Integrand size = 36, antiderivative size = 723 \[ \int \frac {A+B x^2+C x^4}{x^6 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=-\frac {A \sqrt {a+c x^4}}{5 a c x^5}-\frac {(B c-A d) \sqrt {a+c x^4}}{3 a c^2 x^3}-\frac {\left (5 a c (c C-B d)-A \left (3 c^3-5 a d^2\right )\right ) \sqrt {a+c x^4}}{5 a^2 c^3 x}+\frac {\left (5 a c (c C-B d)-A \left (3 c^3-5 a d^2\right )\right ) x \sqrt {a+c x^4}}{5 a^2 c^{5/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {d^{3/2} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {c^3+a d^2} x}{\sqrt {c} \sqrt {d} \sqrt {a+c x^4}}\right )}{2 c^{7/2} \sqrt {c^3+a d^2}}-\frac {\left (5 a c (c C-B d)-A \left (3 c^3-5 a d^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{7/4} c^{11/4} \sqrt {a+c x^4}}-\frac {\left (9 A c^{9/2}+\sqrt {a} c^3 (5 B c-14 A d)+30 a^{3/2} d \left (c^2 C-B c d+A d^2\right )-5 a c^{3/2} \left (3 c^2 C-2 B c d+2 A d^2\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{30 a^{7/4} c^{11/4} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}}+\frac {d \left (c^{3/2}+\sqrt {a} d\right ) \left (c^2 C-B c d+A d^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (c^{3/2}-\sqrt {a} d\right )^2}{4 \sqrt {a} c^{3/2} d},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} c^{17/4} \left (c^{3/2}-\sqrt {a} d\right ) \sqrt {a+c x^4}} \] Output:
-1/5*A*(c*x^4+a)^(1/2)/a/c/x^5-1/3*(-A*d+B*c)*(c*x^4+a)^(1/2)/a/c^2/x^3-1/ 5*(5*a*c*(-B*d+C*c)-A*(-5*a*d^2+3*c^3))*(c*x^4+a)^(1/2)/a^2/c^3/x+1/5*(5*a *c*(-B*d+C*c)-A*(-5*a*d^2+3*c^3))*x*(c*x^4+a)^(1/2)/a^2/c^(5/2)/(a^(1/2)+c ^(1/2)*x^2)-1/2*d^(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))/c^(7/2)/(a*d^2+c^3)^(1/2)-1/5*(5*a*c*(-B*d+C *c)-A*(-5*a*d^2+3*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))/a^(7 /4)/c^(11/4)/(c*x^4+a)^(1/2)-1/30*(9*A*c^(9/2)+a^(1/2)*c^3*(-14*A*d+5*B*c) +30*a^(3/2)*d*(A*d^2-B*c*d+C*c^2)-5*a*c^(3/2)*(2*A*d^2-2*B*c*d+3*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)*InverseJaco biAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/a^(7/4)/c^(11/4)/(c^(3/2)-a^ (1/2)*d)/(c*x^4+a)^(1/2)+1/4*d*(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(si n(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^(17/4)/(c^(3/2)-a^(1/2)*d)/(c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.68 (sec) , antiderivative size = 822, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^2+C x^4}{x^6 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {-3 a^2 A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^3-5 a^2 B \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^3 x^2+5 a^2 A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^2 d x^2+6 a A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^4 x^4-15 a^2 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^3 C x^4+15 a^2 B \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^2 d x^4-15 a^2 A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d^2 x^4-5 a B \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^4 x^6+5 a A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^3 d x^6+9 A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^5 x^8-15 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^4 C x^8+15 a B \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^3 d x^8-15 a A \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^2 d^2 x^8+3 \sqrt {a} c^{3/2} \left (5 a c (c C-B d)+A \left (-3 c^3+5 a d^2\right )\right ) x^5 \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )-\sqrt {a} c^{3/2} \left (-9 A c^3-5 i \sqrt {a} c^{3/2} (B c-A d)+15 a \left (c^2 C-B c d+A d^2\right )\right ) x^5 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+15 i a^2 c^2 C d x^5 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )-15 i a^2 B c d^2 x^5 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+15 i a^2 A d^3 x^5 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {i \sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{15 a^2 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c^4 x^5 \sqrt {a+c x^4}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/(x^6*(c + d*x^2)*Sqrt[a + c*x^4]),x]
Output:
(-3*a^2*A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^3 - 5*a^2*B*Sqrt[(I*Sqrt[c])/Sqrt[a] ]*c^3*x^2 + 5*a^2*A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^2*d*x^2 + 6*a*A*Sqrt[(I*Sq rt[c])/Sqrt[a]]*c^4*x^4 - 15*a^2*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^3*C*x^4 + 15* a^2*B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^2*d*x^4 - 15*a^2*A*Sqrt[(I*Sqrt[c])/Sqrt [a]]*c*d^2*x^4 - 5*a*B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^4*x^6 + 5*a*A*Sqrt[(I*S qrt[c])/Sqrt[a]]*c^3*d*x^6 + 9*A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^5*x^8 - 15*a* Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^4*C*x^8 + 15*a*B*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^3 *d*x^8 - 15*a*A*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^2*d^2*x^8 + 3*Sqrt[a]*c^(3/2)* (5*a*c*(c*C - B*d) + A*(-3*c^3 + 5*a*d^2))*x^5*Sqrt[1 + (c*x^4)/a]*Ellipti cE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - Sqrt[a]*c^(3/2)*(-9*A*c^3 - (5*I)*Sqrt[a]*c^(3/2)*(B*c - A*d) + 15*a*(c^2*C - B*c*d + A*d^2))*x^5*S qrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + (15*I)*a^2*c^2*C*d*x^5*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*d)/c^ (3/2), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - (15*I)*a^2*B*c*d^2*x^ 5*Sqrt[1 + (c*x^4)/a]*EllipticPi[((-I)*Sqrt[a]*d)/c^(3/2), I*ArcSinh[Sqrt[ (I*Sqrt[c])/Sqrt[a]]*x], -1] + (15*I)*a^2*A*d^3*x^5*Sqrt[1 + (c*x^4)/a]*El lipticPi[((-I)*Sqrt[a]*d)/c^(3/2), I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(15*a^2*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^4*x^5*Sqrt[a + c*x^4])
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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{x^6 \sqrt {a+c x^4} \left (c+d x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2245 |
| \(\displaystyle -\frac {\int -\frac {-3 A c d x^4-c (3 A c-5 a C) x^2+5 a (B c-A d)}{x^4 \left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-3 A c d x^4-c (3 A c-5 a C) x^2+5 a (B c-A d)}{x^4 \left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 2245 |
| \(\displaystyle \frac {-\frac {\int \frac {5 a c d (B c-A d) x^4+a c^2 (5 B c+4 A d) x^2+3 a \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right )}{x^2 \left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 2245 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\frac {\int -\frac {-3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (9 A c^3-15 a C c^2+20 a B d c-20 a A d^2\right ) x^2+5 a^2 \left (A d c^3-3 a C d c^2-3 a A d^3-B \left (c^4-3 a c d^2\right )\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {-\frac {\int -\frac {3 a c d \left ((3 A c-5 a C) c^2+5 a d (B c-A d)\right ) x^4-a c^2 \left (5 a c (3 c C-4 B d)-A \left (9 c^3-20 a d^2\right )\right ) x^2+5 a^2 \left (B c^4-A d c^3+3 a C d c^2-3 a B d^2 c+3 a A d^3\right )}{\left (d x^2+c\right ) \sqrt {c x^4+a}}dx}{a c}-\frac {3 \sqrt {a+c x^4} \left (5 a d (B c-A d)+c^2 (3 A c-5 a C)\right )}{c x}}{3 a c}-\frac {5 \sqrt {a+c x^4} (B c-A d)}{3 c x^3}}{5 a c}-\frac {A \sqrt {a+c x^4}}{5 a c x^5}\) |
Input:
Int[(A + B*x^2 + C*x^4)/(x^6*(c + d*x^2)*Sqrt[a + c*x^4]),x]
Output:
$Aborted
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 = 4.45 (sec) , antiderivative size = 494, normalized size of antiderivative = 0.68
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{4}+a}\, \left (15 A a \,d^{2} x^{4}-9 A \,c^{3} x^{4}-15 B a c d \,x^{4}+15 C a \,c^{2} x^{4}-5 A a c d \,x^{2}+5 B a \,c^{2} x^{2}+3 A a \,c^{2}\right )}{15 c^{3} a^{2} x^{5}}+\frac {\frac {3 i \sqrt {c}\, \left (5 A a \,d^{2}-3 A \,c^{3}-5 a B c d +5 C a \,c^{2}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {5 B \,c^{3} a \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {5 A a \,c^{2} d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {15 d \,a^{2} \left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{c \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}}{15 a^{2} c^{3}}\) | \(494\) |
| default | \(\frac {A \left (-\frac {\sqrt {c \,x^{4}+a}}{5 a \,x^{5}}+\frac {3 c \sqrt {c \,x^{4}+a}}{5 a^{2} x}-\frac {3 i c^{\frac {3}{2}} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{5 a^{\frac {3}{2}} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{c}-\frac {\left (A d -B c \right ) \left (-\frac {\sqrt {c \,x^{4}+a}}{3 a \,x^{3}}-\frac {c \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{3 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{c^{2}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \left (-\frac {\sqrt {c \,x^{4}+a}}{a x}+\frac {i \sqrt {c}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{c^{3}}-\frac {d \left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, \frac {i \sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{c^{4} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\) | \(499\) |
| elliptic | \(\text {Expression too large to display}\) | \(1203\) |
Input:
int((C*x^4+B*x^2+A)/x^6/(d*x^2+c)/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/15*(c*x^4+a)^(1/2)*(15*A*a*d^2*x^4-9*A*c^3*x^4-15*B*a*c*d*x^4+15*C*a*c^ 2*x^4-5*A*a*c*d*x^2+5*B*a*c^2*x^2+3*A*a*c^2)/c^3/a^2/x^5+1/15/a^2/c^3*(3*I *c^(1/2)*(5*A*a*d^2-3*A*c^3-5*B*a*c*d+5*C*a*c^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))-5*B*c^3*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)*Ellipti cF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+5*A*a*c^2*d/(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)-15*d*a^2*(A*d^2-B*c*d+C*c^2)/ c/(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)))
\[ \int \frac {A+B x^2+C x^4}{x^6 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )} x^{6}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^6/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="fric as")
Output:
integral((C*x^4 + B*x^2 + A)*sqrt(c*x^4 + a)/(c*d*x^12 + c^2*x^10 + a*d*x^ 8 + a*c*x^6), x)
\[ \int \frac {A+B x^2+C x^4}{x^6 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{x^{6} \sqrt {a + c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:
integrate((C*x**4+B*x**2+A)/x**6/(d*x**2+c)/(c*x**4+a)**(1/2),x)
Output:
Integral((A + B*x**2 + C*x**4)/(x**6*sqrt(a + c*x**4)*(c + d*x**2)), x)
\[ \int \frac {A+B x^2+C x^4}{x^6 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )} x^{6}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^6/(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^6), x)
\[ \int \frac {A+B x^2+C x^4}{x^6 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )} x^{6}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/x^6/(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^6), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{x^6 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{x^6\,\sqrt {c\,x^4+a}\,\left (d\,x^2+c\right )} \,d x \] Input:
int((A + B*x^2 + C*x^4)/(x^6*(a + c*x^4)^(1/2)*(c + d*x^2)),x)
Output:
int((A + B*x^2 + C*x^4)/(x^6*(a + c*x^4)^(1/2)*(c + d*x^2)), x)
\[ \int \frac {A+B x^2+C x^4}{x^6 \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {-3 \sqrt {c \,x^{4}+a}\, a -5 \sqrt {c \,x^{4}+a}\, b \,x^{2}-15 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{10}+c^{2} x^{8}+a d \,x^{6}+a c \,x^{4}}d x \right ) a^{2} d \,x^{5}-15 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{8}+c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a b d \,x^{5}+6 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{8}+c^{2} x^{6}+a d \,x^{4}+a c \,x^{2}}d x \right ) a \,c^{2} x^{5}-9 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) a c d \,x^{5}-5 \left (\int \frac {\sqrt {c \,x^{4}+a}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b \,c^{2} x^{5}-5 \left (\int \frac {\sqrt {c \,x^{4}+a}\, x^{2}}{c d \,x^{6}+c^{2} x^{4}+a d \,x^{2}+a c}d x \right ) b c d \,x^{5}}{15 a c \,x^{5}} \] Input:
int((C*x^4+B*x^2+A)/x^6/(d*x^2+c)/(c*x^4+a)^(1/2),x)
Output:
( - 3*sqrt(a + c*x**4)*a - 5*sqrt(a + c*x**4)*b*x**2 - 15*int(sqrt(a + c*x **4)/(a*c*x**4 + a*d*x**6 + c**2*x**8 + c*d*x**10),x)*a**2*d*x**5 - 15*int (sqrt(a + c*x**4)/(a*c*x**2 + a*d*x**4 + c**2*x**6 + c*d*x**8),x)*a*b*d*x* *5 + 6*int(sqrt(a + c*x**4)/(a*c*x**2 + a*d*x**4 + c**2*x**6 + c*d*x**8),x )*a*c**2*x**5 - 9*int(sqrt(a + c*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x **6),x)*a*c*d*x**5 - 5*int(sqrt(a + c*x**4)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*b*c**2*x**5 - 5*int((sqrt(a + c*x**4)*x**2)/(a*c + a*d*x**2 + c**2*x**4 + c*d*x**6),x)*b*c*d*x**5)/(15*a*c*x**5)