Optimal. Leaf size=69 \[ -\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{2 b}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {52, 56, 222}
\begin {gather*} \frac {3 \text {ArcSin}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}-\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{2 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 56
Rule 222
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {2-b x}} \, dx &=-\frac {x^{3/2} \sqrt {2-b x}}{2 b}+\frac {3 \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx}{2 b}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{2 b}+\frac {3 \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{2 b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{2 b}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=-\frac {3 \sqrt {x} \sqrt {2-b x}}{2 b^2}-\frac {x^{3/2} \sqrt {2-b x}}{2 b}+\frac {3 \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{5/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 63, normalized size = 0.91 \begin {gather*} -\frac {\sqrt {x} \sqrt {2-b x} (3+b x)}{2 b^2}-\frac {3 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{(-b)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 89, normalized size = 1.29
method | result | size |
meijerg | \(-\frac {4 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {5}{2}} \left (5 b x +15\right ) \sqrt {-\frac {b x}{2}+1}}{40 b^{2}}+\frac {3 \sqrt {\pi }\, \left (-b \right )^{\frac {5}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{4 b^{\frac {5}{2}}}\right )}{\left (-b \right )^{\frac {3}{2}} \sqrt {\pi }\, b}\) | \(73\) |
default | \(-\frac {x^{\frac {3}{2}} \sqrt {-b x +2}}{2 b}+\frac {-\frac {3 \sqrt {x}\, \sqrt {-b x +2}}{2 b}+\frac {3 \sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{2 b^{\frac {3}{2}} \sqrt {-b x +2}\, \sqrt {x}}}{b}\) | \(89\) |
risch | \(\frac {\left (b x +3\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{2 b^{2} \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {3 \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{2 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +2}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 85, normalized size = 1.23 \begin {gather*} -\frac {\frac {5 \, \sqrt {-b x + 2} b}{\sqrt {x}} + \frac {3 \, {\left (-b x + 2\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}}{b^{4} - \frac {2 \, {\left (b x - 2\right )} b^{3}}{x} + \frac {{\left (b x - 2\right )}^{2} b^{2}}{x^{2}}} - \frac {3 \, \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.49, size = 107, normalized size = 1.55 \begin {gather*} \left [-\frac {{\left (b^{2} x + 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 3 \, \sqrt {-b} \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right )}{2 \, b^{3}}, -\frac {{\left (b^{2} x + 3 \, b\right )} \sqrt {-b x + 2} \sqrt {x} + 6 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{2 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 2.48, size = 162, normalized size = 2.35 \begin {gather*} \begin {cases} - \frac {i x^{\frac {5}{2}}}{2 \sqrt {b x - 2}} - \frac {i x^{\frac {3}{2}}}{2 b \sqrt {b x - 2}} + \frac {3 i \sqrt {x}}{b^{2} \sqrt {b x - 2}} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} & \text {for}\: \left |{b x}\right | > 2 \\\frac {x^{\frac {5}{2}}}{2 \sqrt {- b x + 2}} + \frac {x^{\frac {3}{2}}}{2 b \sqrt {- b x + 2}} - \frac {3 \sqrt {x}}{b^{2} \sqrt {- b x + 2}} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^{3/2}}{\sqrt {2-b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________