Integrand size = 36, antiderivative size = 702 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {(c C-B d) x \sqrt {a+c x^4}}{3 c d^2}+\frac {C x^3 \sqrt {a+c x^4}}{5 c d}+\frac {\left (5 c^3 C-5 B c^2 d+5 A c d^2-3 a C d^2\right ) x \sqrt {a+c x^4}}{5 c^{3/2} d^3 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {c^{3/2} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {c^3+a d^2} x}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 d^{7/2} \sqrt {c^3+a d^2}}-\frac {\sqrt [4]{a} \left (5 c^3 C-5 B c^2 d+5 A c d^2-3 a C d^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 c^{7/4} d^3 \sqrt {a+c x^4}}+\frac {\sqrt [4]{a} \left (30 c^{9/2} C-30 B c^{7/2} d-10 \sqrt {a} c^3 C d+10 \sqrt {a} B c^2 d^2+30 A c^{5/2} d^2-14 a c^{3/2} C d^2+5 a B \sqrt {c} d^3-15 \sqrt {a} A c d^3+9 a^{3/2} C d^3\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{30 c^{7/4} d^3 \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}-\frac {c^{3/4} \left (c^{3/2}+\sqrt {a} d\right ) \left (c^2 C-B c d+A d^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (c^{3/2}-\sqrt {a} d\right )^2}{4 \sqrt {a} c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} d^4 \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}} \] Output:
-1/3*(-B*d+C*c)*x*(c*x^4+a)^(1/2)/c/d^2+1/5*C*x^3*(c*x^4+a)^(1/2)/c/d+1/5* (5*A*c*d^2-5*B*c^2*d-3*C*a*d^2+5*C*c^3)*x*(c*x^4+a)^(1/2)/c^(3/2)/d^3/(a^( 1/2)+c^(1/2)*x^2)+1/2*c^(3/2)*(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))/d^(7/2)/(a*d^2+c^3)^(1/2)-1/5*a^(1/4)* (5*A*c*d^2-5*B*c^2*d-3*C*a*d^2+5*C*c^3)*(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))/c^(7/4)/d^3/(c*x^4+a)^(1/2)+1/30*a^(1/4)*(30*c^(9/2)*C-30*B*c^ (7/2)*d-10*a^(1/2)*c^3*C*d+10*a^(1/2)*B*c^2*d^2+30*A*c^(5/2)*d^2-14*a*c^(3 /2)*C*d^2+5*a*B*c^(1/2)*d^3-15*a^(1/2)*A*c*d^3+9*a^(3/2)*C*d^3)*(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*ar ctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/c^(7/4)/d^3/(c^(3/2)-a^(1/2)*d)/(c*x^ 4+a)^(1/2)-1/4*c^(3/4)*(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*arct an(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)/d^4/(c^(3/2)-a^(1/2)*d)/(c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.74 (sec) , antiderivative size = 406, normalized size of antiderivative = 0.58 \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {-3 \sqrt {a} d \left (-5 c^3 C+5 B c^2 d-5 A c d^2+3 a C d^2\right ) \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\left (15 i c^{9/2} C-15 i B c^{7/2} d-15 \sqrt {a} c^3 C d+15 \sqrt {a} B c^2 d^2+15 i A c^{5/2} d^2-5 i a c^{3/2} C d^2+5 i a B \sqrt {c} d^3-15 \sqrt {a} A c d^3+9 a^{3/2} C d^3\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {c} \left (-\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} d^2 x \left (5 c C-5 B d-3 C d x^2\right ) \left (a+c x^4\right )-15 i c^2 \left (c^2 C-B c d+A d^2\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{15 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^{3/2} d^4 \sqrt {a+c x^4}} \] Input:
Integrate[(x^4*(A + B*x^2 + C*x^4))/((c + d*x^2)*Sqrt[a + c*x^4]),x]
Output:
(-3*Sqrt[a]*d*(-5*c^3*C + 5*B*c^2*d - 5*A*c*d^2 + 3*a*C*d^2)*Sqrt[1 + (c*x ^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + ((15*I)*c^( 9/2)*C - (15*I)*B*c^(7/2)*d - 15*Sqrt[a]*c^3*C*d + 15*Sqrt[a]*B*c^2*d^2 + (15*I)*A*c^(5/2)*d^2 - (5*I)*a*c^(3/2)*C*d^2 + (5*I)*a*B*Sqrt[c]*d^3 - 15* Sqrt[a]*A*c*d^3 + 9*a^(3/2)*C*d^3)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh [Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[c]*(-(Sqrt[(I*Sqrt[c])/Sqrt[a]]* d^2*x*(5*c*C - 5*B*d - 3*C*d*x^2)*(a + c*x^4)) - (15*I)*c^2*(c^2*C - B*c*d + A*d^2)*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*d)/c^(3/2), I*ArcSi nh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))/(15*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^(3/ 2)*d^4*Sqrt[a + c*x^4])
Time = 1.82 (sec) , antiderivative size = 675, normalized size of antiderivative = 0.96, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {2237, 25, 2237, 25, 2233, 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.
| \(\displaystyle \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\sqrt {a+c x^4} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2237 |
| \(\displaystyle \frac {\int -\frac {C x^2 \left (d x^2+c\right ) \left (5 c x^4+3 a\right )-5 c d x^4 \left (C x^4+B x^2+A\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{5 c d}+\frac {C x^3 \sqrt {a+c x^4}}{5 c d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\int \frac {C x^2 \left (d x^2+c\right ) \left (5 c x^4+3 a\right )-5 c d x^4 \left (C x^4+B x^2+A\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{5 c d}\) |
| \(\Big \downarrow \) 2237 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\frac {\int -\frac {5 c (c C-B d) \left (d x^2+c\right ) \left (3 c x^4+a\right )-3 c d \left (C x^2 \left (d x^2+c\right ) \left (5 c x^4+3 a\right )-5 c d x^4 \left (C x^4+B x^2+A\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{3 c d}+\frac {5 x \sqrt {a+c x^4} (c C-B d)}{3 d}}{5 c d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\frac {5 x \sqrt {a+c x^4} (c C-B d)}{3 d}-\frac {\int \frac {5 c (c C-B d) \left (d x^2+c\right ) \left (3 c x^4+a\right )-3 c d \left (C x^2 \left (d x^2+c\right ) \left (5 c x^4+3 a\right )-5 c d x^4 \left (C x^4+B x^2+A\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{3 c d}}{5 c d}\) |
| \(\Big \downarrow \) 2233 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\frac {5 x \sqrt {a+c x^4} (c C-B d)}{3 d}-\frac {\frac {\int \frac {c \left (\sqrt {a} c^{3/2} \left (5 \sqrt {a} \sqrt {c} d (c C-B d)+3 \left (5 C c^3-5 B d c^2+5 A d^2 c-3 a C d^2\right )\right )-\left (a c (4 c C+5 B d) d^2+3 \left (c^2-\sqrt {a} \sqrt {c} d\right ) \left (5 C c^3-5 B d c^2+5 A d^2 c-3 a C d^2\right )\right ) x^2\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c d}-\frac {3 \sqrt {a} \sqrt {c} \left (-3 a C d^2+5 A c d^2-5 B c^2 d+5 c^3 C\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx}{d}}{3 c d}}{5 c d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\frac {5 x \sqrt {a+c x^4} (c C-B d)}{3 d}-\frac {\frac {\int \frac {\sqrt {a} c^{3/2} \left (5 \sqrt {a} \sqrt {c} d (c C-B d)+3 \left (5 C c^3-5 B d c^2+5 A d^2 c-3 a C d^2\right )\right )-\left (a c (4 c C+5 B d) d^2+3 \left (c^2-\sqrt {a} \sqrt {c} d\right ) \left (5 C c^3-5 B d c^2+5 A d^2 c-3 a C d^2\right )\right ) x^2}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{d}-\frac {3 \sqrt {c} \left (-3 a C d^2+5 A c d^2-5 B c^2 d+5 c^3 C\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{d}}{3 c d}}{5 c d}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\frac {5 x \sqrt {a+c x^4} (c C-B d)}{3 d}-\frac {\frac {\int \frac {\sqrt {a} c^{3/2} \left (5 \sqrt {a} \sqrt {c} d (c C-B d)+3 \left (5 C c^3-5 B d c^2+5 A d^2 c-3 a C d^2\right )\right )-\left (a c (4 c C+5 B d) d^2+3 \left (c^2-\sqrt {a} \sqrt {c} d\right ) \left (5 C c^3-5 B d c^2+5 A d^2 c-3 a C d^2\right )\right ) x^2}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{d}-\frac {3 \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a C d^2+5 A c d^2-5 B c^2 d+5 c^3 C\right )}{d}}{3 c d}}{5 c d}\) |
| \(\Big \downarrow \) 2227 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\frac {5 x \sqrt {a+c x^4} (c C-B d)}{3 d}-\frac {\frac {\frac {\sqrt {a} \sqrt {c} \left (9 a^{3/2} C d^3-5 \sqrt {a} c d \left (3 A d^2-2 B c d+2 c^2 C\right )-a \sqrt {c} d^2 (14 c C-5 B d)+30 c^{5/2} \left (A d^2-B c d+c^2 C\right )\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}-\frac {15 \sqrt {a} c^4 \left (A d^2-B c d+c^2 C\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}}{d}-\frac {3 \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a C d^2+5 A c d^2-5 B c^2 d+5 c^3 C\right )}{d}}{3 c d}}{5 c d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\frac {5 x \sqrt {a+c x^4} (c C-B d)}{3 d}-\frac {\frac {\frac {\sqrt {a} \sqrt {c} \left (9 a^{3/2} C d^3-5 \sqrt {a} c d \left (3 A d^2-2 B c d+2 c^2 C\right )-a \sqrt {c} d^2 (14 c C-5 B d)+30 c^{5/2} \left (A d^2-B c d+c^2 C\right )\right ) \int \frac {1}{\sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}-\frac {15 c^4 \left (A d^2-B c d+c^2 C\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}}{d}-\frac {3 \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a C d^2+5 A c d^2-5 B c^2 d+5 c^3 C\right )}{d}}{3 c d}}{5 c d}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\frac {5 x \sqrt {a+c x^4} (c C-B d)}{3 d}-\frac {\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (9 a^{3/2} C d^3-5 \sqrt {a} c d \left (3 A d^2-2 B c d+2 c^2 C\right )-a \sqrt {c} d^2 (14 c C-5 B d)+30 c^{5/2} \left (A d^2-B c d+c^2 C\right )\right )}{2 \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}-\frac {15 c^4 \left (A d^2-B c d+c^2 C\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{c^{3/2}-\sqrt {a} d}}{d}-\frac {3 \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a C d^2+5 A c d^2-5 B c^2 d+5 c^3 C\right )}{d}}{3 c d}}{5 c d}\) |
| \(\Big \downarrow \) 2221 |
| \(\displaystyle \frac {C x^3 \sqrt {a+c x^4}}{5 c d}-\frac {\frac {5 x \sqrt {a+c x^4} (c C-B d)}{3 d}-\frac {\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right ) \left (9 a^{3/2} C d^3-5 \sqrt {a} c d \left (3 A d^2-2 B c d+2 c^2 C\right )-a \sqrt {c} d^2 (14 c C-5 B d)+30 c^{5/2} \left (A d^2-B c d+c^2 C\right )\right )}{2 \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}-\frac {15 c^4 \left (A d^2-B c d+c^2 C\right ) \left (\frac {\left (\sqrt {a} d+c^{3/2}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \left (\frac {c^{3/2}}{\sqrt {a}}-d\right )^2}{4 c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c^{5/4} d \sqrt {a+c x^4}}-\frac {\left (c^{3/2}-\sqrt {a} d\right ) \arctan \left (\frac {x \sqrt {a d^2+c^3}}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 \sqrt {c} \sqrt {d} \sqrt {a d^2+c^3}}\right )}{c^{3/2}-\sqrt {a} d}}{d}-\frac {3 \sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (-3 a C d^2+5 A c d^2-5 B c^2 d+5 c^3 C\right )}{d}}{3 c d}}{5 c d}\) |
Input:
Int[(x^4*(A + B*x^2 + C*x^4))/((c + d*x^2)*Sqrt[a + c*x^4]),x]
Output:
(C*x^3*Sqrt[a + c*x^4])/(5*c*d) - ((5*(c*C - B*d)*x*Sqrt[a + c*x^4])/(3*d) - ((-3*Sqrt[c]*(5*c^3*C - 5*B*c^2*d + 5*A*c*d^2 - 3*a*C*d^2)*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqr t[(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])))/d + ((a^(1/4)*c^(1/4)*(9*a^(3/2) *C*d^3 - a*Sqrt[c]*d^2*(14*c*C - 5*B*d) + 30*c^(5/2)*(c^2*C - B*c*d + A*d^ 2) - 5*Sqrt[a]*c*d*(2*c^2*C - 2*B*c*d + 3*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*(c^(3/2) - Sqrt[a]*d)*Sqrt[a + c*x^4]) - (15*c^4*(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) + Sqrt[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))/d)/(3*c*d))/(5*c*d)
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_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> W ith[{q = Expon[Px, x]}, Simp[Coeff[Px, x, q]*x^(q - 5)*(Sqrt[a + c*x^4]/(c* e*(q - 3))), x] + Simp[1/(c*e*(q - 3)) Int[(c*e*(q - 3)*Px - Coeff[Px, x, q]*x^(q - 6)*(d + e*x^2)*(a*(q - 5) + c*(q - 3)*x^4))/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; GtQ[q, 4]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x]
Result contains complex when optimal does not.
Time = 6.06 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(\frac {x \left (3 C d \,x^{2}+5 B d -5 C c \right ) \sqrt {c \,x^{4}+a}}{15 c \,d^{2}}+\frac {-\frac {5 \left (3 A \,c^{2} d^{2}+B a \,d^{3}-3 B \,c^{3} d -C a c \,d^{2}+3 C \,c^{4}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{d^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 i \left (5 A c \,d^{2}-5 d \,c^{2} B -3 C a \,d^{2}+5 C \,c^{3}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{d \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+\frac {15 c^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{d^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{15 d^{2} c}\) | \(405\) |
| default | \(\frac {c \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 )}{d^{4} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {\frac {C \,c^{3} \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}}-d^{2} \left (B d -C c \right ) \left (\frac {x \sqrt {c \,x^{4}+a}}{3 c}-\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 c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )-\frac {i d \left (A \,d^{2}-B c d +C \,c^{2}\right ) \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}\, \sqrt {c}}+\frac {A c \,d^{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}}-C \,d^{3} \left (\frac {x^{3} \sqrt {c \,x^{4}+a}}{5 c}-\frac {3 i a^{\frac {3}{2}} \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 )}{5 c^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )-\frac {d \,c^{2} B \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}}}{d^{4}}\) | \(686\) |
| elliptic | \(\text {Expression too large to display}\) | \(1398\) |
Input:
int(x^4*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/15*x*(3*C*d*x^2+5*B*d-5*C*c)/c*(c*x^4+a)^(1/2)/d^2+1/15/d^2/c*(-5*(3*A*c ^2*d^2+B*a*d^3-3*B*c^3*d-C*a*c*d^2+3*C*c^4)/d^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)+3*I/d*(5*A*c*d^2-5*B*c^2*d- 3*C*a*d^2+5*C*c^3)*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)/c^(1/2)*(Ellipt icF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I ))+15*c^2*(A*d^2-B*c*d+C*c^2)/d^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)*Elliptic Pi(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. \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\text {Timed out} \] Input:
integrate(x^4*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="fric as")
Output:
Timed out
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {x^{4} \left (A + B x^{2} + C x^{4}\right )}{\sqrt {a + c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate(x**4*(C*x**4+B*x**2+A)/(d*x**2+c)/(c*x**4+a)**(1/2),x)
Output:
Integral(x**4*(A + B*x**2 + C*x**4)/(sqrt(a + c*x**4)*(c + d*x**2)), x)
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{4}}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^4*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="maxi ma")
Output:
integrate((C*x^4 + B*x^2 + A)*x^4/(sqrt(c*x^4 + a)*(d*x^2 + c)), x)
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} x^{4}}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )}} \,d x } \] Input:
integrate(x^4*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="giac ")
Output:
integrate((C*x^4 + B*x^2 + A)*x^4/(sqrt(c*x^4 + a)*(d*x^2 + c)), x)
Timed out. \[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {x^4\,\left (C\,x^4+B\,x^2+A\right )}{\sqrt {c\,x^4+a}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((x^4*(A + B*x^2 + C*x^4))/((a + c*x^4)^(1/2)*(c + d*x^2)),x)
Output:
int((x^4*(A + B*x^2 + C*x^4))/((a + c*x^4)^(1/2)*(c + d*x^2)), x)
\[ \int \frac {x^4 \left (A+B x^2+C x^4\right )}{\left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {5 \sqrt {c \,x^{4}+a}\, b d x -5 \sqrt {c \,x^{4}+a}\, c^{2} x +3 \sqrt {c \,x^{4}+a}\, c d \,x^{3}-5 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b c d +5 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a \,c^{3}+6 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{4}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a c \,d^{2}-15 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{4}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \,c^{2} d +15 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{4}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) c^{4}-5 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a b \,d^{2}-4 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a \,c^{2} d}{15 c \,d^{2}} \] Input:
int(x^4*(C*x^4+B*x^2+A)/(d*x^2+c)/(c*x^4+a)^(1/2),x)
Output:
(5*sqrt(a + c*x**4)*b*d*x - 5*sqrt(a + c*x**4)*c**2*x + 3*sqrt(a + c*x**4) *c*d*x**3 - 5*int(sqrt(a + c*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6) ,x)*a*b*c*d + 5*int(sqrt(a + c*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x** 6),x)*a*c**3 + 6*int((sqrt(a + c*x**4)*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*a*c*d**2 - 15*int((sqrt(a + c*x**4)*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*b*c**2*d + 15*int((sqrt(a + c*x**4)*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*c**4 - 5*int((sqrt(a + c*x**4)*x**2)/( a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*a*b*d**2 - 4*int((sqrt(a + c*x** 4)*x**2)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*a*c**2*d)/(15*c*d**2)