Integrand size = 29, antiderivative size = 73 \[ \int \frac {\left (d-e x^2\right )^{5/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {x \left (d-e x^2\right )^{3/2}}{3 d \left (d^2-e^2 x^4\right )^{3/2}}+\frac {2 x \sqrt {d-e x^2}}{3 d^2 \sqrt {d^2-e^2 x^4}} \] Output:
1/3*x*(-e*x^2+d)^(3/2)/d/(-e^2*x^4+d^2)^(3/2)+2/3*x*(-e*x^2+d)^(1/2)/d^2/( -e^2*x^4+d^2)^(1/2)
Time = 5.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \frac {\left (d-e x^2\right )^{5/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {x \left (3 d+2 e x^2\right ) \sqrt {d^2-e^2 x^4}}{3 d^2 \sqrt {d-e x^2} \left (d+e x^2\right )^2} \] Input:
Integrate[(d - e*x^2)^(5/2)/(d^2 - e^2*x^4)^(5/2),x]
Output:
(x*(3*d + 2*e*x^2)*Sqrt[d^2 - e^2*x^4])/(3*d^2*Sqrt[d - e*x^2]*(d + e*x^2) ^2)
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1396, 209, 208}
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.
\(\displaystyle \int \frac {\left (d-e x^2\right )^{5/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1396 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \int \frac {1}{\left (e x^2+d\right )^{5/2}}dx}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 209 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 \int \frac {1}{\left (e x^2+d\right )^{3/2}}dx}{3 d}+\frac {x}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 x}{3 d^2 \sqrt {d+e x^2}}+\frac {x}{3 d \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\) |
Input:
Int[(d - e*x^2)^(5/2)/(d^2 - e^2*x^4)^(5/2),x]
Output:
(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(3*d*(d + e*x^2)^(3/2)) + (2*x)/(3*d^2 *Sqrt[d + e*x^2])))/Sqrt[d^2 - e^2*x^4]
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 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 = 48, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(\frac {\left (e \,x^{2}+d \right ) x \left (2 e \,x^{2}+3 d \right ) \left (-e \,x^{2}+d \right )^{\frac {5}{2}}}{3 d^{2} \left (-e^{2} x^{4}+d^{2}\right )^{\frac {5}{2}}}\) | \(48\) |
orering | \(\frac {\left (e \,x^{2}+d \right ) x \left (2 e \,x^{2}+3 d \right ) \left (-e \,x^{2}+d \right )^{\frac {5}{2}}}{3 d^{2} \left (-e^{2} x^{4}+d^{2}\right )^{\frac {5}{2}}}\) | \(48\) |
default | \(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x \left (2 e \,x^{2}+3 d \right )}{3 \sqrt {-e \,x^{2}+d}\, \left (e \,x^{2}+d \right )^{2} d^{2}}\) | \(50\) |
Input:
int((-e*x^2+d)^(5/2)/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*(e*x^2+d)*x*(2*e*x^2+3*d)*(-e*x^2+d)^(5/2)/d^2/(-e^2*x^4+d^2)^(5/2)
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01 \[ \int \frac {\left (d-e x^2\right )^{5/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=-\frac {\sqrt {-e^{2} x^{4} + d^{2}} {\left (2 \, e x^{3} + 3 \, d x\right )} \sqrt {-e x^{2} + d}}{3 \, {\left (d^{2} e^{3} x^{6} + d^{3} e^{2} x^{4} - d^{4} e x^{2} - d^{5}\right )}} \] Input:
integrate((-e*x^2+d)^(5/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
Output:
-1/3*sqrt(-e^2*x^4 + d^2)*(2*e*x^3 + 3*d*x)*sqrt(-e*x^2 + d)/(d^2*e^3*x^6 + d^3*e^2*x^4 - d^4*e*x^2 - d^5)
\[ \int \frac {\left (d-e x^2\right )^{5/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {\left (d - e x^{2}\right )^{\frac {5}{2}}}{\left (- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((-e*x**2+d)**(5/2)/(-e**2*x**4+d**2)**(5/2),x)
Output:
Integral((d - e*x**2)**(5/2)/(-(-d + e*x**2)*(d + e*x**2))**(5/2), x)
\[ \int \frac {\left (d-e x^2\right )^{5/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (-e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^(5/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
Output:
integrate((-e*x^2 + d)^(5/2)/(-e^2*x^4 + d^2)^(5/2), x)
\[ \int \frac {\left (d-e x^2\right )^{5/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int { \frac {{\left (-e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (-e^{2} x^{4} + d^{2}\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((-e*x^2+d)^(5/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
Output:
integrate((-e*x^2 + d)^(5/2)/(-e^2*x^4 + d^2)^(5/2), x)
Time = 17.37 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d-e x^2\right )^{5/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^4}\,\left (\frac {x\,\sqrt {d-e\,x^2}}{d\,e^3}+\frac {2\,x^3\,\sqrt {d-e\,x^2}}{3\,d^2\,e^2}\right )}{x^6-\frac {d^3}{e^3}+\frac {d\,x^4}{e}-\frac {d^2\,x^2}{e^2}} \] Input:
int((d - e*x^2)^(5/2)/(d^2 - e^2*x^4)^(5/2),x)
Output:
-((d^2 - e^2*x^4)^(1/2)*((x*(d - e*x^2)^(1/2))/(d*e^3) + (2*x^3*(d - e*x^2 )^(1/2))/(3*d^2*e^2)))/(x^6 - d^3/e^3 + (d*x^4)/e - (d^2*x^2)/e^2)
Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d-e x^2\right )^{5/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {3 \sqrt {e \,x^{2}+d}\, d e x +2 \sqrt {e \,x^{2}+d}\, e^{2} x^{3}-2 \sqrt {e}\, d^{2}-4 \sqrt {e}\, d e \,x^{2}-2 \sqrt {e}\, e^{2} x^{4}}{3 d^{2} e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:
int((-e*x^2+d)^(5/2)/(-e^2*x^4+d^2)^(5/2),x)
Output:
(3*sqrt(d + e*x**2)*d*e*x + 2*sqrt(d + e*x**2)*e**2*x**3 - 2*sqrt(e)*d**2 - 4*sqrt(e)*d*e*x**2 - 2*sqrt(e)*e**2*x**4)/(3*d**2*e*(d**2 + 2*d*e*x**2 + e**2*x**4))