Integrand size = 26, antiderivative size = 287 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^7} \, dx=\frac {2 x \sqrt {d^2-e^2 x^4}}{9 \left (d+e x^2\right )^5}+\frac {2 x \sqrt {d^2-e^2 x^4}}{21 d \left (d+e x^2\right )^4}+\frac {73 x \sqrt {d^2-e^2 x^4}}{630 d^2 \left (d+e x^2\right )^3}+\frac {16 x \sqrt {d^2-e^2 x^4}}{105 d^3 \left (d+e x^2\right )^2}+\frac {17 x \sqrt {d^2-e^2 x^4}}{60 d^4 \left (d+e x^2\right )}+\frac {17 \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{60 d^{5/2} \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {16 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{105 d^{5/2} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
2/9*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^5+2/21*x*(-e^2*x^4+d^2)^(1/2)/d/(e*x^ 2+d)^4+73/630*x*(-e^2*x^4+d^2)^(1/2)/d^2/(e*x^2+d)^3+16/105*x*(-e^2*x^4+d^ 2)^(1/2)/d^3/(e*x^2+d)^2+17/60*x*(-e^2*x^4+d^2)^(1/2)/d^4/(e*x^2+d)+17/60* (1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/d^(5/2)/e^(1/2)/(-e^2 *x^4+d^2)^(1/2)-16/105*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I )/d^(5/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
Result contains complex when optimal does not.
Time = 11.74 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.57 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^7} \, dx=\frac {-\frac {x \left (-d+e x^2\right ) \left (1095 d^4+2416 d^3 e x^2+2864 d^2 e^2 x^4+1620 d e^3 x^6+357 e^4 x^8\right )}{\left (d+e x^2\right )^4}+\frac {3 i e \sqrt {1-\frac {e^2 x^4}{d^2}} \left (119 E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-64 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\left (-\frac {e}{d}\right )^{3/2}}}{1260 d^4 \sqrt {d^2-e^2 x^4}} \] Input:
Integrate[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^7,x]
Output:
(-((x*(-d + e*x^2)*(1095*d^4 + 2416*d^3*e*x^2 + 2864*d^2*e^2*x^4 + 1620*d* e^3*x^6 + 357*e^4*x^8))/(d + e*x^2)^4) + ((3*I)*e*Sqrt[1 - (e^2*x^4)/d^2]* (119*EllipticE[I*ArcSinh[Sqrt[-(e/d)]*x], -1] - 64*EllipticF[I*ArcSinh[Sqr t[-(e/d)]*x], -1]))/(-(e/d))^(3/2))/(1260*d^4*Sqrt[d^2 - e^2*x^4])
Time = 0.94 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.25, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {1396, 315, 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.
| \(\displaystyle \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^7} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \frac {\left (d-e x^2\right )^{3/2}}{\left (e x^2+d\right )^{11/2}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\int \frac {d e \left (7 d-5 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{9/2}}dx}{9 d e}+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \int \frac {7 d-5 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{9/2}}dx+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}-\frac {\int -\frac {2 d e \left (43 d-30 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{7/2}}dx}{14 d^2 e}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\int \frac {43 d-30 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{7/2}}dx}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}-\frac {\int -\frac {3 d e \left (119 d-73 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{10 d^2 e}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \int \frac {119 d-73 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{5/2}}dx}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {6 d e \left (87 d-32 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\int \frac {87 d-32 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^{3/2}}dx}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\frac {119 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}-\frac {\int -\frac {d e \left (119 e x^2+55 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\frac {\int \frac {d e \left (119 e x^2+55 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d^2 e}+\frac {119 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\frac {\int \frac {119 e x^2+55 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {119 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\frac {119 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-64 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{2 d}+\frac {119 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\frac {119 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {64 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {119 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\frac {\frac {119 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {64 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {119 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\frac {\frac {119 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {64 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {119 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\frac {\frac {119 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {64 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {119 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{9} \left (\frac {\frac {3 \left (\frac {\frac {\frac {119 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {64 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}}{2 d}+\frac {119 x \sqrt {d-e x^2}}{2 d \sqrt {d+e x^2}}}{d}+\frac {32 x \sqrt {d-e x^2}}{d \left (d+e x^2\right )^{3/2}}\right )}{10 d}+\frac {73 x \sqrt {d-e x^2}}{10 d \left (d+e x^2\right )^{5/2}}}{7 d}+\frac {6 x \sqrt {d-e x^2}}{7 d \left (d+e x^2\right )^{7/2}}\right )+\frac {2 x \sqrt {d-e x^2}}{9 \left (d+e x^2\right )^{9/2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[(d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^7,x]
Output:
(Sqrt[d^2 - e^2*x^4]*((2*x*Sqrt[d - e*x^2])/(9*(d + e*x^2)^(9/2)) + ((6*x* Sqrt[d - e*x^2])/(7*d*(d + e*x^2)^(7/2)) + ((73*x*Sqrt[d - e*x^2])/(10*d*( d + e*x^2)^(5/2)) + (3*((32*x*Sqrt[d - e*x^2])/(d*(d + e*x^2)^(3/2)) + ((1 19*x*Sqrt[d - e*x^2])/(2*d*Sqrt[d + e*x^2]) + ((119*d^(3/2)*Sqrt[1 - (e^2* x^4)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e* x^2]*Sqrt[d + e*x^2]) - (64*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcS in[(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))/9))/(Sqrt[d - e*x^2]*Sqrt[d + e*x^2])
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[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[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 = 6.90 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.12
| method | result | size |
| default | \(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{9 e^{5} \left (x^{2}+\frac {d}{e}\right )^{5}}+\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{21 d \,e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {73 x \sqrt {-e^{2} x^{4}+d^{2}}}{630 d^{2} e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {16 x \sqrt {-e^{2} x^{4}+d^{2}}}{105 d^{3} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {17 \left (-e^{2} x^{2}+d e \right ) x}{60 e \,d^{4} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {11 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{84 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {17 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{60 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(321\) |
| elliptic | \(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{9 e^{5} \left (x^{2}+\frac {d}{e}\right )^{5}}+\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{21 d \,e^{4} \left (x^{2}+\frac {d}{e}\right )^{4}}+\frac {73 x \sqrt {-e^{2} x^{4}+d^{2}}}{630 d^{2} e^{3} \left (x^{2}+\frac {d}{e}\right )^{3}}+\frac {16 x \sqrt {-e^{2} x^{4}+d^{2}}}{105 d^{3} e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {17 \left (-e^{2} x^{2}+d e \right ) x}{60 e \,d^{4} \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {11 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{84 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {17 \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{60 d^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(321\) |
Input:
int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^7,x,method=_RETURNVERBOSE)
Output:
2/9*x/e^5*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^5+2/21/d*x/e^4*(-e^2*x^4+d^2)^(1/ 2)/(x^2+d/e)^4+73/630/d^2/e^3*x*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^3+16/105/d^ 3/e^2*x*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+17/60*(-e^2*x^2+d*e)/e/d^4*x/((x^ 2+d/e)*(-e^2*x^2+d*e))^(1/2)+11/84/d^3/(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)-17/60/d^3/(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)*(Ellipti cF(x*(e/d)^(1/2),I)-EllipticE(x*(e/d)^(1/2),I))
Time = 0.11 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.17 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^7} \, dx=\frac {357 \, {\left (e^{6} x^{10} + 5 \, d e^{5} x^{8} + 10 \, d^{2} e^{4} x^{6} + 10 \, d^{3} e^{3} x^{4} + 5 \, d^{4} e^{2} x^{2} + d^{5} e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + 3 \, {\left ({\left (55 \, d e^{5} - 119 \, e^{6}\right )} x^{10} + 5 \, {\left (55 \, d^{2} e^{4} - 119 \, d e^{5}\right )} x^{8} + 10 \, {\left (55 \, d^{3} e^{3} - 119 \, d^{2} e^{4}\right )} x^{6} + 55 \, d^{6} - 119 \, d^{5} e + 10 \, {\left (55 \, d^{4} e^{2} - 119 \, d^{3} e^{3}\right )} x^{4} + 5 \, {\left (55 \, d^{5} e - 119 \, d^{4} e^{2}\right )} x^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left (357 \, e^{5} x^{9} + 1620 \, d e^{4} x^{7} + 2864 \, d^{2} e^{3} x^{5} + 2416 \, d^{3} e^{2} x^{3} + 1095 \, d^{4} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{1260 \, {\left (d^{4} e^{6} x^{10} + 5 \, d^{5} e^{5} x^{8} + 10 \, d^{6} e^{4} x^{6} + 10 \, d^{7} e^{3} x^{4} + 5 \, d^{8} e^{2} x^{2} + d^{9} e\right )}} \] Input:
integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^7,x, algorithm="fricas")
Output:
1/1260*(357*(e^6*x^10 + 5*d*e^5*x^8 + 10*d^2*e^4*x^6 + 10*d^3*e^3*x^4 + 5* d^4*e^2*x^2 + d^5*e)*sqrt(e/d)*elliptic_e(arcsin(x*sqrt(e/d)), -1) + 3*((5 5*d*e^5 - 119*e^6)*x^10 + 5*(55*d^2*e^4 - 119*d*e^5)*x^8 + 10*(55*d^3*e^3 - 119*d^2*e^4)*x^6 + 55*d^6 - 119*d^5*e + 10*(55*d^4*e^2 - 119*d^3*e^3)*x^ 4 + 5*(55*d^5*e - 119*d^4*e^2)*x^2)*sqrt(e/d)*elliptic_f(arcsin(x*sqrt(e/d )), -1) + (357*e^5*x^9 + 1620*d*e^4*x^7 + 2864*d^2*e^3*x^5 + 2416*d^3*e^2* x^3 + 1095*d^4*e*x)*sqrt(-e^2*x^4 + d^2))/(d^4*e^6*x^10 + 5*d^5*e^5*x^8 + 10*d^6*e^4*x^6 + 10*d^7*e^3*x^4 + 5*d^8*e^2*x^2 + d^9*e)
Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^7} \, dx=\text {Timed out} \] Input:
integrate((-e**2*x**4+d**2)**(3/2)/(e*x**2+d)**7,x)
Output:
Timed out
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^7} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{7}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^7,x, algorithm="maxima")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^7, x)
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^7} \, dx=\int { \frac {{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}}}{{\left (e x^{2} + d\right )}^{7}} \,d x } \] Input:
integrate((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^7,x, algorithm="giac")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)/(e*x^2 + d)^7, x)
Timed out. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^7} \, dx=\int \frac {{\left (d^2-e^2\,x^4\right )}^{3/2}}{{\left (e\,x^2+d\right )}^7} \,d x \] Input:
int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^7,x)
Output:
int((d^2 - e^2*x^4)^(3/2)/(d + e*x^2)^7, x)
\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^7} \, dx =\text {Too large to display} \] Input:
int((-e^2*x^4+d^2)^(3/2)/(e*x^2+d)^7,x)
Output:
(sqrt(d**2 - e**2*x**4)*x + 4*int(sqrt(d**2 - e**2*x**4)/(d**7 + 5*d**6*e* x**2 + 9*d**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3*e**4*x**8 - 9*d**2*e** 5*x**10 - 5*d*e**6*x**12 - e**7*x**14),x)*d**7 + 20*int(sqrt(d**2 - e**2*x **4)/(d**7 + 5*d**6*e*x**2 + 9*d**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3* e**4*x**8 - 9*d**2*e**5*x**10 - 5*d*e**6*x**12 - e**7*x**14),x)*d**6*e*x** 2 + 40*int(sqrt(d**2 - e**2*x**4)/(d**7 + 5*d**6*e*x**2 + 9*d**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3*e**4*x**8 - 9*d**2*e**5*x**10 - 5*d*e**6*x**1 2 - e**7*x**14),x)*d**5*e**2*x**4 + 40*int(sqrt(d**2 - e**2*x**4)/(d**7 + 5*d**6*e*x**2 + 9*d**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3*e**4*x**8 - 9 *d**2*e**5*x**10 - 5*d*e**6*x**12 - e**7*x**14),x)*d**4*e**3*x**6 + 20*int (sqrt(d**2 - e**2*x**4)/(d**7 + 5*d**6*e*x**2 + 9*d**5*e**2*x**4 + 5*d**4* e**3*x**6 - 5*d**3*e**4*x**8 - 9*d**2*e**5*x**10 - 5*d*e**6*x**12 - e**7*x **14),x)*d**3*e**4*x**8 + 4*int(sqrt(d**2 - e**2*x**4)/(d**7 + 5*d**6*e*x* *2 + 9*d**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3*e**4*x**8 - 9*d**2*e**5* x**10 - 5*d*e**6*x**12 - e**7*x**14),x)*d**2*e**5*x**10 - 2*int((sqrt(d**2 - e**2*x**4)*x**4)/(d**7 + 5*d**6*e*x**2 + 9*d**5*e**2*x**4 + 5*d**4*e**3 *x**6 - 5*d**3*e**4*x**8 - 9*d**2*e**5*x**10 - 5*d*e**6*x**12 - e**7*x**14 ),x)*d**5*e**2 - 10*int((sqrt(d**2 - e**2*x**4)*x**4)/(d**7 + 5*d**6*e*x** 2 + 9*d**5*e**2*x**4 + 5*d**4*e**3*x**6 - 5*d**3*e**4*x**8 - 9*d**2*e**5*x **10 - 5*d*e**6*x**12 - e**7*x**14),x)*d**4*e**3*x**2 - 20*int((sqrt(d*...