Optimal. Leaf size=65 \[ -\frac {6 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}+\frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}+\frac {2 x^{3/2}}{b \sqrt {2-b x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {47, 50, 54, 216} \[ \frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}-\frac {6 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}+\frac {2 x^{3/2}}{b \sqrt {2-b x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{(2-b x)^{3/2}} \, dx &=\frac {2 x^{3/2}}{b \sqrt {2-b x}}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{b}\\ &=\frac {2 x^{3/2}}{b \sqrt {2-b x}}+\frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}-\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{b^2}\\ &=\frac {2 x^{3/2}}{b \sqrt {2-b x}}+\frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}-\frac {6 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {2 x^{3/2}}{b \sqrt {2-b x}}+\frac {3 \sqrt {x} \sqrt {2-b x}}{b^2}-\frac {6 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 30, normalized size = 0.46 \[ \frac {x^{5/2} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {b x}{2}\right )}{5 \sqrt {2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 138, normalized size = 2.12 \[ \left [-\frac {3 \, {\left (b x - 2\right )} \sqrt {-b} \log \left (-b x - \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) - {\left (b^{2} x - 6 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{b^{4} x - 2 \, b^{3}}, \frac {6 \, {\left (b x - 2\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) + {\left (b^{2} x - 6 \, b\right )} \sqrt {-b x + 2} \sqrt {x}}{b^{4} x - 2 \, b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 10.41, size = 119, normalized size = 1.83 \[ -\frac {{\left (\frac {3 \, \log \left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt {-b}} - \frac {\sqrt {{\left (b x - 2\right )} b + 2 \, b} \sqrt {-b x + 2}}{b} + \frac {16 \, \sqrt {-b}}{{\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b}\right )} {\left | b \right |}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 133, normalized size = 2.05 \[ -\frac {\left (\frac {3 \arctan \left (\frac {\left (x -\frac {1}{b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+2 x}}\right )}{b^{\frac {5}{2}}}+\frac {4 \sqrt {-\left (x -\frac {2}{b}\right )^{2} b -2 x +\frac {4}{b}}}{\left (x -\frac {2}{b}\right ) b^{3}}\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {-b x +2}\, \sqrt {x}}-\frac {\left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}\, \sqrt {x}}{\sqrt {-\left (b x -2\right ) x}\, \sqrt {-b x +2}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 3.00, size = 71, normalized size = 1.09 \[ \frac {2 \, {\left (2 \, b - \frac {3 \, {\left (b x - 2\right )}}{x}\right )}}{\frac {\sqrt {-b x + 2} b^{3}}{\sqrt {x}} + \frac {{\left (-b x + 2\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}}} + \frac {6 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{3/2}}{{\left (2-b\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 3.20, size = 128, normalized size = 1.97 \[ \begin {cases} \frac {i x^{\frac {3}{2}}}{b \sqrt {b x - 2}} - \frac {6 i \sqrt {x}}{b^{2} \sqrt {b x - 2}} + \frac {6 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} & \text {for}\: \frac {\left |{b x}\right |}{2} > 1 \\- \frac {x^{\frac {3}{2}}}{b \sqrt {- b x + 2}} + \frac {6 \sqrt {x}}{b^{2} \sqrt {- b x + 2}} - \frac {6 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________