Integrand size = 27, antiderivative size = 143 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\frac {e^4 \sqrt {d^2-e^2 x^2}}{16 d x^2}-\frac {e^2 \left (d^2-e^2 x^2\right )^{3/2}}{24 d x^4}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}+\frac {e \left (d^2-e^2 x^2\right )^{5/2}}{5 d^2 x^5}-\frac {e^6 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{16 d^2} \]
-1/24*e^2*(-e^2*x^2+d^2)^(3/2)/d/x^4-1/6*(-e^2*x^2+d^2)^(5/2)/d/x^6+1/5*e* (-e^2*x^2+d^2)^(5/2)/d^2/x^5-1/16*e^6*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d^2+ 1/16*e^4*(-e^2*x^2+d^2)^(1/2)/d/x^2
Time = 0.36 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-40 d^5+48 d^4 e x+70 d^3 e^2 x^2-96 d^2 e^3 x^3-15 d e^4 x^4+48 e^5 x^5\right )}{240 d^2 x^6}-\frac {\sqrt {d^2} e^6 \log (x)}{16 d^3}+\frac {\sqrt {d^2} e^6 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{16 d^3} \]
(Sqrt[d^2 - e^2*x^2]*(-40*d^5 + 48*d^4*e*x + 70*d^3*e^2*x^2 - 96*d^2*e^3*x ^3 - 15*d*e^4*x^4 + 48*e^5*x^5))/(240*d^2*x^6) - (Sqrt[d^2]*e^6*Log[x])/(1 6*d^3) + (Sqrt[d^2]*e^6*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(16*d^3)
Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.03, 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, 534, 243, 51, 51, 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.
| \(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 566 |
| \(\displaystyle \int \frac {(d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^7}dx\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle -\frac {\int \frac {d e (6 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^6}dx}{6 d^2}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {(6 d-e x) \left (d^2-e^2 x^2\right )^{3/2}}{x^6}dx}{6 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle -\frac {e \left (-e \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^5}dx-\frac {6 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {e \left (-\frac {1}{2} e \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^6}dx^2-\frac {6 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -\frac {e \left (-\frac {1}{2} e \left (-\frac {3}{4} e^2 \int \frac {\sqrt {d^2-e^2 x^2}}{x^4}dx^2-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {6 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -\frac {e \left (-\frac {1}{2} e \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {6 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {e \left (-\frac {1}{2} e \left (-\frac {3}{4} e^2 \left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {6 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {e \left (-\frac {1}{2} e \left (-\frac {3}{4} e^2 \left (\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {6 \left (d^2-e^2 x^2\right )^{5/2}}{5 d x^5}\right )}{6 d}-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{6 d x^6}\) |
-1/6*(d^2 - e^2*x^2)^(5/2)/(d*x^6) - (e*((-6*(d^2 - e^2*x^2)^(5/2))/(5*d*x ^5) - (e*(-1/2*(d^2 - e^2*x^2)^(3/2)/x^4 - (3*e^2*(-(Sqrt[d^2 - e^2*x^2]/x ^2) + (e^2*ArcTanh[Sqrt[d^2 - e^2*x^2]/d])/d))/4))/2))/(6*d)
3.2.14.3.1 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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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 = 0.54 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.85
| method | result | size |
| risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-48 e^{5} x^{5}+15 d \,e^{4} x^{4}+96 d^{2} e^{3} x^{3}-70 d^{3} e^{2} x^{2}-48 d^{4} e x +40 d^{5}\right )}{240 x^{6} d^{2}}-\frac {e^{6} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{16 d \sqrt {d^{2}}}\) | \(121\) |
| default | \(\text {Expression too large to display}\) | \(1300\) |
-1/240*(-e^2*x^2+d^2)^(1/2)*(-48*e^5*x^5+15*d*e^4*x^4+96*d^2*e^3*x^3-70*d^ 3*e^2*x^2-48*d^4*e*x+40*d^5)/x^6/d^2-1/16/d*e^6/(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 = 108, normalized size of antiderivative = 0.76 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\frac {15 \, e^{6} x^{6} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) + {\left (48 \, e^{5} x^{5} - 15 \, d e^{4} x^{4} - 96 \, d^{2} e^{3} x^{3} + 70 \, d^{3} e^{2} x^{2} + 48 \, d^{4} e x - 40 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{240 \, d^{2} x^{6}} \]
1/240*(15*e^6*x^6*log(-(d - sqrt(-e^2*x^2 + d^2))/x) + (48*e^5*x^5 - 15*d* e^4*x^4 - 96*d^2*e^3*x^3 + 70*d^3*e^2*x^2 + 48*d^4*e*x - 40*d^5)*sqrt(-e^2 *x^2 + d^2))/(d^2*x^6)
Result contains complex when optimal does not.
Time = 9.03 (sec) , antiderivative size = 918, normalized size of antiderivative = 6.42 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=d^{3} \left (\begin {cases} - \frac {d^{2}}{6 e x^{7} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {5 e}{24 x^{5} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{3}}{48 d^{2} x^{3} \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} - \frac {e^{5}}{16 d^{4} x \sqrt {\frac {d^{2}}{e^{2} x^{2}} - 1}} + \frac {e^{6} \operatorname {acosh}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {for}\: \left |{\frac {d^{2}}{e^{2} x^{2}}}\right | > 1 \\\frac {i d^{2}}{6 e x^{7} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {5 i e}{24 x^{5} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{3}}{48 d^{2} x^{3} \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} + \frac {i e^{5}}{16 d^{4} x \sqrt {- \frac {d^{2}}{e^{2} x^{2}} + 1}} - \frac {i e^{6} \operatorname {asin}{\left (\frac {d}{e x} \right )}}{16 d^{5}} & \text {otherwise} \end {cases}\right ) - d^{2} e \left (\begin {cases} \frac {3 i d^{3} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 i d e^{2} x^{2} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 i e^{6} x^{6} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {i e^{4} x^{4} \sqrt {-1 + \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {for}\: \left |{\frac {e^{2} x^{2}}{d^{2}}}\right | > 1 \\\frac {3 d^{3} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} - \frac {4 d e^{2} x^{2} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{2} x^{5} + 15 e^{2} x^{7}} + \frac {2 e^{6} x^{6} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{5} x^{5} + 15 d^{3} e^{2} x^{7}} - \frac {e^{4} x^{4} \sqrt {1 - \frac {e^{2} x^{2}}{d^{2}}}}{- 15 d^{3} x^{5} + 15 d e^{2} x^{7}} & \text {otherwise} \end {cases}\right ) - d e^{2} \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 ) + e^{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**3*Piecewise((-d**2/(6*e*x**7*sqrt(d**2/(e**2*x**2) - 1)) + 5*e/(24*x**5 *sqrt(d**2/(e**2*x**2) - 1)) + e**3/(48*d**2*x**3*sqrt(d**2/(e**2*x**2) - 1)) - e**5/(16*d**4*x*sqrt(d**2/(e**2*x**2) - 1)) + e**6*acosh(d/(e*x))/(1 6*d**5), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(6*e*x**7*sqrt(-d**2/(e**2*x* *2) + 1)) - 5*I*e/(24*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(48*d**2* x**3*sqrt(-d**2/(e**2*x**2) + 1)) + I*e**5/(16*d**4*x*sqrt(-d**2/(e**2*x** 2) + 1)) - I*e**6*asin(d/(e*x))/(16*d**5), True)) - d**2*e*Piecewise((3*I* d**3*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) - 4*I*d*e**2 *x**2*sqrt(-1 + e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2*x**7) + 2*I*e**6* x**6*sqrt(-1 + e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e**2*x**7) - I*e** 4*x**4*sqrt(-1 + e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x**7), Abs(e** 2*x**2/d**2) > 1), (3*d**3*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e* *2*x**7) - 4*d*e**2*x**2*sqrt(1 - e**2*x**2/d**2)/(-15*d**2*x**5 + 15*e**2 *x**7) + 2*e**6*x**6*sqrt(1 - e**2*x**2/d**2)/(-15*d**5*x**5 + 15*d**3*e** 2*x**7) - e**4*x**4*sqrt(1 - e**2*x**2/d**2)/(-15*d**3*x**5 + 15*d*e**2*x* *7), True)) - d*e**2*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*...
Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=-\frac {e^{6} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{16 \, d^{2}} + \frac {\sqrt {-e^{2} x^{2} + d^{2}} e^{6}}{16 \, d^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}{16 \, d^{3} x^{2}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}}{5 \, d^{2} x^{3}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}}{8 \, d x^{4}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e}{5 \, x^{5}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{6 \, x^{6}} \]
-1/16*e^6*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d^2 + 1/16*s qrt(-e^2*x^2 + d^2)*e^6/d^3 + 1/16*(-e^2*x^2 + d^2)^(3/2)*e^4/(d^3*x^2) - 1/5*(-e^2*x^2 + d^2)^(3/2)*e^3/(d^2*x^3) + 1/8*(-e^2*x^2 + d^2)^(3/2)*e^2/ (d*x^4) + 1/5*(-e^2*x^2 + d^2)^(3/2)*e/x^5 - 1/6*(-e^2*x^2 + d^2)^(3/2)*d/ x^6
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (123) = 246\).
Time = 0.29 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.24 \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\frac {{\left (5 \, e^{7} - \frac {12 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{5}}{x} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{3}}{x^{2}} + \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e}{x^{3}} - \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e x^{4}} - \frac {120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{3} x^{5}}\right )} e^{12} x^{6}}{1920 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{2} {\left | e \right |}} - \frac {e^{7} \log \left (\frac {{\left | -2 \, d e - 2 \, \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |} \right |}}{2 \, e^{2} {\left | x \right |}}\right )}{16 \, d^{2} {\left | e \right |}} + \frac {\frac {120 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{10} e^{9} {\left | e \right |}}{x} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{10} e^{7} {\left | e \right |}}{x^{2}} - \frac {60 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{10} e^{5} {\left | e \right |}}{x^{3}} + \frac {15 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{10} e^{3} {\left | e \right |}}{x^{4}} + \frac {12 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{10} e {\left | e \right |}}{x^{5}} - \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{10} {\left | e \right |}}{e x^{6}}}{1920 \, d^{12} e^{6}} \]
1/1920*(5*e^7 - 12*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^5/x - 15*(d*e + s qrt(-e^2*x^2 + d^2)*abs(e))^2*e^3/x^2 + 60*(d*e + sqrt(-e^2*x^2 + d^2)*abs (e))^3*e/x^3 - 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e*x^4) - 120*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e^3*x^5))*e^12*x^6/((d*e + sqrt(-e^2*x^ 2 + d^2)*abs(e))^6*d^2*abs(e)) - 1/16*e^7*log(1/2*abs(-2*d*e - 2*sqrt(-e^2 *x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d^2*abs(e)) + 1/1920*(120*(d*e + sqrt(- e^2*x^2 + d^2)*abs(e))*d^10*e^9*abs(e)/x + 15*(d*e + sqrt(-e^2*x^2 + d^2)* abs(e))^2*d^10*e^7*abs(e)/x^2 - 60*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d ^10*e^5*abs(e)/x^3 + 15*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*d^10*e^3*abs (e)/x^4 + 12*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d^10*e*abs(e)/x^5 - 5*( d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d^10*abs(e)/(e*x^6))/(d^12*e^6)
Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7 (d+e x)} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}}{x^7\,\left (d+e\,x\right )} \,d x \]