Integrand size = 27, antiderivative size = 114 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}+\frac {e \sqrt {d^2-e^2 x^2}}{2 d^2 x^2}-\frac {2 e^2 \sqrt {d^2-e^2 x^2}}{3 d^3 x}+\frac {e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{2 d^3} \]
1/2*e^3*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^3-1/3*(-e^2*x^2+d^2)^(1/2)/d/x^3 +1/2*e*(-e^2*x^2+d^2)^(1/2)/d^2/x^2-2/3*e^2*(-e^2*x^2+d^2)^(1/2)/d^3/x
Time = 0.16 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\frac {\left (-2 d^2+3 d e x-4 e^2 x^2\right ) \sqrt {d^2-e^2 x^2}}{6 d^3 x^3}+\frac {\sqrt {d^2} e^3 \log (x)}{2 d^4}-\frac {\sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{2 d^4} \]
((-2*d^2 + 3*d*e*x - 4*e^2*x^2)*Sqrt[d^2 - e^2*x^2])/(6*d^3*x^3) + (Sqrt[d ^2]*e^3*Log[x])/(2*d^4) - (Sqrt[d^2]*e^3*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^ 2]])/(2*d^4)
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {566, 539, 27, 539, 27, 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.
\(\displaystyle \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx\) |
\(\Big \downarrow \) 566 |
\(\displaystyle \int \frac {d-e x}{x^4 \sqrt {d^2-e^2 x^2}}dx\) |
\(\Big \downarrow \) 539 |
\(\displaystyle -\frac {\int \frac {d e (3 d-2 e x)}{x^3 \sqrt {d^2-e^2 x^2}}dx}{3 d^2}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {e \int \frac {3 d-2 e x}{x^3 \sqrt {d^2-e^2 x^2}}dx}{3 d}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}\) |
\(\Big \downarrow \) 539 |
\(\displaystyle -\frac {e \left (-\frac {\int \frac {d e (4 d-3 e x)}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d^2}-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{3 d}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {e \left (-\frac {e \int \frac {4 d-3 e x}{x^2 \sqrt {d^2-e^2 x^2}}dx}{2 d}-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{3 d}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle -\frac {e \left (-\frac {e \left (-3 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-\frac {4 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{3 d}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\frac {e \left (-\frac {e \left (-\frac {3}{2} e \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {4 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{3 d}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {e \left (-\frac {e \left (\frac {3 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e}-\frac {4 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{3 d}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {e \left (-\frac {e \left (\frac {3 e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {4 \sqrt {d^2-e^2 x^2}}{d x}\right )}{2 d}-\frac {3 \sqrt {d^2-e^2 x^2}}{2 d x^2}\right )}{3 d}-\frac {\sqrt {d^2-e^2 x^2}}{3 d x^3}\) |
-1/3*Sqrt[d^2 - e^2*x^2]/(d*x^3) - (e*((-3*Sqrt[d^2 - e^2*x^2])/(2*d*x^2) - (e*((-4*Sqrt[d^2 - e^2*x^2])/(d*x) + (3*e*ArcTanh[Sqrt[d^2 - e^2*x^2]/d] )/d))/(2*d)))/(3*d)
3.1.100.3.1 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_))*((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_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[((x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] : > Int[x^m*(a/c + b*(x/d))*(a + b*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[p, 0]
Time = 0.43 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.77
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (4 e^{2} x^{2}-3 d e x +2 d^{2}\right )}{6 d^{3} x^{3}}+\frac {e^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 d^{2} \sqrt {d^{2}}}\) | \(88\) |
default | \(-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 d^{3} x^{3}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{d^{2} x}-\frac {2 e^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{d^{2}}\right )}{d^{3}}-\frac {e^{3} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{d^{4}}-\frac {e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{2 d^{2} x^{2}}-\frac {e^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )}{2 d^{2}}\right )}{d^{2}}+\frac {e^{3} \left (\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}+\frac {d e \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{\sqrt {e^{2}}}\right )}{d^{4}}\) | \(351\) |
-1/6*(-e^2*x^2+d^2)^(1/2)*(4*e^2*x^2-3*d*e*x+2*d^2)/d^3/x^3+1/2/d^2*e^3/(d ^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=-\frac {3 \, e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (4 \, e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, d^{3} x^{3}} \]
-1/6*(3*e^3*x^3*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (4*e^2*x^2 - 3*d*e*x + 2*d^2)*sqrt(-e^2*x^2 + d^2))/(d^3*x^3)
\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\int \frac {\sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{x^{4} \left (d + e x\right )}\, dx \]
\[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\int { \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{{\left (e x + d\right )} x^{4}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (98) = 196\).
Time = 0.29 (sec) , antiderivative size = 255, normalized size of antiderivative = 2.24 \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\frac {{\left (e^{4} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{2}}{x} + \frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{x^{2}}\right )} e^{6} x^{3}}{24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{3} {\left | e \right |}} + \frac {e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, d^{3} {\left | e \right |}} - \frac {\frac {9 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{6} e^{4}}{x} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{6} e^{2}}{x^{2}} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{6}}{x^{3}}}{24 \, d^{9} e^{2} {\left | e \right |}} \]
1/24*(e^4 - 3*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^2/x + 9*(d*e + sqrt(-e ^2*x^2 + d^2)*abs(e))^2/x^2)*e^6*x^3/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^ 3*d^3*abs(e)) + 1/2*e^4*log(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e) )/(e^2*abs(x)))/(d^3*abs(e)) - 1/24*(9*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)) *d^6*e^4/x - 3*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^6*e^2/x^2 + (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^6/x^3)/(d^9*e^2*abs(e))
Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^2}}{x^4 (d+e x)} \, dx=\int \frac {\sqrt {d^2-e^2\,x^2}}{x^4\,\left (d+e\,x\right )} \,d x \]