Integrand size = 22, antiderivative size = 563 \[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (c d^2-a e^2\right ) \left (d+e x^2\right )^3}-\frac {5 e^2 \left (3 c d^2-a e^2\right ) x \sqrt {a-c x^4}}{24 d^2 \left (c d^2-a e^2\right )^2 \left (d+e x^2\right )^2}-\frac {e^2 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) x \sqrt {a-c x^4}}{16 d^3 \left (c d^2-a e^2\right )^3 \left (d+e x^2\right )}-\frac {a^{3/4} \sqrt [4]{c} e \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{16 d^3 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} \left (57 c^2 d^4-30 \sqrt {a} c^{3/2} d^3 e-32 a c d^2 e^2+10 a^{3/2} \sqrt {c} d e^3+15 a^2 e^4\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{48 d^3 \left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (\sqrt {c} d+\sqrt {a} e\right )^3 \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (35 c^3 d^6-7 a c^2 d^4 e^2+17 a^2 c d^2 e^4-5 a^3 e^6\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{16 \sqrt [4]{c} d^4 \left (c d^2-a e^2\right )^3 \sqrt {a-c x^4}} \] Output:
-1/6*e^2*x*(-c*x^4+a)^(1/2)/d/(-a*e^2+c*d^2)/(e*x^2+d)^3-5/24*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)^2-1/16*e^2*(5*a ^2*e^4-14*a*c*d^2*e^2+29*c^2*d^4)*x*(-c*x^4+a)^(1/2)/d^3/(-a*e^2+c*d^2)^3/ (e*x^2+d)-1/16*a^(3/4)*c^(1/4)*e*(5*a^2*e^4-14*a*c*d^2*e^2+29*c^2*d^4)*(1- c*x^4/a)^(1/2)*EllipticE(c^(1/4)*x/a^(1/4),I)/d^3/(-a*e^2+c*d^2)^3/(-c*x^4 +a)^(1/2)-1/48*a^(1/4)*c^(1/4)*(57*c^2*d^4-30*a^(1/2)*c^(3/2)*d^3*e-32*a*c *d^2*e^2+10*a^(3/2)*c^(1/2)*d*e^3+15*a^2*e^4)*(1-c*x^4/a)^(1/2)*EllipticF( c^(1/4)*x/a^(1/4),I)/d^3/(c^(1/2)*d-a^(1/2)*e)^2/(c^(1/2)*d+a^(1/2)*e)^3/( -c*x^4+a)^(1/2)+1/16*a^(1/4)*(-5*a^3*e^6+17*a^2*c*d^2*e^4-7*a*c^2*d^4*e^2+ 35*c^3*d^6)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1/4),-a^(1/2)*e/c^(1 /2)/d,I)/c^(1/4)/d^4/(-a*e^2+c*d^2)^3/(-c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.40 (sec) , antiderivative size = 458, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\frac {-\frac {d e^2 x \left (a-c x^4\right ) \left (8 \left (c d^3-a d e^2\right )^2+10 d \left (c d^2-a e^2\right ) \left (3 c d^2-a e^2\right ) \left (d+e x^2\right )+3 \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) \left (d+e x^2\right )^2\right )}{\left (c d^2-a e^2\right )^3 \left (d+e x^2\right )^3}-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (3 \sqrt {a} \sqrt {c} d e \left (29 c^2 d^4-14 a c d^2 e^2+5 a^2 e^4\right ) E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} d \left (57 c^{5/2} d^5-87 \sqrt {a} c^2 d^4 e-2 a c^{3/2} d^3 e^2+42 a^{3/2} c d^2 e^3+5 a^2 \sqrt {c} d e^4-15 a^{5/2} e^5\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+3 \left (-35 c^3 d^6+7 a c^2 d^4 e^2-17 a^2 c d^2 e^4+5 a^3 e^6\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} \left (-c d^2+a e^2\right )^3}}{48 d^4 \sqrt {a-c x^4}} \] Input:
Integrate[1/((d + e*x^2)^4*Sqrt[a - c*x^4]),x]
Output:
(-((d*e^2*x*(a - c*x^4)*(8*(c*d^3 - a*d*e^2)^2 + 10*d*(c*d^2 - a*e^2)*(3*c *d^2 - a*e^2)*(d + e*x^2) + 3*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*(d + e*x^2)^2))/((c*d^2 - a*e^2)^3*(d + e*x^2)^3)) - (I*Sqrt[1 - (c*x^4)/a]* (3*Sqrt[a]*Sqrt[c]*d*e*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*EllipticE [I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a])]*x], -1] + Sqrt[c]*d*(57*c^(5/2)*d^5 - 87*Sqrt[a]*c^2*d^4*e - 2*a*c^(3/2)*d^3*e^2 + 42*a^(3/2)*c*d^2*e^3 + 5*a^2* Sqrt[c]*d*e^4 - 15*a^(5/2)*e^5)*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a] )]*x], -1] + 3*(-35*c^3*d^6 + 7*a*c^2*d^4*e^2 - 17*a^2*c*d^2*e^4 + 5*a^3*e ^6)*EllipticPi[-((Sqrt[a]*e)/(Sqrt[c]*d)), I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a ])]*x], -1]))/(Sqrt[-(Sqrt[c]/Sqrt[a])]*(-(c*d^2) + a*e^2)^3))/(48*d^4*Sqr t[a - c*x^4])
Time = 3.19 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {1552, 2211, 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.
| \(\displaystyle \int \frac {1}{\sqrt {a-c x^4} \left (d+e x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1552 |
| \(\displaystyle \frac {\int \frac {3 c e^2 x^4-6 c d e x^2+6 c d^2-5 a e^2}{\left (e x^2+d\right )^3 \sqrt {a-c x^4}}dx}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 2211 |
| \(\displaystyle \frac {\frac {\int \frac {24 c^2 d^4-29 a c e^2 d^2-8 c e \left (6 c d^2-a e^2\right ) x^2 d+15 a^2 e^4+5 c e^2 \left (3 c d^2-a e^2\right ) x^4}{\left (e x^2+d\right )^2 \sqrt {a-c x^4}}dx}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 2211 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {48 c^3 d^6-19 a c^2 e^2 d^4+46 a^2 c e^4 d^2-4 c e \left (36 c^2 d^4-11 a c e^2 d^2+5 a^2 e^4\right ) x^2 d-15 a^3 e^6-3 c e^2 \left (29 c^2 d^4-14 a c e^2 d^2+5 a^2 e^4\right ) x^4}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 2235 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-\frac {\int \frac {c e^2 \left (3 e \left (29 c^2 d^4-14 a c e^2 d^2+5 a^2 e^4\right ) x^2+d \left (57 c^2 d^4-2 a c e^2 d^2+5 a^2 e^4\right )\right )}{\sqrt {a-c x^4}}dx}{e^2}}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \int \frac {3 e \left (29 c^2 d^4-14 a c e^2 d^2+5 a^2 e^4\right ) x^2+d \left (57 c^2 d^4-2 a c e^2 d^2+5 a^2 e^4\right )}{\sqrt {a-c x^4}}dx}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1513 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{\sqrt {c}}+\left (-\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{\sqrt {c}}+5 a^2 d e^4-2 a c d^3 e^2+57 c^2 d^5\right ) \int \frac {1}{\sqrt {a-c x^4}}dx\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\left (-\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{\sqrt {c}}+5 a^2 d e^4-2 a c d^3 e^2+57 c^2 d^5\right ) \int \frac {1}{\sqrt {a-c x^4}}dx\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt {1-\frac {c x^4}{a}} \left (-\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{\sqrt {c}}+5 a^2 d e^4-2 a c d^3 e^2+57 c^2 d^5\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{\sqrt {c}}+5 a^2 d e^4-2 a c d^3 e^2+57 c^2 d^5\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1390 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 e \sqrt {1-\frac {c x^4}{a}} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{\sqrt {c}}+5 a^2 d e^4-2 a c d^3 e^2+57 c^2 d^5\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1389 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {3 \sqrt {a} e \sqrt {1-\frac {c x^4}{a}} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {c} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{\sqrt {c}}+5 a^2 d e^4-2 a c d^3 e^2+57 c^2 d^5\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {a-c x^4}}dx-c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{\sqrt {c}}+5 a^2 d e^4-2 a c d^3 e^2+57 c^2 d^5\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}+\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1543 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \sqrt {1-\frac {c x^4}{a}} \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \int \frac {1}{\left (e x^2+d\right ) \sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}-c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{\sqrt {c}}+5 a^2 d e^4-2 a c d^3 e^2+57 c^2 d^5\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}+\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1542 |
| \(\displaystyle \frac {\frac {\frac {\frac {3 \sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-5 a^3 e^6+17 a^2 c d^2 e^4-7 a c^2 d^4 e^2+35 c^3 d^6\right ) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} e}{\sqrt {c} d},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4}}-c \left (\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-\frac {3 \sqrt {a} e \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{\sqrt {c}}+5 a^2 d e^4-2 a c d^3 e^2+57 c^2 d^5\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}+\frac {3 a^{3/4} e \sqrt {1-\frac {c x^4}{a}} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right ) E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}\right )}{2 d \left (c d^2-a e^2\right )}-\frac {3 e^2 x \sqrt {a-c x^4} \left (5 a^2 e^4-14 a c d^2 e^2+29 c^2 d^4\right )}{2 d \left (d+e x^2\right ) \left (c d^2-a e^2\right )}}{4 d \left (c d^2-a e^2\right )}-\frac {5 e^2 x \sqrt {a-c x^4} \left (3 c d^2-a e^2\right )}{4 d \left (d+e x^2\right )^2 \left (c d^2-a e^2\right )}}{6 d \left (c d^2-a e^2\right )}-\frac {e^2 x \sqrt {a-c x^4}}{6 d \left (d+e x^2\right )^3 \left (c d^2-a e^2\right )}\) |
Input:
Int[1/((d + e*x^2)^4*Sqrt[a - c*x^4]),x]
Output:
-1/6*(e^2*x*Sqrt[a - c*x^4])/(d*(c*d^2 - a*e^2)*(d + e*x^2)^3) + ((-5*e^2* (3*c*d^2 - a*e^2)*x*Sqrt[a - c*x^4])/(4*d*(c*d^2 - a*e^2)*(d + e*x^2)^2) + ((-3*e^2*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4)*x*Sqrt[a - c*x^4])/(2* d*(c*d^2 - a*e^2)*(d + e*x^2)) + (-(c*((3*a^(3/4)*e*(29*c^2*d^4 - 14*a*c*d ^2*e^2 + 5*a^2*e^4)*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)*(57*c^2*d^5 - 2*a*c*d^3*e^2 + 5*a^2*d*e^4 - (3*Sqrt[a]*e*(29*c^2*d^4 - 14*a*c*d^2*e^2 + 5*a^2*e^4))/S qrt[c])*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c ^(1/4)*Sqrt[a - c*x^4]))) + (3*a^(1/4)*(35*c^3*d^6 - 7*a*c^2*d^4*e^2 + 17* a^2*c*d^2*e^4 - 5*a^3*e^6)*Sqrt[1 - (c*x^4)/a]*EllipticPi[-((Sqrt[a]*e)/(S qrt[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)))/(6*d*(c*d^2 - a*e^2))
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. 1419 vs. \(2 (489 ) = 978\).
Time = 1.59 (sec) , antiderivative size = 1420, normalized size of antiderivative = 2.52
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1420\) |
| elliptic | \(\text {Expression too large to display}\) | \(1420\) |
Input:
int(1/(e*x^2+d)^4/(-c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/6*e^2/d/(a*e^2-c*d^2)*x*(-c*x^4+a)^(1/2)/(e*x^2+d)^3+5/24*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)^2+1/16*e^2*(5*a^2*e ^4-14*a*c*d^2*e^2+29*c^2*d^4)/(a*e^2-c*d^2)^3/d^3*x*(-c*x^4+a)^(1/2)/(e*x^ 2+d)+19/16*c^3*d^2/(a*e^2-c*d^2)^3/(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)*Ellipt icF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-35/16/(a*e^2-c*d^2)^3*d^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)*EllipticPi(x*(1/a^(1/2)*c^(1/2))^(1/2),-a^(1/2)*e/c^(1 /2)/d,(-1/a^(1/2)*c^(1/2))^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2))*c^3+5/48*c/d^2 /(a*e^2-c*d^2)^3/(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^2*e^4-1/24*c^2/(a*e^2-c*d^2)^3/(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^2-5/16*c^(1/2)*e^5/( a*e^2-c*d^2)^3/d^3*a^(5/2)/(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)+7/8*c^(3/2)*e^3/(a*e^2-c*d^2)^3/d*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)-29/16 *c^(5/2)*e/(a*e^2-c*d^2)^3*d*a^(1/2)/(1/a^(1/2)*c^(1/2))^(1/2)*(1-1/a^(...
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x^2+d)^4/(-c*x^4+a)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a - c x^{4}} \left (d + e x^{2}\right )^{4}}\, dx \] Input:
integrate(1/(e*x**2+d)**4/(-c*x**4+a)**(1/2),x)
Output:
Integral(1/(sqrt(a - c*x**4)*(d + e*x**2)**4), x)
\[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{4}} \,d x } \] Input:
integrate(1/(e*x^2+d)^4/(-c*x^4+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(-c*x^4 + a)*(e*x^2 + d)^4), x)
\[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\int { \frac {1}{\sqrt {-c x^{4} + a} {\left (e x^{2} + d\right )}^{4}} \,d x } \] Input:
integrate(1/(e*x^2+d)^4/(-c*x^4+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-c*x^4 + a)*(e*x^2 + d)^4), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\int \frac {1}{\sqrt {a-c\,x^4}\,{\left (e\,x^2+d\right )}^4} \,d x \] Input:
int(1/((a - c*x^4)^(1/2)*(d + e*x^2)^4),x)
Output:
int(1/((a - c*x^4)^(1/2)*(d + e*x^2)^4), x)
\[ \int \frac {1}{\left (d+e x^2\right )^4 \sqrt {a-c x^4}} \, dx=\int \frac {\sqrt {-c \,x^{4}+a}}{-c \,e^{4} x^{12}-4 c d \,e^{3} x^{10}+a \,e^{4} x^{8}-6 c \,d^{2} e^{2} x^{8}+4 a d \,e^{3} x^{6}-4 c \,d^{3} e \,x^{6}+6 a \,d^{2} e^{2} x^{4}-c \,d^{4} x^{4}+4 a \,d^{3} e \,x^{2}+a \,d^{4}}d x \] Input:
int(1/(e*x^2+d)^4/(-c*x^4+a)^(1/2),x)
Output:
int(sqrt(a - c*x**4)/(a*d**4 + 4*a*d**3*e*x**2 + 6*a*d**2*e**2*x**4 + 4*a* d*e**3*x**6 + a*e**4*x**8 - c*d**4*x**4 - 4*c*d**3*e*x**6 - 6*c*d**2*e**2* x**8 - 4*c*d*e**3*x**10 - c*e**4*x**12),x)