Integrand size = 26, antiderivative size = 212 \[ \int \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {4 d^3 x \left (90 d+77 e x^2\right ) \sqrt {d^2-e^2 x^4}}{1155}+\frac {2}{693} d x \left (54 d+77 e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2}-\frac {1}{11} x \left (d^2-e^2 x^4\right )^{5/2}+\frac {8 d^{13/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{15 \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {104 d^{13/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{1155 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
4/1155*d^3*x*(77*e*x^2+90*d)*(-e^2*x^4+d^2)^(1/2)+2/693*d*x*(77*e*x^2+54*d )*(-e^2*x^4+d^2)^(3/2)-1/11*x*(-e^2*x^4+d^2)^(5/2)+8/15*d^(13/2)*(1-e^2*x^ 4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+1 04/1155*d^(13/2)*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/e^(1 /2)/(-e^2*x^4+d^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 9.71 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.59 \[ \int \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {x \sqrt {d^2-e^2 x^4} \left (-3 \left (d^2-e^2 x^4\right )^2 \sqrt {1-\frac {e^2 x^4}{d^2}}+36 d^4 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )+22 d^3 e x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )\right )}{33 \sqrt {1-\frac {e^2 x^4}{d^2}}} \] Input:
Integrate[(d + e*x^2)^2*(d^2 - e^2*x^4)^(3/2),x]
Output:
(x*Sqrt[d^2 - e^2*x^4]*(-3*(d^2 - e^2*x^4)^2*Sqrt[1 - (e^2*x^4)/d^2] + 36* d^4*Hypergeometric2F1[-3/2, 1/4, 5/4, (e^2*x^4)/d^2] + 22*d^3*e*x^2*Hyperg eometric2F1[-3/2, 3/4, 7/4, (e^2*x^4)/d^2]))/(33*Sqrt[1 - (e^2*x^4)/d^2])
Time = 0.90 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.61, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {1396, 318, 27, 403, 25, 27, 403, 27, 403, 27, 403, 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.
| \(\displaystyle \int \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{7/2}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (-\frac {\int -2 d e \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{3/2} \left (11 e x^2+6 d\right )dx}{11 e}-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \int \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^{3/2} \left (11 e x^2+6 d\right )dx-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (-\frac {\int -d e \left (d-e x^2\right )^{3/2} \sqrt {e x^2+d} \left (131 e x^2+65 d\right )dx}{9 e}-\frac {11}{9} x \left (d+e x^2\right )^{3/2} \left (d-e x^2\right )^{5/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {\int d e \left (d-e x^2\right )^{3/2} \sqrt {e x^2+d} \left (131 e x^2+65 d\right )dx}{9 e}-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \int \left (d-e x^2\right )^{3/2} \sqrt {e x^2+d} \left (131 e x^2+65 d\right )dx-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (-\frac {\int -\frac {2 d e \left (d-e x^2\right )^{3/2} \left (424 e x^2+293 d\right )}{\sqrt {e x^2+d}}dx}{7 e}-\frac {131}{7} x \sqrt {d+e x^2} \left (d-e x^2\right )^{5/2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \int \frac {\left (d-e x^2\right )^{3/2} \left (424 e x^2+293 d\right )}{\sqrt {e x^2+d}}dx-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {\int \frac {3 d e \sqrt {d-e x^2} \left (501 e x^2+347 d\right )}{\sqrt {e x^2+d}}dx}{5 e}+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {3}{5} d \int \frac {\sqrt {d-e x^2} \left (501 e x^2+347 d\right )}{\sqrt {e x^2+d}}dx+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {3}{5} d \left (\frac {\int \frac {6 d e \left (77 e x^2+90 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 e}+167 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {3}{5} d \left (2 d \int \frac {77 e x^2+90 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx+167 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {3}{5} d \left (2 d \left (13 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx+77 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx\right )+167 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {3}{5} d \left (2 d \left (\frac {13 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+77 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx\right )+167 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {3}{5} d \left (2 d \left (\frac {77 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {13 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+167 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {3}{5} d \left (2 d \left (\frac {13 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+167 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {3}{5} d \left (2 d \left (\frac {13 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+167 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{11} d \left (\frac {1}{9} d \left (\frac {2}{7} d \left (\frac {3}{5} d \left (2 d \left (\frac {13 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {77 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+167 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )+\frac {424}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {131}{7} x \left (d-e x^2\right )^{5/2} \sqrt {d+e x^2}\right )-\frac {11}{9} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{11} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[(d + e*x^2)^2*(d^2 - e^2*x^4)^(3/2),x]
Output:
(Sqrt[d^2 - e^2*x^4]*(-1/11*(x*(d - e*x^2)^(5/2)*(d + e*x^2)^(5/2)) + (2*d *((-11*x*(d - e*x^2)^(5/2)*(d + e*x^2)^(3/2))/9 + (d*((-131*x*(d - e*x^2)^ (5/2)*Sqrt[d + e*x^2])/7 + (2*d*((424*x*(d - e*x^2)^(3/2)*Sqrt[d + e*x^2]) /5 + (3*d*(167*x*Sqrt[d - e*x^2]*Sqrt[d + e*x^2] + 2*d*((77*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqr t[d - e*x^2]*Sqrt[d + e*x^2]) + (13*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]))))/5))/7))/9))/11))/(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[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 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[(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 = 5.61 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.97
| method | result | size |
| risch | \(\frac {x \left (-315 e^{4} x^{8}-770 d \,e^{3} x^{6}+90 d^{2} e^{2} x^{4}+1694 d^{3} e \,x^{2}+1305 d^{4}\right ) \sqrt {-e^{2} x^{4}+d^{2}}}{3465}+\frac {8 d^{5} \left (\frac {90 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {77 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )}{1155}\) | \(205\) |
| elliptic | \(-\frac {e^{4} x^{9} \sqrt {-e^{2} x^{4}+d^{2}}}{11}-\frac {2 d \,e^{3} x^{7} \sqrt {-e^{2} x^{4}+d^{2}}}{9}+\frac {2 d^{2} e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{77}+\frac {22 d^{3} e \,x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{45}+\frac {29 d^{4} x \sqrt {-e^{2} x^{4}+d^{2}}}{77}+\frac {48 d^{6} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{77 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {8 d^{6} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{15 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(256\) |
| default | \(d^{2} \left (-\frac {e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {3 d^{2} x \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {4 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+e^{2} \left (-\frac {e^{2} x^{9} \sqrt {-e^{2} x^{4}+d^{2}}}{11}+\frac {13 d^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{77}-\frac {4 d^{4} x \sqrt {-e^{2} x^{4}+d^{2}}}{77 e^{2}}+\frac {4 d^{6} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{77 e^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+2 d e \left (-\frac {e^{2} x^{7} \sqrt {-e^{2} x^{4}+d^{2}}}{9}+\frac {11 d^{2} x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{45}-\frac {4 d^{5} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{15 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\) | \(380\) |
Input:
int((e*x^2+d)^2*(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/3465*x*(-315*e^4*x^8-770*d*e^3*x^6+90*d^2*e^2*x^4+1694*d^3*e*x^2+1305*d^ 4)*(-e^2*x^4+d^2)^(1/2)+8/1155*d^5*(90*d/(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)-77*d/(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 = 158, normalized size of antiderivative = 0.75 \[ \int \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2} \, dx=-\frac {1848 \, \sqrt {-e^{2}} d^{6} x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 24 \, {\left (77 \, d^{6} + 90 \, d^{5} e\right )} \sqrt {-e^{2}} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) + {\left (315 \, e^{6} x^{10} + 770 \, d e^{5} x^{8} - 90 \, d^{2} e^{4} x^{6} - 1694 \, d^{3} e^{3} x^{4} - 1305 \, d^{4} e^{2} x^{2} + 1848 \, d^{5} e\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{3465 \, e^{2} x} \] Input:
integrate((e*x^2+d)^2*(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
Output:
-1/3465*(1848*sqrt(-e^2)*d^6*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), - 1) - 24*(77*d^6 + 90*d^5*e)*sqrt(-e^2)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt( d/e)/x), -1) + (315*e^6*x^10 + 770*d*e^5*x^8 - 90*d^2*e^4*x^6 - 1694*d^3*e ^3*x^4 - 1305*d^4*e^2*x^2 + 1848*d^5*e)*sqrt(-e^2*x^4 + d^2))/(e^2*x)
Time = 1.81 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.85 \[ \int \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {d^{5} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {d^{4} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} - \frac {d^{2} e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} - \frac {d e^{4} x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \] Input:
integrate((e*x**2+d)**2*(-e**2*x**4+d**2)**(3/2),x)
Output:
d**5*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), e**2*x**4*exp_polar(2*I*pi)/d **2)/(4*gamma(5/4)) + d**4*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), e* *2*x**4*exp_polar(2*I*pi)/d**2)/(2*gamma(7/4)) - d**2*e**3*x**7*gamma(7/4) *hyper((-1/2, 7/4), (11/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/(2*gamma(11 /4)) - d*e**4*x**9*gamma(9/4)*hyper((-1/2, 9/4), (13/4,), e**2*x**4*exp_po lar(2*I*pi)/d**2)/(4*gamma(13/4))
\[ \int \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int { {\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2} \,d x } \] Input:
integrate((e*x^2+d)^2*(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)*(e*x^2 + d)^2, x)
\[ \int \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int { {\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{2} \,d x } \] Input:
integrate((e*x^2+d)^2*(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
Output:
integrate((-e^2*x^4 + d^2)^(3/2)*(e*x^2 + d)^2, x)
Timed out. \[ \int \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int {\left (d^2-e^2\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^2 \,d x \] Input:
int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^2,x)
Output:
int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^2, x)
\[ \int \left (d+e x^2\right )^2 \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {29 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{4} x}{77}+\frac {22 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{3} e \,x^{3}}{45}+\frac {2 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{2} e^{2} x^{5}}{77}-\frac {2 \sqrt {-e^{2} x^{4}+d^{2}}\, d \,e^{3} x^{7}}{9}-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{4} x^{9}}{11}+\frac {48 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{6}}{77}+\frac {8 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{5} e}{15} \] Input:
int((e*x^2+d)^2*(-e^2*x^4+d^2)^(3/2),x)
Output:
(1305*sqrt(d**2 - e**2*x**4)*d**4*x + 1694*sqrt(d**2 - e**2*x**4)*d**3*e*x **3 + 90*sqrt(d**2 - e**2*x**4)*d**2*e**2*x**5 - 770*sqrt(d**2 - e**2*x**4 )*d*e**3*x**7 - 315*sqrt(d**2 - e**2*x**4)*e**4*x**9 + 2160*int(sqrt(d**2 - e**2*x**4)/(d**2 - e**2*x**4),x)*d**6 + 1848*int((sqrt(d**2 - e**2*x**4) *x**2)/(d**2 - e**2*x**4),x)*d**5*e)/3465