Integrand size = 26, antiderivative size = 258 \[ \int \frac {1}{\left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {3 x \left (5 d-7 e x^2\right )}{56 d^6 \sqrt {d^2-e^2 x^4}}+\frac {x}{14 d^2 \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4}}+\frac {x}{7 d^3 \left (d+e x^2\right )^2 \sqrt {d^2-e^2 x^4}}+\frac {x}{4 d^4 \left (d+e x^2\right ) \sqrt {d^2-e^2 x^4}}+\frac {3 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{8 d^{9/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {3 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{28 d^{9/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
3/56*x*(-7*e*x^2+5*d)/d^6/(-e^2*x^4+d^2)^(1/2)+1/14*x/d^2/(e*x^2+d)^3/(-e^ 2*x^4+d^2)^(1/2)+1/7*x/d^3/(e*x^2+d)^2/(-e^2*x^4+d^2)^(1/2)+1/4*x/d^4/(e*x ^2+d)/(-e^2*x^4+d^2)^(1/2)+3/8*(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d ^(1/2),I)/d^(9/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)-3/28*(1-e^2*x^4/d^2)^(1/2)* EllipticF(e^(1/2)*x/d^(1/2),I)/d^(9/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
Result contains complex when optimal does not.
Time = 10.85 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {\frac {41 d^4 x+60 d^3 e x^3-4 d^2 e^2 x^5-48 d e^3 x^7-21 e^4 x^9}{\left (d+e x^2\right )^3}+\frac {3 i e \sqrt {1-\frac {e^2 x^4}{d^2}} \left (7 E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-2 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\left (-\frac {e}{d}\right )^{3/2}}}{56 d^6 \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[1/((d + e*x^2)^3*(d^2 - e^2*x^4)^(3/2)),x]
Output:
((41*d^4*x + 60*d^3*e*x^3 - 4*d^2*e^2*x^5 - 48*d*e^3*x^7 - 21*e^4*x^9)/(d + e*x^2)^3 + ((3*I)*e*Sqrt[1 - (e^2*x^4)/d^2]*(7*EllipticE[I*ArcSinh[Sqrt[ -(e/d)]*x], -1] - 2*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1]))/(-(e/d))^(3 /2))/(56*d^6*Sqrt[d^2 - e^2*x^4])
Time = 0.93 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.41, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {1396, 316, 27, 402, 27, 402, 27, 402, 27, 402, 25, 27, 399, 289, 329, 327, 765, 762}
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.
| \(\displaystyle \int \frac {1}{\left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {1}{\left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{9/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (7 e x^2+d\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{9/2}}dx}{2 d^2 e}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {7 e x^2+d}{\sqrt {d-e x^2} \left (e x^2+d\right )^{9/2}}dx}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {-\frac {\int -\frac {10 d e \left (3 e x^2+2 d\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{7/2}}dx}{14 d^2 e}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \int \frac {3 e x^2+2 d}{\sqrt {d-e x^2} \left (e x^2+d\right )^{7/2}}dx}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (-\frac {\int -\frac {3 d e \left (e x^2+7 d\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{10 d^2 e}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \int \frac {e x^2+7 d}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {6 d e \left (6 d-e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\int \frac {6 d-e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\frac {7 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\int -\frac {d e \left (7 e x^2+5 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\frac {\int \frac {d e \left (7 e x^2+5 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}+\frac {7 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\frac {\int \frac {7 e x^2+5 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {7 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\frac {7 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-2 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {7 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\frac {7 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {7 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\frac {\frac {7 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {7 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\frac {\frac {7 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {7 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\frac {\frac {7 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {7 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {5 \left (\frac {3 \left (\frac {\frac {\frac {7 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {7 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}-\frac {x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}\right )}{7 d}-\frac {3 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}}{2 d^2}+\frac {x}{2 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^{7/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[1/((d + e*x^2)^3*(d^2 - e^2*x^4)^(3/2)),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(2*d^2*Sqrt[d - e*x^2]*(d + e*x^2)^(7/ 2)) + ((-3*x*Sqrt[d - e*x^2])/(7*d*(d + e*x^2)^(7/2)) + (5*(-1/10*(x*Sqrt[ d - e*x^2])/(d*(d + e*x^2)^(5/2)) + (3*((x*Sqrt[d - e*x^2])/(d*(d + e*x^2) ^(3/2)) + ((7*x*Sqrt[d - e*x^2])/(2*d*Sqrt[d + e*x^2]) + ((7*d^(3/2)*Sqrt[ 1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sq rt[d - e*x^2]*Sqrt[d + e*x^2]) - (2*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*Ellipt icF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e* x^2]))/(2*d))/d))/(10*d)))/(7*d))/(2*d^2)))/Sqrt[d^2 - e^2*x^4]
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)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart [p]) Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* c + a*d, 0] && !IntegerQ[p]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, 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[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])) Int[Sqrt[1 + b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b *c + a*d, 0] && !(LtQ[a*c, 0] && GtQ[a*b, 0])
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
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[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x _Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p]) Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* e^2, 0] && !IntegerQ[p] && !(EqQ[q, 1] && EqQ[n, 2])
Time = 2.43 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.31
| method | result | size |
| default | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{28 d^{3} e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {5 x \sqrt {-e^{2} x^{4}+d^{2}}}{56 d^{4} e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {19 x \sqrt {-e^{2} x^{4}+d^{2}}}{112 d^{5} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {13 \left (-e^{2} x^{2}+d e \right ) x}{32 e \,d^{6} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}-\frac {\left (-e^{2} x^{2}-d e \right ) x}{32 e \,d^{6} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {15 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{56 d^{5} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{8 d^{5} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(338\) |
| elliptic | \(\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{28 d^{3} e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {5 x \sqrt {-e^{2} x^{4}+d^{2}}}{56 d^{4} e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {19 x \sqrt {-e^{2} x^{4}+d^{2}}}{112 d^{5} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {13 \left (-e^{2} x^{2}+d e \right ) x}{32 e \,d^{6} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}-\frac {\left (-e^{2} x^{2}-d e \right ) x}{32 e \,d^{6} \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {15 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{56 d^{5} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {3 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{8 d^{5} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(338\) |
Input:
int(1/(e*x^2+d)^3/(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/28/d^3*x/e^4*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^4+5/56/d^4/e^3*x*(-e^2*x^4+d ^2)^(1/2)/(x^2+d/e)^3+19/112/d^5/e^2*x*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+13 /32*(-e^2*x^2+d*e)/e/d^6*x/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)-1/32*(-e^2*x^2 -d*e)/e/d^6*x/((x^2-d/e)*(-e^2*x^2-d*e))^(1/2)+15/56/d^5/(e/d)^(1/2)*(1-e* x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*EllipticF(x*(e/d)^(1/2 ),I)-3/8/d^5/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2 )^(1/2)*(EllipticF(x*(e/d)^(1/2),I)-EllipticE(x*(e/d)^(1/2),I))
Time = 0.08 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {21 \, {\left (e^{6} x^{10} + 3 \, d e^{5} x^{8} + 2 \, d^{2} e^{4} x^{6} - 2 \, d^{3} e^{3} x^{4} - 3 \, d^{4} e^{2} x^{2} - d^{5} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + 3 \, {\left ({\left (5 \, d e^{5} - 7 \, e^{6}\right )} x^{10} + 3 \, {\left (5 \, d^{2} e^{4} - 7 \, d e^{5}\right )} x^{8} + 2 \, {\left (5 \, d^{3} e^{3} - 7 \, d^{2} e^{4}\right )} x^{6} - 5 \, d^{6} + 7 \, d^{5} e - 2 \, {\left (5 \, d^{4} e^{2} - 7 \, d^{3} e^{3}\right )} x^{4} - 3 \, {\left (5 \, d^{5} e - 7 \, d^{4} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (21 \, e^{5} x^{9} + 48 \, d e^{4} x^{7} + 4 \, d^{2} e^{3} x^{5} - 60 \, d^{3} e^{2} x^{3} - 41 \, d^{4} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{56 \, {\left (d^{6} e^{6} x^{10} + 3 \, d^{7} e^{5} x^{8} + 2 \, d^{8} e^{4} x^{6} - 2 \, d^{9} e^{3} x^{4} - 3 \, d^{10} e^{2} x^{2} - d^{11} e\right )}} \] Input:
integrate(1/(e*x^2+d)^3/(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
Output:
1/56*(21*(e^6*x^10 + 3*d*e^5*x^8 + 2*d^2*e^4*x^6 - 2*d^3*e^3*x^4 - 3*d^4*e ^2*x^2 - d^5*e)*sqrt(e/d)*elliptic_e(arcsin(x*sqrt(e/d)), -1) + 3*((5*d*e^ 5 - 7*e^6)*x^10 + 3*(5*d^2*e^4 - 7*d*e^5)*x^8 + 2*(5*d^3*e^3 - 7*d^2*e^4)* x^6 - 5*d^6 + 7*d^5*e - 2*(5*d^4*e^2 - 7*d^3*e^3)*x^4 - 3*(5*d^5*e - 7*d^4 *e^2)*x^2)*sqrt(e/d)*elliptic_f(arcsin(x*sqrt(e/d)), -1) + (21*e^5*x^9 + 4 8*d*e^4*x^7 + 4*d^2*e^3*x^5 - 60*d^3*e^2*x^3 - 41*d^4*e*x)*sqrt(-e^2*x^4 + d^2))/(d^6*e^6*x^10 + 3*d^7*e^5*x^8 + 2*d^8*e^4*x^6 - 2*d^9*e^3*x^4 - 3*d ^10*e^2*x^2 - d^11*e)
\[ \int \frac {1}{\left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{3}}\, dx \] Input:
integrate(1/(e*x**2+d)**3/(-e**2*x**4+d**2)**(3/2),x)
Output:
Integral(1/((-(-d + e*x**2)*(d + e*x**2))**(3/2)*(d + e*x**2)**3), x)
\[ \int \frac {1}{\left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(1/(e*x^2+d)^3/(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-e^2*x^4 + d^2)^(3/2)*(e*x^2 + d)^3), x)
\[ \int \frac {1}{\left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{3}} \,d x } \] Input:
integrate(1/(e*x^2+d)^3/(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-e^2*x^4 + d^2)^(3/2)*(e*x^2 + d)^3), x)
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (d^2-e^2\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^3} \,d x \] Input:
int(1/((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^3),x)
Output:
int(1/((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^3), x)
\[ \int \frac {1}{\left (d+e x^2\right )^3 \left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e^{7} x^{14}+3 d \,e^{6} x^{12}+d^{2} e^{5} x^{10}-5 d^{3} e^{4} x^{8}-5 d^{4} e^{3} x^{6}+d^{5} e^{2} x^{4}+3 d^{6} e \,x^{2}+d^{7}}d x \] Input:
int(1/(e*x^2+d)^3/(-e^2*x^4+d^2)^(3/2),x)
Output:
int(sqrt(d**2 - e**2*x**4)/(d**7 + 3*d**6*e*x**2 + d**5*e**2*x**4 - 5*d**4 *e**3*x**6 - 5*d**3*e**4*x**8 + d**2*e**5*x**10 + 3*d*e**6*x**12 + e**7*x* *14),x)