Optimal. Leaf size=34 \[ \frac{1}{\sqrt{x} \sqrt{2-b x}}-\frac{\sqrt{2-b x}}{\sqrt{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0028815, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {45, 37} \[ \frac{1}{\sqrt{x} \sqrt{2-b x}}-\frac{\sqrt{2-b x}}{\sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} (2-b x)^{3/2}} \, dx &=\frac{1}{\sqrt{x} \sqrt{2-b x}}+\int \frac{1}{x^{3/2} \sqrt{2-b x}} \, dx\\ &=\frac{1}{\sqrt{x} \sqrt{2-b x}}-\frac{\sqrt{2-b x}}{\sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0068941, size = 21, normalized size = 0.62 \[ \frac{b x-1}{\sqrt{x} \sqrt{2-b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 18, normalized size = 0.5 \begin{align*}{(bx-1){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03326, size = 38, normalized size = 1.12 \begin{align*} \frac{b \sqrt{x}}{2 \, \sqrt{-b x + 2}} - \frac{\sqrt{-b x + 2}}{2 \, \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.40884, size = 66, normalized size = 1.94 \begin{align*} -\frac{{\left (b x - 1\right )} \sqrt{-b x + 2} \sqrt{x}}{b x^{2} - 2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.51213, size = 90, normalized size = 2.65 \begin{align*} \begin{cases} \frac{\sqrt{b}}{\sqrt{-1 + \frac{2}{b x}}} - \frac{1}{\sqrt{b} x \sqrt{-1 + \frac{2}{b x}}} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\- \frac{i b^{\frac{5}{2}} x \sqrt{1 - \frac{2}{b x}}}{b^{2} x - 2 b} + \frac{i b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}}{b^{2} x - 2 b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.09955, size = 112, normalized size = 3.29 \begin{align*} -\frac{\sqrt{-b x + 2} b^{2}}{2 \, \sqrt{{\left (b x - 2\right )} b + 2 \, b}{\left | b \right |}} - \frac{2 \, \sqrt{-b} b^{2}}{{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]