Integrand size = 18, antiderivative size = 177 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=\frac {(7 A b-10 a B) \sqrt {a+b x}}{40 a x^4}+\frac {b (7 A b-10 a B) \sqrt {a+b x}}{240 a^2 x^3}-\frac {b^2 (7 A b-10 a B) \sqrt {a+b x}}{192 a^3 x^2}+\frac {b^3 (7 A b-10 a B) \sqrt {a+b x}}{128 a^4 x}-\frac {A (a+b x)^{3/2}}{5 a x^5}-\frac {b^4 (7 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{128 a^{9/2}} \]
-1/5*A*(b*x+a)^(3/2)/a/x^5-1/128*b^4*(7*A*b-10*B*a)*arctanh((b*x+a)^(1/2)/ a^(1/2))/a^(9/2)+1/40*(7*A*b-10*B*a)*(b*x+a)^(1/2)/a/x^4+1/240*b*(7*A*b-10 *B*a)*(b*x+a)^(1/2)/a^2/x^3-1/192*b^2*(7*A*b-10*B*a)*(b*x+a)^(1/2)/a^3/x^2 +1/128*b^3*(7*A*b-10*B*a)*(b*x+a)^(1/2)/a^4/x
Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=\frac {\frac {\sqrt {a} \sqrt {a+b x} \left (105 A b^4 x^4-16 a^3 b x (3 A+5 B x)-96 a^4 (4 A+5 B x)-10 a b^3 x^3 (7 A+15 B x)+4 a^2 b^2 x^2 (14 A+25 B x)\right )}{x^5}-15 b^4 (7 A b-10 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{1920 a^{9/2}} \]
((Sqrt[a]*Sqrt[a + b*x]*(105*A*b^4*x^4 - 16*a^3*b*x*(3*A + 5*B*x) - 96*a^4 *(4*A + 5*B*x) - 10*a*b^3*x^3*(7*A + 15*B*x) + 4*a^2*b^2*x^2*(14*A + 25*B* x)))/x^5 - 15*b^4*(7*A*b - 10*a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(1920*a ^(9/2))
Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {87, 51, 52, 52, 52, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(7 A b-10 a B) \int \frac {\sqrt {a+b x}}{x^5}dx}{10 a}-\frac {A (a+b x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 51 |
| \(\displaystyle -\frac {(7 A b-10 a B) \left (\frac {1}{8} b \int \frac {1}{x^4 \sqrt {a+b x}}dx-\frac {\sqrt {a+b x}}{4 x^4}\right )}{10 a}-\frac {A (a+b x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(7 A b-10 a B) \left (\frac {1}{8} b \left (-\frac {5 b \int \frac {1}{x^3 \sqrt {a+b x}}dx}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )-\frac {\sqrt {a+b x}}{4 x^4}\right )}{10 a}-\frac {A (a+b x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(7 A b-10 a B) \left (\frac {1}{8} b \left (-\frac {5 b \left (-\frac {3 b \int \frac {1}{x^2 \sqrt {a+b x}}dx}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )-\frac {\sqrt {a+b x}}{4 x^4}\right )}{10 a}-\frac {A (a+b x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(7 A b-10 a B) \left (\frac {1}{8} b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{x \sqrt {a+b x}}dx}{2 a}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )-\frac {\sqrt {a+b x}}{4 x^4}\right )}{10 a}-\frac {A (a+b x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {(7 A b-10 a B) \left (\frac {1}{8} b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )-\frac {\sqrt {a+b x}}{4 x^4}\right )}{10 a}-\frac {A (a+b x)^{3/2}}{5 a x^5}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(7 A b-10 a B) \left (\frac {1}{8} b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b x}}{a x}\right )}{4 a}-\frac {\sqrt {a+b x}}{2 a x^2}\right )}{6 a}-\frac {\sqrt {a+b x}}{3 a x^3}\right )-\frac {\sqrt {a+b x}}{4 x^4}\right )}{10 a}-\frac {A (a+b x)^{3/2}}{5 a x^5}\) |
-1/5*(A*(a + b*x)^(3/2))/(a*x^5) - ((7*A*b - 10*a*B)*(-1/4*Sqrt[a + b*x]/x ^4 + (b*(-1/3*Sqrt[a + b*x]/(a*x^3) - (5*b*(-1/2*Sqrt[a + b*x]/(a*x^2) - ( 3*b*(-(Sqrt[a + b*x]/(a*x)) + (b*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/a^(3/2))) /(4*a)))/(6*a)))/8))/(10*a)
3.4.97.3.1 Defintions of rubi rules used
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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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]
Time = 0.53 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {7 x^{5} b^{4} \left (A b -\frac {10 B a}{7}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128}+\frac {7 \left (-\frac {5 x^{3} \left (\frac {15 B x}{7}+A \right ) b^{3} a^{\frac {3}{2}}}{4}+b^{2} x^{2} \left (\frac {25 B x}{14}+A \right ) a^{\frac {5}{2}}-\frac {6 x \left (\frac {5 B x}{3}+A \right ) b \,a^{\frac {7}{2}}}{7}+\frac {12 \left (-5 B x -4 A \right ) a^{\frac {9}{2}}}{7}+\frac {15 A \sqrt {a}\, b^{4} x^{4}}{8}\right ) \sqrt {b x +a}}{240}}{a^{\frac {9}{2}} x^{5}}\) | \(118\) |
| risch | \(-\frac {\sqrt {b x +a}\, \left (-105 A \,b^{4} x^{4}+150 B a \,b^{3} x^{4}+70 A a \,b^{3} x^{3}-100 B \,a^{2} b^{2} x^{3}-56 A \,a^{2} b^{2} x^{2}+80 B \,a^{3} b \,x^{2}+48 A \,a^{3} b x +480 B \,a^{4} x +384 A \,a^{4}\right )}{1920 x^{5} a^{4}}-\frac {b^{4} \left (7 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 a^{\frac {9}{2}}}\) | \(131\) |
| derivativedivides | \(2 b^{4} \left (-\frac {-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{4}}+\frac {7 \left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{3}}-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{2}}+\frac {\left (79 A b -58 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a}+\left (\frac {7 A b}{256}-\frac {5 B a}{128}\right ) \sqrt {b x +a}}{b^{5} x^{5}}-\frac {\left (7 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {9}{2}}}\right )\) | \(143\) |
| default | \(2 b^{4} \left (-\frac {-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {9}{2}}}{256 a^{4}}+\frac {7 \left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {7}{2}}}{384 a^{3}}-\frac {\left (7 A b -10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{30 a^{2}}+\frac {\left (79 A b -58 B a \right ) \left (b x +a \right )^{\frac {3}{2}}}{384 a}+\left (\frac {7 A b}{256}-\frac {5 B a}{128}\right ) \sqrt {b x +a}}{b^{5} x^{5}}-\frac {\left (7 A b -10 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{256 a^{\frac {9}{2}}}\right )\) | \(143\) |
7/240/a^(9/2)*(-15/8*x^5*b^4*(A*b-10/7*B*a)*arctanh((b*x+a)^(1/2)/a^(1/2)) +(-5/4*x^3*(15/7*B*x+A)*b^3*a^(3/2)+b^2*x^2*(25/14*B*x+A)*a^(5/2)-6/7*x*(5 /3*B*x+A)*b*a^(7/2)+12/7*(-5*B*x-4*A)*a^(9/2)+15/8*A*a^(1/2)*b^4*x^4)*(b*x +a)^(1/2))/x^5
Time = 0.23 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.72 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=\left [-\frac {15 \, {\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} \sqrt {a} x^{5} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (384 \, A a^{5} + 15 \, {\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{4} - 10 \, {\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt {b x + a}}{3840 \, a^{5} x^{5}}, -\frac {15 \, {\left (10 \, B a b^{4} - 7 \, A b^{5}\right )} \sqrt {-a} x^{5} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (384 \, A a^{5} + 15 \, {\left (10 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{4} - 10 \, {\left (10 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (10 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} + A a^{4} b\right )} x\right )} \sqrt {b x + a}}{1920 \, a^{5} x^{5}}\right ] \]
[-1/3840*(15*(10*B*a*b^4 - 7*A*b^5)*sqrt(a)*x^5*log((b*x - 2*sqrt(b*x + a) *sqrt(a) + 2*a)/x) + 2*(384*A*a^5 + 15*(10*B*a^2*b^3 - 7*A*a*b^4)*x^4 - 10 *(10*B*a^3*b^2 - 7*A*a^2*b^3)*x^3 + 8*(10*B*a^4*b - 7*A*a^3*b^2)*x^2 + 48* (10*B*a^5 + A*a^4*b)*x)*sqrt(b*x + a))/(a^5*x^5), -1/1920*(15*(10*B*a*b^4 - 7*A*b^5)*sqrt(-a)*x^5*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (384*A*a^5 + 15 *(10*B*a^2*b^3 - 7*A*a*b^4)*x^4 - 10*(10*B*a^3*b^2 - 7*A*a^2*b^3)*x^3 + 8* (10*B*a^4*b - 7*A*a^3*b^2)*x^2 + 48*(10*B*a^5 + A*a^4*b)*x)*sqrt(b*x + a)) /(a^5*x^5)]
Leaf count of result is larger than twice the leaf count of optimal. 357 vs. \(2 (165) = 330\).
Time = 177.45 (sec) , antiderivative size = 357, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=- \frac {A a}{5 \sqrt {b} x^{\frac {11}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {9 A \sqrt {b}}{40 x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {A b^{\frac {3}{2}}}{240 a x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {7 A b^{\frac {5}{2}}}{960 a^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {7 A b^{\frac {7}{2}}}{384 a^{3} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {7 A b^{\frac {9}{2}}}{128 a^{4} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {7 A b^{5} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{128 a^{\frac {9}{2}}} - \frac {B a}{4 \sqrt {b} x^{\frac {9}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {7 B \sqrt {b}}{24 x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {B b^{\frac {3}{2}}}{96 a x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 B b^{\frac {5}{2}}}{192 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {5 B b^{\frac {7}{2}}}{64 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {5 B b^{4} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{64 a^{\frac {7}{2}}} \]
-A*a/(5*sqrt(b)*x**(11/2)*sqrt(a/(b*x) + 1)) - 9*A*sqrt(b)/(40*x**(9/2)*sq rt(a/(b*x) + 1)) + A*b**(3/2)/(240*a*x**(7/2)*sqrt(a/(b*x) + 1)) - 7*A*b** (5/2)/(960*a**2*x**(5/2)*sqrt(a/(b*x) + 1)) + 7*A*b**(7/2)/(384*a**3*x**(3 /2)*sqrt(a/(b*x) + 1)) + 7*A*b**(9/2)/(128*a**4*sqrt(x)*sqrt(a/(b*x) + 1)) - 7*A*b**5*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(128*a**(9/2)) - B*a/(4*sqrt( b)*x**(9/2)*sqrt(a/(b*x) + 1)) - 7*B*sqrt(b)/(24*x**(7/2)*sqrt(a/(b*x) + 1 )) + B*b**(3/2)/(96*a*x**(5/2)*sqrt(a/(b*x) + 1)) - 5*B*b**(5/2)/(192*a**2 *x**(3/2)*sqrt(a/(b*x) + 1)) - 5*B*b**(7/2)/(64*a**3*sqrt(x)*sqrt(a/(b*x) + 1)) + 5*B*b**4*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/(64*a**(7/2))
Time = 0.29 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=-\frac {1}{3840} \, b^{5} {\left (\frac {2 \, {\left (15 \, {\left (10 \, B a - 7 \, A b\right )} {\left (b x + a\right )}^{\frac {9}{2}} - 70 \, {\left (10 \, B a^{2} - 7 \, A a b\right )} {\left (b x + a\right )}^{\frac {7}{2}} + 128 \, {\left (10 \, B a^{3} - 7 \, A a^{2} b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (58 \, B a^{4} - 79 \, A a^{3} b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - 15 \, {\left (10 \, B a^{5} - 7 \, A a^{4} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{5} a^{4} b - 5 \, {\left (b x + a\right )}^{4} a^{5} b + 10 \, {\left (b x + a\right )}^{3} a^{6} b - 10 \, {\left (b x + a\right )}^{2} a^{7} b + 5 \, {\left (b x + a\right )} a^{8} b - a^{9} b} + \frac {15 \, {\left (10 \, B a - 7 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \]
-1/3840*b^5*(2*(15*(10*B*a - 7*A*b)*(b*x + a)^(9/2) - 70*(10*B*a^2 - 7*A*a *b)*(b*x + a)^(7/2) + 128*(10*B*a^3 - 7*A*a^2*b)*(b*x + a)^(5/2) - 10*(58* B*a^4 - 79*A*a^3*b)*(b*x + a)^(3/2) - 15*(10*B*a^5 - 7*A*a^4*b)*sqrt(b*x + a))/((b*x + a)^5*a^4*b - 5*(b*x + a)^4*a^5*b + 10*(b*x + a)^3*a^6*b - 10* (b*x + a)^2*a^7*b + 5*(b*x + a)*a^8*b - a^9*b) + 15*(10*B*a - 7*A*b)*log(( sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a)))/(a^(9/2)*b))
Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=-\frac {\frac {15 \, {\left (10 \, B a b^{5} - 7 \, A b^{6}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {150 \, {\left (b x + a\right )}^{\frac {9}{2}} B a b^{5} - 700 \, {\left (b x + a\right )}^{\frac {7}{2}} B a^{2} b^{5} + 1280 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{3} b^{5} - 580 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{4} b^{5} - 150 \, \sqrt {b x + a} B a^{5} b^{5} - 105 \, {\left (b x + a\right )}^{\frac {9}{2}} A b^{6} + 490 \, {\left (b x + a\right )}^{\frac {7}{2}} A a b^{6} - 896 \, {\left (b x + a\right )}^{\frac {5}{2}} A a^{2} b^{6} + 790 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{3} b^{6} + 105 \, \sqrt {b x + a} A a^{4} b^{6}}{a^{4} b^{5} x^{5}}}{1920 \, b} \]
-1/1920*(15*(10*B*a*b^5 - 7*A*b^6)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a )*a^4) + (150*(b*x + a)^(9/2)*B*a*b^5 - 700*(b*x + a)^(7/2)*B*a^2*b^5 + 12 80*(b*x + a)^(5/2)*B*a^3*b^5 - 580*(b*x + a)^(3/2)*B*a^4*b^5 - 150*sqrt(b* x + a)*B*a^5*b^5 - 105*(b*x + a)^(9/2)*A*b^6 + 490*(b*x + a)^(7/2)*A*a*b^6 - 896*(b*x + a)^(5/2)*A*a^2*b^6 + 790*(b*x + a)^(3/2)*A*a^3*b^6 + 105*sqr t(b*x + a)*A*a^4*b^6)/(a^4*b^5*x^5))/b
Time = 0.51 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {a+b x} (A+B x)}{x^6} \, dx=\frac {\left (\frac {7\,A\,b^5}{128}-\frac {5\,B\,a\,b^4}{64}\right )\,\sqrt {a+b\,x}-\frac {\left (7\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{5/2}}{15\,a^2}+\frac {7\,\left (7\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{7/2}}{192\,a^3}-\frac {\left (7\,A\,b^5-10\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{9/2}}{128\,a^4}+\frac {\left (79\,A\,b^5-58\,B\,a\,b^4\right )\,{\left (a+b\,x\right )}^{3/2}}{192\,a}}{5\,a\,{\left (a+b\,x\right )}^4-5\,a^4\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^5-10\,a^2\,{\left (a+b\,x\right )}^3+10\,a^3\,{\left (a+b\,x\right )}^2+a^5}-\frac {b^4\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (7\,A\,b-10\,B\,a\right )}{128\,a^{9/2}} \]
(((7*A*b^5)/128 - (5*B*a*b^4)/64)*(a + b*x)^(1/2) - ((7*A*b^5 - 10*B*a*b^4 )*(a + b*x)^(5/2))/(15*a^2) + (7*(7*A*b^5 - 10*B*a*b^4)*(a + b*x)^(7/2))/( 192*a^3) - ((7*A*b^5 - 10*B*a*b^4)*(a + b*x)^(9/2))/(128*a^4) + ((79*A*b^5 - 58*B*a*b^4)*(a + b*x)^(3/2))/(192*a))/(5*a*(a + b*x)^4 - 5*a^4*(a + b*x ) - (a + b*x)^5 - 10*a^2*(a + b*x)^3 + 10*a^3*(a + b*x)^2 + a^5) - (b^4*at anh((a + b*x)^(1/2)/a^(1/2))*(7*A*b - 10*B*a))/(128*a^(9/2))