\(\int \frac {\sqrt {a+b x^4}}{(c+d x^4)^2} \, dx\) [31]

Optimal result

Integrand size = 21, antiderivative size = 694 \[ \int \frac {\sqrt {a+b x^4}}{\left (c+d x^4\right )^2} \, dx=\frac {x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right )}-\frac {(b c-3 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{16 (-c)^{7/4} d^{3/4} \sqrt {b c-a d}}-\frac {(b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{16 (-c)^{7/4} d^{3/4} \sqrt {b c-a d}}+\frac {a^{3/4} b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 c (b c+a d) \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) (b c-3 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{b} c^2 \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) d \sqrt {a+b x^4}}+\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) (b c-3 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{b} c^2 \left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) d \sqrt {a+b x^4}} \] Output:

1/4*x*(b*x^4+a)^(1/2)/c/(d*x^4+c)-1/16*(-3*a*d+b*c)*arctan((-a*d+b*c)^(1/2 
)*x/(-c)^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/(-c)^(7/4)/d^(3/4)/(-a*d+b*c)^(1/2 
)-1/16*(-3*a*d+b*c)*arctanh((-a*d+b*c)^(1/2)*x/(-c)^(1/4)/d^(1/4)/(b*x^4+a 
)^(1/2))/(-c)^(7/4)/d^(3/4)/(-a*d+b*c)^(1/2)+1/2*a^(3/4)*b^(3/4)*(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*a 
rctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/c/(a*d+b*c)/(b*x^4+a)^(1/2)+1/32*(b^ 
(1/2)*(-c)^(1/2)+a^(1/2)*d^(1/2))*(-3*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)*EllipticPi(sin(2*arctan(b^(1/4)*x/a^( 
1/4))),-1/4*(b^(1/2)*(-c)^(1/2)-a^(1/2)*d^(1/2))^2/a^(1/2)/b^(1/2)/(-c)^(1 
/2)/d^(1/2),1/2*2^(1/2))/a^(1/4)/b^(1/4)/c^2/(b^(1/2)*(-c)^(1/2)-a^(1/2)*d 
^(1/2))/d/(b*x^4+a)^(1/2)+1/32*(b^(1/2)*(-c)^(1/2)-a^(1/2)*d^(1/2))*(-3*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)*Ell 
ipticPi(sin(2*arctan(b^(1/4)*x/a^(1/4))),1/4*(b^(1/2)*(-c)^(1/2)+a^(1/2)*d 
^(1/2))^2/a^(1/2)/b^(1/2)/(-c)^(1/2)/d^(1/2),1/2*2^(1/2))/a^(1/4)/b^(1/4)/ 
c^2/(b^(1/2)*(-c)^(1/2)+a^(1/2)*d^(1/2))/d/(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 = 10.39 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {a+b x^4}}{\left (c+d x^4\right )^2} \, dx=\frac {x \left (\frac {b x^4 \sqrt {1+\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{c^2}+\frac {5 \left (\frac {a+b x^4}{c}+\frac {15 a^2 \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )-2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )}\right )}{c+d x^4}\right )}{20 \sqrt {a+b x^4}} \] Input:

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

Output:

(x*((b*x^4*Sqrt[1 + (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), -( 
(d*x^4)/c)])/c^2 + (5*((a + b*x^4)/c + (15*a^2*AppellF1[1/4, 1/2, 1, 5/4, 
-((b*x^4)/a), -((d*x^4)/c)])/(5*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a 
), -((d*x^4)/c)] - 2*x^4*(2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), - 
((d*x^4)/c)] + b*c*AppellF1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)]) 
)))/(c + d*x^4)))/(20*Sqrt[a + b*x^4])
 

Rubi [A] (verified)

Time = 2.26 (sec) , antiderivative size = 1001, normalized size of antiderivative = 1.44, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {929, 25, 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.

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

\(\Big \downarrow \) 929

\(\displaystyle \frac {x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right )}-\frac {\int -\frac {b x^4+3 a}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{4 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {b x^4+3 a}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{4 c}+\frac {x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right )}\)

\(\Big \downarrow \) 1021

\(\displaystyle \frac {\frac {b \int \frac {1}{\sqrt {b x^4+a}}dx}{d}-\frac {(b c-3 a d) \int \frac {1}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{d}}{4 c}+\frac {x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right )}\)

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 1541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

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

\(\Big \downarrow \) 2223

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

Input:

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

Output:

(x*Sqrt[a + b*x^4])/(4*c*(c + d*x^4)) + ((b^(3/4)*(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)*d*Sqrt[a + b*x^4]) - ((b*c - 3*a*d)*(((b^(1/4) 
*c*(Sqrt[b] + (Sqrt[a]*Sqrt[d])/Sqrt[-c])*(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*c + a*d)*Sqrt[a + b*x^4]) + ((Sqrt[b]*Sqrt[-c] + Sq 
rt[a]*Sqrt[d])*Sqrt[d]*(-1/2*((-c)^(3/4)*(Sqrt[b] - (Sqrt[a]*Sqrt[d])/Sqrt 
[-c])*ArcTanh[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/( 
d^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[a] + (Sqrt[b]*Sqrt[-c])/Sqrt[d])*(Sqrt[a 
] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[-1 
/4*(Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])^2/(Sqrt[a]*Sqrt[b]*Sqrt[-c]*Sqrt[d 
]), 2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*Sqrt[a + b*x^4 
])))/(b*c + a*d))/(2*c) + ((b^(1/4)*(Sqrt[b]*c + Sqrt[a]*Sqrt[-c]*Sqrt[d]) 
*(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*c + a*d)*Sqrt[a + 
b*x^4]) - ((Sqrt[b]*Sqrt[-c] - Sqrt[a]*Sqrt[d])*Sqrt[d]*(((-c)^(1/4)*(Sqrt 
[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*ArcTan[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^( 
1/4)*Sqrt[a + b*x^4])])/(2*d^(1/4)*Sqrt[b*c - a*d]) + ((Sqrt[a] - (Sqrt[b] 
*Sqrt[-c])/Sqrt[d])*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sq 
rt[b]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])^2/(4*Sqrt...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 929

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-x)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*n*(p + 1))), x] + Simp[1 
/(a*n*(p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(n*(p + 
 1) + 1) + d*(n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, 
x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, 
 c, d, n, p, q, x]
 

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 = 2.23 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.44

method	result	size
default	\(\frac {x \sqrt {b \,x^{4}+a}}{4 c \left (d \,x^{4}+c \right )}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{4 c d \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d +c \right )}{\sum }\frac {\left (-3 a d +b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c \,d^{2}}\)	\(304\)
elliptic	\(\frac {x \sqrt {b \,x^{4}+a}}{4 c \left (d \,x^{4}+c \right )}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{4 c d \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d +c \right )}{\sum }\frac {\left (-3 a d +b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c \,d^{2}}\)	\(304\)

Input:

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

Output:

1/4*x*(b*x^4+a)^(1/2)/c/(d*x^4+c)+1/4/c/d*b/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I 
*b^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*b^(1/2)*x^2/a^(1/2))^(1/2)/(b*x^4+a)^(1/2 
)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)-1/32/c/d^2*sum((-3*a*d+b*c)/_al 
pha^3*(-1/((a*d-b*c)/d)^(1/2)*arctanh(1/2*(2*_alpha^2*b*x^2+2*a)/((a*d-b*c 
)/d)^(1/2)/(b*x^4+a)^(1/2))+2/(I/a^(1/2)*b^(1/2))^(1/2)*_alpha^3*d/c*(1-I* 
b^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*b^(1/2)*x^2/a^(1/2))^(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/c*d,(-I 
/a^(1/2)*b^(1/2))^(1/2)/(I/a^(1/2)*b^(1/2))^(1/2))),_alpha=RootOf(_Z^4*d+c 
))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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