Integrand size = 21, antiderivative size = 723 \[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )^2} \, dx=-\frac {d x \sqrt {a+b x^4}}{4 c (b c-a d) \left (c+d x^4\right )}+\frac {\sqrt [4]{d} (5 b c-3 a d) \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{16 (-c)^{7/4} (b c-a d)^{3/2}}+\frac {\sqrt [4]{d} (5 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt [4]{-c} \sqrt [4]{d} \sqrt {a+b x^4}}\right )}{16 (-c)^{7/4} (b c-a d)^{3/2}}+\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} c (b c+a d) \sqrt {a+b x^4}}-\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) (5 b c-3 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{b} c^2 \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) (b c-a d) \sqrt {a+b x^4}}-\frac {\left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) (5 b c-3 a d) \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {-c} \sqrt {d}},2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{32 \sqrt [4]{a} \sqrt [4]{b} c^2 \left (\sqrt {b} \sqrt {-c}+\sqrt {a} \sqrt {d}\right ) (b c-a d) \sqrt {a+b x^4}} \] Output:
-1/4*d*x*(b*x^4+a)^(1/2)/c/(-a*d+b*c)/(d*x^4+c)+1/16*d^(1/4)*(-3*a*d+5*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) /(-a*d+b*c)^(3/2)+1/16*d^(1/4)*(-3*a*d+5*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)/(-a*d+b*c)^(3/2)+1/2*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)*InverseJa cobiAM(2*arctan(b^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(1/4)/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+5*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))/(-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+5*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))/(- a*d+b*c)/(b*x^4+a)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.35 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )^2} \, dx=\frac {x \left (\frac {b d x^4 \sqrt {1+\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{-b c+a d}+\frac {c \left (25 a c \left (-4 b c+4 a d+b d x^4\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )-10 d x^4 \left (a+b x^4\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}{(b c-a d) \left (c+d x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )\right )}\right )}{20 c^2 \sqrt {a+b x^4}} \] Input:
Integrate[1/(Sqrt[a + b*x^4]*(c + d*x^4)^2),x]
Output:
(x*((b*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)])/(-(b*c) + a*d) + (c*(25*a*c*(-4*b*c + 4*a*d + b*d*x^4)*Appe llF1[1/4, 1/2, 1, 5/4, -((b*x^4)/a), -((d*x^4)/c)] - 10*d*x^4*(a + b*x^4)* (2*a*d*AppellF1[5/4, 1/2, 2, 9/4, -((b*x^4)/a), -((d*x^4)/c)] + b*c*Appell F1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))/((b*c - a*d)*(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*Appell F1[5/4, 3/2, 1, 9/4, -((b*x^4)/a), -((d*x^4)/c)])))))/(20*c^2*Sqrt[a + b*x ^4])
Time = 2.30 (sec) , antiderivative size = 1016, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {931, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle \frac {\int \frac {-b d x^4+4 b c-3 a d}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {(5 b c-3 a d) \int \frac {1}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx-b \int \frac {1}{\sqrt {b x^4+a}}dx}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(5 b c-3 a d) \int \frac {1}{\sqrt {b x^4+a} \left (d x^4+c\right )}dx-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {(5 b c-3 a d) \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{2 c}+\frac {\int \frac {1}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{2 c}\right )-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1541 |
| \(\displaystyle \frac {(5 b c-3 a d) \left (\frac {\frac {\sqrt {b} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}-\frac {\sqrt {a} \sqrt {d} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}+\frac {\frac {\sqrt {b} c \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}+\frac {\sqrt {a} \sqrt {d} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\sqrt {a} \left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}\right )-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(5 b c-3 a d) \left (\frac {\frac {\sqrt {b} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}-\frac {\sqrt {d} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}+\frac {\frac {\sqrt {b} c \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \int \frac {1}{\sqrt {b x^4+a}}dx}{a d+b c}+\frac {\sqrt {d} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}\right )-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {(5 b c-3 a d) \left (\frac {\frac {\sqrt [4]{b} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\sqrt {a} \sqrt {-c} \sqrt {d}+\sqrt {b} c\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} (a d+b c)}-\frac {\sqrt {d} \left (\sqrt {b} \sqrt {-c}-\sqrt {a} \sqrt {d}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {-c}}\right ) \sqrt {b x^4+a}}dx}{a d+b c}}{2 c}+\frac {\frac {\sqrt {d} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {-c}\right ) \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{a d+b c}+\frac {\sqrt [4]{b} c \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-c}}+\sqrt {b}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4} (a d+b c)}}{2 c}\right )-\frac {b^{3/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt {a+b x^4}}}{4 c (b c-a d)}-\frac {d x \sqrt {a+b x^4}}{4 c \left (c+d x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {(5 b c-3 a d) \left (\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}+\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} \int \frac {\sqrt {b} x^2+\sqrt {a}}{\left (\frac {\sqrt {d} x^2}{\sqrt {-c}}+1\right ) \sqrt {b x^4+a}}dx}{b c+a d}}{2 c}\right )-\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}}}{4 c (b c-a d)}-\frac {d x \sqrt {b x^4+a}}{4 c (b c-a d) \left (d x^4+c\right )}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {(5 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 )-\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}}}{4 c (b c-a d)}-\frac {d x \sqrt {b x^4+a}}{4 c (b c-a d) \left (d x^4+c\right )}\) |
Input:
Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)^2),x]
Output:
-1/4*(d*x*Sqrt[a + b*x^4])/(c*(b*c - a*d)*(c + d*x^4)) + (-1/2*(b^(3/4)*(S qrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Elliptic F[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(a^(1/4)*Sqrt[a + b*x^4]) + (5*b*c - 3*a*d)*(((b^(1/4)*c*(Sqrt[b] + (Sqrt[a]*Sqrt[d])/Sqrt[-c])*(Sqrt[a] + Sq rt[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]) + ((Sq rt[b]*Sqrt[-c] + Sqrt[a]*Sqrt[d])*Sqrt[d]*(-1/2*((-c)^(3/4)*(Sqrt[b] - (Sq rt[a]*Sqrt[d])/Sqrt[-c])*ArcTanh[(Sqrt[b*c - a*d]*x)/((-c)^(1/4)*d^(1/4)*S qrt[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] + Sq rt[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] - 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] + Sqrt[b]*x^2)^2]*EllipticPi[(Sqrt[b]*Sqrt[-c] + Sqrt[a]...
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
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)]
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)]
Result contains complex when optimal does not.
Time = 2.30 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.46
| method | result | size |
| default | \(\frac {d x \sqrt {b \,x^{4}+a}}{4 c \left (a d -b c \right ) \left (d \,x^{4}+c \right )}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{4 c \left (a d -b c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d +c \right )}{\sum }\frac {\left (-3 a d +5 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}\right )}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c d}\) | \(333\) |
| elliptic | \(\frac {d x \sqrt {b \,x^{4}+a}}{4 c \left (a d -b c \right ) \left (d \,x^{4}+c \right )}+\frac {b \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{4 c \left (a d -b c \right ) \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} d +c \right )}{\sum }\frac {\left (-3 a d +5 b c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, \frac {i \sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, c \sqrt {b \,x^{4}+a}}\right )}{\left (a d -b c \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 c d}\) | \(333\) |
Input:
int(1/(b*x^4+a)^(1/2)/(d*x^4+c)^2,x,method=_RETURNVERBOSE)
Output:
1/4/c*d/(a*d-b*c)*x*(b*x^4+a)^(1/2)/(d*x^4+c)+1/4*b/c/(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)-1/32/c/d*su m((-3*a*d+5*b*c)/(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))
Timed out. \[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^4+a)^(1/2)/(d*x^4+c)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )^2} \, dx=\int \frac {1}{\sqrt {a + b x^{4}} \left (c + d x^{4}\right )^{2}}\, dx \] Input:
integrate(1/(b*x**4+a)**(1/2)/(d*x**4+c)**2,x)
Output:
Integral(1/(sqrt(a + b*x**4)*(c + d*x**4)**2), x)
\[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )^2} \, dx=\int { \frac {1}{\sqrt {b x^{4} + a} {\left (d x^{4} + c\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^4+a)^(1/2)/(d*x^4+c)^2,x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x^4 + a)*(d*x^4 + c)^2), x)
\[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )^2} \, dx=\int { \frac {1}{\sqrt {b x^{4} + a} {\left (d x^{4} + c\right )}^{2}} \,d x } \] Input:
integrate(1/(b*x^4+a)^(1/2)/(d*x^4+c)^2,x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x^4 + a)*(d*x^4 + c)^2), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )^2} \, dx=\int \frac {1}{\sqrt {b\,x^4+a}\,{\left (d\,x^4+c\right )}^2} \,d x \] Input:
int(1/((a + b*x^4)^(1/2)*(c + d*x^4)^2),x)
Output:
int(1/((a + b*x^4)^(1/2)*(c + d*x^4)^2), x)
\[ \int \frac {1}{\sqrt {a+b x^4} \left (c+d x^4\right )^2} \, dx=\int \frac {\sqrt {b \,x^{4}+a}}{b \,d^{2} x^{12}+a \,d^{2} x^{8}+2 b c d \,x^{8}+2 a c d \,x^{4}+b \,c^{2} x^{4}+a \,c^{2}}d x \] Input:
int(1/(b*x^4+a)^(1/2)/(d*x^4+c)^2,x)
Output:
int(sqrt(a + b*x**4)/(a*c**2 + 2*a*c*d*x**4 + a*d**2*x**8 + b*c**2*x**4 + 2*b*c*d*x**8 + b*d**2*x**12),x)