Integrand size = 18, antiderivative size = 88 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx=-\frac {3 A b-a B}{a^2 b \sqrt {x}}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}-\frac {(3 A b-a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}} \]
-(3*A*b-B*a)*arctan(b^(1/2)*x^(1/2)/a^(1/2))/a^(5/2)/b^(1/2)+(-3*A*b+B*a)/ a^2/b/x^(1/2)+(A*b-B*a)/a/b/(b*x+a)/x^(1/2)
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.76 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx=\frac {-2 a A-3 A b x+a B x}{a^2 \sqrt {x} (a+b x)}+\frac {(-3 A b+a B) \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2} \sqrt {b}} \]
(-2*a*A - 3*A*b*x + a*B*x)/(a^2*Sqrt[x]*(a + b*x)) + ((-3*A*b + a*B)*ArcTa n[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(a^(5/2)*Sqrt[b])
Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {87, 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.
\(\displaystyle \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {(3 A b-a B) \int \frac {1}{x^{3/2} (a+b x)}dx}{2 a b}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {(3 A b-a B) \left (-\frac {b \int \frac {1}{\sqrt {x} (a+b x)}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a b}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {(3 A b-a B) \left (-\frac {2 b \int \frac {1}{a+b x}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{2 a b}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {(3 A b-a B) \left (-\frac {2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {x}}\right )}{2 a b}+\frac {A b-a B}{a b \sqrt {x} (a+b x)}\) |
(A*b - a*B)/(a*b*Sqrt[x]*(a + b*x)) + ((3*A*b - a*B)*(-2/(a*Sqrt[x]) - (2* Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2)))/(2*a*b)
3.4.59.3.1 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 = 64, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(64\) |
default | \(-\frac {2 \left (\frac {\left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2 A}{a^{2} \sqrt {x}}\) | \(64\) |
risch | \(-\frac {2 A}{a^{2} \sqrt {x}}-\frac {\frac {2 \left (\frac {A b}{2}-\frac {B a}{2}\right ) \sqrt {x}}{b x +a}+\frac {\left (3 A b -B a \right ) \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}}{a^{2}}\) | \(64\) |
-2/a^2*((1/2*A*b-1/2*B*a)*x^(1/2)/(b*x+a)+1/2*(3*A*b-B*a)/(a*b)^(1/2)*arct an(b*x^(1/2)/(a*b)^(1/2)))-2*A/a^2/x^(1/2)
Time = 0.25 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.44 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx=\left [\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {-a b} \log \left (\frac {b x - a + 2 \, \sqrt {-a b} \sqrt {x}}{b x + a}\right ) - 2 \, {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{2 \, {\left (a^{3} b^{2} x^{2} + a^{4} b x\right )}}, -\frac {{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{2} + {\left (B a^{2} - 3 \, A a b\right )} x\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{b \sqrt {x}}\right ) + {\left (2 \, A a^{2} b - {\left (B a^{2} b - 3 \, A a b^{2}\right )} x\right )} \sqrt {x}}{a^{3} b^{2} x^{2} + a^{4} b x}\right ] \]
[1/2*(((B*a*b - 3*A*b^2)*x^2 + (B*a^2 - 3*A*a*b)*x)*sqrt(-a*b)*log((b*x - a + 2*sqrt(-a*b)*sqrt(x))/(b*x + a)) - 2*(2*A*a^2*b - (B*a^2*b - 3*A*a*b^2 )*x)*sqrt(x))/(a^3*b^2*x^2 + a^4*b*x), -(((B*a*b - 3*A*b^2)*x^2 + (B*a^2 - 3*A*a*b)*x)*sqrt(a*b)*arctan(sqrt(a*b)/(b*sqrt(x))) + (2*A*a^2*b - (B*a^2 *b - 3*A*a*b^2)*x)*sqrt(x))/(a^3*b^2*x^2 + a^4*b*x)]
Leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (73) = 146\).
Time = 5.18 (sec) , antiderivative size = 794, normalized size of antiderivative = 9.02 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge b = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{a^{2}} & \text {for}\: b = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{b^{2}} & \text {for}\: a = 0 \\- \frac {3 A a b \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 A a b \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {4 A a b \sqrt {- \frac {a}{b}}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {3 A b^{2} x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {6 A b^{2} x \sqrt {- \frac {a}{b}}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {B a^{2} \sqrt {x} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {B a^{2} \sqrt {x} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {B a b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} - \frac {B a b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {- \frac {a}{b}} \right )}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} + \frac {2 B a b x \sqrt {- \frac {a}{b}}}{2 a^{3} b \sqrt {x} \sqrt {- \frac {a}{b}} + 2 a^{2} b^{2} x^{\frac {3}{2}} \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(b, 0) ), ((-2*A/sqrt(x) + 2*B*sqrt(x))/a**2, Eq(b, 0)), ((-2*A/(5*x**(5/2)) - 2* B/(3*x**(3/2)))/b**2, Eq(a, 0)), (-3*A*a*b*sqrt(x)*log(sqrt(x) - sqrt(-a/b ))/(2*a**3*b*sqrt(x)*sqrt(-a/b) + 2*a**2*b**2*x**(3/2)*sqrt(-a/b)) + 3*A*a *b*sqrt(x)*log(sqrt(x) + sqrt(-a/b))/(2*a**3*b*sqrt(x)*sqrt(-a/b) + 2*a**2 *b**2*x**(3/2)*sqrt(-a/b)) - 4*A*a*b*sqrt(-a/b)/(2*a**3*b*sqrt(x)*sqrt(-a/ b) + 2*a**2*b**2*x**(3/2)*sqrt(-a/b)) - 3*A*b**2*x**(3/2)*log(sqrt(x) - sq rt(-a/b))/(2*a**3*b*sqrt(x)*sqrt(-a/b) + 2*a**2*b**2*x**(3/2)*sqrt(-a/b)) + 3*A*b**2*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/(2*a**3*b*sqrt(x)*sqrt(-a/b) + 2*a**2*b**2*x**(3/2)*sqrt(-a/b)) - 6*A*b**2*x*sqrt(-a/b)/(2*a**3*b*sqrt (x)*sqrt(-a/b) + 2*a**2*b**2*x**(3/2)*sqrt(-a/b)) + B*a**2*sqrt(x)*log(sqr t(x) - sqrt(-a/b))/(2*a**3*b*sqrt(x)*sqrt(-a/b) + 2*a**2*b**2*x**(3/2)*sqr t(-a/b)) - B*a**2*sqrt(x)*log(sqrt(x) + sqrt(-a/b))/(2*a**3*b*sqrt(x)*sqrt (-a/b) + 2*a**2*b**2*x**(3/2)*sqrt(-a/b)) + B*a*b*x**(3/2)*log(sqrt(x) - s qrt(-a/b))/(2*a**3*b*sqrt(x)*sqrt(-a/b) + 2*a**2*b**2*x**(3/2)*sqrt(-a/b)) - B*a*b*x**(3/2)*log(sqrt(x) + sqrt(-a/b))/(2*a**3*b*sqrt(x)*sqrt(-a/b) + 2*a**2*b**2*x**(3/2)*sqrt(-a/b)) + 2*B*a*b*x*sqrt(-a/b)/(2*a**3*b*sqrt(x) *sqrt(-a/b) + 2*a**2*b**2*x**(3/2)*sqrt(-a/b)), True))
Time = 0.29 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx=-\frac {2 \, A a - {\left (B a - 3 \, A b\right )} x}{a^{2} b x^{\frac {3}{2}} + a^{3} \sqrt {x}} + \frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} \]
-(2*A*a - (B*a - 3*A*b)*x)/(a^2*b*x^(3/2) + a^3*sqrt(x)) + (B*a - 3*A*b)*a rctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2)
Time = 0.29 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.68 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx=\frac {{\left (B a - 3 \, A b\right )} \arctan \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {B a x - 3 \, A b x - 2 \, A a}{{\left (b x^{\frac {3}{2}} + a \sqrt {x}\right )} a^{2}} \]
(B*a - 3*A*b)*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*a^2) + (B*a*x - 3*A*b *x - 2*A*a)/((b*x^(3/2) + a*sqrt(x))*a^2)
Time = 0.53 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.74 \[ \int \frac {A+B x}{x^{3/2} (a+b x)^2} \, dx=-\frac {\frac {2\,A}{a}+\frac {x\,\left (3\,A\,b-B\,a\right )}{a^2}}{a\,\sqrt {x}+b\,x^{3/2}}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )\,\left (3\,A\,b-B\,a\right )}{a^{5/2}\,\sqrt {b}} \]