Integrand size = 25, antiderivative size = 118 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\frac {e^2 (3 d+8 e x) \sqrt {d^2-e^2 x^2}}{8 x^2}-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}+e^4 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {3}{8} e^4 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right ) \]
-1/12*(4*e*x+3*d)*(-e^2*x^2+d^2)^(3/2)/x^4+e^4*arctan(e*x/(-e^2*x^2+d^2)^( 1/2))-3/8*e^4*arctanh((-e^2*x^2+d^2)^(1/2)/d)+1/8*e^2*(8*e*x+3*d)*(-e^2*x^ 2+d^2)^(1/2)/x^2
Time = 0.28 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.28 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\frac {1}{24} \left (\frac {\sqrt {d^2-e^2 x^2} \left (-6 d^3-8 d^2 e x+15 d e^2 x^2+32 e^3 x^3\right )}{x^4}-48 e^4 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )-\frac {9 \sqrt {d^2} e^4 \log (x)}{d}+\frac {9 \sqrt {d^2} e^4 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{d}\right ) \]
((Sqrt[d^2 - e^2*x^2]*(-6*d^3 - 8*d^2*e*x + 15*d*e^2*x^2 + 32*e^3*x^3))/x^ 4 - 48*e^4*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])] - (9*Sqrt[d^2]* e^4*Log[x])/d + (9*Sqrt[d^2]*e^4*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/d)/ 24
Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {537, 25, 537, 25, 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.
| \(\displaystyle \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {1}{4} e^2 \int -\frac {(3 d+4 e x) \sqrt {d^2-e^2 x^2}}{x^3}dx-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{4} e^2 \int \frac {(3 d+4 e x) \sqrt {d^2-e^2 x^2}}{x^3}dx-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle -\frac {1}{4} e^2 \left (\frac {1}{2} e^2 \int -\frac {3 d+8 e x}{x \sqrt {d^2-e^2 x^2}}dx-\frac {(3 d+8 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{4} e^2 \left (-\frac {1}{2} e^2 \int \frac {3 d+8 e x}{x \sqrt {d^2-e^2 x^2}}dx-\frac {(3 d+8 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle -\frac {1}{4} e^2 \left (-\frac {1}{2} e^2 \left (8 e \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+3 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx\right )-\frac {(3 d+8 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {1}{4} e^2 \left (-\frac {1}{2} e^2 \left (3 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+8 e \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {(3 d+8 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {1}{4} e^2 \left (-\frac {1}{2} e^2 \left (3 d \int \frac {1}{x \sqrt {d^2-e^2 x^2}}dx+8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )-\frac {(3 d+8 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {1}{4} e^2 \left (-\frac {1}{2} e^2 \left (\frac {3}{2} d \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2+8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )\right )-\frac {(3 d+8 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {1}{4} e^2 \left (-\frac {1}{2} e^2 \left (8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-\frac {3 d \int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}}{e^2}\right )-\frac {(3 d+8 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {1}{4} e^2 \left (-\frac {1}{2} e^2 \left (8 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-3 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )\right )-\frac {(3 d+8 e x) \sqrt {d^2-e^2 x^2}}{2 x^2}\right )-\frac {(3 d+4 e x) \left (d^2-e^2 x^2\right )^{3/2}}{12 x^4}\) |
-1/12*((3*d + 4*e*x)*(d^2 - e^2*x^2)^(3/2))/x^4 - (e^2*(-1/2*((3*d + 8*e*x )*Sqrt[d^2 - e^2*x^2])/x^2 - (e^2*(8*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]] - 3 *ArcTanh[Sqrt[d^2 - e^2*x^2]/d]))/2))/4
3.1.11.3.1 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[(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 = 0.38 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-32 e^{3} x^{3}-15 d \,e^{2} x^{2}+8 d^{2} e x +6 d^{3}\right )}{24 x^{4}}+\frac {e^{5} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}}-\frac {3 e^{4} d \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{8 \sqrt {d^{2}}}\) | \(125\) |
| default | \(d \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{4 d^{2} x^{4}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{2 d^{2} x^{2}}-\frac {3 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )+e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{3 d^{2} x^{3}}-\frac {2 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{d^{2} x}-\frac {4 e^{2} \left (\frac {x \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 d^{2} \left (\frac {x \sqrt {-e^{2} x^{2}+d^{2}}}{2}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )}{d^{2}}\right )}{3 d^{2}}\right )\) | \(281\) |
-1/24*(-e^2*x^2+d^2)^(1/2)*(-32*e^3*x^3-15*d*e^2*x^2+8*d^2*e*x+6*d^3)/x^4+ e^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))-3/8*e^4*d/(d^2) ^(1/2)*ln((2*d^2+2*(d^2)^(1/2)*(-e^2*x^2+d^2)^(1/2))/x)
Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=-\frac {48 \, e^{4} x^{4} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - 9 \, e^{4} x^{4} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (32 \, e^{3} x^{3} + 15 \, d e^{2} x^{2} - 8 \, d^{2} e x - 6 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{24 \, x^{4}} \]
-1/24*(48*e^4*x^4*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) - 9*e^4*x^4*lo g(-(d - sqrt(-e^2*x^2 + d^2))/x) - (32*e^3*x^3 + 15*d*e^2*x^2 - 8*d^2*e*x - 6*d^3)*sqrt(-e^2*x^2 + d^2))/x^4
Result contains complex when optimal does not.
Time = 4.59 (sec) , antiderivative size = 541, normalized size of antiderivative = 4.58 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=d^{3} \left (\begin {cases} - \frac {d^{2}}{4 e x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {3 e}{8 x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{3}}{8 d^{2} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{4} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{4 e x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {3 i e}{8 x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{3}}{8 d^{2} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{4} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{8 d^{3}} & \text {otherwise} \end {cases}\right ) + d^{2} e \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 e^{2} \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 ) - e^{3} \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**3*Piecewise((-d**2/(4*e*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 3*e/(8*x**3* sqrt(d**2/(e**2*x**2) - 1)) - e**3/(8*d**2*x*sqrt(d**2/(e**2*x**2) - 1)) + e**4*acosh(d/(e*x))/(8*d**3), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(4*e*x* *5*sqrt(-d**2/(e**2*x**2) + 1)) - 3*I*e/(8*x**3*sqrt(-d**2/(e**2*x**2) + 1 )) + I*e**3/(8*d**2*x*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**4*asin(d/(e*x))/ (8*d**3), True)) + d**2*e*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*e**2*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)) - e**3*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*sqrt(1 - e**2*x**2/d**2)), True))
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (104) = 208\).
Time = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.88 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\frac {e^{5} \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}}} - \frac {3}{8} \, e^{4} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right ) + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{5} x}{d^{2}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{4}}{8 \, d} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{8 \, d^{3}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{3 \, d^{2} x} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}}{8 \, d^{3} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e}{3 \, d^{2} x^{3}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{4 \, d x^{4}} \]
e^5*arcsin(e^2*x/(d*sqrt(e^2)))/sqrt(e^2) - 3/8*e^4*log(2*d^2/abs(x) + 2*s qrt(-e^2*x^2 + d^2)*d/abs(x)) + sqrt(-e^2*x^2 + d^2)*e^5*x/d^2 + 3/8*sqrt( -e^2*x^2 + d^2)*e^4/d + 1/8*(-e^2*x^2 + d^2)^(3/2)*e^4/d^3 + 2/3*(-e^2*x^2 + d^2)^(3/2)*e^3/(d^2*x) + 1/8*(-e^2*x^2 + d^2)^(5/2)*e^2/(d^3*x^2) - 1/3 *(-e^2*x^2 + d^2)^(5/2)*e/(d^2*x^3) - 1/4*(-e^2*x^2 + d^2)^(5/2)/(d*x^4)
Leaf count of result is larger than twice the leaf count of optimal. 327 vs. \(2 (104) = 208\).
Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.77 \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\frac {{\left (3 \, e^{5} + \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{3}}{x} - \frac {24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e}{x^{2}} - \frac {120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e x^{3}}\right )} e^{8} x^{4}}{192 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} {\left | e \right |}} + \frac {e^{5} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{{\left | e \right |}} - \frac {3 \, e^{5} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{8 \, {\left | e \right |}} + \frac {\frac {120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{5} {\left | e \right |}}{x} + \frac {24 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{3} {\left | e \right |}}{x^{2}} - \frac {8 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e {\left | e \right |}}{x^{3}} - \frac {3 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} {\left | e \right |}}{e x^{4}}}{192 \, e^{4}} \]
1/192*(3*e^5 + 8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^3/x - 24*(d*e + sqr t(-e^2*x^2 + d^2)*abs(e))^2*e/x^2 - 120*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e) )^3/(e*x^3))*e^8*x^4/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*abs(e)) + e^5* arcsin(e*x/d)*sgn(d)*sgn(e)/abs(e) - 3/8*e^5*log(1/2*abs(-2*d*e - 2*sqrt(- e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/abs(e) + 1/192*(120*(d*e + sqrt(-e^2* x^2 + d^2)*abs(e))*e^5*abs(e)/x + 24*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2 *e^3*abs(e)/x^2 - 8*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*e*abs(e)/x^3 - 3 *(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*abs(e)/(e*x^4))/e^4
Timed out. \[ \int \frac {(d+e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^5} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )}{x^5} \,d x \]