Integrand size = 18, antiderivative size = 202 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\frac {35 b^3 (11 A b-8 a B)}{64 a^5 (a+b x)^{3/2}}-\frac {A}{4 a x^4 (a+b x)^{3/2}}+\frac {11 A b-8 a B}{24 a^2 x^3 (a+b x)^{3/2}}-\frac {3 b (11 A b-8 a B)}{32 a^3 x^2 (a+b x)^{3/2}}+\frac {21 b^2 (11 A b-8 a B)}{64 a^4 x (a+b x)^{3/2}}+\frac {105 b^3 (11 A b-8 a B)}{64 a^6 \sqrt {a+b x}}-\frac {105 b^3 (11 A b-8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{64 a^{13/2}} \]
35/64*b^3*(11*A*b-8*B*a)/a^5/(b*x+a)^(3/2)-1/4*A/a/x^4/(b*x+a)^(3/2)+1/24* (11*A*b-8*B*a)/a^2/x^3/(b*x+a)^(3/2)-3/32*b*(11*A*b-8*B*a)/a^3/x^2/(b*x+a) ^(3/2)+21/64*b^2*(11*A*b-8*B*a)/a^4/x/(b*x+a)^(3/2)-105/64*b^3*(11*A*b-8*B *a)*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(13/2)+105/64*b^3*(11*A*b-8*B*a)/a^6/ (b*x+a)^(1/2)
Time = 0.32 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.75 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\frac {\frac {\sqrt {a} \left (3465 A b^5 x^5+21 a^2 b^3 x^3 (33 A-160 B x)+420 a b^4 x^4 (11 A-6 B x)-16 a^5 (3 A+4 B x)+8 a^4 b x (11 A+18 B x)-18 a^3 b^2 x^2 (11 A+28 B x)\right )}{x^4 (a+b x)^{3/2}}+315 b^3 (-11 A b+8 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{192 a^{13/2}} \]
((Sqrt[a]*(3465*A*b^5*x^5 + 21*a^2*b^3*x^3*(33*A - 160*B*x) + 420*a*b^4*x^ 4*(11*A - 6*B*x) - 16*a^5*(3*A + 4*B*x) + 8*a^4*b*x*(11*A + 18*B*x) - 18*a ^3*b^2*x^2*(11*A + 28*B*x)))/(x^4*(a + b*x)^(3/2)) + 315*b^3*(-11*A*b + 8* a*B)*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(192*a^(13/2))
Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {87, 52, 52, 52, 61, 61, 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.
| \(\displaystyle \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {(11 A b-8 a B) \int \frac {1}{x^4 (a+b x)^{5/2}}dx}{8 a}-\frac {A}{4 a x^4 (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(11 A b-8 a B) \left (-\frac {3 b \int \frac {1}{x^3 (a+b x)^{5/2}}dx}{2 a}-\frac {1}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(11 A b-8 a B) \left (-\frac {3 b \left (-\frac {7 b \int \frac {1}{x^2 (a+b x)^{5/2}}dx}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {(11 A b-8 a B) \left (-\frac {3 b \left (-\frac {7 b \left (-\frac {5 b \int \frac {1}{x (a+b x)^{5/2}}dx}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {(11 A b-8 a B) \left (-\frac {3 b \left (-\frac {7 b \left (-\frac {5 b \left (\frac {\int \frac {1}{x (a+b x)^{3/2}}dx}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {(11 A b-8 a B) \left (-\frac {3 b \left (-\frac {7 b \left (-\frac {5 b \left (\frac {\frac {\int \frac {1}{x \sqrt {a+b x}}dx}{a}+\frac {2}{a \sqrt {a+b x}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {(11 A b-8 a B) \left (-\frac {3 b \left (-\frac {7 b \left (-\frac {5 b \left (\frac {\frac {2 \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{a b}+\frac {2}{a \sqrt {a+b x}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 (a+b x)^{3/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {(11 A b-8 a B) \left (-\frac {3 b \left (-\frac {7 b \left (-\frac {5 b \left (\frac {\frac {2}{a \sqrt {a+b x}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{a^{3/2}}}{a}+\frac {2}{3 a (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{a x (a+b x)^{3/2}}\right )}{4 a}-\frac {1}{2 a x^2 (a+b x)^{3/2}}\right )}{2 a}-\frac {1}{3 a x^3 (a+b x)^{3/2}}\right )}{8 a}-\frac {A}{4 a x^4 (a+b x)^{3/2}}\) |
-1/4*A/(a*x^4*(a + b*x)^(3/2)) - ((11*A*b - 8*a*B)*(-1/3*1/(a*x^3*(a + b*x )^(3/2)) - (3*b*(-1/2*1/(a*x^2*(a + b*x)^(3/2)) - (7*b*(-(1/(a*x*(a + b*x) ^(3/2))) - (5*b*(2/(3*a*(a + b*x)^(3/2)) + (2/(a*Sqrt[a + b*x]) - (2*ArcTa nh[Sqrt[a + b*x]/Sqrt[a]])/a^(3/2))/a))/(2*a)))/(4*a)))/(2*a)))/(8*a)
3.5.53.3.1 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 + 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] :> 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] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
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 = 0.59 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.68
| method | result | size |
| pseudoelliptic | \(-\frac {33 \left (\frac {35 x^{4} \left (A b -\frac {8 B a}{11}\right ) b^{3} \left (b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{2}+\left (\frac {32 B x}{99}+\frac {8 A}{33}\right ) a^{\frac {11}{2}}+x b \left (-\frac {70 x^{3} \left (-\frac {6 B x}{11}+A \right ) b^{3} a^{\frac {3}{2}}}{3}-\frac {7 x^{2} \left (-\frac {160 B x}{33}+A \right ) b^{2} a^{\frac {5}{2}}}{2}+b x \left (\frac {28 B x}{11}+A \right ) a^{\frac {7}{2}}+\left (-\frac {8 B x}{11}-\frac {4 A}{9}\right ) a^{\frac {9}{2}}-\frac {35 A \sqrt {a}\, b^{4} x^{4}}{2}\right )\right )}{32 a^{\frac {13}{2}} \left (b x +a \right )^{\frac {3}{2}} x^{4}}\) | \(138\) |
| risch | \(-\frac {\sqrt {b x +a}\, \left (-1545 A \,b^{3} x^{3}+984 B a \,b^{2} x^{3}+518 a A \,b^{2} x^{2}-272 B \,a^{2} b \,x^{2}-184 a^{2} A b x +64 a^{3} B x +48 a^{3} A \right )}{192 a^{6} x^{4}}+\frac {b^{3} \left (-\frac {2 \left (-640 A b +512 B a \right )}{\sqrt {b x +a}}+\frac {256 a \left (A b -B a \right )}{3 \left (b x +a \right )^{\frac {3}{2}}}-\frac {2 \left (1155 A b -840 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{128 a^{6}}\) | \(149\) |
| derivativedivides | \(2 b^{3} \left (-\frac {\frac {\left (-\frac {515 A b}{128}+\frac {41 B a}{16}\right ) \left (b x +a \right )^{\frac {7}{2}}+\left (\frac {5153}{384} a b A -\frac {403}{48} a^{2} B \right ) \left (b x +a \right )^{\frac {5}{2}}+\left (-\frac {5855}{384} a^{2} b A +\frac {445}{48} a^{3} B \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {765}{128} A \,a^{3} b -\frac {55}{16} B \,a^{4}\right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {105 \left (11 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{6}}-\frac {-5 A b +4 B a}{a^{6} \sqrt {b x +a}}-\frac {-A b +B a}{3 a^{5} \left (b x +a \right )^{\frac {3}{2}}}\right )\) | \(169\) |
| default | \(2 b^{3} \left (-\frac {\frac {\left (-\frac {515 A b}{128}+\frac {41 B a}{16}\right ) \left (b x +a \right )^{\frac {7}{2}}+\left (\frac {5153}{384} a b A -\frac {403}{48} a^{2} B \right ) \left (b x +a \right )^{\frac {5}{2}}+\left (-\frac {5855}{384} a^{2} b A +\frac {445}{48} a^{3} B \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {765}{128} A \,a^{3} b -\frac {55}{16} B \,a^{4}\right ) \sqrt {b x +a}}{b^{4} x^{4}}+\frac {105 \left (11 A b -8 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{128 \sqrt {a}}}{a^{6}}-\frac {-5 A b +4 B a}{a^{6} \sqrt {b x +a}}-\frac {-A b +B a}{3 a^{5} \left (b x +a \right )^{\frac {3}{2}}}\right )\) | \(169\) |
-33/32/a^(13/2)/(b*x+a)^(3/2)*(35/2*x^4*(A*b-8/11*B*a)*b^3*(b*x+a)^(3/2)*a rctanh((b*x+a)^(1/2)/a^(1/2))+(32/99*B*x+8/33*A)*a^(11/2)+x*b*(-70/3*x^3*( -6/11*B*x+A)*b^3*a^(3/2)-7/2*x^2*(-160/33*B*x+A)*b^2*a^(5/2)+b*x*(28/11*B* x+A)*a^(7/2)+(-8/11*B*x-4/9*A)*a^(9/2)-35/2*A*a^(1/2)*b^4*x^4))/x^4
Time = 0.24 (sec) , antiderivative size = 495, normalized size of antiderivative = 2.45 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\left [-\frac {315 \, {\left ({\left (8 \, B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 2 \, {\left (8 \, B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + {\left (8 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4}\right )} \sqrt {a} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (48 \, A a^{6} + 315 \, {\left (8 \, B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 420 \, {\left (8 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 63 \, {\left (8 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} - 18 \, {\left (8 \, B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{384 \, {\left (a^{7} b^{2} x^{6} + 2 \, a^{8} b x^{5} + a^{9} x^{4}\right )}}, -\frac {315 \, {\left ({\left (8 \, B a b^{5} - 11 \, A b^{6}\right )} x^{6} + 2 \, {\left (8 \, B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + {\left (8 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (48 \, A a^{6} + 315 \, {\left (8 \, B a^{2} b^{4} - 11 \, A a b^{5}\right )} x^{5} + 420 \, {\left (8 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4}\right )} x^{4} + 63 \, {\left (8 \, B a^{4} b^{2} - 11 \, A a^{3} b^{3}\right )} x^{3} - 18 \, {\left (8 \, B a^{5} b - 11 \, A a^{4} b^{2}\right )} x^{2} + 8 \, {\left (8 \, B a^{6} - 11 \, A a^{5} b\right )} x\right )} \sqrt {b x + a}}{192 \, {\left (a^{7} b^{2} x^{6} + 2 \, a^{8} b x^{5} + a^{9} x^{4}\right )}}\right ] \]
[-1/384*(315*((8*B*a*b^5 - 11*A*b^6)*x^6 + 2*(8*B*a^2*b^4 - 11*A*a*b^5)*x^ 5 + (8*B*a^3*b^3 - 11*A*a^2*b^4)*x^4)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*s qrt(a) + 2*a)/x) + 2*(48*A*a^6 + 315*(8*B*a^2*b^4 - 11*A*a*b^5)*x^5 + 420* (8*B*a^3*b^3 - 11*A*a^2*b^4)*x^4 + 63*(8*B*a^4*b^2 - 11*A*a^3*b^3)*x^3 - 1 8*(8*B*a^5*b - 11*A*a^4*b^2)*x^2 + 8*(8*B*a^6 - 11*A*a^5*b)*x)*sqrt(b*x + a))/(a^7*b^2*x^6 + 2*a^8*b*x^5 + a^9*x^4), -1/192*(315*((8*B*a*b^5 - 11*A* b^6)*x^6 + 2*(8*B*a^2*b^4 - 11*A*a*b^5)*x^5 + (8*B*a^3*b^3 - 11*A*a^2*b^4) *x^4)*sqrt(-a)*arctan(sqrt(b*x + a)*sqrt(-a)/a) + (48*A*a^6 + 315*(8*B*a^2 *b^4 - 11*A*a*b^5)*x^5 + 420*(8*B*a^3*b^3 - 11*A*a^2*b^4)*x^4 + 63*(8*B*a^ 4*b^2 - 11*A*a^3*b^3)*x^3 - 18*(8*B*a^5*b - 11*A*a^4*b^2)*x^2 + 8*(8*B*a^6 - 11*A*a^5*b)*x)*sqrt(b*x + a))/(a^7*b^2*x^6 + 2*a^8*b*x^5 + a^9*x^4)]
Timed out. \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.18 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=-\frac {1}{384} \, b^{4} {\left (\frac {2 \, {\left (128 \, B a^{6} - 128 \, A a^{5} b + 315 \, {\left (8 \, B a - 11 \, A b\right )} {\left (b x + a\right )}^{5} - 1155 \, {\left (8 \, B a^{2} - 11 \, A a b\right )} {\left (b x + a\right )}^{4} + 1533 \, {\left (8 \, B a^{3} - 11 \, A a^{2} b\right )} {\left (b x + a\right )}^{3} - 837 \, {\left (8 \, B a^{4} - 11 \, A a^{3} b\right )} {\left (b x + a\right )}^{2} + 128 \, {\left (8 \, B a^{5} - 11 \, A a^{4} b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {11}{2}} a^{6} b - 4 \, {\left (b x + a\right )}^{\frac {9}{2}} a^{7} b + 6 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{8} b - 4 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{9} b + {\left (b x + a\right )}^{\frac {3}{2}} a^{10} b} + \frac {315 \, {\left (8 \, B a - 11 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {13}{2}} b}\right )} \]
-1/384*b^4*(2*(128*B*a^6 - 128*A*a^5*b + 315*(8*B*a - 11*A*b)*(b*x + a)^5 - 1155*(8*B*a^2 - 11*A*a*b)*(b*x + a)^4 + 1533*(8*B*a^3 - 11*A*a^2*b)*(b*x + a)^3 - 837*(8*B*a^4 - 11*A*a^3*b)*(b*x + a)^2 + 128*(8*B*a^5 - 11*A*a^4 *b)*(b*x + a))/((b*x + a)^(11/2)*a^6*b - 4*(b*x + a)^(9/2)*a^7*b + 6*(b*x + a)^(7/2)*a^8*b - 4*(b*x + a)^(5/2)*a^9*b + (b*x + a)^(3/2)*a^10*b) + 315 *(8*B*a - 11*A*b)*log((sqrt(b*x + a) - sqrt(a))/(sqrt(b*x + a) + sqrt(a))) /(a^(13/2)*b))
Time = 0.29 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=-\frac {105 \, {\left (8 \, B a b^{3} - 11 \, A b^{4}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{64 \, \sqrt {-a} a^{6}} - \frac {2 \, {\left (12 \, {\left (b x + a\right )} B a b^{3} + B a^{2} b^{3} - 15 \, {\left (b x + a\right )} A b^{4} - A a b^{4}\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6}} - \frac {984 \, {\left (b x + a\right )}^{\frac {7}{2}} B a b^{3} - 3224 \, {\left (b x + a\right )}^{\frac {5}{2}} B a^{2} b^{3} + 3560 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{3} b^{3} - 1320 \, \sqrt {b x + a} B a^{4} b^{3} - 1545 \, {\left (b x + a\right )}^{\frac {7}{2}} A b^{4} + 5153 \, {\left (b x + a\right )}^{\frac {5}{2}} A a b^{4} - 5855 \, {\left (b x + a\right )}^{\frac {3}{2}} A a^{2} b^{4} + 2295 \, \sqrt {b x + a} A a^{3} b^{4}}{192 \, a^{6} b^{4} x^{4}} \]
-105/64*(8*B*a*b^3 - 11*A*b^4)*arctan(sqrt(b*x + a)/sqrt(-a))/(sqrt(-a)*a^ 6) - 2/3*(12*(b*x + a)*B*a*b^3 + B*a^2*b^3 - 15*(b*x + a)*A*b^4 - A*a*b^4) /((b*x + a)^(3/2)*a^6) - 1/192*(984*(b*x + a)^(7/2)*B*a*b^3 - 3224*(b*x + a)^(5/2)*B*a^2*b^3 + 3560*(b*x + a)^(3/2)*B*a^3*b^3 - 1320*sqrt(b*x + a)*B *a^4*b^3 - 1545*(b*x + a)^(7/2)*A*b^4 + 5153*(b*x + a)^(5/2)*A*a*b^4 - 585 5*(b*x + a)^(3/2)*A*a^2*b^4 + 2295*sqrt(b*x + a)*A*a^3*b^4)/(a^6*b^4*x^4)
Time = 0.53 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x}{x^5 (a+b x)^{5/2}} \, dx=\frac {\frac {2\,\left (A\,b^4-B\,a\,b^3\right )}{3\,a}+\frac {2\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,\left (a+b\,x\right )}{3\,a^2}-\frac {279\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^2}{64\,a^3}+\frac {511\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^3}{64\,a^4}-\frac {385\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^4}{64\,a^5}+\frac {105\,\left (11\,A\,b^4-8\,B\,a\,b^3\right )\,{\left (a+b\,x\right )}^5}{64\,a^6}}{{\left (a+b\,x\right )}^{11/2}-4\,a\,{\left (a+b\,x\right )}^{9/2}+a^4\,{\left (a+b\,x\right )}^{3/2}-4\,a^3\,{\left (a+b\,x\right )}^{5/2}+6\,a^2\,{\left (a+b\,x\right )}^{7/2}}-\frac {105\,b^3\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (11\,A\,b-8\,B\,a\right )}{64\,a^{13/2}} \]
((2*(A*b^4 - B*a*b^3))/(3*a) + (2*(11*A*b^4 - 8*B*a*b^3)*(a + b*x))/(3*a^2 ) - (279*(11*A*b^4 - 8*B*a*b^3)*(a + b*x)^2)/(64*a^3) + (511*(11*A*b^4 - 8 *B*a*b^3)*(a + b*x)^3)/(64*a^4) - (385*(11*A*b^4 - 8*B*a*b^3)*(a + b*x)^4) /(64*a^5) + (105*(11*A*b^4 - 8*B*a*b^3)*(a + b*x)^5)/(64*a^6))/((a + b*x)^ (11/2) - 4*a*(a + b*x)^(9/2) + a^4*(a + b*x)^(3/2) - 4*a^3*(a + b*x)^(5/2) + 6*a^2*(a + b*x)^(7/2)) - (105*b^3*atanh((a + b*x)^(1/2)/a^(1/2))*(11*A* b - 8*B*a))/(64*a^(13/2))