Integrand size = 27, antiderivative size = 187 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {55 e^7 \sqrt {d^2-e^2 x^2}}{128 x^2}+\frac {55 e^5 \left (d^2-e^2 x^2\right )^{3/2}}{192 x^4}-\frac {11 e^3 \left (d^2-e^2 x^2\right )^{5/2}}{48 x^6}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}-\frac {3 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}-\frac {29 e^2 \left (d^2-e^2 x^2\right )^{7/2}}{63 d x^7}+\frac {55 e^9 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{128 d} \]
55/192*e^5*(-e^2*x^2+d^2)^(3/2)/x^4-11/48*e^3*(-e^2*x^2+d^2)^(5/2)/x^6-1/9 *d*(-e^2*x^2+d^2)^(7/2)/x^9-3/8*e*(-e^2*x^2+d^2)^(7/2)/x^8-29/63*e^2*(-e^2 *x^2+d^2)^(7/2)/d/x^7+55/128*e^9*arctanh((-e^2*x^2+d^2)^(1/2)/d)/d-55/128* e^7*(-e^2*x^2+d^2)^(1/2)/x^2
Time = 0.49 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.90 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-896 d^8-3024 d^7 e x-1024 d^6 e^2 x^2+7224 d^5 e^3 x^3+8448 d^4 e^4 x^4-3066 d^3 e^5 x^5-10240 d^2 e^6 x^6-4599 d e^7 x^7+3712 e^8 x^8\right )}{8064 d x^9}+\frac {55 e^9 \log (x)}{128 \sqrt {d^2}}-\frac {55 e^9 \log \left (\sqrt {d^2}-\sqrt {d^2-e^2 x^2}\right )}{128 \sqrt {d^2}} \]
(Sqrt[d^2 - e^2*x^2]*(-896*d^8 - 3024*d^7*e*x - 1024*d^6*e^2*x^2 + 7224*d^ 5*e^3*x^3 + 8448*d^4*e^4*x^4 - 3066*d^3*e^5*x^5 - 10240*d^2*e^6*x^6 - 4599 *d*e^7*x^7 + 3712*e^8*x^8))/(8064*d*x^9) + (55*e^9*Log[x])/(128*Sqrt[d^2]) - (55*e^9*Log[Sqrt[d^2] - Sqrt[d^2 - e^2*x^2]])/(128*Sqrt[d^2])
Time = 0.42 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {540, 25, 2338, 25, 27, 534, 243, 51, 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.
\(\displaystyle \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx\) |
\(\Big \downarrow \) 540 |
\(\displaystyle -\frac {\int -\frac {\left (d^2-e^2 x^2\right )^{5/2} \left (27 e d^4+29 e^2 x d^3+9 e^3 x^2 d^2\right )}{x^9}dx}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\left (d^2-e^2 x^2\right )^{5/2} \left (27 e d^4+29 e^2 x d^3+9 e^3 x^2 d^2\right )}{x^9}dx}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle \frac {-\frac {\int -\frac {d^4 e^2 (232 d+99 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^8}dx}{8 d^2}-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\int \frac {d^4 e^2 (232 d+99 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^8}dx}{8 d^2}-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{8} d^2 e^2 \int \frac {(232 d+99 e x) \left (d^2-e^2 x^2\right )^{5/2}}{x^8}dx-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {\frac {1}{8} d^2 e^2 \left (99 e \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^7}dx-\frac {232 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{8} d^2 e^2 \left (\frac {99}{2} e \int \frac {\left (d^2-e^2 x^2\right )^{5/2}}{x^8}dx^2-\frac {232 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\frac {1}{8} d^2 e^2 \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \int \frac {\left (d^2-e^2 x^2\right )^{3/2}}{x^6}dx^2-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {232 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\frac {1}{8} d^2 e^2 \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \left (-\frac {3}{4} e^2 \int \frac {\sqrt {d^2-e^2 x^2}}{x^4}dx^2-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {232 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\frac {1}{8} d^2 e^2 \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \left (-\frac {3}{4} e^2 \left (-\frac {1}{2} e^2 \int \frac {1}{x^2 \sqrt {d^2-e^2 x^2}}dx^2-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {232 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {1}{8} d^2 e^2 \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \left (-\frac {3}{4} e^2 \left (\int \frac {1}{\frac {d^2}{e^2}-\frac {x^4}{e^2}}d\sqrt {d^2-e^2 x^2}-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {232 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {1}{8} d^2 e^2 \left (\frac {99}{2} e \left (-\frac {5}{6} e^2 \left (-\frac {3}{4} e^2 \left (\frac {e^2 \text {arctanh}\left (\frac {\sqrt {d^2-e^2 x^2}}{d}\right )}{d}-\frac {\sqrt {d^2-e^2 x^2}}{x^2}\right )-\frac {\left (d^2-e^2 x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (d^2-e^2 x^2\right )^{5/2}}{3 x^6}\right )-\frac {232 \left (d^2-e^2 x^2\right )^{7/2}}{7 d x^7}\right )-\frac {27 d^2 e \left (d^2-e^2 x^2\right )^{7/2}}{8 x^8}}{9 d^2}-\frac {d \left (d^2-e^2 x^2\right )^{7/2}}{9 x^9}\) |
-1/9*(d*(d^2 - e^2*x^2)^(7/2))/x^9 + ((-27*d^2*e*(d^2 - e^2*x^2)^(7/2))/(8 *x^8) + (d^2*e^2*((-232*(d^2 - e^2*x^2)^(7/2))/(7*d*x^7) + (99*e*(-1/3*(d^ 2 - e^2*x^2)^(5/2)/x^6 - (5*e^2*(-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))/ 6))/2))/8)/(9*d^2)
3.1.80.3.1 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_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*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*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 0.58 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.81
method | result | size |
risch | \(-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, \left (-3712 e^{8} x^{8}+4599 d \,e^{7} x^{7}+10240 d^{2} e^{6} x^{6}+3066 d^{3} e^{5} x^{5}-8448 d^{4} x^{4} e^{4}-7224 d^{5} e^{3} x^{3}+1024 d^{6} e^{2} x^{2}+3024 d^{7} e x +896 d^{8}\right )}{8064 x^{9} d}+\frac {55 e^{9} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{128 \sqrt {d^{2}}}\) | \(151\) |
default | \(e^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )+d^{3} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{9 d^{2} x^{9}}-\frac {2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{63 d^{4} x^{7}}\right )-\frac {3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 d \,x^{7}}+3 d^{2} e \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{8 d^{2} x^{8}}+\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{6 d^{2} x^{6}}-\frac {e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{4 d^{2} x^{4}}-\frac {3 e^{2} \left (-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{2 d^{2} x^{2}}-\frac {5 e^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{5}+d^{2} \left (\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3}+d^{2} \left (\sqrt {-e^{2} x^{2}+d^{2}}-\frac {d^{2} \ln \left (\frac {2 d^{2}+2 \sqrt {d^{2}}\, \sqrt {-e^{2} x^{2}+d^{2}}}{x}\right )}{\sqrt {d^{2}}}\right )\right )\right )}{2 d^{2}}\right )}{4 d^{2}}\right )}{6 d^{2}}\right )}{8 d^{2}}\right )\) | \(504\) |
-1/8064*(-e^2*x^2+d^2)^(1/2)*(-3712*e^8*x^8+4599*d*e^7*x^7+10240*d^2*e^6*x ^6+3066*d^3*e^5*x^5-8448*d^4*e^4*x^4-7224*d^5*e^3*x^3+1024*d^6*e^2*x^2+302 4*d^7*e*x+896*d^8)/x^9/d+55/128*e^9/(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 = 142, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {3465 \, e^{9} x^{9} \log \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{x}\right ) - {\left (3712 \, e^{8} x^{8} - 4599 \, d e^{7} x^{7} - 10240 \, d^{2} e^{6} x^{6} - 3066 \, d^{3} e^{5} x^{5} + 8448 \, d^{4} e^{4} x^{4} + 7224 \, d^{5} e^{3} x^{3} - 1024 \, d^{6} e^{2} x^{2} - 3024 \, d^{7} e x - 896 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8064 \, d x^{9}} \]
-1/8064*(3465*e^9*x^9*log(-(d - sqrt(-e^2*x^2 + d^2))/x) - (3712*e^8*x^8 - 4599*d*e^7*x^7 - 10240*d^2*e^6*x^6 - 3066*d^3*e^5*x^5 + 8448*d^4*e^4*x^4 + 7224*d^5*e^3*x^3 - 1024*d^6*e^2*x^2 - 3024*d^7*e*x - 896*d^8)*sqrt(-e^2* x^2 + d^2))/(d*x^9)
Result contains complex when optimal does not.
Time = 31.65 (sec) , antiderivative size = 1889, normalized size of antiderivative = 10.10 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\text {Too large to display} \]
d**7*Piecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(9*x**8) + e**3*sqrt(d**2/(e **2*x**2) - 1)/(63*d**2*x**6) + 2*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105*d** 4*x**4) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(315*d**6*x**2) + 16*e**9*sqrt (d**2/(e**2*x**2) - 1)/(315*d**8), Abs(d**2/(e**2*x**2)) > 1), (-I*e*sqrt( -d**2/(e**2*x**2) + 1)/(9*x**8) + I*e**3*sqrt(-d**2/(e**2*x**2) + 1)/(63*d **2*x**6) + 2*I*e**5*sqrt(-d**2/(e**2*x**2) + 1)/(105*d**4*x**4) + 8*I*e** 7*sqrt(-d**2/(e**2*x**2) + 1)/(315*d**6*x**2) + 16*I*e**9*sqrt(-d**2/(e**2 *x**2) + 1)/(315*d**8), True)) + 3*d**6*e*Piecewise((-d**2/(8*e*x**9*sqrt( d**2/(e**2*x**2) - 1)) + 7*e/(48*x**7*sqrt(d**2/(e**2*x**2) - 1)) + e**3/( 192*d**2*x**5*sqrt(d**2/(e**2*x**2) - 1)) + 5*e**5/(384*d**4*x**3*sqrt(d** 2/(e**2*x**2) - 1)) - 5*e**7/(128*d**6*x*sqrt(d**2/(e**2*x**2) - 1)) + 5*e **8*acosh(d/(e*x))/(128*d**7), Abs(d**2/(e**2*x**2)) > 1), (I*d**2/(8*e*x* *9*sqrt(-d**2/(e**2*x**2) + 1)) - 7*I*e/(48*x**7*sqrt(-d**2/(e**2*x**2) + 1)) - I*e**3/(192*d**2*x**5*sqrt(-d**2/(e**2*x**2) + 1)) - 5*I*e**5/(384*d **4*x**3*sqrt(-d**2/(e**2*x**2) + 1)) + 5*I*e**7/(128*d**6*x*sqrt(-d**2/(e **2*x**2) + 1)) - 5*I*e**8*asin(d/(e*x))/(128*d**7), True)) + d**5*e**2*Pi ecewise((-e*sqrt(d**2/(e**2*x**2) - 1)/(7*x**6) + e**3*sqrt(d**2/(e**2*x** 2) - 1)/(35*d**2*x**4) + 4*e**5*sqrt(d**2/(e**2*x**2) - 1)/(105*d**4*x**2) + 8*e**7*sqrt(d**2/(e**2*x**2) - 1)/(105*d**6), Abs(d**2/(e**2*x**2)) > 1 ), (-I*e*sqrt(-d**2/(e**2*x**2) + 1)/(7*x**6) + I*e**3*sqrt(-d**2/(e**2...
Time = 0.28 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.32 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {55 \, e^{9} \log \left (\frac {2 \, d^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{{\left | x \right |}}\right )}{128 \, d} - \frac {55 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{9}}{128 \, d^{2}} - \frac {55 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{9}}{384 \, d^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{9}}{128 \, d^{6}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{7}}{128 \, d^{6} x^{2}} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{5}}{192 \, d^{4} x^{4}} - \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{3}}{48 \, d^{2} x^{6}} - \frac {29 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e^{2}}{63 \, d x^{7}} - \frac {3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e}{8 \, x^{8}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d}{9 \, x^{9}} \]
55/128*e^9*log(2*d^2/abs(x) + 2*sqrt(-e^2*x^2 + d^2)*d/abs(x))/d - 55/128* sqrt(-e^2*x^2 + d^2)*e^9/d^2 - 55/384*(-e^2*x^2 + d^2)^(3/2)*e^9/d^4 - 11/ 128*(-e^2*x^2 + d^2)^(5/2)*e^9/d^6 - 11/128*(-e^2*x^2 + d^2)^(7/2)*e^7/(d^ 6*x^2) + 11/192*(-e^2*x^2 + d^2)^(7/2)*e^5/(d^4*x^4) - 11/48*(-e^2*x^2 + d ^2)^(7/2)*e^3/(d^2*x^6) - 29/63*(-e^2*x^2 + d^2)^(7/2)*e^2/(d*x^7) - 3/8*( -e^2*x^2 + d^2)^(7/2)*e/x^8 - 1/9*(-e^2*x^2 + d^2)^(7/2)*d/x^9
Leaf count of result is larger than twice the leaf count of optimal. 648 vs. \(2 (159) = 318\).
Time = 0.32 (sec) , antiderivative size = 648, normalized size of antiderivative = 3.47 \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\frac {{\left (28 \, e^{10} + \frac {189 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} e^{8}}{x} + \frac {324 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} e^{6}}{x^{2}} - \frac {672 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} e^{4}}{x^{3}} - \frac {3024 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} e^{2}}{x^{4}} - \frac {1512 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{x^{5}} + \frac {9744 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{2} x^{6}} + \frac {18144 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{4} x^{7}} - \frac {16632 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{6} x^{8}}\right )} e^{18} x^{9}}{129024 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9} d {\left | e \right |}} + \frac {55 \, e^{10} \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 )}{128 \, d {\left | e \right |}} + \frac {\frac {16632 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )} d^{8} e^{16}}{x} - \frac {18144 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2} d^{8} e^{14}}{x^{2}} - \frac {9744 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3} d^{8} e^{12}}{x^{3}} + \frac {1512 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4} d^{8} e^{10}}{x^{4}} + \frac {3024 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5} d^{8} e^{8}}{x^{5}} + \frac {672 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6} d^{8} e^{6}}{x^{6}} - \frac {324 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7} d^{8} e^{4}}{x^{7}} - \frac {189 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8} d^{8} e^{2}}{x^{8}} - \frac {28 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9} d^{8}}{x^{9}}}{129024 \, d^{9} e^{8} {\left | e \right |}} \]
1/129024*(28*e^10 + 189*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*e^8/x + 324*(d *e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*e^6/x^2 - 672*(d*e + sqrt(-e^2*x^2 + d ^2)*abs(e))^3*e^4/x^3 - 3024*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4*e^2/x^4 - 1512*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/x^5 + 9744*(d*e + sqrt(-e^2* x^2 + d^2)*abs(e))^6/(e^2*x^6) + 18144*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)) ^7/(e^4*x^7) - 16632*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8/(e^6*x^8))*e^18 *x^9/((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9*d*abs(e)) + 55/128*e^10*log(1/ 2*abs(-2*d*e - 2*sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*abs(x)))/(d*abs(e)) + 1 /129024*(16632*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))*d^8*e^16/x - 18144*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^2*d^8*e^14/x^2 - 9744*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3*d^8*e^12/x^3 + 1512*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^ 4*d^8*e^10/x^4 + 3024*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5*d^8*e^8/x^5 + 672*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^6*d^8*e^6/x^6 - 324*(d*e + sqrt(-e ^2*x^2 + d^2)*abs(e))^7*d^8*e^4/x^7 - 189*(d*e + sqrt(-e^2*x^2 + d^2)*abs( e))^8*d^8*e^2/x^8 - 28*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9*d^8/x^9)/(d^9 *e^8*abs(e))
Timed out. \[ \int \frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2}}{x^{10}} \, dx=\int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3}{x^{10}} \,d x \]