Integrand size = 24, antiderivative size = 79 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2}{5 e (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{5 d^4 \sqrt {d^2-e^2 x^2}} \] Output:
1/5*x/d^2/(-e^2*x^2+d^2)^(3/2)-2/5/e/(e*x+d)/(-e^2*x^2+d^2)^(3/2)+2/5*x/d^ 4/(-e^2*x^2+d^2)^(1/2)
Time = 0.35 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^3+d^2 e x+4 d e^2 x^2+2 e^3 x^3\right )}{5 d^4 e (d-e x) (d+e x)^3} \] Input:
Integrate[1/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]
Output:
(Sqrt[d^2 - e^2*x^2]*(-2*d^3 + d^2*e*x + 4*d*e^2*x^2 + 2*e^3*x^3))/(5*d^4* e*(d - e*x)*(d + e*x)^3)
Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.25, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {461, 470, 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.
\(\displaystyle \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {3 \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}dx}{5 d}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 470 |
\(\displaystyle \frac {3 \left (\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d}-\frac {1}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}}\right )}{5 d}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {3 \left (\frac {2 x}{3 d^3 \sqrt {d^2-e^2 x^2}}-\frac {1}{3 d e (d+e x) \sqrt {d^2-e^2 x^2}}\right )}{5 d}-\frac {1}{5 d e (d+e x)^2 \sqrt {d^2-e^2 x^2}}\) |
Input:
Int[1/((d + e*x)^2*(d^2 - e^2*x^2)^(3/2)),x]
Output:
-1/5*1/(d*e*(d + e*x)^2*Sqrt[d^2 - e^2*x^2]) + (3*((2*x)/(3*d^3*Sqrt[d^2 - e^2*x^2]) - 1/(3*d*e*(d + e*x)*Sqrt[d^2 - e^2*x^2])))/(5*d)
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl ify[n + 2*p + 2]/(2*c*(n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[(n + 2*p + 2)/(2*c*(n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[n, 0] && NeQ[n + p + 1, 0] && IntegerQ[2*p]
Time = 0.26 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (-2 e^{3} x^{3}-4 d \,e^{2} x^{2}-d^{2} e x +2 d^{3}\right )}{5 \left (e x +d \right ) d^{4} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(66\) |
orering | \(-\frac {\left (-e x +d \right ) \left (-2 e^{3} x^{3}-4 d \,e^{2} x^{2}-d^{2} e x +2 d^{3}\right )}{5 \left (e x +d \right ) d^{4} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(66\) |
trager | \(-\frac {\left (-2 e^{3} x^{3}-4 d \,e^{2} x^{2}-d^{2} e x +2 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{4} \left (e x +d \right )^{3} e \left (-e x +d \right )}\) | \(68\) |
default | \(\frac {-\frac {1}{5 d e \left (x +\frac {d}{e}\right )^{2} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}+\frac {3 e \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{3 e \,d^{3} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}}{e^{2}}\) | \(156\) |
Input:
int(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/5*(-e*x+d)*(-2*e^3*x^3-4*d*e^2*x^2-d^2*e*x+2*d^3)/(e*x+d)/d^4/e/(-e^2*x ^2+d^2)^(3/2)
Time = 0.12 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.46 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, e^{4} x^{4} + 4 \, d e^{3} x^{3} - 4 \, d^{3} e x - 2 \, d^{4} + {\left (2 \, e^{3} x^{3} + 4 \, d e^{2} x^{2} + d^{2} e x - 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{4} e^{5} x^{4} + 2 \, d^{5} e^{4} x^{3} - 2 \, d^{7} e^{2} x - d^{8} e\right )}} \] Input:
integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="fricas")
Output:
-1/5*(2*e^4*x^4 + 4*d*e^3*x^3 - 4*d^3*e*x - 2*d^4 + (2*e^3*x^3 + 4*d*e^2*x ^2 + d^2*e*x - 2*d^3)*sqrt(-e^2*x^2 + d^2))/(d^4*e^5*x^4 + 2*d^5*e^4*x^3 - 2*d^7*e^2*x - d^8*e)
\[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \] Input:
integrate(1/(e*x+d)**2/(-e**2*x**2+d**2)**(3/2),x)
Output:
Integral(1/((-(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)**2), x)
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (67) = 134\).
Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {1}{5 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} d e^{3} x^{2} + 2 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e\right )}} - \frac {1}{5 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{3} e\right )}} + \frac {2 \, x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4}} \] Input:
integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="maxima")
Output:
-1/5/(sqrt(-e^2*x^2 + d^2)*d*e^3*x^2 + 2*sqrt(-e^2*x^2 + d^2)*d^2*e^2*x + sqrt(-e^2*x^2 + d^2)*d^3*e) - 1/5/(sqrt(-e^2*x^2 + d^2)*d^2*e^2*x + sqrt(- e^2*x^2 + d^2)*d^3*e) + 2/5*x/(sqrt(-e^2*x^2 + d^2)*d^4)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 193, normalized size of antiderivative = 2.44 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {e^{3} {\left (\frac {5}{d^{4} e^{3} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} - \frac {d^{16} e^{12} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4} + 5 \, d^{16} e^{12} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4} + 15 \, d^{16} e^{12} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4}}{d^{20} e^{15} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{5} \mathrm {sgn}\left (e\right )^{5}}\right )} + \frac {16 i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{4}}}{40 \, {\left | e \right |}} \] Input:
integrate(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x, algorithm="giac")
Output:
1/40*(e^3*(5/(d^4*e^3*sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))*sgn(e)) - ( d^16*e^12*(2*d/(e*x + d) - 1)^(5/2)*sgn(1/(e*x + d))^4*sgn(e)^4 + 5*d^16*e ^12*(2*d/(e*x + d) - 1)^(3/2)*sgn(1/(e*x + d))^4*sgn(e)^4 + 15*d^16*e^12*s qrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))^4*sgn(e)^4)/(d^20*e^15*sgn(1/(e*x + d))^5*sgn(e)^5)) + 16*I*sgn(1/(e*x + d))*sgn(e)/d^4)/abs(e)
Time = 6.55 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (-2\,d^3+d^2\,e\,x+4\,d\,e^2\,x^2+2\,e^3\,x^3\right )}{5\,d^4\,e\,{\left (d+e\,x\right )}^3\,\left (d-e\,x\right )} \] Input:
int(1/((d^2 - e^2*x^2)^(3/2)*(d + e*x)^2),x)
Output:
((d^2 - e^2*x^2)^(1/2)*(2*e^3*x^3 - 2*d^3 + 4*d*e^2*x^2 + d^2*e*x))/(5*d^4 *e*(d + e*x)^3*(d - e*x))
Time = 0.24 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d^{2}+2 \sqrt {-e^{2} x^{2}+d^{2}}\, d e x +\sqrt {-e^{2} x^{2}+d^{2}}\, e^{2} x^{2}-4 d^{3}+2 d^{2} e x +8 d \,e^{2} x^{2}+4 e^{3} x^{3}}{10 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:
int(1/(e*x+d)^2/(-e^2*x^2+d^2)^(3/2),x)
Output:
(sqrt(d**2 - e**2*x**2)*d**2 + 2*sqrt(d**2 - e**2*x**2)*d*e*x + sqrt(d**2 - e**2*x**2)*e**2*x**2 - 4*d**3 + 2*d**2*e*x + 8*d*e**2*x**2 + 4*e**3*x**3 )/(10*sqrt(d**2 - e**2*x**2)*d**4*e*(d**2 + 2*d*e*x + e**2*x**2))