\(\int \frac {1}{(a+b x^4)^{3/2} (c-d x^4)} \, dx\) [51]

Optimal result

Integrand size = 22, antiderivative size = 620 \[ \int \frac {1}{\left (a+b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\frac {b x}{2 a (b c+a d) \sqrt {a+b x^4}}+\frac {d^{5/4} \arctan \left (\frac {\sqrt {b c+a d} x}{\sqrt [4]{c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 c^{3/4} (b c+a d)^{3/2}}+\frac {d^{5/4} \text {arctanh}\left (\frac {\sqrt {b c+a d} x}{\sqrt [4]{c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{4 c^{3/4} (b c+a d)^{3/2}}+\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} (b c-a d) \sqrt {a+b x^4}}-\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2 d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a d) (b c+a d) \sqrt {a+b x^4}}-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {a} \sqrt {d}\right )^2 d \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{8 \sqrt [4]{a} \sqrt [4]{b} c (b c-a d) (b c+a d) \sqrt {a+b x^4}} \] Output:

1/2*b*x/a/(a*d+b*c)/(b*x^4+a)^(1/2)+1/4*d^(5/4)*arctan((a*d+b*c)^(1/2)*x/c 
^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/c^(3/4)/(a*d+b*c)^(3/2)+1/4*d^(5/4)*arctan 
h((a*d+b*c)^(1/2)*x/c^(1/4)/d^(1/4)/(b*x^4+a)^(1/2))/c^(3/4)/(a*d+b*c)^(3/ 
2)+1/4*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*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(5/4)/(-a* 
d+b*c)/(b*x^4+a)^(1/2)-1/8*(b^(1/2)*c^(1/2)+a^(1/2)*d^(1/2))^2*d*(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*ar 
ctan(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/(-a*d+b*c)/(a*d+b*c 
)/(b*x^4+a)^(1/2)-1/8*(b^(1/2)*c^(1/2)-a^(1/2)*d^(1/2))^2*d*(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/(-a*d+b*c)/(a*d+b*c)/(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 = 10.27 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\left (a+b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\frac {-5 a c x \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right ) \left (5 c \left (2 b c+2 a d-b d x^4\right )+b d x^4 \sqrt {1+\frac {b x^4}{a}} \left (-c+d x^4\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )+2 b x^5 \left (c-d x^4\right ) \left (5 c-d x^4 \sqrt {1+\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )\right ) \left (-2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )}{10 a c (b c+a d) \sqrt {a+b x^4} \left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )-b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )} \] Input:

Integrate[1/((a + b*x^4)^(3/2)*(c - d*x^4)),x]
 

Output:

(-5*a*c*x*AppellF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), (d*x^4)/c]*(5*c*(2*b*c 
+ 2*a*d - b*d*x^4) + b*d*x^4*Sqrt[1 + (b*x^4)/a]*(-c + d*x^4)*AppellF1[5/4 
, 1/2, 1, 9/4, -((b*x^4)/a), (d*x^4)/c]) + 2*b*x^5*(c - d*x^4)*(5*c - d*x^ 
4*Sqrt[1 + (b*x^4)/a]*AppellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), (d*x^4)/c]) 
*(-2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), (d*x^4)/c] + b*c*AppellF 
1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), (d*x^4)/c]))/(10*a*c*(b*c + a*d)*Sqrt[a 
+ b*x^4]*(-c + d*x^4)*(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])))
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 942, normalized size of antiderivative = 1.52, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {931, 25, 1021, 761, 925, 27, 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.

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

\(\Big \downarrow \) 931

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1021

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

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

\(\Big \downarrow \) 2223

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

Input:

Int[1/((a + b*x^4)^(3/2)*(c - d*x^4)),x]
 

Output:

(b*x)/(2*a*(b*c + a*d)*Sqrt[a + b*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)*Sqrt[a + b*x^4]) + 2*a*d*(((b^(1/4)*(Sqrt[a] 
 + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*Ar 
cTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[b]*Sqrt[c] - Sqrt[a]*Sqr 
t[d])*Sqrt[a + b*x^4]) - (Sqrt[d]*(-1/2*((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])])/(c^(1/4) 
*d^(1/4)*Sqrt[b*c + a*d]) + ((Sqrt[a]/Sqrt[c] + Sqrt[b]/Sqrt[d])*(Sqrt[a] 
+ Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[-1/4 
*(Sqrt[a]*((Sqrt[b]*Sqrt[c])/Sqrt[a] - Sqrt[d])^2)/(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 
])))/(Sqrt[b]*Sqrt[c] - Sqrt[a]*Sqrt[d]))/(2*Sqrt[c]) + ((b^(1/4)*(Sqrt[a] 
 + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*Ar 
cTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[a]*Sqr 
t[d])*Sqrt[a + b*x^4]) + (Sqrt[d]*(((Sqrt[b]*Sqrt[c] + Sqrt[a]*Sqrt[d])*Ar 
cTanh[(Sqrt[b*c + a*d]*x)/(c^(1/4)*d^(1/4)*Sqrt[a + b*x^4])])/(2*c^(1/4)*d 
^(1/4)*Sqrt[b*c + a*d]) + ((Sqrt[a]/Sqrt[c] - Sqrt[b]/Sqrt[d])*(Sqrt[a] + 
Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticPi[(Sqrt[ 
b]*Sqrt[c] + Sqrt[a]*Sqrt[d])^2/(4*Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]), 2*Arc 
Tan[(b^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*b^(1/4)*Sqrt[a + b*x^4])))/(...
 

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 931

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - 
 a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d))   Int[(a + b*x^n)^(p + 1)*(c 
 + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, 
 x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, 
-1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[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] (verified)

Result contains complex when optimal does not.

Time = 1.65 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.50

method	result	size
default	\(\frac {b x}{2 a \left (a d +b c \right ) \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \left (a d +b c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d +b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d +b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}}{\left (a d +b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\)	\(310\)
elliptic	\(\frac {b x}{2 a \left (a d +b c \right ) \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \left (a d +b c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d -c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d +b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d +b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}}{\left (a d +b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}\)	\(310\)

Input:

int(1/(b*x^4+a)^(3/2)/(-d*x^4+c),x,method=_RETURNVERBOSE)
 

Output:

1/2*b*x/a/(a*d+b*c)/((x^4+a/b)*b)^(1/2)+1/2*b/a/(a*d+b*c)/(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)-1/8*sum(1/(a*d+b* 
c)/_alpha^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. \[ \int \frac {1}{\left (a+b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^4+a)^(3/2)/(-d*x^4+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=- \int \frac {1}{- a c \sqrt {a + b x^{4}} + a d x^{4} \sqrt {a + b x^{4}} - b c x^{4} \sqrt {a + b x^{4}} + b d x^{8} \sqrt {a + b x^{4}}}\, dx \] Input:

integrate(1/(b*x**4+a)**(3/2)/(-d*x**4+c),x)
 

Output:

-Integral(1/(-a*c*sqrt(a + b*x**4) + a*d*x**4*sqrt(a + b*x**4) - b*c*x**4* 
sqrt(a + b*x**4) + b*d*x**8*sqrt(a + b*x**4)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}} \,d x } \] Input:

integrate(1/(b*x^4+a)^(3/2)/(-d*x^4+c),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\int { -\frac {1}{{\left (b x^{4} + a\right )}^{\frac {3}{2}} {\left (d x^{4} - c\right )}} \,d x } \] Input:

integrate(1/(b*x^4+a)^(3/2)/(-d*x^4+c),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\int \frac {1}{{\left (b\,x^4+a\right )}^{3/2}\,\left (c-d\,x^4\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b x^4\right )^{3/2} \left (c-d x^4\right )} \, dx=\int \frac {\sqrt {b \,x^{4}+a}}{-b^{2} d \,x^{12}-2 a b d \,x^{8}+b^{2} c \,x^{8}-a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \] Input:

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

Output:

int(sqrt(a + b*x**4)/(a**2*c - a**2*d*x**4 + 2*a*b*c*x**4 - 2*a*b*d*x**8 + 
 b**2*c*x**8 - b**2*d*x**12),x)