Integrand size = 20, antiderivative size = 140 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^6} \, dx=-\frac {c^2 (8 A-15 B x) \sqrt {a+c x^2}}{8 x}-\frac {c (8 A+15 B x) \left (a+c x^2\right )^{3/2}}{24 x^3}-\frac {(4 A+5 B x) \left (a+c x^2\right )^{5/2}}{20 x^5}+A c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {15}{8} \sqrt {a} B c^2 \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right ) \]
-1/24*c*(15*B*x+8*A)*(c*x^2+a)^(3/2)/x^3-1/20*(5*B*x+4*A)*(c*x^2+a)^(5/2)/ x^5+A*c^(5/2)*arctanh(x*c^(1/2)/(c*x^2+a)^(1/2))-15/8*B*c^2*arctanh((c*x^2 +a)^(1/2)/a^(1/2))*a^(1/2)-1/8*c^2*(-15*B*x+8*A)*(c*x^2+a)^(1/2)/x
Time = 0.47 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.06 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^6} \, dx=-\frac {\sqrt {a+c x^2} \left (8 c^2 x^4 (23 A-15 B x)+6 a^2 (4 A+5 B x)+a c x^2 (88 A+135 B x)\right )}{120 x^5}+2 A c^{5/2} \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+c x^2}}\right )-\frac {15}{8} \sqrt {a} B c^2 \log (x)+\frac {15}{8} \sqrt {a} B c^2 \log \left (-\sqrt {a}+\sqrt {a+c x^2}\right ) \]
-1/120*(Sqrt[a + c*x^2]*(8*c^2*x^4*(23*A - 15*B*x) + 6*a^2*(4*A + 5*B*x) + a*c*x^2*(88*A + 135*B*x)))/x^5 + 2*A*c^(5/2)*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a ] + Sqrt[a + c*x^2])] - (15*Sqrt[a]*B*c^2*Log[x])/8 + (15*Sqrt[a]*B*c^2*Lo g[-Sqrt[a] + Sqrt[a + c*x^2]])/8
Time = 0.31 (sec) , antiderivative size = 142, 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 = {537, 25, 537, 25, 536, 538, 224, 219, 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.
| \(\displaystyle \int \frac {\left (a+c x^2\right )^{5/2} (A+B x)}{x^6} \, dx\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle -\frac {1}{4} c \int -\frac {(4 A+5 B x) \left (c x^2+a\right )^{3/2}}{x^4}dx-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} c \int \frac {(4 A+5 B x) \left (c x^2+a\right )^{3/2}}{x^4}dx-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 537 |
| \(\displaystyle \frac {1}{4} c \left (-\frac {1}{2} c \int -\frac {(8 A+15 B x) \sqrt {c x^2+a}}{x^2}dx-\frac {\left (a+c x^2\right )^{3/2} (8 A+15 B x)}{6 x^3}\right )-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} c \left (\frac {1}{2} c \int \frac {(8 A+15 B x) \sqrt {c x^2+a}}{x^2}dx-\frac {\left (a+c x^2\right )^{3/2} (8 A+15 B x)}{6 x^3}\right )-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 536 |
| \(\displaystyle \frac {1}{4} c \left (\frac {1}{2} c \left (\int \frac {15 a B+8 A c x}{x \sqrt {c x^2+a}}dx-\frac {\sqrt {a+c x^2} (8 A-15 B x)}{x}\right )-\frac {\left (a+c x^2\right )^{3/2} (8 A+15 B x)}{6 x^3}\right )-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {1}{4} c \left (\frac {1}{2} c \left (8 A c \int \frac {1}{\sqrt {c x^2+a}}dx+15 a B \int \frac {1}{x \sqrt {c x^2+a}}dx-\frac {\sqrt {a+c x^2} (8 A-15 B x)}{x}\right )-\frac {\left (a+c x^2\right )^{3/2} (8 A+15 B x)}{6 x^3}\right )-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {1}{4} c \left (\frac {1}{2} c \left (8 A c \int \frac {1}{1-\frac {c x^2}{c x^2+a}}d\frac {x}{\sqrt {c x^2+a}}+15 a B \int \frac {1}{x \sqrt {c x^2+a}}dx-\frac {\sqrt {a+c x^2} (8 A-15 B x)}{x}\right )-\frac {\left (a+c x^2\right )^{3/2} (8 A+15 B x)}{6 x^3}\right )-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} c \left (\frac {1}{2} c \left (15 a B \int \frac {1}{x \sqrt {c x^2+a}}dx+8 A \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {\sqrt {a+c x^2} (8 A-15 B x)}{x}\right )-\frac {\left (a+c x^2\right )^{3/2} (8 A+15 B x)}{6 x^3}\right )-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{4} c \left (\frac {1}{2} c \left (\frac {15}{2} a B \int \frac {1}{x^2 \sqrt {c x^2+a}}dx^2+8 A \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {\sqrt {a+c x^2} (8 A-15 B x)}{x}\right )-\frac {\left (a+c x^2\right )^{3/2} (8 A+15 B x)}{6 x^3}\right )-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{4} c \left (\frac {1}{2} c \left (\frac {15 a B \int \frac {1}{\frac {x^4}{c}-\frac {a}{c}}d\sqrt {c x^2+a}}{c}+8 A \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {\sqrt {a+c x^2} (8 A-15 B x)}{x}\right )-\frac {\left (a+c x^2\right )^{3/2} (8 A+15 B x)}{6 x^3}\right )-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{4} c \left (\frac {1}{2} c \left (8 A \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {a+c x^2}}\right )-\frac {\sqrt {a+c x^2} (8 A-15 B x)}{x}-15 \sqrt {a} B \text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )\right )-\frac {\left (a+c x^2\right )^{3/2} (8 A+15 B x)}{6 x^3}\right )-\frac {\left (a+c x^2\right )^{5/2} (4 A+5 B x)}{20 x^5}\) |
-1/20*((4*A + 5*B*x)*(a + c*x^2)^(5/2))/x^5 + (c*(-1/6*((8*A + 15*B*x)*(a + c*x^2)^(3/2))/x^3 + (c*(-(((8*A - 15*B*x)*Sqrt[a + c*x^2])/x) + 8*A*Sqrt [c]*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]] - 15*Sqrt[a]*B*ArcTanh[Sqrt[a + c *x^2]/Sqrt[a]]))/2))/4
3.4.49.3.1 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]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[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 = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.87
| method | result | size |
| risch | \(-\frac {\sqrt {c \,x^{2}+a}\, \left (184 A \,c^{2} x^{4}+135 a B c \,x^{3}+88 a A c \,x^{2}+30 a^{2} B x +24 A \,a^{2}\right )}{120 x^{5}}+A \,c^{\frac {5}{2}} \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )-\frac {15 B \sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right ) c^{2}}{8}+B \,c^{2} \sqrt {c \,x^{2}+a}\) | \(122\) |
| default | \(B \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{4 a \,x^{4}}+\frac {3 c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{2 a \,x^{2}}+\frac {5 c \left (\frac {\left (c \,x^{2}+a \right )^{\frac {5}{2}}}{5}+a \left (\frac {\left (c \,x^{2}+a \right )^{\frac {3}{2}}}{3}+a \left (\sqrt {c \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )\right )\right )\right )}{2 a}\right )}{4 a}\right )+A \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{5 a \,x^{5}}+\frac {2 c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{3 a \,x^{3}}+\frac {4 c \left (-\frac {\left (c \,x^{2}+a \right )^{\frac {7}{2}}}{a x}+\frac {6 c \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 a \left (\frac {x \left (c \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {c \,x^{2}+a}}{2}+\frac {a \ln \left (x \sqrt {c}+\sqrt {c \,x^{2}+a}\right )}{2 \sqrt {c}}\right )}{4}\right )}{6}\right )}{a}\right )}{3 a}\right )}{5 a}\right )\) | \(259\) |
-1/120*(c*x^2+a)^(1/2)*(184*A*c^2*x^4+135*B*a*c*x^3+88*A*a*c*x^2+30*B*a^2* x+24*A*a^2)/x^5+A*c^(5/2)*ln(x*c^(1/2)+(c*x^2+a)^(1/2))-15/8*B*a^(1/2)*ln( (2*a+2*a^(1/2)*(c*x^2+a)^(1/2))/x)*c^2+B*c^2*(c*x^2+a)^(1/2)
Time = 0.32 (sec) , antiderivative size = 534, normalized size of antiderivative = 3.81 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^6} \, dx=\left [\frac {120 \, A c^{\frac {5}{2}} x^{5} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + 225 \, B \sqrt {a} c^{2} x^{5} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{240 \, x^{5}}, -\frac {240 \, A \sqrt {-c} c^{2} x^{5} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 225 \, B \sqrt {a} c^{2} x^{5} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{240 \, x^{5}}, \frac {225 \, B \sqrt {-a} c^{2} x^{5} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) + 60 \, A c^{\frac {5}{2}} x^{5} \log \left (-2 \, c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {c} x - a\right ) + {\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{120 \, x^{5}}, -\frac {120 \, A \sqrt {-c} c^{2} x^{5} \arctan \left (\frac {\sqrt {-c} x}{\sqrt {c x^{2} + a}}\right ) - 225 \, B \sqrt {-a} c^{2} x^{5} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right ) - {\left (120 \, B c^{2} x^{5} - 184 \, A c^{2} x^{4} - 135 \, B a c x^{3} - 88 \, A a c x^{2} - 30 \, B a^{2} x - 24 \, A a^{2}\right )} \sqrt {c x^{2} + a}}{120 \, x^{5}}\right ] \]
[1/240*(120*A*c^(5/2)*x^5*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + 225*B*sqrt(a)*c^2*x^5*log(-(c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2 ) + 2*(120*B*c^2*x^5 - 184*A*c^2*x^4 - 135*B*a*c*x^3 - 88*A*a*c*x^2 - 30*B *a^2*x - 24*A*a^2)*sqrt(c*x^2 + a))/x^5, -1/240*(240*A*sqrt(-c)*c^2*x^5*ar ctan(sqrt(-c)*x/sqrt(c*x^2 + a)) - 225*B*sqrt(a)*c^2*x^5*log(-(c*x^2 - 2*s qrt(c*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(120*B*c^2*x^5 - 184*A*c^2*x^4 - 13 5*B*a*c*x^3 - 88*A*a*c*x^2 - 30*B*a^2*x - 24*A*a^2)*sqrt(c*x^2 + a))/x^5, 1/120*(225*B*sqrt(-a)*c^2*x^5*arctan(sqrt(-a)/sqrt(c*x^2 + a)) + 60*A*c^(5 /2)*x^5*log(-2*c*x^2 - 2*sqrt(c*x^2 + a)*sqrt(c)*x - a) + (120*B*c^2*x^5 - 184*A*c^2*x^4 - 135*B*a*c*x^3 - 88*A*a*c*x^2 - 30*B*a^2*x - 24*A*a^2)*sqr t(c*x^2 + a))/x^5, -1/120*(120*A*sqrt(-c)*c^2*x^5*arctan(sqrt(-c)*x/sqrt(c *x^2 + a)) - 225*B*sqrt(-a)*c^2*x^5*arctan(sqrt(-a)/sqrt(c*x^2 + a)) - (12 0*B*c^2*x^5 - 184*A*c^2*x^4 - 135*B*a*c*x^3 - 88*A*a*c*x^2 - 30*B*a^2*x - 24*A*a^2)*sqrt(c*x^2 + a))/x^5]
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (128) = 256\).
Time = 5.73 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.10 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^6} \, dx=- \frac {A \sqrt {a} c^{2}}{x \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {A a^{2} \sqrt {c} \sqrt {\frac {a}{c x^{2}} + 1}}{5 x^{4}} - \frac {11 A a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15 x^{2}} - \frac {8 A c^{\frac {5}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{15} + A c^{\frac {5}{2}} \operatorname {asinh}{\left (\frac {\sqrt {c} x}{\sqrt {a}} \right )} - \frac {A c^{3} x}{\sqrt {a} \sqrt {1 + \frac {c x^{2}}{a}}} - \frac {15 B \sqrt {a} c^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {c} x} \right )}}{8} - \frac {B a^{3}}{4 \sqrt {c} x^{5} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {3 B a^{2} \sqrt {c}}{8 x^{3} \sqrt {\frac {a}{c x^{2}} + 1}} - \frac {B a c^{\frac {3}{2}} \sqrt {\frac {a}{c x^{2}} + 1}}{x} + \frac {7 B a c^{\frac {3}{2}}}{8 x \sqrt {\frac {a}{c x^{2}} + 1}} + \frac {B c^{\frac {5}{2}} x}{\sqrt {\frac {a}{c x^{2}} + 1}} \]
-A*sqrt(a)*c**2/(x*sqrt(1 + c*x**2/a)) - A*a**2*sqrt(c)*sqrt(a/(c*x**2) + 1)/(5*x**4) - 11*A*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/(15*x**2) - 8*A*c**(5/2 )*sqrt(a/(c*x**2) + 1)/15 + A*c**(5/2)*asinh(sqrt(c)*x/sqrt(a)) - A*c**3*x /(sqrt(a)*sqrt(1 + c*x**2/a)) - 15*B*sqrt(a)*c**2*asinh(sqrt(a)/(sqrt(c)*x ))/8 - B*a**3/(4*sqrt(c)*x**5*sqrt(a/(c*x**2) + 1)) - 3*B*a**2*sqrt(c)/(8* x**3*sqrt(a/(c*x**2) + 1)) - B*a*c**(3/2)*sqrt(a/(c*x**2) + 1)/x + 7*B*a*c **(3/2)/(8*x*sqrt(a/(c*x**2) + 1)) + B*c**(5/2)*x/sqrt(a/(c*x**2) + 1)
Time = 0.24 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.56 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^6} \, dx=\frac {2 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} A c^{3} x}{3 \, a^{2}} + \frac {\sqrt {c x^{2} + a} A c^{3} x}{a} + A c^{\frac {5}{2}} \operatorname {arsinh}\left (\frac {c x}{\sqrt {a c}}\right ) - \frac {15}{8} \, B \sqrt {a} c^{2} \operatorname {arsinh}\left (\frac {a}{\sqrt {a c} {\left | x \right |}}\right ) + \frac {15}{8} \, \sqrt {c x^{2} + a} B c^{2} + \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} B c^{2}}{8 \, a^{2}} + \frac {5 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} B c^{2}}{8 \, a} - \frac {8 \, {\left (c x^{2} + a\right )}^{\frac {5}{2}} A c^{2}}{15 \, a^{2} x} - \frac {3 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} B c}{8 \, a^{2} x^{2}} - \frac {2 \, {\left (c x^{2} + a\right )}^{\frac {7}{2}} A c}{15 \, a^{2} x^{3}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} B}{4 \, a x^{4}} - \frac {{\left (c x^{2} + a\right )}^{\frac {7}{2}} A}{5 \, a x^{5}} \]
2/3*(c*x^2 + a)^(3/2)*A*c^3*x/a^2 + sqrt(c*x^2 + a)*A*c^3*x/a + A*c^(5/2)* arcsinh(c*x/sqrt(a*c)) - 15/8*B*sqrt(a)*c^2*arcsinh(a/(sqrt(a*c)*abs(x))) + 15/8*sqrt(c*x^2 + a)*B*c^2 + 3/8*(c*x^2 + a)^(5/2)*B*c^2/a^2 + 5/8*(c*x^ 2 + a)^(3/2)*B*c^2/a - 8/15*(c*x^2 + a)^(5/2)*A*c^2/(a^2*x) - 3/8*(c*x^2 + a)^(7/2)*B*c/(a^2*x^2) - 2/15*(c*x^2 + a)^(7/2)*A*c/(a^2*x^3) - 1/4*(c*x^ 2 + a)^(7/2)*B/(a*x^4) - 1/5*(c*x^2 + a)^(7/2)*A/(a*x^5)
Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (114) = 228\).
Time = 0.31 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.36 \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^6} \, dx=\frac {15 \, B a c^{2} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a}} - A c^{\frac {5}{2}} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + a} \right |}\right ) + \sqrt {c x^{2} + a} B c^{2} + \frac {135 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{9} B a c^{2} + 360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{8} A a c^{\frac {5}{2}} - 150 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{7} B a^{2} c^{2} - 720 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{6} A a^{2} c^{\frac {5}{2}} + 1120 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{4} A a^{3} c^{\frac {5}{2}} + 150 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{3} B a^{4} c^{2} - 560 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} A a^{4} c^{\frac {5}{2}} - 135 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )} B a^{5} c^{2} + 184 \, A a^{5} c^{\frac {5}{2}}}{60 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + a}\right )}^{2} - a\right )}^{5}} \]
15/4*B*a*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) - A* c^(5/2)*log(abs(-sqrt(c)*x + sqrt(c*x^2 + a))) + sqrt(c*x^2 + a)*B*c^2 + 1 /60*(135*(sqrt(c)*x - sqrt(c*x^2 + a))^9*B*a*c^2 + 360*(sqrt(c)*x - sqrt(c *x^2 + a))^8*A*a*c^(5/2) - 150*(sqrt(c)*x - sqrt(c*x^2 + a))^7*B*a^2*c^2 - 720*(sqrt(c)*x - sqrt(c*x^2 + a))^6*A*a^2*c^(5/2) + 1120*(sqrt(c)*x - sqr t(c*x^2 + a))^4*A*a^3*c^(5/2) + 150*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B*a^4* c^2 - 560*(sqrt(c)*x - sqrt(c*x^2 + a))^2*A*a^4*c^(5/2) - 135*(sqrt(c)*x - sqrt(c*x^2 + a))*B*a^5*c^2 + 184*A*a^5*c^(5/2))/((sqrt(c)*x - sqrt(c*x^2 + a))^2 - a)^5
Timed out. \[ \int \frac {(A+B x) \left (a+c x^2\right )^{5/2}}{x^6} \, dx=\int \frac {{\left (c\,x^2+a\right )}^{5/2}\,\left (A+B\,x\right )}{x^6} \,d x \]