Integrand size = 26, antiderivative size = 205 \[ \int \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4} \, dx=\frac {2}{21} d^2 x \left (5 d+7 e x^2\right ) \sqrt {d^2-e^2 x^4}-\frac {3}{7} d x \left (d^2-e^2 x^4\right )^{3/2}-\frac {1}{9} e x^3 \left (d^2-e^2 x^4\right )^{3/2}+\frac {4 d^{11/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{3 \sqrt {e} \sqrt {d^2-e^2 x^4}}-\frac {8 d^{11/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{21 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:
2/21*d^2*x*(7*e*x^2+5*d)*(-e^2*x^4+d^2)^(1/2)-3/7*d*x*(-e^2*x^4+d^2)^(3/2) -1/9*e*x^3*(-e^2*x^4+d^2)^(3/2)+4/3*d^(11/2)*(1-e^2*x^4/d^2)^(1/2)*Ellipti cE(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)-8/21*d^(11/2)*(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 = 8.42 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.70 \[ \int \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4} \, dx=\frac {x \sqrt {d^2-e^2 x^4} \left (\sqrt {1-\frac {e^2 x^4}{d^2}} \left (-27 d^3-7 d^2 e x^2+27 d e^2 x^4+7 e^3 x^6\right )+90 d^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )+70 d^2 e x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )\right )}{63 \sqrt {1-\frac {e^2 x^4}{d^2}}} \] Input:
Integrate[(d + e*x^2)^3*Sqrt[d^2 - e^2*x^4],x]
Output:
(x*Sqrt[d^2 - e^2*x^4]*(Sqrt[1 - (e^2*x^4)/d^2]*(-27*d^3 - 7*d^2*e*x^2 + 2 7*d*e^2*x^4 + 7*e^3*x^6) + 90*d^3*Hypergeometric2F1[-1/2, 1/4, 5/4, (e^2*x ^4)/d^2] + 70*d^2*e*x^2*Hypergeometric2F1[-1/2, 3/4, 7/4, (e^2*x^4)/d^2])) /(63*Sqrt[1 - (e^2*x^4)/d^2])
Time = 0.80 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.49, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.577, Rules used = {1396, 318, 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.
| \(\displaystyle \int \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4} \, dx\) |
| \(\Big \downarrow \) 1396 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \sqrt {d-e x^2} \left (e x^2+d\right )^{7/2}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (-\frac {\int -10 d e \sqrt {d-e x^2} \left (e x^2+d\right )^{3/2} \left (2 e x^2+d\right )dx}{9 e}-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \int \sqrt {d-e x^2} \left (e x^2+d\right )^{3/2} \left (2 e x^2+d\right )dx-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (-\frac {\int -3 d e \sqrt {d-e x^2} \sqrt {e x^2+d} \left (7 e x^2+3 d\right )dx}{7 e}-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \int \sqrt {d-e x^2} \sqrt {e x^2+d} \left (7 e x^2+3 d\right )dx-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (-\frac {\int -\frac {2 d e \sqrt {d-e x^2} \left (18 e x^2+11 d\right )}{\sqrt {e x^2+d}}dx}{5 e}-\frac {7}{5} x \sqrt {d+e x^2} \left (d-e x^2\right )^{3/2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (\frac {2}{5} d \int \frac {\sqrt {d-e x^2} \left (18 e x^2+11 d\right )}{\sqrt {e x^2+d}}dx-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (\frac {2}{5} d \left (\frac {\int \frac {3 d e \left (7 e x^2+5 d\right )}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx}{3 e}+6 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (\frac {2}{5} d \left (d \int \frac {7 e x^2+5 d}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx+6 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (\frac {2}{5} d \left (d \left (7 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-2 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx\right )+6 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 289 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (\frac {2}{5} d \left (d \left (7 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx-\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+6 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 329 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (\frac {2}{5} d \left (d \left (\frac {7 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+6 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (\frac {2}{5} d \left (d \left (\frac {7 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+6 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 765 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (\frac {2}{5} d \left (d \left (\frac {7 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+6 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {10}{9} d \left (\frac {3}{7} d \left (\frac {2}{5} d \left (d \left (\frac {7 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+6 x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {7}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )-\frac {2}{7} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{3/2}\right )-\frac {1}{9} x \left (d-e x^2\right )^{3/2} \left (d+e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[(d + e*x^2)^3*Sqrt[d^2 - e^2*x^4],x]
Output:
(Sqrt[d^2 - e^2*x^4]*(-1/9*(x*(d - e*x^2)^(3/2)*(d + e*x^2)^(5/2)) + (10*d *((-2*x*(d - e*x^2)^(3/2)*(d + e*x^2)^(3/2))/7 + (3*d*((-7*x*(d - e*x^2)^( 3/2)*Sqrt[d + e*x^2])/5 + (2*d*(6*x*Sqrt[d - e*x^2]*Sqrt[d + e*x^2] + d*(( 7*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]) - (2*d^(3/2)*Sqrt[1 - (e^2*x ^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x ^2]*Sqrt[d + e*x^2]))))/5))/7))/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[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 = 7.88 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(\frac {x \left (7 e^{3} x^{6}+27 d \,e^{2} x^{4}+35 d^{2} e \,x^{2}+3 d^{3}\right ) \sqrt {-e^{2} x^{4}+d^{2}}}{63}+\frac {4 d^{4} \left (\frac {5 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {7 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )}{21}\) | \(194\) |
| elliptic | \(\frac {e^{3} x^{7} \sqrt {-e^{2} x^{4}+d^{2}}}{9}+\frac {3 d \,e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {5 d^{2} e \,x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{9}+\frac {d^{3} x \sqrt {-e^{2} x^{4}+d^{2}}}{21}+\frac {20 d^{5} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{21 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {4 d^{5} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(231\) |
| default | \(d^{3} \left (\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {2 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+e^{3} \left (\frac {x^{7} \sqrt {-e^{2} x^{4}+d^{2}}}{9}-\frac {2 d^{2} x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{45 e^{2}}-\frac {2 d^{5} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{15 e^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+3 d \,e^{2} \left (\frac {x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}-\frac {2 d^{2} x \sqrt {-e^{2} x^{4}+d^{2}}}{21 e^{2}}+\frac {2 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{21 e^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+3 d^{2} e \left (\frac {x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{5}-\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\) | \(439\) |
Input:
int((e*x^2+d)^3*(-e^2*x^4+d^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/63*x*(7*e^3*x^6+27*d*e^2*x^4+35*d^2*e*x^2+3*d^3)*(-e^2*x^4+d^2)^(1/2)+4/ 21*d^4*(5*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)-7*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 = 148, normalized size of antiderivative = 0.72 \[ \int \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4} \, dx=-\frac {84 \, \sqrt {-e^{2}} d^{5} x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 12 \, {\left (7 \, d^{5} + 5 \, d^{4} e\right )} \sqrt {-e^{2}} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - {\left (7 \, e^{5} x^{8} + 27 \, d e^{4} x^{6} + 35 \, d^{2} e^{3} x^{4} + 3 \, d^{3} e^{2} x^{2} - 84 \, d^{4} e\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{63 \, e^{2} x} \] Input:
integrate((e*x^2+d)^3*(-e^2*x^4+d^2)^(1/2),x, algorithm="fricas")
Output:
-1/63*(84*sqrt(-e^2)*d^5*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), -1) - 12*(7*d^5 + 5*d^4*e)*sqrt(-e^2)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e)/x ), -1) - (7*e^5*x^8 + 27*d*e^4*x^6 + 35*d^2*e^3*x^4 + 3*d^3*e^2*x^2 - 84*d ^4*e)*sqrt(-e^2*x^4 + d^2))/(e^2*x)
Time = 1.76 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.90 \[ \int \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4} \, dx=\frac {d^{4} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {3 d^{3} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} + \frac {3 d^{2} e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {d e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate((e*x**2+d)**3*(-e**2*x**4+d**2)**(1/2),x)
Output:
d**4*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)) + 3*d**3*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/(4*gamma(7/4)) + 3*d**2*e**2*x**5*gamma( 5/4)*hyper((-1/2, 5/4), (9/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/(4*gamma (9/4)) + d*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)/(4*gamma(11/4))
\[ \int \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4} \, dx=\int { \sqrt {-e^{2} x^{4} + d^{2}} {\left (e x^{2} + d\right )}^{3} \,d x } \] Input:
integrate((e*x^2+d)^3*(-e^2*x^4+d^2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-e^2*x^4 + d^2)*(e*x^2 + d)^3, x)
\[ \int \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4} \, dx=\int { \sqrt {-e^{2} x^{4} + d^{2}} {\left (e x^{2} + d\right )}^{3} \,d x } \] Input:
integrate((e*x^2+d)^3*(-e^2*x^4+d^2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-e^2*x^4 + d^2)*(e*x^2 + d)^3, x)
Timed out. \[ \int \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4} \, dx=\int \sqrt {d^2-e^2\,x^4}\,{\left (e\,x^2+d\right )}^3 \,d x \] Input:
int((d^2 - e^2*x^4)^(1/2)*(d + e*x^2)^3,x)
Output:
int((d^2 - e^2*x^4)^(1/2)*(d + e*x^2)^3, x)
\[ \int \left (d+e x^2\right )^3 \sqrt {d^2-e^2 x^4} \, dx=\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, d^{3} x}{21}+\frac {5 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{2} e \,x^{3}}{9}+\frac {3 \sqrt {-e^{2} x^{4}+d^{2}}\, d \,e^{2} x^{5}}{7}+\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{3} x^{7}}{9}+\frac {20 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{5}}{21}+\frac {4 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{4} e}{3} \] Input:
int((e*x^2+d)^3*(-e^2*x^4+d^2)^(1/2),x)
Output:
(3*sqrt(d**2 - e**2*x**4)*d**3*x + 35*sqrt(d**2 - e**2*x**4)*d**2*e*x**3 + 27*sqrt(d**2 - e**2*x**4)*d*e**2*x**5 + 7*sqrt(d**2 - e**2*x**4)*e**3*x** 7 + 60*int(sqrt(d**2 - e**2*x**4)/(d**2 - e**2*x**4),x)*d**5 + 84*int((sqr t(d**2 - e**2*x**4)*x**2)/(d**2 - e**2*x**4),x)*d**4*e)/63