Integrand size = 25, antiderivative size = 121 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=-\frac {3 d e (d+e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}-\frac {3}{2} d^2 e^2 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d^2 e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
-1/2*(-e*x+d)*(-e^2*x^2+d^2)^(3/2)/x^2-3/2*d^2*e^2*arctan(e*x/(-e^2*x^2+d^ 2)^(1/2))+3/2*d^2*e^2*arctanh((-e^2*x^2+d^2)^(1/2)/d)-3/2*d*e*(e*x+d)*(-e^ 2*x^2+d^2)^(1/2)/x
Time = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.22 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=\frac {1}{2} \left (-\frac {\sqrt {d^2-e^2 x^2} \left (d^3+2 d^2 e x+2 d e^2 x^2+e^3 x^3\right )}{x^2}+6 d^2 e^2 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+3 d \sqrt {d^2} e^2 \log (x)-3 d \sqrt {d^2} e^2 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )\right ) \]
(-((Sqrt[d^2 - e^2*x^2]*(d^3 + 2*d^2*e*x + 2*d*e^2*x^2 + e^3*x^3))/x^2) + 6*d^2*e^2*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] + 3*d*Sqrt[d^2]* e^2*Log[x] - 3*d*Sqrt[d^2]*e^2*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/2
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.89, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {537, 25, 535, 27, 538, 224, 216, 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.
| \(\displaystyle \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {3}{2} e^2 \int -\frac {(d+2 e x) \sqrt {d^2-e^2 x^2}}{x}dx-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {3}{2} e^2 \int \frac {(d+2 e x) \sqrt {d^2-e^2 x^2}}{x}dx-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 535 |
| \(\displaystyle -\frac {3}{2} e^2 \left (\frac {1}{2} d^2 \int \frac {2 (d+e x)}{x \sqrt {d^2-e^2 x^2}}dx+(d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{2} e^2 \left (d^2 \int \frac {d+e x}{x \sqrt {d^2-e^2 x^2}}dx+(d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {3}{2} e^2 \left (d^2 \left (e \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx\right )+(d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {3}{2} e^2 \left (d^2 \left (d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+e \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}\right )+(d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {3}{2} e^2 \left (d^2 \left (d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+(d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {3}{2} e^2 \left (d^2 \left (\frac {1}{2} d \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2+\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )+(d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {3}{2} e^2 \left (d^2 \left (\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {d \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e^2}\right )+(d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {3}{2} e^2 \left (d^2 \left (\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )+(d+e x) \sqrt {d^2-e^2 x^2}\right )-\frac {(d+2 e x) \left (d^2-e^2 x^2\right )^{3/2}}{2 x^2}\) |
-1/2*((d + 2*e*x)*(d^2 - e^2*x^2)^(3/2))/x^2 - (3*e^2*((d + e*x)*Sqrt[d^2 - e^2*x^2] + d^2*(ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - ArcTanh[Sqrt[d^2 - e ^2*x^2]/d])))/2
3.1.9.3.1 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[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p + 1) Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*(c*(m + 2) + d*(m + 1)*x)*((a + b*x^2)^p/((m + 1)*(m + 2))), x] - Simp[2*b*(p/((m + 1)*(m + 2))) Int[x^(m + 2)*(c*(m + 2) + d*(m + 1) *x)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -2] && GtQ[p, 0] && !ILtQ[m + 2*p + 3, 0] && IntegerQ[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Time = 0.36 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(-\frac {d^{2} \sqrt {-e^{2} x^{2}+d^{2}}\, \left (2 e x +d \right )}{2 x^{2}}-\frac {3 e^{3} d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}-\frac {e^{3} x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {3 e^{2} d^{3} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}-d \,e^{2} \sqrt {-e^{2} x^{2}+d^{2}}\) | \(150\) |
| default | \(d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )+e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{d^{2} x}-\frac {4 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{d^{2}}\right )\) | \(219\) |
-1/2*d^2*(-e^2*x^2+d^2)^(1/2)*(2*e*x+d)/x^2-3/2*e^3*d^2/(e^2)^(1/2)*arctan ((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-1/2*e^3*x*(-e^2*x^2+d^2)^(1/2)+3/2*e^ 2*d^3/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)-d*e^2*( -e^2*x^2+d^2)^(1/2)
Time = 0.34 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=\frac {6 \, d^{2} e^{2} x^{2} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 3 \, d^{2} e^{2} x^{2} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - 2 \, d^{2} e^{2} x^{2} - {\left (e^{3} x^{3} + 2 \, d e^{2} x^{2} + 2 \, d^{2} e x + d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{2 \, x^{2}} \]
1/2*(6*d^2*e^2*x^2*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 3*d^2*e^2*x ^2*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - 2*d^2*e^2*x^2 - (e^3*x^3 + 2*d*e^2 *x^2 + 2*d^2*e*x + d^3)*sqrt(-e^2*x^2 + d^2))/x^2
Result contains complex when optimal does not.
Time = 3.22 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.67 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=d^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \frac {d^{2} \left (\begin {cases} \frac {\log {\left (- 2 e^{2} x + 2 \sqrt {- e^{2}} \sqrt {d^{2} - e^{2} x^{2}} \right )}}{\sqrt {- e^{2}}} & \text {for}\: d^{2} \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- e^{2} x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {d^{2} - e^{2} x^{2}}}{2} & \text {for}\: e^{2} \neq 0 \\x \sqrt {d^{2}} & \text {otherwise} \end {cases}\right ) \]
d**3*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(2*x) + e**2*acosh(d/(e*x))/ (2*d), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(2*e*x**3*sqrt(-d**2/(e**2*x**2 ) + 1)) - I*e/(2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**2*asin(d/(e*x))/(2* d), True)) + d**2*e*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I*e*aco sh(e*x/d) - I*e**2*x/(d*sqrt(-1 + e**2*x**2/d**2)), Abs(e**2*x**2/d**2) > 1), (-d/(x*sqrt(1 - e**2*x**2/d**2)) - e*asin(e*x/d) + e**2*x/(d*sqrt(1 - e**2*x**2/d**2)), True)) - d*e**2*Piecewise((d**2/(e*x*sqrt(d**2/(e**2*x** 2) - 1)) - d*acosh(d/(e*x)) - e*x/sqrt(d**2/(e**2*x**2) - 1), Abs(d**2/(e* *2*x**2)) > 1), (-I*d**2/(e*x*sqrt(-d**2/(e**2*x**2) + 1)) + I*d*asin(d/(e *x)) + I*e*x/sqrt(-d**2/(e**2*x**2) + 1), True)) - e**3*Piecewise((d**2*Pi ecewise((log(-2*e**2*x + 2*sqrt(-e**2)*sqrt(d**2 - e**2*x**2))/sqrt(-e**2) , Ne(d**2, 0)), (x*log(x)/sqrt(-e**2*x**2), True))/2 + x*sqrt(d**2 - e**2* x**2)/2, Ne(e**2, 0)), (x*sqrt(d**2), True))
Time = 0.28 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.42 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=-\frac {3 \, d^{2} e^{3} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{2 \, \sqrt {e^{2}}} + \frac {3}{2} \, d^{2} e^{2} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3} x - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{2} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{2 \, d} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{x} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{2 \, d x^{2}} \]
-3/2*d^2*e^3*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 3/2*d^2*e^2*log(2*d^2 /abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) - 3/2*sqrt(-e^2*x^2 + d^2)*e^3* x - 3/2*sqrt(-e^2*x^2 + d^2)*d*e^2 - 1/2*(-e^2*x^2 + d^2)^(3/2)*e^2/d - (- e^2*x^2 + d^2)^(3/2)*e/x - 1/2*(-e^2*x^2 + d^2)^(5/2)/(d*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (106) = 212\).
Time = 0.29 (sec) , antiderivative size = 244, normalized size of antiderivative = 2.02 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=-\frac {3 \, d^{2} e^{3} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{2 \, {\left | e \right |}} + \frac {3 \, d^{2} e^{3} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, {\left | e \right |}} + \frac {{\left (d^{2} e^{3} + \frac {4 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} e}{x}\right )} e^{4} x^{2}}{8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} {\left | e \right |}} - \frac {1}{2} \, {\left (e^{3} x + 2 \, d e^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}} - \frac {\frac {4 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{2} e {\left | e \right |}}{x} + \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{2} {\left | e \right |}}{e x^{2}}}{8 \, e^{2}} \]
-3/2*d^2*e^3*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 3/2*d^2*e^3*log(1/2*abs( -2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) + 1/8*(d^2*e^ 3 + 4*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*e/x)*e^4*x^2/((d*e + sqrt(-e ^2*x^2 + d^2)*abs(e))^2*abs(e)) - 1/2*(e^3*x + 2*d*e^2)*sqrt(-e^2*x^2 + d^ 2) - 1/8*(4*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^2*e*abs(e)/x + (d*e + sq rt(-e^2*x^2 + d^2)*abs(e))^2*d^2*abs(e)/(e*x^2))/e^2
Time = 12.53 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^3} \, dx=\frac {3\,d^2\,e^2\,\mathrm {atanh}\left (\frac {\sqrt {d^2-e^2\,x^2}}{d}\right )}{2}-\frac {d^3\,\sqrt {d^2-e^2\,x^2}}{2\,x^2}-d\,e^2\,\sqrt {d^2-e^2\,x^2}-\frac {e\,{\left (d^2-e^2\,x^2\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ \frac {e^2\,x^2}{d^2}\right )}{x\,{\left (1-\frac {e^2\,x^2}{d^2}\right )}^{3/2}} \]