Integrand size = 25, antiderivative size = 120 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\frac {e^2 (2 d-3 e x) \sqrt {d^2-e^2 x^2}}{2 x}-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}+d e^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} d e^3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
-1/6*(3*e*x+2*d)*(-e^2*x^2+d^2)^(3/2)/x^3+d*e^3*arctan(e*x/(-e^2*x^2+d^2)^ (1/2))+3/2*d*e^3*arctanh((-e^2*x^2+d^2)^(1/2)/d)+1/2*e^2*(-3*e*x+2*d)*(-e^ 2*x^2+d^2)^(1/2)/x
Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^3-3 d^2 e x+8 d e^2 x^2-6 e^3 x^3\right )}{6 x^3}-2 d e^3 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )+\frac {3}{2} \sqrt {d^2} e^3 \log (x)-\frac {3}{2} \sqrt {d^2} e^3 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right ) \]
(Sqrt[d^2 - e^2*x^2]*(-2*d^3 - 3*d^2*e*x + 8*d*e^2*x^2 - 6*e^3*x^3))/(6*x^ 3) - 2*d*e^3*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] + (3*Sqrt[d^2 ]*e^3*Log[x])/2 - (3*Sqrt[d^2]*e^3*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/2
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {537, 25, 536, 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.
| \(\displaystyle \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {1}{2} e^2 \int -\frac {(2 d+3 e x) \sqrt {d^2-e^2 x^2}}{x^2}dx-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} e^2 \int \frac {(2 d+3 e x) \sqrt {d^2-e^2 x^2}}{x^2}dx-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle -\frac {1}{2} e^2 \left (\int \frac {3 d^2 e-2 d e^2 x}{x \sqrt {d^2-e^2 x^2}}dx-\frac {(2 d-3 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {1}{2} e^2 \left (3 d^2 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-2 d e^2 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {(2 d-3 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {1}{2} e^2 \left (3 d^2 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-2 d e^2 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {(2 d-3 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {1}{2} e^2 \left (3 d^2 e \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx-2 d e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {(2 d-3 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {1}{2} e^2 \left (\frac {3}{2} d^2 e \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-2 d e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {(2 d-3 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {1}{2} e^2 \left (-\frac {3 d^2 \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e}-2 d e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {(2 d-3 e x) \sqrt {d^2-e^2 x^2}}{x}\right )-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {1}{2} e^2 \left (-2 d e \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-3 d e \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )-\frac {\sqrt {d^2-e^2 x^2} (2 d-3 e x)}{x}\right )-\frac {(2 d+3 e x) \left (d^2-e^2 x^2\right )^{3/2}}{6 x^3}\) |
-1/6*((2*d + 3*e*x)*(d^2 - e^2*x^2)^(3/2))/x^3 - (e^2*(-(((2*d - 3*e*x)*Sq rt[d^2 - e^2*x^2])/x) - 2*d*e*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 3*d*e*Ar cTanh[Sqrt[d^2 - e^2*x^2]/d]))/2
3.1.10.3.1 Defintions of rubi rules used
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_)^2, x_Symbol] :> S imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer Q[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 = 0.37 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, d \left (-8 e^{2} x^{2}+3 d e x +2 d^{2}\right )}{6 x^{3}}+\frac {e^{4} d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-e^{3} \sqrt {-e^{2} x^{2}+d^{2}}+\frac {3 e^{3} d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{2 \sqrt {d^{2}}}\) | \(137\) |
| default | \(e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )+d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{d^{2} x}-\frac {4 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{d^{2}}\right )}{3 d^{2}}\right )\) | \(250\) |
-1/6*(-e^2*x^2+d^2)^(1/2)*d*(-8*e^2*x^2+3*d*e*x+2*d^2)/x^3+e^4*d/(e^2)^(1/ 2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-e^3*(-e^2*x^2+d^2)^(1/2)+3/2 *e^3*d^2/(d^2)^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)
Time = 0.35 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=-\frac {12 \, d e^{3} x^{3} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + 9 \, d e^{3} x^{3} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + 6 \, d e^{3} x^{3} + {\left (6 \, e^{3} x^{3} - 8 \, d e^{2} x^{2} + 3 \, d^{2} e x + 2 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{6 \, x^{3}} \]
-1/6*(12*d*e^3*x^3*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + 9*d*e^3*x^3 *log(-(d - sqrt(-e^2*x^2 + d^2))/x) + 6*d*e^3*x^3 + (6*e^3*x^3 - 8*d*e^2*x ^2 + 3*d^2*e*x + 2*d^3)*sqrt(-e^2*x^2 + d^2))/x^3
Result contains complex when optimal does not.
Time = 3.46 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.81 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=d^{3} \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 x^{2}} + \frac {e^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{3 d^{2}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i e \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 x^{2}} + \frac {i e^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}}{3 d^{2}} & \text {otherwise} \end {cases}\right ) + d^{2} e \left (\begin {cases} - \frac {e \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}}{2 x} + \frac {e^{2} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{2 d} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{2 e x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e}{2 x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{2} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{2 d} & \text {otherwise} \end {cases}\right ) - d e^{2} \left (\begin {cases} \frac {i d}{x \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} + i e \operatorname {acosh}{\left (\frac {e x}{d} \right )} - \frac {i e^{2} x}{d \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\- \frac {d}{x \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} - e \operatorname {asin}{\left (\frac {e x}{d} \right )} + \frac {e^{2} x}{d \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}} & \text {otherwise} \end {cases}\right ) - e^{3} \left (\begin {cases} \frac {d^{2}}{e x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - d \operatorname {acosh}{\left (\frac {d}{e x} \right )} - \frac {e x}{\sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\- \frac {i d^{2}}{e x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + i d \operatorname {asin}{\left (\frac {d}{e x} \right )} + \frac {i e x}{\sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} & \text {otherwise} \end {cases}\right ) \]
d**3*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(3*x**2) + e**3*sqrt(d**2/(e **2*x**2) - 1)/(3*d**2), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt(-d**2/(e** 2*x**2) + 1)/(3*x**2) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(3*d**2), True) ) + d**2*e*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*e**2*Piecewise((I*d/(x*sqrt(-1 + e**2*x**2/d**2)) + I *e*acosh(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*sqr t(1 - e**2*x**2/d**2)), True)) - e**3*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))
Time = 0.29 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.63 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\frac {d e^{4} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} + \frac {3}{2} \, d e^{3} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{4} x}{d} - \frac {3}{2} \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{2 \, d^{2}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{3 \, d x} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{2 \, d^{2} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{3 \, d x^{3}} \]
d*e^4*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) + 3/2*d*e^3*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x)) + sqrt(-e^2*x^2 + d^2)*e^4*x/d - 3/2*sqr t(-e^2*x^2 + d^2)*e^3 - 1/2*(-e^2*x^2 + d^2)^(3/2)*e^3/d^2 + 2/3*(-e^2*x^2 + d^2)^(3/2)*e^2/(d*x) - 1/2*(-e^2*x^2 + d^2)^(5/2)*e/(d^2*x^2) - 1/3*(-e ^2*x^2 + d^2)^(5/2)/(d*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (106) = 212\).
Time = 0.28 (sec) , antiderivative size = 285, normalized size of antiderivative = 2.38 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\frac {d e^{4} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} + \frac {{\left (d e^{4} + \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e^{2}}{x} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d}{x^{2}}\right )} e^{6} x^{3}}{24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} {\left | e \right |}} + \frac {3 \, d e^{4} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{2 \, {\left | e \right |}} - \sqrt {-e^{2} x^{2} + d^{2}} e^{3} + \frac {\frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d e^{4}}{x} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d e^{2}}{x^{2}} - \frac {{\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d}{x^{3}}}{24 \, e^{2} {\left | e \right |}} \]
d*e^4*arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) + 1/24*(d*e^4 + 3*(d*e + sqrt(-e^ 2*x^2 + d^2)*abs(e))*d*e^2/x - 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d/ x^2)*e^6*x^3/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*abs(e)) + 3/2*d*e^4*lo g(1/2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) - s qrt(-e^2*x^2 + d^2)*e^3 + 1/24*(15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d*e ^4/x - 3*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d*e^2/x^2 - (d*e + sqrt(-e^ 2*x^2 + d^2)*abs(e))^3*d/x^3)/(e^2*abs(e))
Timed out. \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^4} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )}{x^4} \,d x \]