Integrand size = 36, antiderivative size = 593 \[ \int \frac {A+B x^2+C x^4}{x^4 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {A \sqrt {a+c x^4}}{3 a c x^3}-\frac {(B c-A d) \sqrt {a+c x^4}}{a c^2 x}+\frac {(B c-A d) x \sqrt {a+c x^4}}{a c^{3/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt {d} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {c^3+a d^2} x}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 c^{5/2} \sqrt {c^3+a d^2}}-\frac {(B c-A d) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{a^{3/4} c^{7/4} \sqrt {a+c x^4}}-\frac {\left (A c^3-\sqrt {a} c^{3/2} (3 B c-2 A d)-3 a \left (c^2 C-2 B c d+2 A d^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{6 a^{5/4} c^{7/4} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}-\frac {\left (c^{3/2}+\sqrt {a} d\right ) \left (c^2 C-B c d+A d^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (c^{3/2}-\sqrt {a} d\right )^2}{4 \sqrt {a} c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c^{13/4} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}} \] Output:
-1/3*A*(c*x^4+a)^(1/2)/a/c/x^3-(-A*d+B*c)*(c*x^4+a)^(1/2)/a/c^2/x+(-A*d+B* c)*x*(c*x^4+a)^(1/2)/a/c^(3/2)/(a^(1/2)+c^(1/2)*x^2)+1/2*d^(1/2)*(A*d^2-B* c*d+C*c^2)*arctan((a*d^2+c^3)^(1/2)*x/c^(1/2)/d^(1/2)/(c*x^4+a)^(1/2))/c^( 5/2)/(a*d^2+c^3)^(1/2)-(-A*d+B*c)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2 )+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^( 1/2))/a^(3/4)/c^(7/4)/(c*x^4+a)^(1/2)-1/6*(A*c^3-a^(1/2)*c^(3/2)*(-2*A*d+3 *B*c)-3*a*(2*A*d^2-2*B*c*d+C*c^2))*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/ 2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2 ^(1/2))/a^(5/4)/c^(7/4)/(c^(3/2)-a^(1/2)*d)/(c*x^4+a)^(1/2)-1/4*(c^(3/2)+a ^(1/2)*d)*(A*d^2-B*c*d+C*c^2)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^ (1/2)*x^2)^2)^(1/2)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(c^(3 /2)-a^(1/2)*d)^2/a^(1/2)/c^(3/2)/d,1/2*2^(1/2))/a^(1/4)/c^(13/4)/(c^(3/2)- a^(1/2)*d)/(c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.15 (sec) , antiderivative size = 549, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x^2+C x^4}{x^4 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {-a A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^2-3 a B \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^2 x^2+3 a A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d x^2-A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^3 x^4-3 B \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^3 x^6+3 A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^2 d x^6+3 \sqrt {a} c^{3/2} (B c-A d) x^3 \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-c^{3/2} \left (-i A c^{3/2}+3 \sqrt {a} (B c-A d)\right ) x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i a c^2 C x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 i a B c d x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-3 i a A d^2 x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{3 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^3 x^3 \sqrt {a+c x^4}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(x^4*(c + d*x^2)*Sqrt[a + c*x^4]),x]
Output:
(-(a*A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^2) - 3*a*B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^ 2*x^2 + 3*a*A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d*x^2 - A*Sqrt[(I*Sqrt[c])/Sqrt[ a]]*c^3*x^4 - 3*B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^3*x^6 + 3*A*Sqrt[(I*Sqrt[c]) /Sqrt[a]]*c^2*d*x^6 + 3*Sqrt[a]*c^(3/2)*(B*c - A*d)*x^3*Sqrt[1 + (c*x^4)/a ]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - c^(3/2)*((-I)*A* c^(3/2) + 3*Sqrt[a]*(B*c - A*d))*x^3*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSi nh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (3*I)*a*c^2*C*x^3*Sqrt[1 + (c*x^4)/ a]*EllipticPi[((-I)*Sqrt[a]*d)/c^(3/2), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a] ]*x], -1] + (3*I)*a*B*c*d*x^3*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a] *d)/c^(3/2), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (3*I)*a*A*d^2*x ^3*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*d)/c^(3/2), I*ArcSinh[Sqrt [(I*Sqrt[c])/Sqrt[a]]*x], -1])/(3*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^3*x^3*Sqrt [a + c*x^4])
Time = 1.67 (sec) , antiderivative size = 613, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2245, 25, 2245, 25, 2233, 25, 27, 1510, 2227, 27, 761, 2221}
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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{x^4 \sqrt {a+c x^4} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2245 |
| \(\displaystyle -\frac {\int -\frac {-A c d x^4-c (A c-3 a C) x^2+3 a (B c-A d)}{x^2 \left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-A c d x^4-c (A c-3 a C) x^2+3 a (B c-A d)}{x^2 \left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 2245 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 a c d (B c-A d) x^4+a c^2 (3 B c-4 A d) x^2+a \left (3 a c (c C-B d)-A \left (c^3-3 a d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {3 a c d (B c-A d) x^4+a c^2 (3 B c-4 A d) x^2+a \left (3 a c (c C-B d)-A \left (c^3-3 a d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 2233 |
| \(\displaystyle \frac {\frac {\frac {\int -\frac {a c d \left (A c^3-3 \sqrt {a} (B c-A d) c^{3/2}+d \left (A c^{3/2}-\sqrt {a} (3 B c-3 A d)\right ) x^2 \sqrt {c}-3 a \left (C c^2-B d c+A d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c d}-3 a^{3/2} \sqrt {c} (B c-A d) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-3 a^{3/2} \sqrt {c} (B c-A d) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx-\frac {\int \frac {a c d \left (A c^3-3 \sqrt {a} (B c-A d) c^{3/2}+d \left (A c^{3/2}-\sqrt {a} (3 B c-3 A d)\right ) x^2 \sqrt {c}-3 a \left (C c^2-B d c+A d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c d}}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-a \int \frac {A c^3-3 \sqrt {a} (B c-A d) c^{3/2}+d \left (A c^{3/2}-\sqrt {a} (3 B c-3 A d)\right ) x^2 \sqrt {c}-3 a \left (C c^2-B d c+A d^2\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx-3 a \sqrt {c} (B c-A d) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {\frac {-a \int \frac {A c^3-3 \sqrt {a} (B c-A d) c^{3/2}+d \left (A c^{3/2}-\sqrt {a} (3 B c-3 A d)\right ) x^2 \sqrt {c}-3 a \left (C c^2-B d c+A d^2\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx-3 a \sqrt {c} (B c-A d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 2227 |
| \(\displaystyle \frac {\frac {-a \left (\frac {3 a^{3/2} d \left (A d^2-B c d+c^2 C\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}+\frac {\sqrt {c} \left (-\sqrt {a} c^{3/2} (3 B c-2 A d)-3 a \left (2 A d^2-2 B c d+c^2 C\right )+A c^3\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}\right )-3 a \sqrt {c} (B c-A d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-a \left (\frac {3 a d \left (A d^2-B c d+c^2 C\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}+\frac {\sqrt {c} \left (-\sqrt {a} c^{3/2} (3 B c-2 A d)-3 a \left (2 A d^2-2 B c d+c^2 C\right )+A c^3\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}\right )-3 a \sqrt {c} (B c-A d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {\frac {-a \left (\frac {3 a d \left (A d^2-B c d+c^2 C\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (-\sqrt {a} c^{3/2} (3 B c-2 A d)-3 a \left (2 A d^2-2 B c d+c^2 C\right )+A c^3\right )}{2 \sqrt [4]{a} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}\right )-3 a \sqrt {c} (B c-A d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {\frac {-a \left (\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (-\sqrt {a} c^{3/2} (3 B c-2 A d)-3 a \left (2 A d^2-2 B c d+c^2 C\right )+A c^3\right )}{2 \sqrt [4]{a} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}+\frac {3 a d \left (A d^2-B c d+c^2 C\right ) \left (\frac {\left (\sqrt {a} d+c^{3/2}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {c^{3/2}}{\sqrt {a}}-d\right )^2}{4 c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c^{5/4} d \sqrt {a+c x^4}}-\frac {\left (c^{3/2}-\sqrt {a} d\right ) \arctan \left (\frac {x \sqrt {a d^2+c^3}}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+c^3}}\right )}{c^{3/2}-\sqrt {a} d}\right )-3 a \sqrt {c} (B c-A d) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{a c}-\frac {3 \sqrt {a+c x^4} (B c-A d)}{c x}}{3 a c}-\frac {A \sqrt {a+c x^4}}{3 a c x^3}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(x^4*(c + d*x^2)*Sqrt[a + c*x^4]),x]
Output:
-1/3*(A*Sqrt[a + c*x^4])/(a*c*x^3) + ((-3*(B*c - A*d)*Sqrt[a + c*x^4])/(c* x) + (-3*a*Sqrt[c]*(B*c - A*d)*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x ^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c ]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])) - a*((c^(1/4)*(A*c^3 - Sqrt[a]*c^(3/2)*(3*B*c - 2*A*d) - 3*a*(c^ 2*C - 2*B*c*d + 2*A*d^2))*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a ] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1 /4)*(c^(3/2) - Sqrt[a]*d)*Sqrt[a + c*x^4]) + (3*a*d*(c^2*C - B*c*d + A*d^2 )*(-1/2*((c^(3/2) - Sqrt[a]*d)*ArcTan[(Sqrt[c^3 + a*d^2]*x)/(Sqrt[c]*Sqrt[ d]*Sqrt[a + c*x^4])])/(Sqrt[c]*Sqrt[d]*Sqrt[c^3 + a*d^2]) + ((c^(3/2) + Sq rt[a]*d)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^ 2]*EllipticPi[-1/4*(Sqrt[a]*(c^(3/2)/Sqrt[a] - d)^2)/(c^(3/2)*d), 2*ArcTan [(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(5/4)*d*Sqrt[a + c*x^4])))/(c^(3 /2) - Sqrt[a]*d)))/(a*c))/(3*a*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[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && 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[c/a, 2]}, Simp[(A*(c*d + a*e*q) - a*B*(e + d*q) )/(c*d^2 - a*e^2) Int[1/Sqrt[a + c*x^4], x], x] + Simp[a*(B*d - 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, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && NeQ[c*A^2 - a*B^2, 0]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff [P4x, x, 4]}, Simp[-C/(e*q) Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] + Sim p[1/(c*e) Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - a*e*q))*x^2)/((d + e*x ^2)*Sqrt[a + c*x^4]), x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[((Px_)*(x_)^(m_))/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_ Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 2], C = Coeff[Px, x, 4]}, Simp[A*x^(m + 1)*(Sqrt[a + c*x^4]/(a*d*(m + 1))), x] + Simp[1/(a*d*(m + 1)) Int[(x^(m + 2)/((d + e*x^2)*Sqrt[a + c*x^4]))*Simp[a*B*d*(m + 1) - A*a*e*(m + 1) + (a*C*d*(m + 1) - A*c*d*(m + 3))*x^2 - A*c*e*(m + 3)*x^4, x ], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x^2, 2] && ILtQ[m/2, 0]
Result contains complex when optimal does not.
Time = 3.98 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.58
| method | result | size |
| default | \(\frac {A \left (-\frac {\sqrt {c \,x^{4}+a}}{3 a \,x^{3}}-\frac {c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{c}-\frac {\left (A d -B c \right ) \left (-\frac {\sqrt {c \,x^{4}+a}}{a x}+\frac {i \sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{c^{2}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{c^{3} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(345\) |
| risch | \(-\frac {\sqrt {c \,x^{4}+a}\, \left (-3 A d \,x^{2}+3 B c \,x^{2}+A c \right )}{3 c^{2} a \,x^{3}}-\frac {\frac {A \,c^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 i B \,c^{\frac {3}{2}} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 \left (A \,d^{2}-B c d +C \,c^{2}\right ) a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 i A \sqrt {c}\, d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{3 a \,c^{2}}\) | \(438\) |
| elliptic | \(-\frac {A \sqrt {c \,x^{4}+a}}{3 a c \,x^{3}}+\frac {\left (A d -B c \right ) \sqrt {c \,x^{4}+a}}{a \,c^{2} x}-\frac {A \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right ) B}{\sqrt {c}\, \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right ) A d}{c^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right ) B}{\sqrt {c}\, \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right ) A d}{c^{\frac {3}{2}} \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) A}{c^{3} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) B}{c^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right ) C}{c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(755\) |
Input:
int((C*x^4+B*x^2+A)/x^4/(d*x^2+c)/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
A/c*(-1/3/a*(c*x^4+a)^(1/2)/x^3-1/3*c/a/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^( 1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*El lipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I))-(A*d-B*c)/c^2*(-1/a*(c*x^4+a)^(1/2 )/x+I*c^(1/2)/a^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^ (1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*(EllipticF(x*(I*c^(1 /2)/a^(1/2))^(1/2),I)-EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I)))+(A*d^2-B* c*d+C*c^2)/c^3/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*( 1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I*c^(1/2)/a^( 1/2))^(1/2),I/c^(3/2)*a^(1/2)*d,(-I/a^(1/2)*c^(1/2))^(1/2)/(I*c^(1/2)/a^(1 /2))^(1/2))
Timed out. \[ \int \frac {A+B x^2+C x^4}{x^4 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\text {Timed out} \] Input:
integrate((C*x^4+B*x^2+A)/x^4/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{x^4 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{x^{4} \sqrt {a + c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((C*x**4+B*x**2+A)/x**4/(d*x**2+c)/(c*x**4+a)**(1/2),x)
Output:
Integral((A + B*x**2 + C*x**4)/(x**4*sqrt(a + c*x**4)*(c + d*x**2)), x)
\[ \int \frac {A+B x^2+C x^4}{x^4 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )} x^{4}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^4/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="maxi ma")
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(c*x^4 + a)*(d*x^2 + c)*x^4), x)
\[ \int \frac {A+B x^2+C x^4}{x^4 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )} x^{4}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^4/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="giac ")
Output:
integrate((C*x^4 + B*x^2 + A)/(sqrt(c*x^4 + a)*(d*x^2 + c)*x^4), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{x^4 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{x^4\,\sqrt {c\,x^4+a}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((A + B*x^2 + C*x^4)/(x^4*(a + c*x^4)^(1/2)*(c + d*x^2)),x)
Output:
int((A + B*x^2 + C*x^4)/(x^4*(a + c*x^4)^(1/2)*(c + d*x^2)), x)
\[ \int \frac {A+B x^2+C x^4}{x^4 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {-\sqrt {c \,x^{4}+a}-3 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{8}+c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a d \,x^{3}+3 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{8}+c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) b c \,x^{3}+2 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) c^{2} x^{3}-\left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) c d \,x^{3}}{3 c \,x^{3}} \] Input:
int((C*x^4+B*x^2+A)/x^4/(d*x^2+c)/(c*x^4+a)^(1/2),x)
Output:
( - sqrt(a + c*x**4) - 3*int(sqrt(a + c*x**4)/(a*c*x**2 + a*d*x**4 + c**2* x**6 + c*d*x**8),x)*a*d*x**3 + 3*int(sqrt(a + c*x**4)/(a*c*x**2 + a*d*x**4 + c**2*x**6 + c*d*x**8),x)*b*c*x**3 + 2*int(sqrt(a + c*x**4)/(a*c + a*d*x **2 + c**2*x**4 + c*d*x**6),x)*c**2*x**3 - int((sqrt(a + c*x**4)*x**2)/(a* c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*c*d*x**3)/(3*c*x**3)