Integrand size = 24, antiderivative size = 852 \[ \int \frac {x^{16}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=-\frac {\left (20 b^2 c^2+36 a b c d-77 a^2 d^2\right ) x \sqrt {c+d x^4}}{84 b^3 d^2 (b c-a d)}+\frac {(4 b c-11 a d) x^5 \sqrt {c+d x^4}}{28 b^2 d (b c-a d)}+\frac {a x^9 \sqrt {c+d x^4}}{4 b (b c-a d) \left (a+b x^4\right )}+\frac {(-a)^{9/4} (13 b c-11 a d) \arctan \left (\frac {\sqrt {-b c+a d} x}{\sqrt [4]{-a} \sqrt [4]{b} \sqrt {c+d x^4}}\right )}{16 b^{15/4} (-b c+a d)^{3/2}}+\frac {(-a)^{9/4} (13 b c-11 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 b^{15/4} (-b c+a d)^{3/2}}+\frac {c^{3/4} \left (5 b^2 c^2+19 a b c d+77 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 b^3 d^{9/4} (b c+a d) \sqrt {c+d x^4}}-\frac {a^2 \left (\sqrt {b} \sqrt {c}+\sqrt {-a} \sqrt {d}\right ) (13 b c-11 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 b^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 {a^2 \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) (13 b c-11 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 b^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:
-1/84*(-77*a^2*d^2+36*a*b*c*d+20*b^2*c^2)*x*(d*x^4+c)^(1/2)/b^3/d^2/(-a*d+ b*c)+1/28*(-11*a*d+4*b*c)*x^5*(d*x^4+c)^(1/2)/b^2/d/(-a*d+b*c)+1/4*a*x^9*( d*x^4+c)^(1/2)/b/(-a*d+b*c)/(b*x^4+a)+1/16*(-a)^(9/4)*(-11*a*d+13*b*c)*arc tan((a*d-b*c)^(1/2)*x/(-a)^(1/4)/b^(1/4)/(d*x^4+c)^(1/2))/b^(15/4)/(a*d-b* c)^(3/2)+1/16*(-a)^(9/4)*(-11*a*d+13*b*c)*arctanh((a*d-b*c)^(1/2)*x/(-a)^( 1/4)/b^(1/4)/(d*x^4+c)^(1/2))/b^(15/4)/(a*d-b*c)^(3/2)+1/42*c^(3/4)*(77*a^ 2*d^2+19*a*b*c*d+5*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)) /b^3/d^(9/4)/(a*d+b*c)/(d*x^4+c)^(1/2)-1/32*a^2*(b^(1/2)*c^(1/2)+(-a)^(1/2 )*d^(1/2))*(-11*a*d+13*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))/b^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*a^2*(b^(1/2)*c^(1/2)-(-a)^(1/2)*d^(1/2))*(-11*a*d+13* 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)*Ellip ticPi(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))/b^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 = 10.68 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.47 \[ \int \frac {x^{16}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\frac {x \left (-\frac {\left (20 b^3 c^3+36 a b^2 c^2 d+196 a^2 b c d^2-231 a^3 d^3\right ) x^4 \sqrt {1+\frac {d x^4}{c}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{a d^2 (-b c+a d)}+5 \left (-\frac {20 b c^2}{d^2}-\frac {56 a c}{d}-56 a x^4-\frac {8 b c x^4}{d}+12 b x^8+\frac {21 a^3 c}{(b c-a d) \left (a+b x^4\right )}+\frac {21 a^3 d x^4}{(b c-a d) \left (a+b x^4\right )}-\frac {5 a^2 c^2 \left (-20 b^2 c^2-36 a b c d+77 a^2 d^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )}{d^2 (-b c+a d) \left (a+b x^4\right ) \left (-5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+2 x^4 \left (2 b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )+a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},-\frac {d x^4}{c},-\frac {b x^4}{a}\right )\right )\right )}\right )\right )}{420 b^3 \sqrt {c+d x^4}} \] Input:
Integrate[x^16/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
Output:
(x*(-(((20*b^3*c^3 + 36*a*b^2*c^2*d + 196*a^2*b*c*d^2 - 231*a^3*d^3)*x^4*S qrt[1 + (d*x^4)/c]*AppellF1[5/4, 1/2, 1, 9/4, -((d*x^4)/c), -((b*x^4)/a)]) /(a*d^2*(-(b*c) + a*d))) + 5*((-20*b*c^2)/d^2 - (56*a*c)/d - 56*a*x^4 - (8 *b*c*x^4)/d + 12*b*x^8 + (21*a^3*c)/((b*c - a*d)*(a + b*x^4)) + (21*a^3*d* x^4)/((b*c - a*d)*(a + b*x^4)) - (5*a^2*c^2*(-20*b^2*c^2 - 36*a*b*c*d + 77 *a^2*d^2)*AppellF1[1/4, 1/2, 1, 5/4, -((d*x^4)/c), -((b*x^4)/a)])/(d^2*(-( b*c) + a*d)*(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)]))))) )/(420*b^3*Sqrt[c + d*x^4])
Time = 2.97 (sec) , antiderivative size = 1162, normalized size of antiderivative = 1.36, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {970, 1052, 25, 1052, 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.
| \(\displaystyle \int \frac {x^{16}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx\) |
| \(\Big \downarrow \) 970 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {\int \frac {x^8 \left (9 a c-(4 b c-11 a d) x^4\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {-\frac {\int -\frac {x^4 \left (\left (20 b^2 c^2+36 a b d c-77 a^2 d^2\right ) x^4+5 a c (4 b c-11 a d)\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{7 b d}-\frac {x^5 \sqrt {c+d x^4} (4 b c-11 a d)}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\int \frac {x^4 \left (\left (20 b^2 c^2+36 a b d c-77 a^2 d^2\right ) x^4+5 a c (4 b c-11 a d)\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{7 b d}-\frac {x^5 \sqrt {c+d x^4} (4 b c-11 a d)}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {1}{3} x \sqrt {c+d x^4} \left (-\frac {77 a^2 d}{b}+36 a c+\frac {20 b c^2}{d}\right )-\frac {\int \frac {\left (20 b^3 c^3+36 a b^2 d c^2+196 a^2 b d^2 c-231 a^3 d^3\right ) x^4+a c \left (20 b^2 c^2+36 a b d c-77 a^2 d^2\right )}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{3 b d}}{7 b d}-\frac {x^5 \sqrt {c+d x^4} (4 b c-11 a d)}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {1}{3} x \sqrt {c+d x^4} \left (-\frac {77 a^2 d}{b}+36 a c+\frac {20 b c^2}{d}\right )-\frac {\frac {\left (-231 a^3 d^3+196 a^2 b c d^2+36 a b^2 c^2 d+20 b^3 c^3\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{b}-\frac {21 a^3 d^2 (13 b c-11 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{b}}{3 b d}}{7 b d}-\frac {x^5 \sqrt {c+d x^4} (4 b c-11 a d)}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {1}{3} x \sqrt {c+d x^4} \left (-\frac {77 a^2 d}{b}+36 a c+\frac {20 b c^2}{d}\right )-\frac {\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-231 a^3 d^3+196 a^2 b c d^2+36 a b^2 c^2 d+20 b^3 c^3\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {21 a^3 d^2 (13 b c-11 a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx}{b}}{3 b d}}{7 b d}-\frac {x^5 \sqrt {c+d x^4} (4 b c-11 a d)}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 925 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {1}{3} x \sqrt {c+d x^4} \left (-\frac {77 a^2 d}{b}+36 a c+\frac {20 b c^2}{d}\right )-\frac {\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-231 a^3 d^3+196 a^2 b c d^2+36 a b^2 c^2 d+20 b^3 c^3\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {21 a^3 d^2 (13 b c-11 a d) \left (\frac {\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{2 a}+\frac {\int \frac {1}{\left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{2 a}\right )}{b}}{3 b d}}{7 b d}-\frac {x^5 \sqrt {c+d x^4} (4 b c-11 a d)}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 1541 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {1}{3} x \sqrt {c+d x^4} \left (-\frac {77 a^2 d}{b}+36 a c+\frac {20 b c^2}{d}\right )-\frac {\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-231 a^3 d^3+196 a^2 b c d^2+36 a b^2 c^2 d+20 b^3 c^3\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {21 a^3 d^2 (13 b c-11 a d) \left (\frac {\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\sqrt {c} \left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}+\frac {\frac {a \sqrt {d} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \sqrt {c} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\sqrt {c} \left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}\right )}{b}}{3 b d}}{7 b d}-\frac {x^5 \sqrt {c+d x^4} (4 b c-11 a d)}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {1}{3} x \sqrt {c+d x^4} \left (-\frac {77 a^2 d}{b}+36 a c+\frac {20 b c^2}{d}\right )-\frac {\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-231 a^3 d^3+196 a^2 b c d^2+36 a b^2 c^2 d+20 b^3 c^3\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {21 a^3 d^2 (13 b c-11 a d) \left (\frac {\frac {\sqrt {d} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}+\frac {\frac {a \sqrt {d} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \int \frac {1}{\sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{a d+b c}}{2 a}\right )}{b}}{3 b d}}{7 b d}-\frac {x^5 \sqrt {c+d x^4} (4 b c-11 a d)}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {a x^9 \sqrt {c+d x^4}}{4 b \left (a+b x^4\right ) (b c-a d)}-\frac {\frac {\frac {1}{3} x \sqrt {c+d x^4} \left (-\frac {77 a^2 d}{b}+36 a c+\frac {20 b c^2}{d}\right )-\frac {\frac {\left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (-231 a^3 d^3+196 a^2 b c d^2+36 a b^2 c^2 d+20 b^3 c^3\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {c+d x^4}}-\frac {21 a^3 d^2 (13 b c-11 a d) \left (\frac {\frac {\sqrt {b} \left (\sqrt {b} \sqrt {c}-\sqrt {-a} \sqrt {d}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {-a}}\right ) \sqrt {d x^4+c}}dx}{a d+b c}+\frac {\sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\sqrt {-a} \sqrt {b} \sqrt {c}+a \sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}}{2 a}+\frac {\frac {\sqrt {b} \left (\sqrt {-a} \sqrt {d}+\sqrt {b} \sqrt {c}\right ) \int \frac {\sqrt {d} x^2+\sqrt {c}}{\left (\frac {\sqrt {b} x^2}{\sqrt {-a}}+1\right ) \sqrt {d x^4+c}}dx}{a d+b c}+\frac {a \sqrt [4]{d} \left (\sqrt {c}+\sqrt {d} x^2\right ) \sqrt {\frac {c+d x^4}{\left (\sqrt {c}+\sqrt {d} x^2\right )^2}} \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {-a}}+\sqrt {d}\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{d} x}{\sqrt [4]{c}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{c} \sqrt {c+d x^4} (a d+b c)}}{2 a}\right )}{b}}{3 b d}}{7 b d}-\frac {x^5 \sqrt {c+d x^4} (4 b c-11 a d)}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {a x^9 \sqrt {d x^4+c}}{4 b (b c-a d) \left (b x^4+a\right )}-\frac {\frac {\frac {1}{3} \left (-\frac {77 d a^2}{b}+36 c a+\frac {20 b c^2}{d}\right ) x \sqrt {d x^4+c}-\frac {\frac {\left (20 b^3 c^3+36 a b^2 d c^2+196 a^2 b d^2 c-231 a^3 d^3\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 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^4+c}}-\frac {21 a^3 d^2 (13 b c-11 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 )}{b}}{3 b d}}{7 b d}-\frac {(4 b c-11 a d) x^5 \sqrt {d x^4+c}}{7 b d}}{4 b (b c-a d)}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {a x^9 \sqrt {d x^4+c}}{4 b (b c-a d) \left (b x^4+a\right )}-\frac {\frac {\frac {1}{3} \left (-\frac {77 d a^2}{b}+36 c a+\frac {20 b c^2}{d}\right ) x \sqrt {d x^4+c}-\frac {\frac {\left (20 b^3 c^3+36 a b^2 d c^2+196 a^2 b d^2 c-231 a^3 d^3\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 b \sqrt [4]{c} \sqrt [4]{d} \sqrt {d x^4+c}}-\frac {21 a^3 d^2 (13 b c-11 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 )}{b}}{3 b d}}{7 b d}-\frac {(4 b c-11 a d) x^5 \sqrt {d x^4+c}}{7 b d}}{4 b (b c-a d)}\) |
Input:
Int[x^16/((a + b*x^4)^2*Sqrt[c + d*x^4]),x]
Output:
(a*x^9*Sqrt[c + d*x^4])/(4*b*(b*c - a*d)*(a + b*x^4)) - (-1/7*((4*b*c - 11 *a*d)*x^5*Sqrt[c + d*x^4])/(b*d) + (((36*a*c + (20*b*c^2)/d - (77*a^2*d)/b )*x*Sqrt[c + d*x^4])/3 - (((20*b^3*c^3 + 36*a*b^2*c^2*d + 196*a^2*b*c*d^2 - 231*a^3*d^3)*(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*b*c^(1/4)*d^(1/ 4)*Sqrt[c + d*x^4]) - (21*a^3*d^2*(13*b*c - 11*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)/(Sq rt[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[(Sqr t[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]*(Sqrt[b]*Sqrt[c] - Sqrt[-a]*Sqrt[d])*(((-a)^(1/4)*(Sqrt[b]*Sqrt[c] + Sq rt[-a]*Sqrt[d])*ArcTanh[(Sqrt[b*c - a*d]*x)/((-a)^(1/4)*b^(1/4)*Sqrt[c ...
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[(-a)*e^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n) ^(p + 1)*((c + d*x^n)^(q + 1)/(b*n*(b*c - a*d)*(p + 1))), x] + Simp[e^(2*n) /(b*n*(b*c - a*d)*(p + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d *x^n)^q*Simp[a*c*(m - 2*n + 1) + (a*d*(m - n + n*q + 1) + b*c*n*(p + 1))*x^ n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[ n, 0] && LtQ[p, -1] && GtQ[m - n + 1, n] && 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[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q + 1) + 1))), x] - Simp[g^n/(b*d*(m + n*(p + q + 1) + 1)) Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*( f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 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 = 13.64 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.49
| method | result | size |
| elliptic | \(-\frac {a^{3} x \sqrt {d \,x^{4}+c}}{4 b^{3} \left (a d -c b \right ) \left (b \,x^{4}+a \right )}+\frac {x^{5} \sqrt {d \,x^{4}+c}}{7 b^{2} d}+\frac {\left (-\frac {2 a}{b^{3}}-\frac {5 c}{7 b^{2} d}\right ) x \sqrt {d \,x^{4}+c}}{3 d}+\frac {\left (\frac {3 a^{2}}{b^{4}}-\frac {d \,a^{3}}{4 b^{4} \left (a d -c b \right )}-\frac {\left (-\frac {2 a}{b^{3}}-\frac {5 c}{7 b^{2} d}\right ) c}{3 d}\right ) \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, i\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {a^{3} \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (11 a d -13 c b \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {2 d \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 c}{2 \sqrt {\frac {-a d +c b}{b}}\, \sqrt {d \,x^{4}+c}}\right )}{\sqrt {\frac {-a d +c b}{b}}}+\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} b \sqrt {1-\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \sqrt {1+\frac {i \sqrt {d}\, x^{2}}{\sqrt {c}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}, \frac {i \sqrt {c}\, \underline {\hspace {1.25 ex}}\alpha ^{2} b}{\sqrt {d}\, a}, \frac {\sqrt {-\frac {i \sqrt {d}}{\sqrt {c}}}}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}}\right )}{\sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, a \sqrt {d \,x^{4}+c}}\right )}{\left (a d -c b \right ) \underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{32 b^{5}}\) | \(421\) |
| risch | \(-\frac {x \left (-3 d b \,x^{4}+14 a d +5 c b \right ) \sqrt {d \,x^{4}+c}}{21 d^{2} b^{3}}+\frac {\frac {\left (63 a^{2} d^{2}+14 a b c d +5 b^{2} c^{2}\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 )}{b \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {21 a^{3} d^{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 )}{2 b^{2}}+\frac {21 a^{4} d^{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 )}{b}}{21 d^{2} b^{3}}\) | \(679\) |
| default | \(\frac {\frac {x^{5} \sqrt {d \,x^{4}+c}}{7 d}-\frac {5 c x \sqrt {d \,x^{4}+c}}{21 d^{2}}+\frac {5 c^{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 d^{2} \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}}{b^{2}}+\frac {a^{4} \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 )}{b^{4}}+\frac {3 a^{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 )}{b^{4} \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}-\frac {2 a \left (\frac {x \sqrt {d \,x^{4}+c}}{3 d}-\frac {c \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 d \sqrt {\frac {i \sqrt {d}}{\sqrt {c}}}\, \sqrt {d \,x^{4}+c}}\right )}{b^{3}}-\frac {a^{3} \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 )}{2 b^{5}}\) | \(820\) |
Input:
int(x^16/(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/b^2/d*x^5*(d*x^4+c) ^(1/2)+1/3*(-2*a/b^3-5/7/b^2/d*c)/d*x*(d*x^4+c)^(1/2)+(3*a^2/b^4-1/4/b^4*d *a^3/(a*d-b*c)-1/3*(-2*a/b^3-5/7/b^2/d*c)/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)*EllipticF(x*(I/c^(1/2)*d^(1/2))^(1/2),I)-1/32*a^3/b^5*sum((11*a*d-13* 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))),_a lpha=RootOf(_Z^4*b+a))
Timed out. \[ \int \frac {x^{16}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {Timed out} \] Input:
integrate(x^16/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^{16}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {x^{16}}{\left (a + b x^{4}\right )^{2} \sqrt {c + d x^{4}}}\, dx \] Input:
integrate(x**16/(b*x**4+a)**2/(d*x**4+c)**(1/2),x)
Output:
Integral(x**16/((a + b*x**4)**2*sqrt(c + d*x**4)), x)
\[ \int \frac {x^{16}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {x^{16}}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}} \,d x } \] Input:
integrate(x^16/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="maxima")
Output:
integrate(x^16/((b*x^4 + a)^2*sqrt(d*x^4 + c)), x)
\[ \int \frac {x^{16}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int { \frac {x^{16}}{{\left (b x^{4} + a\right )}^{2} \sqrt {d x^{4} + c}} \,d x } \] Input:
integrate(x^16/(b*x^4+a)^2/(d*x^4+c)^(1/2),x, algorithm="giac")
Output:
integrate(x^16/((b*x^4 + a)^2*sqrt(d*x^4 + c)), x)
Timed out. \[ \int \frac {x^{16}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\int \frac {x^{16}}{{\left (b\,x^4+a\right )}^2\,\sqrt {d\,x^4+c}} \,d x \] Input:
int(x^16/((a + b*x^4)^2*(c + d*x^4)^(1/2)),x)
Output:
int(x^16/((a + b*x^4)^2*(c + d*x^4)^(1/2)), x)
\[ \int \frac {x^{16}}{\left (a+b x^4\right )^2 \sqrt {c+d x^4}} \, dx=\text {too large to display} \] Input:
int(x^16/(b*x^4+a)^2/(d*x^4+c)^(1/2),x)
Output:
(55*sqrt(c + d*x**4)*a**2*c*d*x - 33*sqrt(c + d*x**4)*a**2*d**2*x**5 + 25* sqrt(c + d*x**4)*a*b*c**2*x + 18*sqrt(c + d*x**4)*a*b*c*d*x**5 + 9*sqrt(c + d*x**4)*a*b*d**2*x**9 + 15*sqrt(c + d*x**4)*b**2*c**2*x**5 - 9*sqrt(c + d*x**4)*b**2*c*d*x**9 - 55*int(sqrt(c + d*x**4)/(a**3*c*d + a**3*d**2*x**4 - a**2*b*c**2 + a**2*b*c*d*x**4 + 2*a**2*b*d**2*x**8 - 2*a*b**2*c**2*x**4 - a*b**2*c*d*x**8 + a*b**2*d**2*x**12 - b**3*c**2*x**8 - b**3*c*d*x**12), x)*a**5*c**2*d**2 + 30*int(sqrt(c + d*x**4)/(a**3*c*d + a**3*d**2*x**4 - a **2*b*c**2 + a**2*b*c*d*x**4 + 2*a**2*b*d**2*x**8 - 2*a*b**2*c**2*x**4 - a *b**2*c*d*x**8 + a*b**2*d**2*x**12 - b**3*c**2*x**8 - b**3*c*d*x**12),x)*a **4*b*c**3*d - 55*int(sqrt(c + d*x**4)/(a**3*c*d + a**3*d**2*x**4 - a**2*b *c**2 + a**2*b*c*d*x**4 + 2*a**2*b*d**2*x**8 - 2*a*b**2*c**2*x**4 - a*b**2 *c*d*x**8 + a*b**2*d**2*x**12 - b**3*c**2*x**8 - b**3*c*d*x**12),x)*a**4*b *c**2*d**2*x**4 + 25*int(sqrt(c + d*x**4)/(a**3*c*d + a**3*d**2*x**4 - a** 2*b*c**2 + a**2*b*c*d*x**4 + 2*a**2*b*d**2*x**8 - 2*a*b**2*c**2*x**4 - a*b **2*c*d*x**8 + a*b**2*d**2*x**12 - b**3*c**2*x**8 - b**3*c*d*x**12),x)*a** 3*b**2*c**4 + 30*int(sqrt(c + d*x**4)/(a**3*c*d + a**3*d**2*x**4 - a**2*b* c**2 + a**2*b*c*d*x**4 + 2*a**2*b*d**2*x**8 - 2*a*b**2*c**2*x**4 - a*b**2* c*d*x**8 + a*b**2*d**2*x**12 - b**3*c**2*x**8 - b**3*c*d*x**12),x)*a**3*b* *2*c**3*d*x**4 + 25*int(sqrt(c + d*x**4)/(a**3*c*d + a**3*d**2*x**4 - a**2 *b*c**2 + a**2*b*c*d*x**4 + 2*a**2*b*d**2*x**8 - 2*a*b**2*c**2*x**4 - a...