Integrand size = 27, antiderivative size = 89 \[ \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {x}{15 d^2 e^2 \sqrt {d^2-e^2 x^2}}-\frac {d}{5 e^3 (d+e x)^2 \sqrt {d^2-e^2 x^2}}+\frac {7}{15 e^3 (d+e x) \sqrt {d^2-e^2 x^2}} \]
1/15*x/d^2/e^2/(-e^2*x^2+d^2)^(1/2)-1/5*d/e^3/(e*x+d)^2/(-e^2*x^2+d^2)^(1/ 2)+7/15/e^3/(e*x+d)/(-e^2*x^2+d^2)^(1/2)
Time = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.79 \[ \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (4 d^3+8 d^2 e x+2 d e^2 x^2+e^3 x^3\right )}{15 d^2 e^3 (d-e x) (d+e x)^3} \]
(Sqrt[d^2 - e^2*x^2]*(4*d^3 + 8*d^2*e*x + 2*d*e^2*x^2 + e^3*x^3))/(15*d^2* e^3*(d - e*x)*(d + e*x)^3)
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {570, 529, 27, 669, 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.
\(\displaystyle \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 570 |
\(\displaystyle \int \frac {x^2 (d-e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}}dx\) |
\(\Big \downarrow \) 529 |
\(\displaystyle -\frac {\int \frac {d (2 d-5 e x) (d-e x)}{e^2 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {(2 d-5 e x) (d-e x)}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 e^2}-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 669 |
\(\displaystyle -\frac {-\frac {1}{3} \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx-\frac {7 (d-e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle -\frac {-\frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}}-\frac {7 (d-e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}}{5 e^2}-\frac {d (d-e x)^2}{5 e^3 \left (d^2-e^2 x^2\right )^{5/2}}\) |
-1/5*(d*(d - e*x)^2)/(e^3*(d^2 - e^2*x^2)^(5/2)) - ((-7*(d - e*x))/(3*e*(d ^2 - e^2*x^2)^(3/2)) - x/(3*d^2*Sqrt[d^2 - e^2*x^2]))/(5*e^2)
3.2.73.3.1 Defintions of rubi rules used
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)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^( p_), x_Symbol] :> Simp[(d*g + e*f)*(d + e*x)^m*((a + c*x^2)^(p + 1)/(2*c*d* (p + 1))), x] - Simp[e*((m*(d*g + e*f) + 2*e*f*(p + 1))/(2*c*d*(p + 1))) Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && EqQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0]
Time = 0.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(\frac {\left (-e x +d \right ) \left (e^{3} x^{3}+2 d \,e^{2} x^{2}+8 d^{2} e x +4 d^{3}\right )}{15 \left (e x +d \right ) d^{2} e^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}\) | \(65\) |
trager | \(\frac {\left (e^{3} x^{3}+2 d \,e^{2} x^{2}+8 d^{2} e x +4 d^{3}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{15 d^{2} e^{3} \left (e x +d \right )^{3} \left (-e x +d \right )}\) | \(67\) |
default | \(\frac {x}{d^{2} e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}+\frac {d^{2} \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right )^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}+\frac {3 e \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{e^{4}}-\frac {2 d \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{e^{3}}\) | \(287\) |
Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.33 \[ \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {4 \, e^{4} x^{4} + 8 \, d e^{3} x^{3} - 8 \, d^{3} e x - 4 \, d^{4} - {\left (e^{3} x^{3} + 2 \, d e^{2} x^{2} + 8 \, d^{2} e x + 4 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{7} x^{4} + 2 \, d^{3} e^{6} x^{3} - 2 \, d^{5} e^{4} x - d^{6} e^{3}\right )}} \]
1/15*(4*e^4*x^4 + 8*d*e^3*x^3 - 8*d^3*e*x - 4*d^4 - (e^3*x^3 + 2*d*e^2*x^2 + 8*d^2*e*x + 4*d^3)*sqrt(-e^2*x^2 + d^2))/(d^2*e^7*x^4 + 2*d^3*e^6*x^3 - 2*d^5*e^4*x - d^6*e^3)
\[ \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \]
Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.53 \[ \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {d}{5 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{5} x^{2} + 2 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{4} x + \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{3}\right )}} + \frac {7}{15 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{4} x + \sqrt {-e^{2} x^{2} + d^{2}} d e^{3}\right )}} + \frac {x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{2}} \]
-1/5*d/(sqrt(-e^2*x^2 + d^2)*e^5*x^2 + 2*sqrt(-e^2*x^2 + d^2)*d*e^4*x + sq rt(-e^2*x^2 + d^2)*d^2*e^3) + 7/15/(sqrt(-e^2*x^2 + d^2)*e^4*x + sqrt(-e^2 *x^2 + d^2)*d*e^3) + 1/15*x/(sqrt(-e^2*x^2 + d^2)*d^2*e^2)
Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.15 \[ \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {-\frac {8 i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{2} e^{2}} - \frac {15}{d^{2} e^{2} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )} + \frac {3 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4} - 5 \, d^{8} e^{8} {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4} - 15 \, d^{8} e^{8} \sqrt {\frac {2 \, d}{e x + d} - 1} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{4} \mathrm {sgn}\left (e\right )^{4}}{d^{10} e^{10} \mathrm {sgn}\left (\frac {1}{e x + d}\right )^{5} \mathrm {sgn}\left (e\right )^{5}}}{120 \, {\left | e \right |}} \]
-1/120*(-8*I*sgn(1/(e*x + d))*sgn(e)/(d^2*e^2) - 15/(d^2*e^2*sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d))*sgn(e)) + (3*d^8*e^8*(2*d/(e*x + d) - 1)^(5/2) *sgn(1/(e*x + d))^4*sgn(e)^4 - 5*d^8*e^8*(2*d/(e*x + d) - 1)^(3/2)*sgn(1/( e*x + d))^4*sgn(e)^4 - 15*d^8*e^8*sqrt(2*d/(e*x + d) - 1)*sgn(1/(e*x + d)) ^4*sgn(e)^4)/(d^10*e^10*sgn(1/(e*x + d))^5*sgn(e)^5))/abs(e)
Time = 11.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \frac {x^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (4\,d^3+8\,d^2\,e\,x+2\,d\,e^2\,x^2+e^3\,x^3\right )}{15\,d^2\,e^3\,{\left (d+e\,x\right )}^3\,\left (d-e\,x\right )} \]