Optimal. Leaf size=82 \[ \frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{2-b x}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0165851, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {50, 54, 216} \[ \frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{2-b x}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 50
Rule 54
Rule 216
Rubi steps
\begin{align*} \int \frac{(2-b x)^{5/2}}{\sqrt{x}} \, dx &=\frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5}{3} \int \frac{(2-b x)^{3/2}}{\sqrt{x}} \, dx\\ &=\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5}{2} \int \frac{\sqrt{2-b x}}{\sqrt{x}} \, dx\\ &=\frac{5}{2} \sqrt{x} \sqrt{2-b x}+\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5}{2} \int \frac{1}{\sqrt{x} \sqrt{2-b x}} \, dx\\ &=\frac{5}{2} \sqrt{x} \sqrt{2-b x}+\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{1}{3} \sqrt{x} (2-b x)^{5/2}+5 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-b x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{5}{2} \sqrt{x} \sqrt{2-b x}+\frac{5}{6} \sqrt{x} (2-b x)^{3/2}+\frac{1}{3} \sqrt{x} (2-b x)^{5/2}+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0367934, size = 58, normalized size = 0.71 \[ \frac{1}{6} \sqrt{x} \sqrt{2-b x} \left (2 b^2 x^2-13 b x+33\right )+\frac{5 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 91, normalized size = 1.1 \begin{align*}{\frac{1}{3} \left ( -bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{5}{6} \left ( -bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5}{2}\sqrt{x}\sqrt{-bx+2}}+{\frac{5}{2}\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.84821, size = 329, normalized size = 4.01 \begin{align*} \left [\frac{{\left (2 \, b^{3} x^{2} - 13 \, b^{2} x + 33 \, b\right )} \sqrt{-b x + 2} \sqrt{x} - 15 \, \sqrt{-b} \log \left (-b x + \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right )}{6 \, b}, \frac{{\left (2 \, b^{3} x^{2} - 13 \, b^{2} x + 33 \, b\right )} \sqrt{-b x + 2} \sqrt{x} - 30 \, \sqrt{b} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{6 \, b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 9.95409, size = 209, normalized size = 2.55 \begin{align*} \begin{cases} \frac{i b^{3} x^{\frac{7}{2}}}{3 \sqrt{b x - 2}} - \frac{17 i b^{2} x^{\frac{5}{2}}}{6 \sqrt{b x - 2}} + \frac{59 i b x^{\frac{3}{2}}}{6 \sqrt{b x - 2}} - \frac{11 i \sqrt{x}}{\sqrt{b x - 2}} - \frac{5 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{b^{3} x^{\frac{7}{2}}}{3 \sqrt{- b x + 2}} + \frac{17 b^{2} x^{\frac{5}{2}}}{6 \sqrt{- b x + 2}} - \frac{59 b x^{\frac{3}{2}}}{6 \sqrt{- b x + 2}} + \frac{11 \sqrt{x}}{\sqrt{- b x + 2}} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]