Integrand size = 24, antiderivative size = 854 \[ \int \frac {1}{x^8 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {(11 b c-4 a d) \sqrt {c+d x^4}}{28 a^2 c (b c-a d) x^7}+\frac {\left (77 b^2 c^2-36 a b c d-20 a^2 d^2\right ) \sqrt {c+d x^4}}{84 a^3 c^2 (b c-a d) x^3}+\frac {b \sqrt {c+d x^4}}{4 a (b c-a d) x^7 \left (a+b x^4\right )}-\frac {b^{9/4} (11 b c-13 a d) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{16 (-a)^{15/4} (-b c+a d)^{3/2}}-\frac {b^{9/4} (11 b c-13 a d) \text {arctanh}\left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{16 (-a)^{15/4} (-b c+a d)^{3/2}}+\frac {d^{3/4} \left (77 b^2 c^2+19 a b c d+5 a^2 d^2\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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{42 a^3 c^{9/4} (b c+a d) \sqrt {c+d x^4}}+\frac {b^2 \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (11 b c-13 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 {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 )}{32 a^4 \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \sqrt {c+d x^4}}+\frac {b^2 \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (11 b c-13 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 {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 )}{32 a^4 \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) \sqrt [4]{d} (b c-a d) \sqrt {c+d x^4}} \] Output:
-1/28*(-4*a*d+11*b*c)*(d*x^4+c)^(1/2)/a^2/c/(-a*d+b*c)/x^7+1/84*(-20*a^2*d ^2-36*a*b*c*d+77*b^2*c^2)*(d*x^4+c)^(1/2)/a^3/c^2/(-a*d+b*c)/x^3+1/4*b*(d* x^4+c)^(1/2)/a/(-a*d+b*c)/x^7/(b*x^4+a)-1/16*b^(9/4)*(-13*a*d+11*b*c)*arct an((a*d-b*c)^(1/2)*x/(-a)^(1/4)/b^(1/4)/(d*x^4+c)^(1/2))/(-a)^(15/4)/(a*d- b*c)^(3/2)-1/16*b^(9/4)*(-13*a*d+11*b*c)*arctanh((a*d-b*c)^(1/2)*x/(-a)^(1 /4)/b^(1/4)/(d*x^4+c)^(1/2))/(-a)^(15/4)/(a*d-b*c)^(3/2)+1/42*d^(3/4)*(5*a ^2*d^2+19*a*b*c*d+77*b^2*c^2)*(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^3/c^(9/4)/(a*d+b*c)/(d*x^4+c)^(1/2)+1/32*b^2*(b^(1/2)*c^(1/2)+(-a)^(1 /2)*d^(1/2))*(-13*a*d+11*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)*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^4/c^(1/4)/(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))/d^(1/4)/(-a*d+b*c )/(d*x^4+c)^(1/2)+1/32*b^2*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*(-13*a*d+1 1*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)*Ell ipticPi(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^4/c^(1/4)/(b^( 1/2)*c^(1/2)+(-a)^(1/2)*d^(1/2))/d^(1/4)/(-a*d+b*c)/(d*x^4+c)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.67 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.45 \[ \int \frac {1}{x^8 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {b d \left (77 b^2 c^2-36 a b c d-20 a^2 d^2\right ) x^{12} \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+\frac {5 a \left (\left (c+d x^4\right ) \left (77 b^3 c^2 x^8+4 a b^2 c x^4 \left (11 c-9 d x^4\right )+4 a^3 d \left (3 c-5 d x^4\right )-4 a^2 b \left (3 c^2+6 c d x^4+5 d^2 x^8\right )\right )+\frac {5 a c \left (-231 b^3 c^3+196 a b^2 c^2 d+36 a^2 b c d^2+20 a^3 d^3\right ) x^8 \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+2 x^4 \left (2 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )}\right )}{a+b x^4}}{420 a^4 c^2 (b c-a d) x^7 \sqrt {c+d x^4}} \] Input:
Integrate[1/(x^8*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]
Output:
(b*d*(77*b^2*c^2 - 36*a*b*c*d - 20*a^2*d^2)*x^12*Sqrt[1 + (d*x^4)/c]*Appel lF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)] + (5*a*((c + d*x^4)*(77* b^3*c^2*x^8 + 4*a*b^2*c*x^4*(11*c - 9*d*x^4) + 4*a^3*d*(3*c - 5*d*x^4) - 4 *a^2*b*(3*c^2 + 6*c*d*x^4 + 5*d^2*x^8)) + (5*a*c*(-231*b^3*c^3 + 196*a*b^2 *c^2*d + 36*a^2*b*c*d^2 + 20*a^3*d^3)*x^8*AppellF1[1/4, 1/2, 1, 5/4, -((d* x^4)/c), -((b*x^4)/a)])/(-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)]))))/ (a + b*x^4))/(420*a^4*c^2*(b*c - a*d)*x^7*Sqrt[c + d*x^4])
Time = 2.79 (sec) , antiderivative size = 1144, normalized size of antiderivative = 1.34, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {972, 25, 1053, 1053, 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.
| \(\displaystyle \int \frac {1}{x^8 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}-\frac {\int -\frac {9 b d x^4+11 b c-4 a d}{x^8 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {9 b d x^4+11 b c-4 a d}{x^8 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {\int \frac {5 b d (11 b c-4 a d) x^4+77 b^2 c^2-20 a^2 d^2-36 a b c d}{x^4 \left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{7 a c}-\frac {\sqrt {c+d x^4} (11 b c-4 a d)}{7 a c x^7}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {b d \left (77 b^2 c^2-36 a b d c-20 a^2 d^2\right ) x^4+231 b^3 c^3-20 a^3 d^3-36 a^2 b c d^2-196 a b^2 c^2 d}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{3 a c}-\frac {\sqrt {c+d x^4} \left (\frac {77 b^2 c}{a}-\frac {20 a d^2}{c}-36 b d\right )}{3 x^3}}{7 a c}-\frac {\sqrt {c+d x^4} (11 b c-4 a d)}{7 a c x^7}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {-\frac {-\frac {d \left (-20 a^2 d^2-36 a b c d+77 b^2 c^2\right ) \int \frac {1}{\sqrt {d x^4+c}}dx+21 b^2 c^2 (11 b c-13 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{3 a c}-\frac {\sqrt {c+d x^4} \left (\frac {77 b^2 c}{a}-\frac {20 a d^2}{c}-36 b d\right )}{3 x^3}}{7 a c}-\frac {\sqrt {c+d x^4} (11 b c-4 a d)}{7 a c x^7}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {-\frac {-\frac {21 b^2 c^2 (11 b c-13 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-20 a^2 d^2-36 a b c d+77 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4}}}{3 a c}-\frac {\sqrt {c+d x^4} \left (\frac {77 b^2 c}{a}-\frac {20 a d^2}{c}-36 b d\right )}{3 x^3}}{7 a c}-\frac {\sqrt {c+d x^4} (11 b c-4 a d)}{7 a c x^7}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {-\frac {-\frac {21 b^2 c^2 (11 b c-13 a d) \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{2 a}+\frac {\int \frac {1}{\left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{2 a}\right )+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-20 a^2 d^2-36 a b c d+77 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4}}}{3 a c}-\frac {\sqrt {c+d x^4} \left (\frac {77 b^2 c}{a}-\frac {20 a d^2}{c}-36 b d\right )}{3 x^3}}{7 a c}-\frac {\sqrt {c+d x^4} (11 b c-4 a d)}{7 a c x^7}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 1541 |
| \(\displaystyle \frac {-\frac {-\frac {21 b^2 c^2 (11 b c-13 a d) \left (\frac {\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\sqrt {c} \left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}+\frac {\frac {a \sqrt {d} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\sqrt {c} \left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}\right )+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-20 a^2 d^2-36 a b c d+77 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4}}}{3 a c}-\frac {\sqrt {c+d x^4} \left (\frac {77 b^2 c}{a}-\frac {20 a d^2}{c}-36 b d\right )}{3 x^3}}{7 a c}-\frac {\sqrt {c+d x^4} (11 b c-4 a d)}{7 a c x^7}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {21 b^2 c^2 (11 b c-13 a d) \left (\frac {\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}+\frac {\frac {a \sqrt {d} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}\right )+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-20 a^2 d^2-36 a b c d+77 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4}}}{3 a c}-\frac {\sqrt {c+d x^4} \left (\frac {77 b^2 c}{a}-\frac {20 a d^2}{c}-36 b d\right )}{3 x^3}}{7 a c}-\frac {\sqrt {c+d x^4} (11 b c-4 a d)}{7 a c x^7}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {-\frac {-\frac {21 b^2 c^2 (11 b c-13 a d) \left (\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}}{2 a}+\frac {\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{a d+b c}+\frac {a \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}}{2 a}\right )+\frac {d^{3/4} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-20 a^2 d^2-36 a b c d+77 b^2 c^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4}}}{3 a c}-\frac {\sqrt {c+d x^4} \left (\frac {77 b^2 c}{a}-\frac {20 a d^2}{c}-36 b d\right )}{3 x^3}}{7 a c}-\frac {\sqrt {c+d x^4} (11 b c-4 a d)}{7 a c x^7}}{4 a (b c-a d)}+\frac {b \sqrt {c+d x^4}}{4 a x^7 \left (a+b x^4\right ) (b c-a d)}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {\sqrt {d x^4+c} b}{4 a (b c-a d) x^7 \left (b x^4+a\right )}+\frac {-\frac {\sqrt {d x^4+c} (11 b c-4 a d)}{7 a c x^7}-\frac {-\frac {\sqrt {d x^4+c} \left (\frac {77 c b^2}{a}-36 d b-\frac {20 a d^2}{c}\right )}{3 x^3}-\frac {21 b^2 (11 b c-13 a d) \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 ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{b c+a d}}{2 a}\right ) c^2+\frac {d^{3/4} \left (77 b^2 c^2-36 a b d c-20 a^2 d^2\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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt {d x^4+c} \sqrt [4]{c}}}{3 a c}}{7 a c}}{4 a (b c-a d)}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {\sqrt {d x^4+c} b}{4 a (b c-a d) x^7 \left (b x^4+a\right )}+\frac {-\frac {\sqrt {d x^4+c} (11 b c-4 a d)}{7 a c x^7}-\frac {-\frac {\sqrt {d x^4+c} \left (\frac {77 c b^2}{a}-36 d b-\frac {20 a d^2}{c}\right )}{3 x^3}-\frac {21 b^2 (11 b c-13 a d) \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 ) c^2+\frac {d^{3/4} \left (77 b^2 c^2-36 a b d c-20 a^2 d^2\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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt {d x^4+c} \sqrt [4]{c}}}{3 a c}}{7 a c}}{4 a (b c-a d)}\) |
Input:
Int[1/(x^8*(a + b*x^4)^2*Sqrt[c + d*x^4]),x]
Output:
(b*Sqrt[c + d*x^4])/(4*a*(b*c - a*d)*x^7*(a + b*x^4)) + (-1/7*((11*b*c - 4 *a*d)*Sqrt[c + d*x^4])/(a*c*x^7) - (-1/3*(((77*b^2*c)/a - 36*b*d - (20*a*d ^2)/c)*Sqrt[c + d*x^4])/x^3 - ((d^(3/4)*(77*b^2*c^2 - 36*a*b*c*d - 20*a^2* d^2)*(Sqrt[c] + Sqrt[d]*x^2)*Sqrt[(c + d*x^4)/(Sqrt[c] + Sqrt[d]*x^2)^2]*E llipticF[2*ArcTan[(d^(1/4)*x)/c^(1/4)], 1/2])/(2*c^(1/4)*Sqrt[c + d*x^4]) + 21*b^2*c^2*(11*b*c - 13*a*d)*(((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)*S qrt[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)/(Sq rt[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]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(((-a)^(1/4)*(Sqrt[b]*Sqrt[c] + Sqrt[-a]*Sqrt[d])*ArcTa nh[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c + d*x^4])])/(2*b^(1/4...
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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, 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[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
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 = 10.16 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.47
| method | result | size |
| elliptic | \(-\frac {b^{3} x \sqrt {d \,x^{4}+c}}{4 a^{3} \left (a d -c b \right ) \left (b \,x^{4}+a \right )}-\frac {\sqrt {d \,x^{4}+c}}{7 c \,a^{2} x^{7}}+\frac {\left (5 a d +14 c b \right ) \sqrt {d \,x^{4}+c}}{21 c^{2} a^{3} x^{3}}+\frac {\left (-\frac {b^{2} d}{4 a^{3} \left (a d -c b \right )}+\frac {d \left (5 a d +14 c b \right )}{21 c^{2} a^{3}}\right ) \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}+\frac {b \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (13 a d -11 c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\left (a d -c b \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 a^{3}}\) | \(405\) |
| risch | \(-\frac {\sqrt {d \,x^{4}+c}\, \left (-5 a d \,x^{4}-14 b c \,x^{4}+3 a c \right )}{21 c^{2} a^{3} x^{7}}+\frac {\frac {5 a \,d^{2} \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 {14 b c 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 {21 b \,c^{2} \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 )}{4}+21 a \,c^{2} b^{2} \left (-\frac {b x \sqrt {d \,x^{4}+c}}{4 a \left (a d -c b \right ) \left (b \,x^{4}+a \right )}-\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 )}{4 \left (a d -c b \right ) 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 {\left (-5 a d +3 c b \right ) \left (-\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}}\right )}{\left (a d -c b \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 b a}\right )}{21 a^{3} c^{2}}\) | \(729\) |
| default | \(\frac {-\frac {\sqrt {d \,x^{4}+c}}{7 c \,x^{7}}+\frac {5 d \sqrt {d \,x^{4}+c}}{21 c^{2} x^{3}}+\frac {5 d^{2} \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 )}{21 c^{2} \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}}{a^{2}}+\frac {b^{2} \left (-\frac {b x \sqrt {d \,x^{4}+c}}{4 a \left (a d -c b \right ) \left (b \,x^{4}+a \right )}-\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 )}{4 \left (a d -c b \right ) 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 {\left (-5 a d +3 c b \right ) \left (-\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}}\right )}{\left (a d -c b \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}}{32 b a}\right )}{a^{2}}-\frac {2 b \left (-\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}}\right )}{a^{3}}+\frac {b \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 )}{4 a^{3}}\) | \(746\) |
Input:
int(1/x^8/(b*x^4+a)^2/(d*x^4+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4*b^3/a^3/(a*d-b*c)*x*(d*x^4+c)^(1/2)/(b*x^4+a)-1/7/c/a^2*(d*x^4+c)^(1/ 2)/x^7+1/21/c^2*(5*a*d+14*b*c)/a^3*(d*x^4+c)^(1/2)/x^3+(-1/4*b^2*d/a^3/(a* d-b*c)+1/21/c^2*d*(5*a*d+14*b*c)/a^3)/(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)*Elli pticF(x*(I/c^(1/2)*d^(1/2))^(1/2),I)+1/32*b/a^3*sum((13*a*d-11*b*c)/(a*d-b *c)/_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))
Timed out. \[ \int \frac {1}{x^8 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Timed out} \] Input:
integrate(1/x^8/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^8 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^{8} \left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \] Input:
integrate(1/x**8/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
Output:
Integral(1/(x**8*(a + b*x**4)**2*sqrt(c + d*x**4)), x)
\[ \int \frac {1}{x^8 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c} x^{8}} \,d x } \] Input:
integrate(1/x^8/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^8), x)
\[ \int \frac {1}{x^8 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {1}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c} x^{8}} \,d x } \] Input:
integrate(1/x^8/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^4 + a)^2*sqrt(d*x^4 + c)*x^8), x)
Timed out. \[ \int \frac {1}{x^8 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {1}{x^8\,{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \] Input:
int(1/(x^8*(a + b*x^4)^2*(c + d*x^4)^(1/2)),x)
Output:
int(1/(x^8*(a + b*x^4)^2*(c + d*x^4)^(1/2)), x)
\[ \int \frac {1}{x^8 \left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {-3 \sqrt {d \,x^{4}+c}\, a c +5 \sqrt {d \,x^{4}+c}\, a d \,x^{4}+11 \sqrt {d \,x^{4}+c}\, b c \,x^{4}+5 \left (\int \frac {\sqrt {d \,x^{4}+c}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a^{3} d^{2} x^{7}+19 \left (\int \frac {\sqrt {d \,x^{4}+c}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a^{2} b c d \,x^{7}+5 \left (\int \frac {\sqrt {d \,x^{4}+c}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a^{2} b \,d^{2} x^{11}+77 \left (\int \frac {\sqrt {d \,x^{4}+c}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a \,b^{2} c^{2} x^{7}+19 \left (\int \frac {\sqrt {d \,x^{4}+c}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a \,b^{2} c d \,x^{11}+77 \left (\int \frac {\sqrt {d \,x^{4}+c}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) b^{3} c^{2} x^{11}+25 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{4}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a^{2} b \,d^{2} x^{7}+55 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{4}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a \,b^{2} c d \,x^{7}+25 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{4}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) a \,b^{2} d^{2} x^{11}+55 \left (\int \frac {\sqrt {d \,x^{4}+c}\, x^{4}}{b^{2} d \,x^{12}+2 a b d \,x^{8}+b^{2} c \,x^{8}+a^{2} d \,x^{4}+2 a b c \,x^{4}+a^{2} c}d x \right ) b^{3} c d \,x^{11}}{21 a^{2} c^{2} x^{7} \left (b \,x^{4}+a \right )} \] Input:
int(1/x^8/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)
Output:
( - 3*sqrt(c + d*x**4)*a*c + 5*sqrt(c + d*x**4)*a*d*x**4 + 11*sqrt(c + d*x **4)*b*c*x**4 + 5*int(sqrt(c + d*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)*a**3*d**2*x**7 + 19*int( sqrt(c + d*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)*a**2*b*c*d*x**7 + 5*int(sqrt(c + d*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**1 2),x)*a**2*b*d**2*x**11 + 77*int(sqrt(c + d*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)*a*b**2*c**2*x **7 + 19*int(sqrt(c + d*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)*a*b**2*c*d*x**11 + 77*int(sqrt(c + d*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)*b**3*c**2*x**11 + 25*int((sqrt(c + d*x**4)*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)*a**2*b*d**2*x**7 + 55*int((sqrt(c + d*x**4)*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)*a*b**2 *c*d*x**7 + 25*int((sqrt(c + d*x**4)*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)*a*b**2*d**2*x**11 + 55*int((sqrt(c + d*x**4)*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)*b**3*c*d*x**11)/(21*a**2*c**2*x* *7*(a + b*x**4))