Integrand size = 16, antiderivative size = 80 \[ \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx=-\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {3 a^2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{5/2}} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223, 209} \[ \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx=\frac {3 a^2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b} \]
[In]
[Out]
Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {(3 a) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{4 b} \\ & = -\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {\left (3 a^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{8 b^2} \\ & = -\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^2} \\ & = -\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^2} \\ & = -\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{5/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.94 \[ \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx=-\frac {\sqrt {x} \sqrt {a-b x} (3 a+2 b x)}{4 b^2}+\frac {3 a^2 \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right )}{2 b^{5/2}} \]
[In]
[Out]
Time = 0.09 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {\left (2 b x +3 a \right ) \sqrt {x}\, \sqrt {-b x +a}}{4 b^{2}}+\frac {3 a^{2} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{8 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(80\) |
default | \(-\frac {x^{\frac {3}{2}} \sqrt {-b x +a}}{2 b}+\frac {3 a \left (-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\right )}{4 b}\) | \(93\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.49 \[ \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx=\left [-\frac {3 \, a^{2} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x + 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{8 \, b^{3}}, -\frac {3 \, a^{2} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (2 \, b^{2} x + 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{4 \, b^{3}}\right ] \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 4.53 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.68 \[ \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx=\begin {cases} \frac {3 i a^{\frac {3}{2}} \sqrt {x}}{4 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i \sqrt {a} x^{\frac {3}{2}}}{4 b \sqrt {-1 + \frac {b x}{a}}} - \frac {3 i a^{2} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}}} - \frac {i x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {3 a^{\frac {3}{2}} \sqrt {x}}{4 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {\sqrt {a} x^{\frac {3}{2}}}{4 b \sqrt {1 - \frac {b x}{a}}} + \frac {3 a^{2} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.22 \[ \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx=-\frac {3 \, a^{2} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {5}{2}}} - \frac {\frac {5 \, \sqrt {-b x + a} a^{2} b}{\sqrt {x}} + \frac {3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {3}{2}}}}{4 \, {\left (b^{4} - \frac {2 \, {\left (b x - a\right )} b^{3}}{x} + \frac {{\left (b x - a\right )}^{2} b^{2}}{x^{2}}\right )}} \]
[In]
[Out]
none
Time = 78.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.06 \[ \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx=\frac {{\left (\frac {3 \, a^{2} b \log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} - \sqrt {{\left (b x - a\right )} b + a b} {\left (2 \, b x + 3 \, a\right )} \sqrt {-b x + a}\right )} {\left | b \right |}}{4 \, b^{4}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx=\int \frac {x^{3/2}}{\sqrt {a-b\,x}} \,d x \]
[In]
[Out]