Integrand size = 28, antiderivative size = 236 \[ \int \left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {91 d^4 x \sqrt {d^2-e^2 x^4}}{256 \sqrt {d+e x^2}}+\frac {73 d^3 e x^3 \sqrt {d^2-e^2 x^4}}{128 \sqrt {d+e x^2}}+\frac {9 d^2 e^2 x^5 \sqrt {d^2-e^2 x^4}}{160 \sqrt {d+e x^2}}-\frac {19 d e^3 x^7 \sqrt {d^2-e^2 x^4}}{80 \sqrt {d+e x^2}}-\frac {e^4 x^9 \sqrt {d^2-e^2 x^4}}{10 \sqrt {d+e x^2}}+\frac {165 d^5 \arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{256 \sqrt {e}} \] Output:
91/256*d^4*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+73/128*d^3*e*x^3*(-e^2*x ^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+9/160*d^2*e^2*x^5*(-e^2*x^4+d^2)^(1/2)/(e*x^ 2+d)^(1/2)-19/80*d*e^3*x^7*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)-1/10*e^4*x ^9*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+165/256*d^5*arctan(e^(1/2)*x*(e*x^ 2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2))/e^(1/2)
Result contains complex when optimal does not.
Time = 4.60 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.55 \[ \int \left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {\frac {x \sqrt {d^2-e^2 x^4} \left (455 d^4+730 d^3 e x^2+72 d^2 e^2 x^4-304 d e^3 x^6-128 e^4 x^8\right )}{\sqrt {d+e x^2}}+\frac {825 i d^5 \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}}{1280} \] Input:
Integrate[(d + e*x^2)^(3/2)*(d^2 - e^2*x^4)^(3/2),x]
Output:
((x*Sqrt[d^2 - e^2*x^4]*(455*d^4 + 730*d^3*e*x^2 + 72*d^2*e^2*x^4 - 304*d* e^3*x^6 - 128*e^4*x^8))/Sqrt[d + e*x^2] + ((825*I)*d^5*Log[(-2*I)*Sqrt[e]* x + (2*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^2]])/Sqrt[e])/1280
Time = 0.50 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.83, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1396, 318, 25, 27, 403, 25, 27, 299, 211, 211, 224, 216}
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.
\(\displaystyle \int \left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \left (d-e x^2\right )^{3/2} \left (e x^2+d\right )^3dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 318 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (-\frac {\int -d e \left (d-e x^2\right )^{3/2} \left (e x^2+d\right ) \left (19 e x^2+11 d\right )dx}{10 e}-\frac {1}{10} x \left (d+e x^2\right )^2 \left (d-e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\int d e \left (d-e x^2\right )^{3/2} \left (e x^2+d\right ) \left (19 e x^2+11 d\right )dx}{10 e}-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{10} d \int \left (d-e x^2\right )^{3/2} \left (e x^2+d\right ) \left (19 e x^2+11 d\right )dx-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 403 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{10} d \left (-\frac {\int -d e \left (d-e x^2\right )^{3/2} \left (183 e x^2+107 d\right )dx}{8 e}-\frac {19}{8} x \left (d+e x^2\right ) \left (d-e x^2\right )^{5/2}\right )-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{10} d \left (\frac {\int d e \left (d-e x^2\right )^{3/2} \left (183 e x^2+107 d\right )dx}{8 e}-\frac {19}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{10} d \left (\frac {1}{8} d \int \left (d-e x^2\right )^{3/2} \left (183 e x^2+107 d\right )dx-\frac {19}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{10} d \left (\frac {1}{8} d \left (\frac {275}{2} d \int \left (d-e x^2\right )^{3/2}dx-\frac {61}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {19}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{10} d \left (\frac {1}{8} d \left (\frac {275}{2} d \left (\frac {3}{4} d \int \sqrt {d-e x^2}dx+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {61}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {19}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{10} d \left (\frac {1}{8} d \left (\frac {275}{2} d \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\sqrt {d-e x^2}}dx+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {61}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {19}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{10} d \left (\frac {1}{8} d \left (\frac {275}{2} d \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\frac {e x^2}{d-e x^2}+1}d\frac {x}{\sqrt {d-e x^2}}+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {61}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {19}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{10} d \left (\frac {1}{8} d \left (\frac {275}{2} d \left (\frac {3}{4} d \left (\frac {d \arctan \left (\frac {\sqrt {e} x}{\sqrt {d-e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {61}{2} x \left (d-e x^2\right )^{5/2}\right )-\frac {19}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )-\frac {1}{10} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )^2\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\) |
Input:
Int[(d + e*x^2)^(3/2)*(d^2 - e^2*x^4)^(3/2),x]
Output:
(Sqrt[d^2 - e^2*x^4]*(-1/10*(x*(d - e*x^2)^(5/2)*(d + e*x^2)^2) + (d*((-19 *x*(d - e*x^2)^(5/2)*(d + e*x^2))/8 + (d*((-61*x*(d - e*x^2)^(5/2))/2 + (2 75*d*((x*(d - e*x^2)^(3/2))/4 + (3*d*((x*Sqrt[d - e*x^2])/2 + (d*ArcTan[(S qrt[e]*x)/Sqrt[d - e*x^2]])/(2*Sqrt[e])))/4))/2))/8))/10))/(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_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
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[((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[(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 = 0.20 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.67
method | result | size |
default | \(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (-128 e^{\frac {9}{2}} x^{9} \sqrt {-e \,x^{2}+d}-304 d \,e^{\frac {7}{2}} x^{7} \sqrt {-e \,x^{2}+d}+72 d^{2} e^{\frac {5}{2}} x^{5} \sqrt {-e \,x^{2}+d}+730 d^{3} e^{\frac {3}{2}} x^{3} \sqrt {-e \,x^{2}+d}+455 \sqrt {e}\, \sqrt {-e \,x^{2}+d}\, d^{4} x +825 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d^{5}\right )}{1280 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \sqrt {e}}\) | \(159\) |
risch | \(\frac {x \left (-128 e^{4} x^{8}-304 d \,e^{3} x^{6}+72 d^{2} e^{2} x^{4}+730 d^{3} e \,x^{2}+455 d^{4}\right ) \sqrt {-e \,x^{2}+d}\, \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{1280 \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {165 d^{5} \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{256 \sqrt {e}\, \sqrt {-e^{2} x^{4}+d^{2}}}\) | \(176\) |
Input:
int((e*x^2+d)^(3/2)*(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/1280*(-e^2*x^4+d^2)^(1/2)*(-128*e^(9/2)*x^9*(-e*x^2+d)^(1/2)-304*d*e^(7/ 2)*x^7*(-e*x^2+d)^(1/2)+72*d^2*e^(5/2)*x^5*(-e*x^2+d)^(1/2)+730*d^3*e^(3/2 )*x^3*(-e*x^2+d)^(1/2)+455*e^(1/2)*(-e*x^2+d)^(1/2)*d^4*x+825*arctan(e^(1/ 2)*x/(-e*x^2+d)^(1/2))*d^5)/(e*x^2+d)^(1/2)/(-e*x^2+d)^(1/2)/e^(1/2)
Time = 0.14 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.35 \[ \int \left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\left [-\frac {825 \, {\left (d^{5} e x^{2} + d^{6}\right )} \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right ) + 2 \, {\left (128 \, e^{5} x^{9} + 304 \, d e^{4} x^{7} - 72 \, d^{2} e^{3} x^{5} - 730 \, d^{3} e^{2} x^{3} - 455 \, d^{4} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{2560 \, {\left (e^{2} x^{2} + d e\right )}}, -\frac {825 \, {\left (d^{5} e x^{2} + d^{6}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) + {\left (128 \, e^{5} x^{9} + 304 \, d e^{4} x^{7} - 72 \, d^{2} e^{3} x^{5} - 730 \, d^{3} e^{2} x^{3} - 455 \, d^{4} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{1280 \, {\left (e^{2} x^{2} + d e\right )}}\right ] \] Input:
integrate((e*x^2+d)^(3/2)*(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
Output:
[-1/2560*(825*(d^5*e*x^2 + d^6)*sqrt(-e)*log(-(2*e^2*x^4 + d*e*x^2 - 2*sqr t(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(-e)*x - d^2)/(e*x^2 + d)) + 2*(128* e^5*x^9 + 304*d*e^4*x^7 - 72*d^2*e^3*x^5 - 730*d^3*e^2*x^3 - 455*d^4*e*x)* sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d))/(e^2*x^2 + d*e), -1/1280*(825*(d^5*e *x^2 + d^6)*sqrt(e)*arctan(sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(e)*x/ (e^2*x^4 - d^2)) + (128*e^5*x^9 + 304*d*e^4*x^7 - 72*d^2*e^3*x^5 - 730*d^3 *e^2*x^3 - 455*d^4*e*x)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d))/(e^2*x^2 + d *e)]
\[ \int \left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int \left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {3}{2}} \left (d + e x^{2}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((e*x**2+d)**(3/2)*(-e**2*x**4+d**2)**(3/2),x)
Output:
Integral((-(-d + e*x**2)*(d + e*x**2))**(3/2)*(d + e*x**2)**(3/2), x)
\[ \int \left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int { {\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {3}{2}} {\left (e x^{2} + d\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)*(-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)^(3/2), x)
Time = 0.16 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.41 \[ \int \left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=-\frac {165 \, d^{5} \log \left ({\left | -\sqrt {-e} x + \sqrt {-e x^{2} + d} \right |}\right )}{256 \, \sqrt {-e}} + \frac {1}{1280} \, {\left (455 \, d^{4} + 2 \, {\left (365 \, d^{3} e + 4 \, {\left (9 \, d^{2} e^{2} - 2 \, {\left (8 \, e^{4} x^{2} + 19 \, d e^{3}\right )} x^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {-e x^{2} + d} x \] Input:
integrate((e*x^2+d)^(3/2)*(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
Output:
-165/256*d^5*log(abs(-sqrt(-e)*x + sqrt(-e*x^2 + d)))/sqrt(-e) + 1/1280*(4 55*d^4 + 2*(365*d^3*e + 4*(9*d^2*e^2 - 2*(8*e^4*x^2 + 19*d*e^3)*x^2)*x^2)* x^2)*sqrt(-e*x^2 + d)*x
Timed out. \[ \int \left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\int {\left (d^2-e^2\,x^4\right )}^{3/2}\,{\left (e\,x^2+d\right )}^{3/2} \,d x \] Input:
int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^(3/2),x)
Output:
int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^(3/2), x)
Time = 0.18 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.48 \[ \int \left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {825 \sqrt {e}\, \mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right ) d^{5}+455 \sqrt {-e \,x^{2}+d}\, d^{4} e x +730 \sqrt {-e \,x^{2}+d}\, d^{3} e^{2} x^{3}+72 \sqrt {-e \,x^{2}+d}\, d^{2} e^{3} x^{5}-304 \sqrt {-e \,x^{2}+d}\, d \,e^{4} x^{7}-128 \sqrt {-e \,x^{2}+d}\, e^{5} x^{9}}{1280 e} \] Input:
int((e*x^2+d)^(3/2)*(-e^2*x^4+d^2)^(3/2),x)
Output:
(825*sqrt(e)*asin((sqrt(e)*x)/sqrt(d))*d**5 + 455*sqrt(d - e*x**2)*d**4*e* x + 730*sqrt(d - e*x**2)*d**3*e**2*x**3 + 72*sqrt(d - e*x**2)*d**2*e**3*x* *5 - 304*sqrt(d - e*x**2)*d*e**4*x**7 - 128*sqrt(d - e*x**2)*e**5*x**9)/(1 280*e)