\(\int \frac {c+d x^4}{\sqrt {a+b x^4} (e+f x^4)} \, dx\) [5]

Optimal result

Integrand size = 28, antiderivative size = 670 \[ \int \frac {c+d x^4}{\sqrt {a+b x^4} \left (e+f x^4\right )} \, dx=\frac {(d e-c f) \arctan \left (\frac {\sqrt {b e-a f} x}{\sqrt [4]{-e} \sqrt [4]{f} \sqrt {a+b x^4}}\right )}{4 (-e)^{3/4} f^{3/4} \sqrt {b e-a f}}+\frac {(d e-c f) \text {arctanh}\left (\frac {\sqrt {b e-a f} x}{\sqrt [4]{-e} \sqrt [4]{f} \sqrt {a+b x^4}}\right )}{4 (-e)^{3/4} f^{3/4} \sqrt {b e-a f}}+\frac {(b c+a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} (b e+a f) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-e}+\sqrt {a} \sqrt {f}\right ) (d e-c f) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {-e}-\sqrt {a} \sqrt {f}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-e} \sqrt {f}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} e \left (\sqrt {b} \sqrt {-e}-\sqrt {a} \sqrt {f}\right ) f \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-e}-\sqrt {a} \sqrt {f}\right ) (d e-c f) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {-e}+\sqrt {a} \sqrt {f}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-e} \sqrt {f}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} e \left (\sqrt {b} \sqrt {-e}+\sqrt {a} \sqrt {f}\right ) f \sqrt {a+b x^4}} \] Output:

1/4*(-c*f+d*e)*arctan((-a*f+b*e)^(1/2)*x/(-e)^(1/4)/f^(1/4)/(b*x^4+a)^(1/2 
))/(-e)^(3/4)/f^(3/4)/(-a*f+b*e)^(1/2)+1/4*(-c*f+d*e)*arctanh((-a*f+b*e)^( 
1/2)*x/(-e)^(1/4)/f^(1/4)/(b*x^4+a)^(1/2))/(-e)^(3/4)/f^(3/4)/(-a*f+b*e)^( 
1/2)+1/2*(a*d+b*c)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^ 
2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/ 
b^(1/4)/(a*f+b*e)/(b*x^4+a)^(1/2)+1/8*(b^(1/2)*(-e)^(1/2)+a^(1/2)*f^(1/2)) 
*(-c*f+d*e)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/(a^(1/2)+b^(1/2)*x^2)^2)^(1/2 
)*EllipticPi(sin(2*arctan(b^(1/4)*x/a^(1/4))),-1/4*(b^(1/2)*(-e)^(1/2)-a^( 
1/2)*f^(1/2))^2/a^(1/2)/b^(1/2)/(-e)^(1/2)/f^(1/2),1/2*2^(1/2))/a^(1/4)/b^ 
(1/4)/e/(b^(1/2)*(-e)^(1/2)-a^(1/2)*f^(1/2))/f/(b*x^4+a)^(1/2)+1/8*(b^(1/2 
)*(-e)^(1/2)-a^(1/2)*f^(1/2))*(-c*f+d*e)*(a^(1/2)+b^(1/2)*x^2)*((b*x^4+a)/ 
(a^(1/2)+b^(1/2)*x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(b^(1/4)*x/a^(1/4))) 
,1/4*(b^(1/2)*(-e)^(1/2)+a^(1/2)*f^(1/2))^2/a^(1/2)/b^(1/2)/(-e)^(1/2)/f^( 
1/2),1/2*2^(1/2))/a^(1/4)/b^(1/4)/e/(b^(1/2)*(-e)^(1/2)+a^(1/2)*f^(1/2))/f 
/(b*x^4+a)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.46 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.33 \[ \int \frac {c+d x^4}{\sqrt {a+b x^4} \left (e+f x^4\right )} \, dx=\frac {x \left (d x^4 \sqrt {1+\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {f x^4}{e}\right )-\frac {25 a c e^2 \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {f x^4}{e}\right )}{\left (e+f x^4\right ) \left (-5 a e \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {f x^4}{e}\right )+2 x^4 \left (2 a f \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {f x^4}{e}\right )+b e \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {f x^4}{e}\right )\right )\right )}\right )}{5 e \sqrt {a+b x^4}} \] Input:

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

Output:

(x*(d*x^4*Sqrt[1 + (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -(( 
f*x^4)/e)] - (25*a*c*e^2*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((f*x^4 
)/e)])/((e + f*x^4)*(-5*a*e*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((f* 
x^4)/e)] + 2*x^4*(2*a*f*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((f*x^4) 
/e)] + b*e*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((f*x^4)/e)])))))/(5* 
e*Sqrt[a + b*x^4])
 

Rubi [A] (verified)

Time = 2.11 (sec) , antiderivative size = 966, normalized size of antiderivative = 1.44, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1021, 761, 925, 1541, 27, 761, 2221, 2223}

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.

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

\(\Big \downarrow \) 1021

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 1541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

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

\(\Big \downarrow \) 2223

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

Input:

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

Output:

(d*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ell 
ipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*b^(1/4)*f*Sqrt[a + 
b*x^4]) - ((d*e - c*f)*(((b^(1/4)*e*(Sqrt[b] + (Sqrt[a]*Sqrt[f])/Sqrt[-e]) 
*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Ellip 
ticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(b*e + a*f)*Sqrt[a + 
b*x^4]) + ((Sqrt[b]*Sqrt[-e] + Sqrt[a]*Sqrt[f])*Sqrt[f]*(-1/2*((-e)^(3/4)* 
(Sqrt[b] - (Sqrt[a]*Sqrt[f])/Sqrt[-e])*ArcTanh[(Sqrt[b*e - a*f]*x)/((-e)^( 
1/4)*f^(1/4)*Sqrt[a + b*x^4])])/(f^(1/4)*Sqrt[b*e - a*f]) + ((Sqrt[a] + (S 
qrt[b]*Sqrt[-e])/Sqrt[f])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a 
] + Sqrt[b]*x^2)^2]*EllipticPi[-1/4*(Sqrt[b]*Sqrt[-e] - Sqrt[a]*Sqrt[f])^2 
/(Sqrt[a]*Sqrt[b]*Sqrt[-e]*Sqrt[f]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/ 
(4*a^(1/4)*b^(1/4)*Sqrt[a + b*x^4])))/(b*e + a*f))/(2*e) + ((b^(1/4)*(Sqrt 
[b]*e + Sqrt[a]*Sqrt[-e]*Sqrt[f])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4) 
/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2]) 
/(2*a^(1/4)*(b*e + a*f)*Sqrt[a + b*x^4]) - ((Sqrt[b]*Sqrt[-e] - Sqrt[a]*Sq 
rt[f])*Sqrt[f]*(((-e)^(1/4)*(Sqrt[b]*Sqrt[-e] + Sqrt[a]*Sqrt[f])*ArcTan[(S 
qrt[b*e - a*f]*x)/((-e)^(1/4)*f^(1/4)*Sqrt[a + b*x^4])])/(2*f^(1/4)*Sqrt[b 
*e - a*f]) + ((Sqrt[a] - (Sqrt[b]*Sqrt[-e])/Sqrt[f])*(Sqrt[a] + Sqrt[b]*x^ 
2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[-e 
] + Sqrt[a]*Sqrt[f])^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[-e]*Sqrt[f]), 2*ArcTan[(...
 

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 761

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

rule 925

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

rule 1021

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x 
_)^(n_)]), x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* 
e - a*f)/b   Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, 
 d, e, f, n}, x]
 

rule 1541

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

rule 2221

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTan[Rt[c*(d/e 
) + a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[c*(d/e) + a*(e/d), 2])), x] 
+ Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4* 
d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x 
], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && Po 
sQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && PosQ[c*(d/e) + a*(e/d)]
 

rule 2223

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTanh[Rt[(-c)* 
(d/e) - a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[(-c)*(d/e) - a*(e/d), 2] 
)), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^ 
2)]/(4*d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*Arc 
Tan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0 
] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[c*(d/e) + a*(e 
/d)]
 

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.13 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.41

method	result	size
default	\(\frac {d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{f \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}+\frac {\left (c f -d e \right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (f \,\textit {\_Z}^{4}+e \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a f -b e}{f}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} f}{\sqrt {b}\, e}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, e \sqrt {b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 f^{2}}\)	\(273\)
elliptic	\(\frac {d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{f \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (f \,\textit {\_Z}^{4}+e \right )}{\sum }\frac {\left (-c f +d e \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a f -b e}{f}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a f -b e}{f}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} f \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} f}{\sqrt {b}\, e}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, e \sqrt {b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 f^{2}}\)	\(273\)

Input:

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

Output:

d/f/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2) 
*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2), 
I)+1/8*(c*f-d*e)/f^2*sum(1/_alpha^3*(-1/((a*f-b*e)/f)^(1/2)*arctanh(1/2*(2 
*_alpha^2*b*x^2+2*a)/((a*f-b*e)/f)^(1/2)/(b*x^4+a)^(1/2))+2/(I/a^(1/2)*b^( 
1/2))^(1/2)*_alpha^3*f/e*(1-I/a^(1/2)*b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1 
/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*b^(1/2))^(1/2),I*a^ 
(1/2)/b^(1/2)*_alpha^2/e*f,(-I/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^ 
(1/2))),_alpha=RootOf(_Z^4*f+e))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(a + b*x**4)/(a*e + a*f*x**4 + b*e*x**4 + b*f*x**8),x)*c + int((sq 
rt(a + b*x**4)*x**4)/(a*e + a*f*x**4 + b*e*x**4 + b*f*x**8),x)*d