Integrand size = 20, antiderivative size = 172 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^{10}} \, dx=\frac {5 B c^3 \sqrt {a+c x^2}}{128 a x^2}+\frac {5 B c^2 \left (a+c x^2\right )^{3/2}}{192 a x^4}+\frac {B c \left (a+c x^2\right )^{5/2}}{48 a x^6}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}-\frac {B \left (a+c x^2\right )^{7/2}}{8 a x^8}+\frac {2 A c \left (a+c x^2\right )^{7/2}}{63 a^2 x^7}+\frac {5 B c^4 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{128 a^{3/2}} \]
5/192*B*c^2*(c*x^2+a)^(3/2)/a/x^4+1/48*B*c*(c*x^2+a)^(5/2)/a/x^6-1/9*A*(c* x^2+a)^(7/2)/a/x^9-1/8*B*(c*x^2+a)^(7/2)/a/x^8+2/63*A*c*(c*x^2+a)^(7/2)/a^ 2/x^7+5/128*B*c^4*arctanh((c*x^2+a)^(1/2)/a^(1/2))/a^(3/2)+5/128*B*c^3*(c* x^2+a)^(1/2)/a/x^2
Time = 0.84 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.80 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {\sqrt {a+c x^2} \left (-256 A c^4 x^8+112 a^4 (8 A+9 B x)+a c^3 x^6 (128 A+315 B x)+8 a^3 c x^2 (304 A+357 B x)+6 a^2 c^2 x^4 (320 A+413 B x)\right )}{8064 a^2 x^9}-\frac {5 B c^4 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{64 a^{3/2}} \]
-1/8064*(Sqrt[a + c*x^2]*(-256*A*c^4*x^8 + 112*a^4*(8*A + 9*B*x) + a*c^3*x ^6*(128*A + 315*B*x) + 8*a^3*c*x^2*(304*A + 357*B*x) + 6*a^2*c^2*x^4*(320* A + 413*B*x)))/(a^2*x^9) - (5*B*c^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + c*x^2])/ Sqrt[a]])/(64*a^(3/2))
Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {539, 25, 539, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2} (A+B x)}{x^{10}} \, dx\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle -\frac {\int -\frac {(9 a B-2 A c x) \left (c x^2+a\right )^{5/2}}{x^9}dx}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {(9 a B-2 A c x) \left (c x^2+a\right )^{5/2}}{x^9}dx}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 539 |
| \(\displaystyle \frac {-\frac {\int \frac {a c (16 A+9 B x) \left (c x^2+a\right )^{5/2}}{x^8}dx}{8 a}-\frac {9 B \left (a+c x^2\right )^{7/2}}{8 x^8}}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {1}{8} c \int \frac {(16 A+9 B x) \left (c x^2+a\right )^{5/2}}{x^8}dx-\frac {9 B \left (a+c x^2\right )^{7/2}}{8 x^8}}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {-\frac {1}{8} c \left (9 B \int \frac {\left (c x^2+a\right )^{5/2}}{x^7}dx-\frac {16 A \left (a+c x^2\right )^{7/2}}{7 a x^7}\right )-\frac {9 B \left (a+c x^2\right )^{7/2}}{8 x^8}}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {-\frac {1}{8} c \left (\frac {9}{2} B \int \frac {\left (c x^2+a\right )^{5/2}}{x^8}dx^2-\frac {16 A \left (a+c x^2\right )^{7/2}}{7 a x^7}\right )-\frac {9 B \left (a+c x^2\right )^{7/2}}{8 x^8}}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {-\frac {1}{8} c \left (\frac {9}{2} B \left (\frac {5}{6} c \int \frac {\left (c x^2+a\right )^{3/2}}{x^6}dx^2-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {16 A \left (a+c x^2\right )^{7/2}}{7 a x^7}\right )-\frac {9 B \left (a+c x^2\right )^{7/2}}{8 x^8}}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {-\frac {1}{8} c \left (\frac {9}{2} B \left (\frac {5}{6} c \left (\frac {3}{4} c \int \frac {\sqrt {c x^2+a}}{x^4}dx^2-\frac {\left (a+c x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {16 A \left (a+c x^2\right )^{7/2}}{7 a x^7}\right )-\frac {9 B \left (a+c x^2\right )^{7/2}}{8 x^8}}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle \frac {-\frac {1}{8} c \left (\frac {9}{2} B \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{x^2 \sqrt {c x^2+a}}dx^2-\frac {\sqrt {a+c x^2}}{x^2}\right )-\frac {\left (a+c x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {16 A \left (a+c x^2\right )^{7/2}}{7 a x^7}\right )-\frac {9 B \left (a+c x^2\right )^{7/2}}{8 x^8}}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {-\frac {1}{8} c \left (\frac {9}{2} B \left (\frac {5}{6} c \left (\frac {3}{4} c \left (\int \frac {1}{\frac {x^4}{c}-\frac {a}{c}}d\sqrt {c x^2+a}-\frac {\sqrt {a+c x^2}}{x^2}\right )-\frac {\left (a+c x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {16 A \left (a+c x^2\right )^{7/2}}{7 a x^7}\right )-\frac {9 B \left (a+c x^2\right )^{7/2}}{8 x^8}}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {1}{8} c \left (\frac {9}{2} B \left (\frac {5}{6} c \left (\frac {3}{4} c \left (-\frac {c \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {\sqrt {a+c x^2}}{x^2}\right )-\frac {\left (a+c x^2\right )^{3/2}}{2 x^4}\right )-\frac {\left (a+c x^2\right )^{5/2}}{3 x^6}\right )-\frac {16 A \left (a+c x^2\right )^{7/2}}{7 a x^7}\right )-\frac {9 B \left (a+c x^2\right )^{7/2}}{8 x^8}}{9 a}-\frac {A \left (a+c x^2\right )^{7/2}}{9 a x^9}\) |
-1/9*(A*(a + c*x^2)^(7/2))/(a*x^9) + ((-9*B*(a + c*x^2)^(7/2))/(8*x^8) - ( c*((-16*A*(a + c*x^2)^(7/2))/(7*a*x^7) + (9*B*(-1/3*(a + c*x^2)^(5/2)/x^6 + (5*c*(-1/2*(a + c*x^2)^(3/2)/x^4 + (3*c*(-(Sqrt[a + c*x^2]/x^2) - (c*Arc Tanh[Sqrt[a + c*x^2]/Sqrt[a]])/Sqrt[a]))/4))/6))/2))/8)/(9*a)
3.4.53.3.1 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]
Time = 0.38 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (-256 A \,c^{4} x^{8}+315 a B \,c^{3} x^{7}+128 a A \,c^{3} x^{6}+2478 a^{2} B \,c^{2} x^{5}+1920 a^{2} A \,c^{2} x^{4}+2856 a^{3} B c \,x^{3}+2432 a^{3} A c \,x^{2}+1008 a^{4} B x +896 A \,a^{4}\right )}{8064 x^{9} a^{2}}+\frac {5 B \,c^{4} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{128 a^{\frac {3}{2}}}\) | \(138\) |
| default | \(B \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{8 a \,x^{8}}-\frac {c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{6 a \,x^{6}}+\frac {c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 c \left (\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {c \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )}{6 a}\right )}{8 a}\right )+A \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{9 a \,x^{9}}+\frac {2 c \left (c \,x^{2}+a \right )^{\frac {7}{2}}}{63 a^{2} x^{7}}\right )\) | \(204\) |
-1/8064*(c*x^2+a)^(1/2)*(-256*A*c^4*x^8+315*B*a*c^3*x^7+128*A*a*c^3*x^6+24 78*B*a^2*c^2*x^5+1920*A*a^2*c^2*x^4+2856*B*a^3*c*x^3+2432*A*a^3*c*x^2+1008 *B*a^4*x+896*A*a^4)/x^9/a^2+5/128*B/a^(3/2)*c^4*ln((2*a+2*a^(1/2)*(c*x^2+a )^(1/2))/x)
Time = 0.34 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^{10}} \, dx=\left [\frac {315 \, B \sqrt {a} c^{4} x^{9} \log \left (-\frac {c x^{2} + 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (256 \, A c^{4} x^{8} - 315 \, B a c^{3} x^{7} - 128 \, A a c^{3} x^{6} - 2478 \, B a^{2} c^{2} x^{5} - 1920 \, A a^{2} c^{2} x^{4} - 2856 \, B a^{3} c x^{3} - 2432 \, A a^{3} c x^{2} - 1008 \, B a^{4} x - 896 \, A a^{4}\right )} \sqrt {c x^{2} + a}}{16128 \, a^{2} x^{9}}, -\frac {315 \, B \sqrt {-a} c^{4} x^{9} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (256 \, A c^{4} x^{8} - 315 \, B a c^{3} x^{7} - 128 \, A a c^{3} x^{6} - 2478 \, B a^{2} c^{2} x^{5} - 1920 \, A a^{2} c^{2} x^{4} - 2856 \, B a^{3} c x^{3} - 2432 \, A a^{3} c x^{2} - 1008 \, B a^{4} x - 896 \, A a^{4}\right )} \sqrt {c x^{2} + a}}{8064 \, a^{2} x^{9}}\right ] \]
[1/16128*(315*B*sqrt(a)*c^4*x^9*log(-(c*x^2 + 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(256*A*c^4*x^8 - 315*B*a*c^3*x^7 - 128*A*a*c^3*x^6 - 2478*B* a^2*c^2*x^5 - 1920*A*a^2*c^2*x^4 - 2856*B*a^3*c*x^3 - 2432*A*a^3*c*x^2 - 1 008*B*a^4*x - 896*A*a^4)*sqrt(c*x^2 + a))/(a^2*x^9), -1/8064*(315*B*sqrt(- a)*c^4*x^9*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (256*A*c^4*x^8 - 315*B*a*c^3 *x^7 - 128*A*a*c^3*x^6 - 2478*B*a^2*c^2*x^5 - 1920*A*a^2*c^2*x^4 - 2856*B* a^3*c*x^3 - 2432*A*a^3*c*x^2 - 1008*B*a^4*x - 896*A*a^4)*sqrt(c*x^2 + a))/ (a^2*x^9)]
Leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (158) = 316\).
Time = 27.95 (sec) , antiderivative size = 1202, normalized size of antiderivative = 6.99 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^{10}} \, dx=- \frac {35 A a^{9} c^{\frac {19}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {110 A a^{8} c^{\frac {21}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {114 A a^{7} c^{\frac {23}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {40 A a^{6} c^{\frac {25}{2}} x^{6} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {30 A a^{6} c^{\frac {11}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} + \frac {5 A a^{5} c^{\frac {27}{2}} x^{8} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {66 A a^{5} c^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} + \frac {30 A a^{4} c^{\frac {29}{2}} x^{10} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {34 A a^{4} c^{\frac {15}{2}} x^{4} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} + \frac {40 A a^{3} c^{\frac {31}{2}} x^{12} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {6 A a^{3} c^{\frac {17}{2}} x^{6} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} + \frac {16 A a^{2} c^{\frac {33}{2}} x^{14} \sqrt {\frac {a}{c x^{2}} + 1}}{315 a^{7} c^{9} x^{8} + 945 a^{6} c^{10} x^{10} + 945 a^{5} c^{11} x^{12} + 315 a^{4} c^{12} x^{14}} - \frac {24 A a^{2} c^{\frac {19}{2}} x^{8} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {16 A a c^{\frac {21}{2}} x^{10} \sqrt {\frac {a}{c x^{2}} + 1}}{105 a^{5} c^{4} x^{6} + 210 a^{4} c^{5} x^{8} + 105 a^{3} c^{6} x^{10}} - \frac {A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {A c^{\frac {7}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a x^{2}} + \frac {2 A c^{\frac {9}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 a^{2}} - \frac {B a^{3}}{8 \sqrt {c} x^{9} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {23 B a^{2} \sqrt {c}}{48 x^{7} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {127 B a c^{\frac {3}{2}}}{192 x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {133 B c^{\frac {5}{2}}}{384 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {5 B c^{\frac {7}{2}}}{128 a x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {5 B c^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{128 a^{\frac {3}{2}}} \]
-35*A*a**9*c**(19/2)*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c **10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 110*A*a**8*c** (21/2)*x**2*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**1 0 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 114*A*a**7*c**(23/2)*x* *4*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a **5*c**11*x**12 + 315*a**4*c**12*x**14) - 40*A*a**6*c**(25/2)*x**6*sqrt(a/ (c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11* x**12 + 315*a**4*c**12*x**14) - 30*A*a**6*c**(11/2)*sqrt(a/(c*x**2) + 1)/( 105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) + 5*A*a**5* c**(27/2)*x**8*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x **10 + 945*a**5*c**11*x**12 + 315*a**4*c**12*x**14) - 66*A*a**5*c**(13/2)* x**2*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a **3*c**6*x**10) + 30*A*a**4*c**(29/2)*x**10*sqrt(a/(c*x**2) + 1)/(315*a**7 *c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5*c**11*x**12 + 315*a**4*c**12* x**14) - 34*A*a**4*c**(15/2)*x**4*sqrt(a/(c*x**2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) + 40*A*a**3*c**(31/2)*x**12*s qrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 945*a**6*c**10*x**10 + 945*a**5* c**11*x**12 + 315*a**4*c**12*x**14) - 6*A*a**3*c**(17/2)*x**6*sqrt(a/(c*x* *2) + 1)/(105*a**5*c**4*x**6 + 210*a**4*c**5*x**8 + 105*a**3*c**6*x**10) + 16*A*a**2*c**(33/2)*x**14*sqrt(a/(c*x**2) + 1)/(315*a**7*c**9*x**8 + 9...
Time = 0.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.12 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^{10}} \, dx=\frac {5 \, B c^{4} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right )}{128 \, a^{\frac {3}{2}}} - \frac {{\left (c x^{2} + a\right )}^{\frac {5}{2}} B c^{4}}{128 \, a^{4}} - \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{4}}{384 \, a^{3}} - \frac {5 \, \sqrt {c x^{2} + a} B c^{4}}{128 \, a^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c^{3}}{128 \, a^{4} x^{2}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c^{2}}{192 \, a^{3} x^{4}} + \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B c}{48 \, a^{2} x^{6}} + \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} A c}{63 \, a^{2} x^{7}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{8 \, a x^{8}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{9 \, a x^{9}} \]
5/128*B*c^4*arcsinh(a/(sqrt(a*c)*abs(x)))/a^(3/2) - 1/128*(c*x^2 + a)^(5/2 )*B*c^4/a^4 - 5/384*(c*x^2 + a)^(3/2)*B*c^4/a^3 - 5/128*sqrt(c*x^2 + a)*B* c^4/a^2 + 1/128*(c*x^2 + a)^(7/2)*B*c^3/(a^4*x^2) + 1/192*(c*x^2 + a)^(7/2 )*B*c^2/(a^3*x^4) + 1/48*(c*x^2 + a)^(7/2)*B*c/(a^2*x^6) + 2/63*(c*x^2 + a )^(7/2)*A*c/(a^2*x^7) - 1/8*(c*x^2 + a)^(7/2)*B/(a*x^8) - 1/9*(c*x^2 + a)^ (7/2)*A/(a*x^9)
Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (140) = 280\).
Time = 0.30 (sec) , antiderivative size = 491, normalized size of antiderivative = 2.85 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^{10}} \, dx=-\frac {5 \, B c^{4} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a} + \frac {315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{17} B c^{4} + 8022 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{15} B a c^{4} + 16128 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{14} A a c^{\frac {9}{2}} + 10458 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{13} B a^{2} c^{4} + 26880 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{12} A a^{2} c^{\frac {9}{2}} + 18270 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{11} B a^{3} c^{4} + 80640 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{10} A a^{3} c^{\frac {9}{2}} + 48384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} A a^{4} c^{\frac {9}{2}} - 18270 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} B a^{5} c^{4} + 48384 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} A a^{5} c^{\frac {9}{2}} - 10458 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{5} B a^{6} c^{4} + 6912 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{6} c^{\frac {9}{2}} - 8022 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} B a^{7} c^{4} + 2304 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a^{7} c^{\frac {9}{2}} - 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{8} c^{4} - 256 \, A a^{8} c^{\frac {9}{2}}}{4032 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{9} a} \]
-5/64*B*c^4*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/(sqrt(-a)*a) + 1/4032*(315*(sqrt(c)*x - sqrt(c*x^2 + a))^17*B*c^4 + 8022*(sqrt(c)*x - sq rt(c*x^2 + a))^15*B*a*c^4 + 16128*(sqrt(c)*x - sqrt(c*x^2 + a))^14*A*a*c^( 9/2) + 10458*(sqrt(c)*x - sqrt(c*x^2 + a))^13*B*a^2*c^4 + 26880*(sqrt(c)*x - sqrt(c*x^2 + a))^12*A*a^2*c^(9/2) + 18270*(sqrt(c)*x - sqrt(c*x^2 + a)) ^11*B*a^3*c^4 + 80640*(sqrt(c)*x - sqrt(c*x^2 + a))^10*A*a^3*c^(9/2) + 483 84*(sqrt(c)*x - sqrt(c*x^2 + a))^8*A*a^4*c^(9/2) - 18270*(sqrt(c)*x - sqrt (c*x^2 + a))^7*B*a^5*c^4 + 48384*(sqrt(c)*x - sqrt(c*x^2 + a))^6*A*a^5*c^( 9/2) - 10458*(sqrt(c)*x - sqrt(c*x^2 + a))^5*B*a^6*c^4 + 6912*(sqrt(c)*x - sqrt(c*x^2 + a))^4*A*a^6*c^(9/2) - 8022*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B *a^7*c^4 + 2304*(sqrt(c)*x - sqrt(c*x^2 + a))^2*A*a^7*c^(9/2) - 315*(sqrt( c)*x - sqrt(c*x^2 + a))*B*a^8*c^4 - 256*A*a^8*c^(9/2))/(((sqrt(c)*x - sqrt (c*x^2 + a))^2 - a)^9*a)
Time = 15.14 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.10 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^{10}} \, dx=\frac {55\,B\,a\,{\left (c\,x^2+a\right )}^{3/2}}{384\,x^8}-\frac {73\,B\,{\left (c\,x^2+a\right )}^{5/2}}{384\,x^8}-\frac {A\,a^2\,\sqrt {c\,x^2+a}}{9\,x^9}-\frac {5\,B\,a^2\,\sqrt {c\,x^2+a}}{128\,x^8}-\frac {5\,B\,{\left (c\,x^2+a\right )}^{7/2}}{128\,a\,x^8}-\frac {5\,A\,c^2\,\sqrt {c\,x^2+a}}{21\,x^5}-\frac {A\,c^3\,\sqrt {c\,x^2+a}}{63\,a\,x^3}+\frac {2\,A\,c^4\,\sqrt {c\,x^2+a}}{63\,a^2\,x}-\frac {19\,A\,a\,c\,\sqrt {c\,x^2+a}}{63\,x^7}-\frac {B\,c^4\,\mathrm {atan}\left (\frac {\sqrt {c\,x^2+a}\,1{}\mathrm {i}}{\sqrt {a}}\right )\,5{}\mathrm {i}}{128\,a^{3/2}} \]
(55*B*a*(a + c*x^2)^(3/2))/(384*x^8) - (B*c^4*atan(((a + c*x^2)^(1/2)*1i)/ a^(1/2))*5i)/(128*a^(3/2)) - (73*B*(a + c*x^2)^(5/2))/(384*x^8) - (A*a^2*( a + c*x^2)^(1/2))/(9*x^9) - (5*B*a^2*(a + c*x^2)^(1/2))/(128*x^8) - (5*B*( a + c*x^2)^(7/2))/(128*a*x^8) - (5*A*c^2*(a + c*x^2)^(1/2))/(21*x^5) - (A* c^3*(a + c*x^2)^(1/2))/(63*a*x^3) + (2*A*c^4*(a + c*x^2)^(1/2))/(63*a^2*x) - (19*A*a*c*(a + c*x^2)^(1/2))/(63*x^7)