\(\int \frac {1}{x^4 (a+b x^4) \sqrt {c+d x^4}} \, dx\) [248]

Optimal result

Integrand size = 24, antiderivative size = 662 \[ \int \frac {1}{x^4 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=-\frac {\sqrt {c+d x^4}}{3 a c x^3}-\frac {b^{5/4} \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 (-a)^{7/4} \sqrt {-b c+a d}}-\frac {b^{5/4} \text {arctanh}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{4 (-a)^{7/4} \sqrt {-b c+a d}}-\frac {d^{3/4} (4 b c+a d) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{6 a c^{5/4} (b c+a d) \sqrt {c+d x^4}}-\frac {b \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 a^2 \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {b \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right )^2}{4 \sqrt {-a} \sqrt {b} \sqrt {c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{8 a^2 \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} \sqrt {c+d x^4}} \] Output:

-1/3*(d*x^4+c)^(1/2)/a/c/x^3-1/4*b^(5/4)*arctan((a*d-b*c)^(1/2)*x/(-a)^(1/ 
4)/b^(1/4)/(d*x^4+c)^(1/2))/(-a)^(7/4)/(a*d-b*c)^(1/2)-1/4*b^(5/4)*arctanh 
((a*d-b*c)^(1/2)*x/(-a)^(1/4)/b^(1/4)/(d*x^4+c)^(1/2))/(-a)^(7/4)/(a*d-b*c 
)^(1/2)-1/6*d^(3/4)*(a*d+4*b*c)*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+ 
d^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(d^(1/4)*x/c^(1/4)),1/2*2^(1 
/2))/a/c^(5/4)/(a*d+b*c)/(d*x^4+c)^(1/2)-1/8*b*(b^(1/2)*c^(1/2)+(-a)^(1/2) 
*d^(1/2))*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^(1/2)*x^2)^2)^(1/2)* 
EllipticPi(sin(2*arctan(d^(1/4)*x/c^(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^2/c^(1/4)/ 
(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))/d^(1/4)/(d*x^4+c)^(1/2)-1/8*b*(b^(1/2 
)*c^(1/2)-(-a)^(1/2)*d^(1/2))*(c^(1/2)+d^(1/2)*x^2)*((d*x^4+c)/(c^(1/2)+d^ 
(1/2)*x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(d^(1/4)*x/c^(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^2/c^(1/4)/(b^(1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))/d^(1/4)/(d*x^4+c)^ 
(1/2)
 

Mathematica [C] (warning: unable to verify)

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

Time = 10.22 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.51 \[ \int \frac {1}{x^4 \left (a+b x^4\right ) \sqrt {c+d x^4}} \, dx=\frac {-b d x^8 \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+\frac {5 a \left (-5 a c \left (a c+4 b c x^4+2 a d x^4+b d x^8\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+2 x^4 \left (a+b x^4\right ) \left (c+d x^4\right ) \left (2 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )\right )}{\left (a+b x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )-2 x^4 \left (2 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )\right )}}{15 a^2 c x^3 \sqrt {c+d x^4}} \] Input:

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

Output:

(-(b*d*x^8*Sqrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -( 
(b*x^4)/a)]) + (5*a*(-5*a*c*(a*c + 4*b*c*x^4 + 2*a*d*x^4 + b*d*x^8)*Appell 
F1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] + 2*x^4*(a + b*x^4)*(c + 
d*x^4)*(2*b*c*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d 
*AppellF1[5/4, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)])))/((a + b*x^4)*(5 
*a*c*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)] - 2*x^4*(2*b*c 
*AppellF1[5/4, 1/2, 2, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + a*d*AppellF1[5/4 
, 3/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]))))/(15*a^2*c*x^3*Sqrt[c + d*x^ 
4])
 

Rubi [A] (warning: unable to verify)

Time = 2.30 (sec) , antiderivative size = 988, normalized size of antiderivative = 1.49, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {980, 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.

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

\(\Big \downarrow \) 980

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1021

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 925

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

\(\Big \downarrow \) 1541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 2221

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

\(\Big \downarrow \) 2223

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

Input:

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

Output:

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

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 980

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q 
 + 1)/(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^n*(m + 1))   Int[(e*x)^(m + n)*( 
a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) 
 + b*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, 
q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, 
b, c, d, e, m, 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 = 4.18 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.44

method	result	size
default	\(\frac {-\frac {\sqrt {d \,x^{4}+c}}{3 c \,x^{3}}-\frac {d \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{3 c \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}}{a}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 a}\)	\(288\)
risch	\(-\frac {\sqrt {d \,x^{4}+c}}{3 a c \,x^{3}}-\frac {\frac {d \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}+\frac {3 c \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8}}{3 a c}\)	\(289\)
elliptic	\(-\frac {\sqrt {d \,x^{4}+c}}{3 a c \,x^{3}}-\frac {d \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{3 c a \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 a}\)	\(289\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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