Integrand size = 27, antiderivative size = 154 \[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\frac {6 d-5 e x}{15 d^4 x \left (d^2-e^2 x^2\right )^{3/2}}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d-5 e x}{5 d^6 x \sqrt {d^2-e^2 x^2}}-\frac {16 \sqrt {d^2-e^2 x^2}}{5 d^7 x}+\frac {e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d^7} \]
1/15*(-5*e*x+6*d)/d^4/x/(-e^2*x^2+d^2)^(3/2)+1/5/d^2/x/(e*x+d)/(-e^2*x^2+d ^2)^(3/2)+e*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^7+1/5*(-5*e*x+8*d)/d^6/x/(-e ^2*x^2+d^2)^(1/2)-16/5*(-e^2*x^2+d^2)^(1/2)/d^7/x
Time = 0.43 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {\frac {d \sqrt {d^2-e^2 x^2} \left (15 d^5+38 d^4 e x-52 d^3 e^2 x^2-87 d^2 e^3 x^3+33 d e^4 x^4+48 e^5 x^5\right )}{x (d-e x)^2 (d+e x)^3}-15 \sqrt {d^2} e \log (x)+15 \sqrt {d^2} e \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{15 d^8} \]
-1/15*((d*Sqrt[d^2 - e^2*x^2]*(15*d^5 + 38*d^4*e*x - 52*d^3*e^2*x^2 - 87*d ^2*e^3*x^3 + 33*d*e^4*x^4 + 48*e^5*x^5))/(x*(d - e*x)^2*(d + e*x)^3) - 15* Sqrt[d^2]*e*Log[x] + 15*Sqrt[d^2]*e*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/ d^8
Time = 0.44 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {569, 25, 532, 27, 2336, 25, 534, 243, 73, 221}
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.
| \(\displaystyle \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 569 |
| \(\displaystyle \frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int -\frac {6 d-5 e x}{x^2 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {6 d-5 e x}{x^2 \left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {-\frac {\int -\frac {3 \left (\frac {4 e^2 x^2}{d}-5 e x+6 d\right )}{x^2 \left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}-\frac {e (5 d-6 e x)}{3 d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {4 e^2 x^2}{d}-5 e x+6 d}{x^2 \left (d^2-e^2 x^2\right )^{3/2}}dx}{d^2}-\frac {e (5 d-6 e x)}{3 d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {6 d-5 e x}{x^2 \sqrt {d^2-e^2 x^2}}dx}{d^2}-\frac {5 e (d-2 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}-\frac {e (5 d-6 e x)}{3 d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {6 d-5 e x}{x^2 \sqrt {d^2-e^2 x^2}}dx}{d^2}-\frac {5 e (d-2 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}-\frac {e (5 d-6 e x)}{3 d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {\frac {-5 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-\frac {6 \sqrt {d^2-e^2 x^2}}{d x}}{d^2}-\frac {5 e (d-2 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}-\frac {e (5 d-6 e x)}{3 d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {\frac {-\frac {5}{2} e \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {6 \sqrt {d^2-e^2 x^2}}{d x}}{d^2}-\frac {5 e (d-2 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}-\frac {e (5 d-6 e x)}{3 d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {\frac {5 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e}-\frac {6 \sqrt {d^2-e^2 x^2}}{d x}}{d^2}-\frac {5 e (d-2 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}-\frac {e (5 d-6 e x)}{3 d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\frac {5 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {6 \sqrt {d^2-e^2 x^2}}{d x}}{d^2}-\frac {5 e (d-2 e x)}{d^3 \sqrt {d^2-e^2 x^2}}}{d^2}-\frac {e (5 d-6 e x)}{3 d^3 \left (d^2-e^2 x^2\right )^{3/2}}}{5 d^2}+\frac {1}{5 d^2 x (d+e x) \left (d^2-e^2 x^2\right )^{3/2}}\) |
1/(5*d^2*x*(d + e*x)*(d^2 - e^2*x^2)^(3/2)) + (-1/3*(e*(5*d - 6*e*x))/(d^3 *(d^2 - e^2*x^2)^(3/2)) + ((-5*e*(d - 2*e*x))/(d^3*Sqrt[d^2 - e^2*x^2]) + ((-6*Sqrt[d^2 - e^2*x^2])/(d*x) + (5*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d)/ d^2)/d^2)/(5*d^2)
3.2.45.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[((x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] : > Simp[(-x^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*p*(c + d*x))), x] + Simp[1/(2 *c^2*p) Int[x^m*(a + b*x^2)^p*(c*(m + 2*p + 1) - d*(m + 2*p + 2)*x), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[m + 2*p, 0]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a* b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(290\) vs. \(2(136)=272\).
Time = 0.44 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.89
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{d^{7} x}+\frac {e \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{6} \sqrt {d^{2}}}-\frac {17 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{60 d^{6} e \left (x +\frac {d}{e}\right )^{2}}-\frac {413 \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{240 d^{7} \left (x +\frac {d}{e}\right )}+\frac {\sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{24 d^{6} e \left (x -\frac {d}{e}\right )^{2}}-\frac {23 \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{48 d^{7} \left (x -\frac {d}{e}\right )}-\frac {\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{20 d^{5} e^{2} \left (x +\frac {d}{e}\right )^{3}}\) | \(291\) |
| default | \(\frac {-\frac {1}{d^{2} x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {4 e^{2} \left (\frac {x}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {2 x}{3 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}\right )}{d^{2}}}{d}-\frac {e \left (\frac {1}{3 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {\frac {1}{d^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{d^{2} \sqrt {d^{2}}}}{d^{2}}\right )}{d^{2}}+\frac {e \left (-\frac {1}{5 d e \left (x +\frac {d}{e}\right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}+\frac {4 e \left (-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{6 d^{2} e^{2} \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e^{2} d^{4} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{5 d}\right )}{d^{2}}\) | \(333\) |
-(-e^2*x^2+d^2)^(1/2)/d^7/x+1/d^6*e/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(- e^2*x^2+d^2)^(1/2))/x)-17/60/d^6/e/(x+d/e)^2*(-(x+d/e)^2*e^2+2*d*e*(x+d/e) )^(1/2)-413/240/d^7/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)+1/24/d^6/ e/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-23/48/d^7/(x-d/e)*(-(x-d/ e)^2*e^2-2*d*e*(x-d/e))^(1/2)-1/20/d^5/e^2/(x+d/e)^3*(-(x+d/e)^2*e^2+2*d*e *(x+d/e))^(1/2)
Time = 0.41 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.72 \[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=-\frac {23 \, e^{6} x^{6} + 23 \, d e^{5} x^{5} - 46 \, d^{2} e^{4} x^{4} - 46 \, d^{3} e^{3} x^{3} + 23 \, d^{4} e^{2} x^{2} + 23 \, d^{5} e x + 15 \, {\left (e^{6} x^{6} + d e^{5} x^{5} - 2 \, d^{2} e^{4} x^{4} - 2 \, d^{3} e^{3} x^{3} + d^{4} e^{2} x^{2} + d^{5} e x\right )} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (48 \, e^{5} x^{5} + 33 \, d e^{4} x^{4} - 87 \, d^{2} e^{3} x^{3} - 52 \, d^{3} e^{2} x^{2} + 38 \, d^{4} e x + 15 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{7} e^{5} x^{6} + d^{8} e^{4} x^{5} - 2 \, d^{9} e^{3} x^{4} - 2 \, d^{10} e^{2} x^{3} + d^{11} e x^{2} + d^{12} x\right )}} \]
-1/15*(23*e^6*x^6 + 23*d*e^5*x^5 - 46*d^2*e^4*x^4 - 46*d^3*e^3*x^3 + 23*d^ 4*e^2*x^2 + 23*d^5*e*x + 15*(e^6*x^6 + d*e^5*x^5 - 2*d^2*e^4*x^4 - 2*d^3*e ^3*x^3 + d^4*e^2*x^2 + d^5*e*x)*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (48*e ^5*x^5 + 33*d*e^4*x^4 - 87*d^2*e^3*x^3 - 52*d^3*e^2*x^2 + 38*d^4*e*x + 15* d^5)*sqrt(-e^2*x^2 + d^2))/(d^7*e^5*x^6 + d^8*e^4*x^5 - 2*d^9*e^3*x^4 - 2* d^10*e^2*x^3 + d^11*e*x^2 + d^12*x)
\[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x^{2} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
\[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{2}} \,d x } \]
\[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 (d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx=\int \frac {1}{x^2\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,\left (d+e\,x\right )} \,d x \]