Integrand size = 19, antiderivative size = 659 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^4}} \, dx=-\frac {e^3 \sqrt {a+c x^4}}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {3 c d^3 e^3 \sqrt {a+c x^4}}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {3 c^{3/2} d^3 e^2 x \sqrt {a+c x^4}}{\left (c d^4+a e^4\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {3 c d^2 e \left (c d^4-a e^4\right ) \arctan \left (\frac {\sqrt {-c d^4-a e^4} x}{d e \sqrt {a+c x^4}}\right )}{2 \left (-c d^4-a e^4\right )^{5/2}}-\frac {3 c d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {a+c x^4}}\right )}{2 \left (c d^4+a e^4\right )^{5/2}}-\frac {3 \sqrt [4]{a} c^{5/4} d^3 e^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\left (c d^4+a e^4\right )^2 \sqrt {a+c x^4}}+\frac {c^{3/4} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \left (c d^4+a e^4\right ) \sqrt {a+c x^4}}-\frac {3 c^{3/4} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \left (c d^4+a e^4\right )^2 \sqrt {a+c x^4}} \]
3/2*c*d^2*e*(-a*e^4+c*d^4)*arctan(x*(-a*e^4-c*d^4)^(1/2)/d/e/(c*x^4+a)^(1/ 2))/(-a*e^4-c*d^4)^(5/2)-3/2*c*d^2*e*(-a*e^4+c*d^4)*arctanh((c*d^2*x^2+a*e ^2)/(a*e^4+c*d^4)^(1/2)/(c*x^4+a)^(1/2))/(a*e^4+c*d^4)^(5/2)-1/2*e^3*(c*x^ 4+a)^(1/2)/(a*e^4+c*d^4)/(e*x+d)^2-3*c*d^3*e^3*(c*x^4+a)^(1/2)/(a*e^4+c*d^ 4)^2/(e*x+d)+3*c^(3/2)*d^3*e^2*x*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)^2/(a^(1/2)+ x^2*c^(1/2))-3*a^(1/4)*c^(5/4)*d^3*e^2*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2 )^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/ a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/ 2))^2)^(1/2)/(a*e^4+c*d^4)^2/(c*x^4+a)^(1/2)+1/2*c^(3/4)*d*(cos(2*arctan(c ^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin (2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a )/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(1/4)/(a*e^4+c*d^4)/(c*x^4+a)^(1/2)-3/4 *c^(3/4)*d*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4) *x/a^(1/4)))*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(e^2*a^(1/2)+ d^2*c^(1/2))^2/d^2/e^2/a^(1/2)/c^(1/2),1/2*2^(1/2))*(-e^2*a^(1/2)+d^2*c^(1 /2))^2*(a^(1/2)+x^2*c^(1/2))*((c*x^4+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^( 1/4)/(a*e^4+c*d^4)^2/(c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.43 (sec) , antiderivative size = 614, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^4}} \, dx=-\frac {\frac {c d^4 e^3 \sqrt {a+c x^4}}{(d+e x)^2}+\frac {a e^7 \sqrt {a+c x^4}}{(d+e x)^2}+\frac {6 c d^3 e^3 \sqrt {a+c x^4}}{d+e x}-\frac {6 c^2 d^6 e \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )}{\sqrt {-c d^4-a e^4}}+\frac {6 a c d^2 e^5 \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )}{\sqrt {-c d^4-a e^4}}+\frac {6 i a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} c d^3 e^2 \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {a+c x^4}}+\frac {2 i c d \left (-2 c d^4-3 i \sqrt {a} \sqrt {c} d^2 e^2+a e^4\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \sqrt {a+c x^4}}+\frac {6 \sqrt [4]{-1} \sqrt [4]{a} c^{7/4} d^5 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a+c x^4}}-\frac {6 \sqrt [4]{-1} a^{5/4} c^{3/4} d e^4 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {a+c x^4}}}{2 \left (c d^4+a e^4\right )^2} \]
-1/2*((c*d^4*e^3*Sqrt[a + c*x^4])/(d + e*x)^2 + (a*e^7*Sqrt[a + c*x^4])/(d + e*x)^2 + (6*c*d^3*e^3*Sqrt[a + c*x^4])/(d + e*x) - (6*c^2*d^6*e*ArcTan[ (Sqrt[c]*(d^2 - e^2*x^2) + e^2*Sqrt[a + c*x^4])/Sqrt[-(c*d^4) - a*e^4]])/S qrt[-(c*d^4) - a*e^4] + (6*a*c*d^2*e^5*ArcTan[(Sqrt[c]*(d^2 - e^2*x^2) + e ^2*Sqrt[a + c*x^4])/Sqrt[-(c*d^4) - a*e^4]])/Sqrt[-(c*d^4) - a*e^4] + ((6* I)*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c*d^3*e^2*Sqrt[1 + (c*x^4)/a]*EllipticE[I*A rcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[a + c*x^4] + ((2*I)*c*d*(-2 *c*d^4 - (3*I)*Sqrt[a]*Sqrt[c]*d^2*e^2 + a*e^4)*Sqrt[1 + (c*x^4)/a]*Ellipt icF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/(Sqrt[(I*Sqrt[c])/Sqrt[a] ]*Sqrt[a + c*x^4]) + (6*(-1)^(1/4)*a^(1/4)*c^(7/4)*d^5*Sqrt[1 + (c*x^4)/a] *EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4)*c^(1/4)*x)/a ^(1/4)], -1])/Sqrt[a + c*x^4] - (6*(-1)^(1/4)*a^(5/4)*c^(3/4)*d*e^4*Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1)^(3/4) *c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a + c*x^4])/(c*d^4 + a*e^4)^2
Time = 1.78 (sec) , antiderivative size = 677, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.895, Rules used = {2265, 27, 2277, 25, 2280, 27, 1577, 488, 219, 2233, 25, 27, 1510, 2227, 27, 761, 2223}
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.
| \(\displaystyle \int \frac {1}{\sqrt {a+c x^4} (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2265 |
| \(\displaystyle -\frac {c \int -\frac {2 \left (d^3-e x d^2+e^2 x^2 d\right )}{(d+e x)^2 \sqrt {c x^4+a}}dx}{2 \left (a e^4+c d^4\right )}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \int \frac {d^3-e x d^2+e^2 x^2 d}{(d+e x)^2 \sqrt {c x^4+a}}dx}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 2277 |
| \(\displaystyle \frac {c \left (-\frac {\int -\frac {3 c e^2 x^2 d^4+3 c e^3 x^3 d^3+\left (c d^4-2 a e^4\right ) d^2-e \left (2 c d^4-a e^4\right ) x d}{(d+e x) \sqrt {c x^4+a}}dx}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {3 c e^2 x^2 d^4+3 c e^3 x^3 d^3+\left (c d^4-2 a e^4\right ) d^2-e \left (2 c d^4-a e^4\right ) x d}{(d+e x) \sqrt {c x^4+a}}dx}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 2280 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {\left (-e \left (c d^4-2 a e^4\right ) d^2-e \left (2 c d^4-a e^4\right ) d^2\right ) x}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx+\int \frac {-3 c d^3 e^4 x^4+\left (3 c e^2 d^5+e^2 \left (2 c d^4-a e^4\right ) d\right ) x^2+d^3 \left (c d^4-2 a e^4\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {-3 c d^3 e^4 x^4+\left (3 c e^2 d^5+e^2 \left (2 c d^4-a e^4\right ) d\right ) x^2+d^3 \left (c d^4-2 a e^4\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx-3 d^2 e \left (c d^4-a e^4\right ) \int \frac {x}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 1577 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {-3 c d^3 e^4 x^4+\left (3 c e^2 d^5+e^2 \left (2 c d^4-a e^4\right ) d\right ) x^2+d^3 \left (c d^4-2 a e^4\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx-\frac {3}{2} d^2 e \left (c d^4-a e^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {c \left (\frac {\frac {3}{2} d^2 e \left (c d^4-a e^4\right ) \int \frac {1}{c d^4+a e^4-x^4}d\frac {-a e^2-c d^2 x^2}{\sqrt {c x^4+a}}+\int \frac {-3 c d^3 e^4 x^4+\left (3 c e^2 d^5+e^2 \left (2 c d^4-a e^4\right ) d\right ) x^2+d^3 \left (c d^4-2 a e^4\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {-3 c d^3 e^4 x^4+\left (3 c e^2 d^5+e^2 \left (2 c d^4-a e^4\right ) d\right ) x^2+d^3 \left (c d^4-2 a e^4\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx+\frac {3 d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{2 \sqrt {a e^4+c d^4}}}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 2233 |
| \(\displaystyle \frac {c \left (\frac {-3 \sqrt {a} \sqrt {c} d^3 e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx-\frac {\int -\frac {c d e^2 \left (\left (c d^4+3 \sqrt {a} \sqrt {c} e^2 d^2-2 a e^4\right ) d^2+e^2 \left (2 c d^4-3 \sqrt {a} \sqrt {c} e^2 d^2-a e^4\right ) x^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{c e^2}+\frac {3 d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{2 \sqrt {a e^4+c d^4}}}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {-3 \sqrt {a} \sqrt {c} d^3 e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx+\frac {\int \frac {c d e^2 \left (\left (c d^4+3 \sqrt {a} \sqrt {c} e^2 d^2-2 a e^4\right ) d^2+e^2 \left (2 c d^4-3 \sqrt {a} \sqrt {c} e^2 d^2-a e^4\right ) x^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{c e^2}+\frac {3 d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{2 \sqrt {a e^4+c d^4}}}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {-3 \sqrt {c} d^3 e^2 \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx+d \int \frac {\left (c d^4+3 \sqrt {a} \sqrt {c} e^2 d^2-2 a e^4\right ) d^2+e^2 \left (2 c d^4-3 \sqrt {a} \sqrt {c} e^2 d^2-a e^4\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx+\frac {3 d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{2 \sqrt {a e^4+c d^4}}}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle \frac {c \left (\frac {d \int \frac {\left (c d^4+3 \sqrt {a} \sqrt {c} e^2 d^2-2 a e^4\right ) d^2+e^2 \left (2 c d^4-3 \sqrt {a} \sqrt {c} e^2 d^2-a e^4\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx-3 \sqrt {c} d^3 e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )+\frac {3 d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{2 \sqrt {a e^4+c d^4}}}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 2227 |
| \(\displaystyle \frac {c \left (\frac {d \left (\left (a e^4+c d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx+3 \sqrt {a} d^2 e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx\right )-3 \sqrt {c} d^3 e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )+\frac {3 d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{2 \sqrt {a e^4+c d^4}}}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {d \left (\left (a e^4+c d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx+3 d^2 e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx\right )-3 \sqrt {c} d^3 e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )+\frac {3 d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{2 \sqrt {a e^4+c d^4}}}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle \frac {c \left (\frac {d \left (3 d^2 e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^4+c d^4\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}\right )-3 \sqrt {c} d^3 e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )+\frac {3 d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{2 \sqrt {a e^4+c d^4}}}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle \frac {c \left (\frac {d \left (3 d^2 e^2 \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )+\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^4+c d^4\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}\right )-3 \sqrt {c} d^3 e^2 \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )+\frac {3 d^2 e \left (c d^4-a e^4\right ) \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{2 \sqrt {a e^4+c d^4}}}{a e^4+c d^4}-\frac {3 d^3 e^3 \sqrt {a+c x^4}}{(d+e x) \left (a e^4+c d^4\right )}\right )}{a e^4+c d^4}-\frac {e^3 \sqrt {a+c x^4}}{2 (d+e x)^2 \left (a e^4+c d^4\right )}\) |
-1/2*(e^3*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)*(d + e*x)^2) + (c*((-3*d^3*e^3 *Sqrt[a + c*x^4])/((c*d^4 + a*e^4)*(d + e*x)) + ((3*d^2*e*(c*d^4 - a*e^4)* ArcTanh[(-(a*e^2) - c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(2* Sqrt[c*d^4 + a*e^4]) - 3*Sqrt[c]*d^3*e^2*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a ] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4 )*Sqrt[a + c*x^4])) + d*(((c*d^4 + a*e^4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4) ], 1/2])/(2*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4]) + 3*d^2*e^2*(Sqrt[c]*d^2 - Sq rt[a]*e^2)*(((Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTanh[(Sqrt[c*d^4 + a*e^4]*x)/( d*e*Sqrt[a + c*x^4])])/(2*d*e*Sqrt[c*d^4 + a*e^4]) + ((Sqrt[a]/d^2 - Sqrt[ c]/e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2 ]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2* ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4]))))/ (c*d^4 + a*e^4)))/(c*d^4 + a*e^4)
3.3.18.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, 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[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, p, q}, x]
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))*(ArcTanh[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*Arc Tan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0 ] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[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[((d_) + (e_.)*(x_))^(q_)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[e ^3*(d + e*x)^(q + 1)*(Sqrt[a + c*x^4]/((q + 1)*(c*d^4 + a*e^4))), x] + Simp [c/((q + 1)*(c*d^4 + a*e^4)) Int[((d + e*x)^(q + 1)/Sqrt[a + c*x^4])*Simp [d^3*(q + 1) - d^2*e*(q + 1)*x + d*e^2*(q + 1)*x^2 - e^3*(q + 3)*x^3, x], x ], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^4 + a*e^4, 0] && ILtQ[q, -1]
Int[((Px_)*((d_) + (e_.)*(x_))^(q_))/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] : > With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff[Px, x, 3]}, Simp[(-(d^3*D - C*d^2*e + B*d*e^2 - A*e^3))*(d + e*x)^(q + 1)*(Sqrt[a + c*x^4]/((q + 1)*(c*d^4 + a*e^4))), x] + Simp[1/((q + 1)*(c*d ^4 + a*e^4)) Int[((d + e*x)^(q + 1)/Sqrt[a + c*x^4])*Simp[(q + 1)*(a*e*(d ^2*D - C*d*e + B*e^2) + A*d*(c*d^2)) - (e*(q + 1)*(A*c*d^2 + a*e*(d*D - C*e )) - B*d*(c*d^2*(q + 1)))*x + (q + 1)*(D*e*(a*e^2) + c*d*(C*d^2 - e*(B*d - A*e)))*x^2 + c*(q + 3)*(d^3*D - C*d^2*e + B*d*e^2 - A*e^3)*x^3, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px, x], 3] && NeQ[c *d^4 + a*e^4, 0] && LtQ[q, -1]
Int[(Px_)/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Wit h[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2], D = Coeff [Px, x, 3]}, Int[(x*(B*d - A*e + (d*D - C*e)*x^2))/((d^2 - e^2*x^2)*Sqrt[a + c*x^4]), x] + Int[(A*d + (C*d - B*e)*x^2 - D*e*x^4)/((d^2 - e^2*x^2)*Sqrt [a + c*x^4]), x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && LeQ[Expon[Px , x], 3] && NeQ[c*d^4 + a*e^4, 0]
Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 483, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {e^{3} \sqrt {c \,x^{4}+a}}{2 \left (e^{4} a +d^{4} c \right ) \left (e x +d \right )^{2}}-\frac {3 c \,d^{3} e^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )}+\frac {d c \left (e^{4} a -2 d^{4} c \right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\left (e^{4} a +d^{4} c \right )^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 i c^{\frac {3}{2}} d^{3} e^{2} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\left (e^{4} a +d^{4} c \right )^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 c \,d^{2} \left (e^{4} a -d^{4} c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +d^{4} c \right )^{2} e}\) | \(483\) |
| elliptic | \(-\frac {e^{3} \sqrt {c \,x^{4}+a}}{2 \left (e^{4} a +d^{4} c \right ) \left (e x +d \right )^{2}}-\frac {3 c \,d^{3} e^{3} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )}+\frac {d c \left (e^{4} a -2 d^{4} c \right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\left (e^{4} a +d^{4} c \right )^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {3 i c^{\frac {3}{2}} d^{3} e^{2} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (F\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-E\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\left (e^{4} a +d^{4} c \right )^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {3 c \,d^{2} \left (e^{4} a -d^{4} c \right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {\frac {c \,d^{4}}{e^{4}}+a}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{\left (e^{4} a +d^{4} c \right )^{2} e}\) | \(483\) |
-1/2*e^3*(c*x^4+a)^(1/2)/(a*e^4+c*d^4)/(e*x+d)^2-3*c*d^3*e^3*(c*x^4+a)^(1/ 2)/(a*e^4+c*d^4)^2/(e*x+d)+d*c*(a*e^4-2*c*d^4)/(a*e^4+c*d^4)^2/(I/a^(1/2)* c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^( 1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)+3*I*c^(3/2)* d^3*e^2/(a*e^4+c*d^4)^2*a^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^( 1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)*(EllipticF (x*(I/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))- 3*c*d^2*(a*e^4-c*d^4)/(a*e^4+c*d^4)^2/e*(-1/2/(c/e^4*d^4+a)^(1/2)*arctanh( 1/2*(2*c*x^2/e^2*d^2+2*a)/(c/e^4*d^4+a)^(1/2)/(c*x^4+a)^(1/2))+1/(I/a^(1/2 )*c^(1/2))^(1/2)*e/d*(1-I/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)* x^2)^(1/2)/(c*x^4+a)^(1/2)*EllipticPi(x*(I/a^(1/2)*c^(1/2))^(1/2),-I*a^(1/ 2)/c^(1/2)*e^2/d^2,(-I/a^(1/2)*c^(1/2))^(1/2)/(I/a^(1/2)*c^(1/2))^(1/2)))
Timed out. \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {a + c x^{4}} \left (d + e x\right )^{3}}\, dx \]
\[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x + d\right )}^{3}} \,d x } \]
\[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + a} {\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x)^3 \sqrt {a+c x^4}} \, dx=\int \frac {1}{\sqrt {c\,x^4+a}\,{\left (d+e\,x\right )}^3} \,d x \]