Integrand size = 18, antiderivative size = 107 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=-\frac {5 A b-3 a B}{3 a^2 b x^{3/2}}+\frac {5 A b-3 a B}{a^3 \sqrt {x}}+\frac {A b-a B}{a b x^{3/2} (a+b x)}+\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
1/3*(-5*A*b+3*B*a)/a^2/b/x^(3/2)+(A*b-B*a)/a/b/x^(3/2)/(b*x+a)+(5*A*b-3*B* a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))*b^(1/2)/a^(7/2)+(5*A*b-3*B*a)/a^3/x^(1/ 2)
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=\frac {15 A b^2 x^2+a b x (10 A-9 B x)-2 a^2 (A+3 B x)}{3 a^3 x^{3/2} (a+b x)}+\frac {\sqrt {b} (5 A b-3 a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{7/2}} \]
(15*A*b^2*x^2 + a*b*x*(10*A - 9*B*x) - 2*a^2*(A + 3*B*x))/(3*a^3*x^(3/2)*( a + b*x)) + (Sqrt[b]*(5*A*b - 3*a*B)*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^ (7/2)
Time = 0.21 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {87, 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.
| \(\displaystyle \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(5 A b-3 a B) \int \frac {1}{x^{5/2} (a+b x)}dx}{2 a b}+\frac {A b-a B}{a b x^{3/2} (a+b x)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(5 A b-3 a B) \left (-\frac {b \int \frac {1}{x^{3/2} (a+b x)}dx}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a b}+\frac {A b-a B}{a b x^{3/2} (a+b x)}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(5 A b-3 a B) \left (-\frac {b \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a b}+\frac {A b-a B}{a b x^{3/2} (a+b x)}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(5 A b-3 a B) \left (-\frac {b \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a b}+\frac {A b-a B}{a b x^{3/2} (a+b x)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {(5 A b-3 a B) \left (-\frac {b \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{a}-\frac {2}{3 a x^{3/2}}\right )}{2 a b}+\frac {A b-a B}{a b x^{3/2} (a+b x)}\) |
(A*b - a*B)/(a*b*x^(3/2)*(a + b*x)) + ((5*A*b - 3*a*B)*(-2/(3*a*x^(3/2)) - (b*(-2/(a*Sqrt[x]) - (2*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2 )))/a))/(2*a*b)
3.4.60.3.1 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 = 1.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(-\frac {2 \left (-6 A b x +3 B a x +A a \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {b \left (\frac {2 \left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{3}}\) | \(77\) |
| derivativedivides | \(\frac {2 b \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}\) | \(81\) |
| default | \(\frac {2 b \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (5 A b -3 B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{3}}-\frac {2 A}{3 a^{2} x^{\frac {3}{2}}}-\frac {2 \left (-2 A b +B a \right )}{a^{3} \sqrt {x}}\) | \(81\) |
-2/3*(-6*A*b*x+3*B*a*x+A*a)/a^3/x^(3/2)+1/a^3*b*(2*(1/2*A*b-1/2*B*a)*x^(1/ 2)/(b*x+a)+(5*A*b-3*B*a)/(a*b)^(1/2)*arctan(b*x^(1/2)/(a*b)^(1/2)))
Time = 0.24 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.45 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=\left [-\frac {3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {-\frac {b}{a}} - a}{b x + a}\right ) + 2 \, {\left (2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt {x}}{6 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}, \frac {3 \, {\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{3} + {\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a \sqrt {\frac {b}{a}}}{b \sqrt {x}}\right ) - {\left (2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} + a^{4} x^{2}\right )}}\right ] \]
[-1/6*(3*((3*B*a*b - 5*A*b^2)*x^3 + (3*B*a^2 - 5*A*a*b)*x^2)*sqrt(-b/a)*lo g((b*x + 2*a*sqrt(x)*sqrt(-b/a) - a)/(b*x + a)) + 2*(2*A*a^2 + 3*(3*B*a*b - 5*A*b^2)*x^2 + 2*(3*B*a^2 - 5*A*a*b)*x)*sqrt(x))/(a^3*b*x^3 + a^4*x^2), 1/3*(3*((3*B*a*b - 5*A*b^2)*x^3 + (3*B*a^2 - 5*A*a*b)*x^2)*sqrt(b/a)*arcta n(a*sqrt(b/a)/(b*sqrt(x))) - (2*A*a^2 + 3*(3*B*a*b - 5*A*b^2)*x^2 + 2*(3*B *a^2 - 5*A*a*b)*x)*sqrt(x))/(a^3*b*x^3 + a^4*x^2)]
Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (95) = 190\).
Time = 14.76 (sec) , antiderivative size = 882, normalized size of antiderivative = 8.24 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{7 x^{\frac {7}{2}}} - \frac {2 B}{5 x^{\frac {5}{2}}}}{b^{2}} & \text {for}\: a = 0 \\- \frac {4 A a^{2} \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {15 A a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {15 A a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {20 A a b x \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {15 A b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {15 A b^{2} x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {30 A b^{2} x^{2} \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {9 B a^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {9 B a^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {12 B a^{2} x \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {9 B a b x^{\frac {5}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} + \frac {9 B a b x^{\frac {5}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} - \frac {18 B a b x^{2} \sqrt {- \frac {a}{b}}}{6 a^{4} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}} + 6 a^{3} b x^{\frac {5}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2))), Eq(a, 0) & Eq(b, 0) ), ((-2*A/(3*x**(3/2)) - 2*B/sqrt(x))/a**2, Eq(b, 0)), ((-2*A/(7*x**(7/2)) - 2*B/(5*x**(5/2)))/b**2, Eq(a, 0)), (-4*A*a**2*sqrt(-a/b)/(6*a**4*x**(3/ 2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) + 15*A*a*b*x**(3/2)*log(sqrt (x) - sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/ b)) - 15*A*a*b*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt(-a /b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) + 20*A*a*b*x*sqrt(-a/b)/(6*a**4*x**(3/ 2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) + 15*A*b**2*x**(5/2)*log(sqr t(x) - sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a /b)) - 15*A*b**2*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt( -a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) + 30*A*b**2*x**2*sqrt(-a/b)/(6*a**4* x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) - 9*B*a**2*x**(3/2)*lo g(sqrt(x) - sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sq rt(-a/b)) + 9*B*a**2*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/(6*a**4*x**(3/2)*s qrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) - 12*B*a**2*x*sqrt(-a/b)/(6*a**4 *x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) - 9*B*a*b*x**(5/2)*lo g(sqrt(x) - sqrt(-a/b))/(6*a**4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sq rt(-a/b)) + 9*B*a*b*x**(5/2)*log(sqrt(x) + sqrt(-a/b))/(6*a**4*x**(3/2)*sq rt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)) - 18*B*a*b*x**2*sqrt(-a/b)/(6*a** 4*x**(3/2)*sqrt(-a/b) + 6*a**3*b*x**(5/2)*sqrt(-a/b)), True))
Time = 0.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=-\frac {2 \, A a^{2} + 3 \, {\left (3 \, B a b - 5 \, A b^{2}\right )} x^{2} + 2 \, {\left (3 \, B a^{2} - 5 \, A a b\right )} x}{3 \, {\left (a^{3} b x^{\frac {5}{2}} + a^{4} x^{\frac {3}{2}}\right )}} - \frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} \]
-1/3*(2*A*a^2 + 3*(3*B*a*b - 5*A*b^2)*x^2 + 2*(3*B*a^2 - 5*A*a*b)*x)/(a^3* b*x^(5/2) + a^4*x^(3/2)) - (3*B*a*b - 5*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b)) /(sqrt(a*b)*a^3)
Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.79 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=-\frac {{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {B a b \sqrt {x} - A b^{2} \sqrt {x}}{{\left (b x + a\right )} a^{3}} - \frac {2 \, {\left (3 \, B a x - 6 \, A b x + A a\right )}}{3 \, a^{3} x^{\frac {3}{2}}} \]
-(3*B*a*b - 5*A*b^2)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^3) - (B*a*b* sqrt(x) - A*b^2*sqrt(x))/((b*x + a)*a^3) - 2/3*(3*B*a*x - 6*A*b*x + A*a)/( a^3*x^(3/2))
Time = 0.44 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x}{x^{5/2} (a+b x)^2} \, dx=\frac {\frac {2\,x\,\left (5\,A\,b-3\,B\,a\right )}{3\,a^2}-\frac {2\,A}{3\,a}+\frac {b\,x^2\,\left (5\,A\,b-3\,B\,a\right )}{a^3}}{a\,x^{3/2}+b\,x^{5/2}}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (5\,A\,b-3\,B\,a\right )}{a^{7/2}} \]