Integrand size = 22, antiderivative size = 425 \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^2}-\frac {3 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{8 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )}-\frac {3 a^{3/4} \sqrt [4]{c} e \left (3 c d^2-a e^2\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{8 d^2 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (7 c d^2-2 \sqrt {a} \sqrt {c} d e-3 a e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 d^2 \left (\sqrt {c} d+\sqrt {a} e\right ) \left (c d^2-a e^2\right ) \sqrt {a-c x^4}}+\frac {3 \sqrt [4]{a} \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{8 \sqrt [4]{c} d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \]
-1/4*e^2*x*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/(e*x^2+d)^2-3/8*e^2*(-a*e^2+3 *c*d^2)*x*(-c*x^4+a)^(1/2)/d^2/(-a*e^2+c*d^2)^2/(e*x^2+d)-3/8*a^(3/4)*c^(1 /4)*e*(-a*e^2+3*c*d^2)*EllipticE(c^(1/4)*x/a^(1/4),I)*(1-c*x^4/a)^(1/2)/d^ 2/(-a*e^2+c*d^2)^2/(-c*x^4+a)^(1/2)+3/8*a^(1/4)*(a^2*e^4-2*a*c*d^2*e^2+5*c ^2*d^4)*EllipticPi(c^(1/4)*x/a^(1/4),-e*a^(1/2)/d/c^(1/2),I)*(1-c*x^4/a)^( 1/2)/c^(1/4)/d^3/(-a*e^2+c*d^2)^2/(-c*x^4+a)^(1/2)-1/8*a^(1/4)*c^(1/4)*Ell ipticF(c^(1/4)*x/a^(1/4),I)*(7*c*d^2-3*a*e^2-2*d*e*a^(1/2)*c^(1/2))*(1-c*x ^4/a)^(1/2)/d^2/(-a*e^2+c*d^2)/(e*a^(1/2)+d*c^(1/2))/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 10.98 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\frac {\frac {d e^2 x \left (a-c x^4\right ) \left (a e^2 \left (5 d+3 e x^2\right )-c d^2 \left (11 d+9 e x^2\right )\right )}{\left (d+e x^2\right )^2}-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d e \left (-3 c d^2+a e^2\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\left (-7 c^2 d^4+9 \sqrt {a} c^{3/2} d^3 e+a c d^2 e^2-3 a^{3/2} \sqrt {c} d e^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 \left (5 c^2 d^4-2 a c d^2 e^2+a^2 e^4\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}}{8 d^3 \left (c d^2-a e^2\right )^2 \sqrt {a-c x^4}} \]
((d*e^2*x*(a - c*x^4)*(a*e^2*(5*d + 3*e*x^2) - c*d^2*(11*d + 9*e*x^2)))/(d + e*x^2)^2 - (I*Sqrt[1 - (c*x^4)/a]*(3*Sqrt[a]*Sqrt[c]*d*e*(-3*c*d^2 + a* e^2)*EllipticE[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + (-7*c^2*d^4 + 9*Sqrt[a]*c^(3/2)*d^3*e + a*c*d^2*e^2 - 3*a^(3/2)*Sqrt[c]*d*e^3)*EllipticF [I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + 3*(5*c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c ]/Sqrt[a])]*x], -1]))/Sqrt[-(Sqrt[c]/Sqrt[a])])/(8*d^3*(c*d^2 - a*e^2)^2*S qrt[a - c*x^4])
Time = 1.20 (sec) , antiderivative size = 417, normalized size of antiderivative = 0.98, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {1552, 2211, 2235, 27, 1513, 27, 765, 762, 1390, 1389, 327, 1543, 1542}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\sqrt {a-c x^4} \left (d+e x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 1552 |
\(\displaystyle \frac {\int \frac {c e^2 x^4-4 c d e x^2+4 c d^2-3 a e^2}{\left (e x^2+d\right )^2 \sqrt {a-c x^4}}dx}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 2211 |
\(\displaystyle \frac {\frac {\int \frac {8 c^2 d^4-5 a c e^2 d^2-4 c e \left (4 c d^2-a e^2\right ) x^2 d+3 a^2 e^4-3 c e^2 \left (3 c d^2-a e^2\right ) x^4}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 2235 |
\(\displaystyle \frac {\frac {3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-\frac {\int \frac {c e^2 \left (3 e \left (3 c d^2-a e^2\right ) x^2+d \left (7 c d^2-a e^2\right )\right )}{\sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \int \frac {3 e \left (3 c d^2-a e^2\right ) x^2+d \left (7 c d^2-a e^2\right )}{\sqrt {a-c x^4}}dx}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1513 |
\(\displaystyle \frac {\frac {3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \int \frac {1}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {3 \sqrt {a} e \left (3 c d^2-a e^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \int \frac {1}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {3 e \left (3 c d^2-a e^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 765 |
\(\displaystyle \frac {\frac {3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {\sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {3 e \left (3 c d^2-a e^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {\frac {3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 e \left (3 c d^2-a e^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1390 |
\(\displaystyle \frac {\frac {3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 e \sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1389 |
\(\displaystyle \frac {\frac {3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 \sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\frac {3 \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1543 |
\(\displaystyle \frac {\frac {\frac {3 \sqrt {1-\frac {c x^4}{a}} \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}-c \left (\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
\(\Big \downarrow \) 1542 |
\(\displaystyle \frac {\frac {\frac {3 \sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (a^2 e^4-2 a c d^2 e^2+5 c^2 d^4\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4}}-c \left (\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (3 c d^2-a e^2\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-2 \sqrt {a} \sqrt {c} d e-3 a e^2+7 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}\) |
-1/4*(e^2*x*Sqrt[a - c*x^4])/(d*(c*d^2 - a*e^2)*(d + e*x^2)^2) + ((-3*e^2* (3*c*d^2 - a*e^2)*x*Sqrt[a - c*x^4])/(2*d*(c*d^2 - a*e^2)*(d + e*x^2)) + ( -(c*((3*a^(3/4)*e*(3*c*d^2 - a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcSin[( c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]) + (a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*(7*c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 3*a*e^2)*Sqrt[1 - (c*x^4)/ a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(3/4)*Sqrt[a - c*x^4]))) + (3*a^(1/4)*(5*c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*Sqrt[1 - (c*x^4)/a]*El lipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^ (1/4)*d*Sqrt[a - c*x^4]))/(2*d*(c*d^2 - a*e^2)))/(4*d*(c*d^2 - a*e^2))
3.2.62.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt [a + b*x^4] Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ [b/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq rt[a] Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt [1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && !GtQ [a, 0] && !(LtQ[a, 0] && GtQ[c, 0])
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-c/a, 2]}, Simp[(d*q - e)/q Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ {q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x ], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4] Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) ]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && !GtQ[a, 0]
Int[((d_) + (e_.)*(x_)^2)^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp [(-e^2)*x*(d + e*x^2)^(q + 1)*(Sqrt[a + c*x^4]/(2*d*(q + 1)*(c*d^2 + a*e^2) )), x] + Simp[1/(2*d*(q + 1)*(c*d^2 + a*e^2)) Int[((d + e*x^2)^(q + 1)/Sq rt[a + c*x^4])*Simp[a*e^2*(2*q + 3) + 2*c*d^2*(q + 1) - 2*e*c*d*(q + 1)*x^2 + c*e^2*(2*q + 5)*x^4, x], x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && ILtQ[q, -1]
Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol ] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4] }, Simp[(-(C*d^2 - B*d*e + A*e^2))*x*(d + e*x^2)^(q + 1)*(Sqrt[a + c*x^4]/( 2*d*(q + 1)*(c*d^2 + a*e^2))), x] + Simp[1/(2*d*(q + 1)*(c*d^2 + a*e^2)) Int[((d + e*x^2)^(q + 1)/Sqrt[a + c*x^4])*Simp[a*d*(C*d - B*e) + A*(a*e^2*( 2*q + 3) + 2*c*d^2*(q + 1)) + 2*d*(B*c*d - A*c*e + a*C*e)*(q + 1)*x^2 + c*( C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && ILtQ[q, -1]
Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = Coeff[P4x, x, 4]}, Si mp[-(e^2)^(-1) Int[(C*d - B*e - C*e*x^2)/Sqrt[a + c*x^4], x], x] + Simp[( C*d^2 - B*d*e + A*e^2)/e^2 Int[1/((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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (363 ) = 726\).
Time = 1.82 (sec) , antiderivative size = 961, normalized size of antiderivative = 2.26
method | result | size |
default | \(\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{4 \left (a \,e^{2}-c \,d^{2}\right ) d \left (e \,x^{2}+d \right )^{2}}+\frac {3 e^{2} \left (a \,e^{2}-3 c \,d^{2}\right ) x \sqrt {-c \,x^{4}+a}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \left (e \,x^{2}+d \right )}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) a \,e^{2}}{8 d \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {7 c^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {c}\, e^{3} a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {9 c^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 \sqrt {c}\, e^{3} a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {9 c^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 e^{4} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a^{2}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{3} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a c}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {15 d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c^{2}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(961\) |
elliptic | \(\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{4 \left (a \,e^{2}-c \,d^{2}\right ) d \left (e \,x^{2}+d \right )^{2}}+\frac {3 e^{2} \left (a \,e^{2}-3 c \,d^{2}\right ) x \sqrt {-c \,x^{4}+a}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \left (e \,x^{2}+d \right )}+\frac {c \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right ) a \,e^{2}}{8 d \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {7 c^{2} d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 \sqrt {c}\, e^{3} a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {9 c^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 \sqrt {c}\, e^{3} a^{\frac {3}{2}} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {9 c^{\frac {3}{2}} e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, E\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 e^{4} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a^{2}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} d^{3} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {3 e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) a c}{4 \left (a \,e^{2}-c \,d^{2}\right )^{2} d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {15 d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {e \sqrt {a}}{d \sqrt {c}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) c^{2}}{8 \left (a \,e^{2}-c \,d^{2}\right )^{2} \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\) | \(961\) |
1/4*e^2/(a*e^2-c*d^2)/d*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^2+3/8*e^2*(a*e^2-3*c* d^2)/(a*e^2-c*d^2)^2/d^2*x*(-c*x^4+a)^(1/2)/(e*x^2+d)+1/8*c/d/(a*e^2-c*d^2 )^2/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2) *c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2) ,I)*a*e^2-7/8*c^2*d/(a*e^2-c*d^2)^2/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2) *c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*Ellip ticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-3/8*c^(1/2)*e^3/(a*e^2-c*d^2)^2/d^2*a^ (3/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/ 2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/ 2),I)+9/8*c^(3/2)*e/(a*e^2-c*d^2)^2*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1 /a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/ 2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+3/8*c^(1/2)*e^3/(a*e^2-c*d^2)^ 2/d^2*a^(3/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1 +1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticE(x*(1/a^(1/2)*c^(1 /2))^(1/2),I)-9/8*c^(3/2)*e/(a*e^2-c*d^2)^2*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1 /2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^ 4+a)^(1/2)*EllipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I)+3/8/(a*e^2-c*d^2)^2/d^ 3*e^4/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/ 2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1 /2),-e*a^(1/2)/d/c^(1/2),(-1/a^(1/2)*c^(1/2))^(1/2)/(1/a^(1/2)*c^(1/2))...
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{3}}\, dx \]
\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{3}} \,d x } \]
\[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^3} \,d x \]