Integrand size = 18, antiderivative size = 131 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=-\frac {7 A b-5 a B}{5 a^2 b x^{5/2}}+\frac {7 A b-5 a B}{3 a^3 x^{3/2}}-\frac {b (7 A b-5 a B)}{a^4 \sqrt {x}}+\frac {A b-a B}{a b x^{5/2} (a+b x)}-\frac {b^{3/2} (7 A b-5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}} \]
1/5*(-7*A*b+5*B*a)/a^2/b/x^(5/2)+1/3*(7*A*b-5*B*a)/a^3/x^(3/2)+(A*b-B*a)/a /b/x^(5/2)/(b*x+a)-b^(3/2)*(7*A*b-5*B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a ^(9/2)-b*(7*A*b-5*B*a)/a^4/x^(1/2)
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=\frac {-105 A b^3 x^3-2 a^3 (3 A+5 B x)+5 a b^2 x^2 (-14 A+15 B x)+2 a^2 b x (7 A+25 B x)}{15 a^4 x^{5/2} (a+b x)}+\frac {b^{3/2} (-7 A b+5 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{9/2}} \]
(-105*A*b^3*x^3 - 2*a^3*(3*A + 5*B*x) + 5*a*b^2*x^2*(-14*A + 15*B*x) + 2*a ^2*b*x*(7*A + 25*B*x))/(15*a^4*x^(5/2)*(a + b*x)) + (b^(3/2)*(-7*A*b + 5*a *B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(9/2)
Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.95, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {87, 61, 61, 61, 73, 218}
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.
| \(\displaystyle \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(7 A b-5 a B) \int \frac {1}{x^{7/2} (a+b x)}dx}{2 a b}+\frac {A b-a B}{a b x^{5/2} (a+b x)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(7 A b-5 a B) \left (-\frac {b \int \frac {1}{x^{5/2} (a+b x)}dx}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a b}+\frac {A b-a B}{a b x^{5/2} (a+b x)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(7 A b-5 a B) \left (-\frac {b \left (-\frac {b \int \frac {1}{x^{3/2} (a+b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a b}+\frac {A b-a B}{a b x^{5/2} (a+b x)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(7 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a b}+\frac {A b-a B}{a b x^{5/2} (a+b x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(7 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a b}+\frac {A b-a B}{a b x^{5/2} (a+b x)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(7 A b-5 a B) \left (-\frac {b \left (-\frac {b \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{a}-\frac {2}{5 a x^{5/2}}\right )}{2 a b}+\frac {A b-a B}{a b x^{5/2} (a+b x)}\) |
(A*b - a*B)/(a*b*x^(5/2)*(a + b*x)) + ((7*A*b - 5*a*B)*(-2/(5*a*x^(5/2)) - (b*(-2/(3*a*x^(3/2)) - (b*(-2/(a*Sqrt[x]) - (2*Sqrt[b]*ArcTan[(Sqrt[b]*Sq rt[x])/Sqrt[a]])/a^(3/2)))/a))/a))/(2*a*b)
3.4.61.3.1 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] && 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)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Time = 0.48 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(-\frac {2 b^{2} \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4}}-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 A b +B a \right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 b \left (3 A b -2 B a \right )}{a^{4} \sqrt {x}}\) | \(101\) |
| default | \(-\frac {2 b^{2} \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{4}}-\frac {2 A}{5 a^{2} x^{\frac {5}{2}}}-\frac {2 \left (-2 A b +B a \right )}{3 a^{3} x^{\frac {3}{2}}}-\frac {2 b \left (3 A b -2 B a \right )}{a^{4} \sqrt {x}}\) | \(101\) |
| risch | \(-\frac {2 \left (45 A \,b^{2} x^{2}-30 B a b \,x^{2}-10 a A b x +5 a^{2} B x +3 a^{2} A \right )}{15 a^{4} x^{\frac {5}{2}}}-\frac {b^{2} \left (\frac {2 \left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (7 A b -5 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{4}}\) | \(103\) |
-2/a^4*b^2*((1/2*A*b-1/2*B*a)*x^(1/2)/(b*x+a)+1/2*(7*A*b-5*B*a)/(a*b)^(1/2 )*arctan(b*x^(1/2)/(a*b)^(1/2)))-2/5*A/a^2/x^(5/2)-2/3*(-2*A*b+B*a)/a^3/x^ (3/2)-2*b*(3*A*b-2*B*a)/a^4/x^(1/2)
Time = 0.23 (sec) , antiderivative size = 319, normalized size of antiderivative = 2.44 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=\left [-\frac {15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{30 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}, -\frac {15 \, {\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{4} + {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt {x}}{15 \, {\left (a^{4} b x^{4} + a^{5} x^{3}\right )}}\right ] \]
[-1/30*(15*((5*B*a*b^2 - 7*A*b^3)*x^4 + (5*B*a^2*b - 7*A*a*b^2)*x^3)*sqrt( -b/a)*log((b*x - 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(6*A*a^3 - 15* (5*B*a*b^2 - 7*A*b^3)*x^3 - 10*(5*B*a^2*b - 7*A*a*b^2)*x^2 + 2*(5*B*a^3 - 7*A*a^2*b)*x)*sqrt(x))/(a^4*b*x^4 + a^5*x^3), -1/15*(15*((5*B*a*b^2 - 7*A* b^3)*x^4 + (5*B*a^2*b - 7*A*a*b^2)*x^3)*sqrt(b/a)*arctan(a*sqrt(b/a)/(b*sq rt(x))) + (6*A*a^3 - 15*(5*B*a*b^2 - 7*A*b^3)*x^3 - 10*(5*B*a^2*b - 7*A*a* b^2)*x^2 + 2*(5*B*a^3 - 7*A*a^2*b)*x)*sqrt(x))/(a^4*b*x^4 + a^5*x^3)]
Leaf count of result is larger than twice the leaf count of optimal. 1017 vs. \(2 (119) = 238\).
Time = 42.15 (sec) , antiderivative size = 1017, normalized size of antiderivative = 7.76 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{9 x^{\frac {9}{2}}} - \frac {2 B}{7 x^{\frac {7}{2}}}}{b^{2}} & \text {for}\: a = 0 \\- \frac {12 A a^{3} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {28 A a^{2} b x \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 A a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 A a b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {140 A a b^{2} x^{2} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {105 A b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {105 A b^{3} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {210 A b^{3} x^{3} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {20 B a^{3} x \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {75 B a^{2} b x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {75 B a^{2} b x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {100 B a^{2} b x^{2} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {75 B a b^{2} x^{\frac {7}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} - \frac {75 B a b^{2} x^{\frac {7}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} + \frac {150 B a b^{2} x^{3} \sqrt {- \frac {a}{b}}}{30 a^{5} x^{\frac {5}{2}} \sqrt {- \frac {a}{b}} + 30 a^{4} b x^{\frac {7}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(9*x**(9/2)) - 2*B/(7*x**(7/2))), Eq(a, 0) & Eq(b, 0) ), ((-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2)))/a**2, Eq(b, 0)), ((-2*A/(9*x**( 9/2)) - 2*B/(7*x**(7/2)))/b**2, Eq(a, 0)), (-12*A*a**3*sqrt(-a/b)/(30*a**5 *x**(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt(-a/b)) + 28*A*a**2*b*x*sqrt (-a/b)/(30*a**5*x**(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt(-a/b)) - 105 *A*a*b**2*x**(5/2)*log(sqrt(x) - sqrt(-a/b))/(30*a**5*x**(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt(-a/b)) + 105*A*a*b**2*x**(5/2)*log(sqrt(x) + sqr t(-a/b))/(30*a**5*x**(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt(-a/b)) - 1 40*A*a*b**2*x**2*sqrt(-a/b)/(30*a**5*x**(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7 /2)*sqrt(-a/b)) - 105*A*b**3*x**(7/2)*log(sqrt(x) - sqrt(-a/b))/(30*a**5*x **(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt(-a/b)) + 105*A*b**3*x**(7/2)* log(sqrt(x) + sqrt(-a/b))/(30*a**5*x**(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7/2 )*sqrt(-a/b)) - 210*A*b**3*x**3*sqrt(-a/b)/(30*a**5*x**(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt(-a/b)) - 20*B*a**3*x*sqrt(-a/b)/(30*a**5*x**(5/2)* sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt(-a/b)) + 75*B*a**2*b*x**(5/2)*log(sqr t(x) - sqrt(-a/b))/(30*a**5*x**(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt( -a/b)) - 75*B*a**2*b*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(30*a**5*x**(5/2)* sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt(-a/b)) + 100*B*a**2*b*x**2*sqrt(-a/b) /(30*a**5*x**(5/2)*sqrt(-a/b) + 30*a**4*b*x**(7/2)*sqrt(-a/b)) + 75*B*a*b* *2*x**(7/2)*log(sqrt(x) - sqrt(-a/b))/(30*a**5*x**(5/2)*sqrt(-a/b) + 30...
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=-\frac {6 \, A a^{3} - 15 \, {\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{3} - 10 \, {\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{2} + 2 \, {\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x}{15 \, {\left (a^{4} b x^{\frac {7}{2}} + a^{5} x^{\frac {5}{2}}\right )}} + \frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} \]
-1/15*(6*A*a^3 - 15*(5*B*a*b^2 - 7*A*b^3)*x^3 - 10*(5*B*a^2*b - 7*A*a*b^2) *x^2 + 2*(5*B*a^3 - 7*A*a^2*b)*x)/(a^4*b*x^(7/2) + a^5*x^(5/2)) + (5*B*a*b ^2 - 7*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4)
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=\frac {{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {B a b^{2} \sqrt {x} - A b^{3} \sqrt {x}}{{\left (b x + a\right )} a^{4}} + \frac {2 \, {\left (30 \, B a b x^{2} - 45 \, A b^{2} x^{2} - 5 \, B a^{2} x + 10 \, A a b x - 3 \, A a^{2}\right )}}{15 \, a^{4} x^{\frac {5}{2}}} \]
(5*B*a*b^2 - 7*A*b^3)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^4) + (B*a*b ^2*sqrt(x) - A*b^3*sqrt(x))/((b*x + a)*a^4) + 2/15*(30*B*a*b*x^2 - 45*A*b^ 2*x^2 - 5*B*a^2*x + 10*A*a*b*x - 3*A*a^2)/(a^4*x^(5/2))
Time = 0.44 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{7/2} (a+b x)^2} \, dx=-\frac {\frac {2\,A}{5\,a}-\frac {2\,x\,\left (7\,A\,b-5\,B\,a\right )}{15\,a^2}+\frac {b^2\,x^3\,\left (7\,A\,b-5\,B\,a\right )}{a^4}+\frac {2\,b\,x^2\,\left (7\,A\,b-5\,B\,a\right )}{3\,a^3}}{a\,x^{5/2}+b\,x^{7/2}}-\frac {b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (7\,A\,b-5\,B\,a\right )}{a^{9/2}} \]