*d*e*x^4+a*e^2*x^2+c*d^2*x^2+a*d*e)-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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (120) = 240\).

Time = 26.85 (sec) , antiderivative size = 827, normalized size of antiderivative = 5.91 \[ \int \frac {\left (a-c x^4\right ) \sqrt {a+b x^2+c x^4}}{\left (a d e+\left (c d^2+a e^2\right ) x^2+c d e x^4\right )^2} \, dx=\left [\frac {4 \, {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} \sqrt {c x^{4} + b x^{2} + a} x - {\left (c d e x^{4} + a d e + {\left (c d^{2} + a e^{2}\right )} x^{2}\right )} \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 )}{8 \, {\left (a c d^{5} e^{3} - a b d^{4} e^{4} + a^{2} d^{3} e^{5} + {\left (c^{2} d^{5} e^{3} - b c d^{4} e^{4} + a c d^{3} e^{5}\right )} x^{4} + {\left (c^{2} d^{6} e^{2} - b c d^{5} e^{3} + 2 \, a c d^{4} e^{4} - a b d^{3} e^{5} + a^{2} d^{2} e^{6}\right )} x^{2}\right )}}, \frac {2 \, {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} \sqrt {c x^{4} + b x^{2} + a} x + {\left (c d e x^{4} + a d e + {\left (c d^{2} + a e^{2}\right )} x^{2}\right )} \sqrt {c d^{3} e - b d^{2} e^{2} + a d e^{3}} \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 )}{4 \, {\left (a c d^{5} e^{3} - a b d^{4} e^{4} + a^{2} d^{3} e^{5} + {\left (c^{2} d^{5} e^{3} - b c d^{4} e^{4} + a c d^{3} e^{5}\right )} x^{4} + {\left (c^{2} d^{6} e^{2} - b c d^{5} e^{3} + 2 \, a c d^{4} e^{4} - a b d^{3} e^{5} + a^{2} d^{2} e^{6}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(a + b*x**2 + c*x**4)*b*x + int(sqrt(a + b*x**2 + c*x**4)/(a**4*d* 
*2*e**4 + 2*a**4*d*e**5*x**2 + a**4*e**6*x**4 - 2*a**3*b*d**3*e**3 - 3*a** 
3*b*d**2*e**4*x**2 + a**3*b*e**6*x**6 + a**3*c*d**4*e**2 + 4*a**3*c*d**3*e 
**3*x**2 + 6*a**3*c*d**2*e**4*x**4 + 4*a**3*c*d*e**5*x**6 + a**3*c*e**6*x* 
*8 - 2*a**2*b**2*d**3*e**3*x**2 - 4*a**2*b**2*d**2*e**4*x**4 - 2*a**2*b**2 
*d*e**5*x**6 - 3*a**2*b*c*d**4*e**2*x**2 - 6*a**2*b*c*d**3*e**3*x**4 - 3*a 
**2*b*c*d**2*e**4*x**6 + 2*a**2*c**2*d**5*e*x**2 + 6*a**2*c**2*d**4*e**2*x 
**4 + 8*a**2*c**2*d**3*e**3*x**6 + 6*a**2*c**2*d**2*e**4*x**8 + 2*a**2*c** 
2*d*e**5*x**10 - 4*a*b**2*c*d**4*e**2*x**4 - 8*a*b**2*c*d**3*e**3*x**6 - 4 
*a*b**2*c*d**2*e**4*x**8 - 3*a*b*c**2*d**4*e**2*x**6 - 6*a*b*c**2*d**3*e** 
3*x**8 - 3*a*b*c**2*d**2*e**4*x**10 + a*c**3*d**6*x**4 + 4*a*c**3*d**5*e*x 
**6 + 6*a*c**3*d**4*e**2*x**8 + 4*a*c**3*d**3*e**3*x**10 + a*c**3*d**2*e** 
4*x**12 - 2*b**2*c**2*d**5*e*x**6 - 4*b**2*c**2*d**4*e**2*x**8 - 2*b**2*c* 
*2*d**3*e**3*x**10 + b*c**3*d**6*x**6 - 3*b*c**3*d**4*e**2*x**10 - 2*b*c** 
3*d**3*e**3*x**12 + c**4*d**6*x**8 + 2*c**4*d**5*e*x**10 + c**4*d**4*e**2* 
x**12),x)*a**5*d*e**5 + int(sqrt(a + b*x**2 + c*x**4)/(a**4*d**2*e**4 + 2* 
a**4*d*e**5*x**2 + a**4*e**6*x**4 - 2*a**3*b*d**3*e**3 - 3*a**3*b*d**2*e** 
4*x**2 + a**3*b*e**6*x**6 + a**3*c*d**4*e**2 + 4*a**3*c*d**3*e**3*x**2 + 6 
*a**3*c*d**2*e**4*x**4 + 4*a**3*c*d*e**5*x**6 + a**3*c*e**6*x**8 - 2*a**2* 
b**2*d**3*e**3*x**2 - 4*a**2*b**2*d**2*e**4*x**4 - 2*a**2*b**2*d*e**5*x...