Optimal. Leaf size=84 \[ -\frac {\sqrt {x} \sqrt {2-b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {1}{3} x^{3/2} (2-b x)^{3/2}+\frac {\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 84, normalized size of antiderivative = 1.00, 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 = {52, 56, 222}
\begin {gather*} \frac {\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}+\frac {1}{3} x^{3/2} (2-b x)^{3/2}+\frac {1}{2} x^{3/2} \sqrt {2-b x}-\frac {\sqrt {x} \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 \sqrt {x} (2-b x)^{3/2} \, dx &=\frac {1}{3} x^{3/2} (2-b x)^{3/2}+\int \sqrt {x} \sqrt {2-b x} \, dx\\ &=\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {1}{3} x^{3/2} (2-b x)^{3/2}+\frac {1}{2} \int \frac {\sqrt {x}}{\sqrt {2-b x}} \, dx\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {1}{3} x^{3/2} (2-b x)^{3/2}+\frac {\int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx}{2 b}\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {1}{3} x^{3/2} (2-b x)^{3/2}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right )}{b}\\ &=-\frac {\sqrt {x} \sqrt {2-b x}}{2 b}+\frac {1}{2} x^{3/2} \sqrt {2-b x}+\frac {1}{3} x^{3/2} (2-b x)^{3/2}+\frac {\sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )}{b^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.09, size = 71, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {x} \sqrt {2-b x} \left (3-7 b x+2 b^2 x^2\right )}{6 b}+\frac {\log \left (-\sqrt {-b} \sqrt {x}+\sqrt {2-b x}\right )}{(-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.11, size = 94, normalized size = 1.12
method | result | size |
meijerg | \(-\frac {6 \left (-\frac {\sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (-b \right )^{\frac {3}{2}} \left (2 x^{2} b^{2}-7 b x +3\right ) \sqrt {-\frac {b x}{2}+1}}{36 b}+\frac {\sqrt {\pi }\, \left (-b \right )^{\frac {3}{2}} \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{6 b^{\frac {3}{2}}}\right )}{\sqrt {-b}\, \sqrt {\pi }\, b}\) | \(81\) |
default | \(\frac {x^{\frac {3}{2}} \left (-b x +2\right )^{\frac {3}{2}}}{3}+\frac {x^{\frac {3}{2}} \sqrt {-b x +2}}{2}-\frac {\sqrt {x}\, \sqrt {-b x +2}}{2 b}+\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{2 b^{\frac {3}{2}} \sqrt {-b x +2}\, \sqrt {x}}\) | \(94\) |
risch | \(\frac {\left (2 x^{2} b^{2}-7 b x +3\right ) \sqrt {x}\, \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{6 b \sqrt {-x \left (b x -2\right )}\, \sqrt {-b x +2}}+\frac {\sqrt {\left (-b x +2\right ) x}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-x^{2} b +2 x}}\right )}{2 b^{\frac {3}{2}} \sqrt {-b x +2}\, \sqrt {x}}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.34, size = 115, normalized size = 1.37 \begin {gather*} \frac {\frac {3 \, \sqrt {-b x + 2} b^{2}}{\sqrt {x}} + \frac {8 \, {\left (-b x + 2\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (-b x + 2\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}}{3 \, {\left (b^{4} - \frac {3 \, {\left (b x - 2\right )} b^{3}}{x} + \frac {3 \, {\left (b x - 2\right )}^{2} b^{2}}{x^{2}} - \frac {{\left (b x - 2\right )}^{3} b}{x^{3}}\right )}} - \frac {\arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.32, size = 125, normalized size = 1.49 \begin {gather*} \left [-\frac {{\left (2 \, b^{3} x^{2} - 7 \, 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 )}{6 \, b^{2}}, -\frac {{\left (2 \, b^{3} x^{2} - 7 \, 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 )}{6 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 3.58, size = 197, normalized size = 2.35 \begin {gather*} \begin {cases} - \frac {i b^{2} x^{\frac {7}{2}}}{3 \sqrt {b x - 2}} + \frac {11 i b x^{\frac {5}{2}}}{6 \sqrt {b x - 2}} - \frac {17 i x^{\frac {3}{2}}}{6 \sqrt {b x - 2}} + \frac {i \sqrt {x}}{b \sqrt {b x - 2}} - \frac {i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {3}{2}}} & \text {for}\: \left |{b x}\right | > 2 \\\frac {b^{2} x^{\frac {7}{2}}}{3 \sqrt {- b x + 2}} - \frac {11 b x^{\frac {5}{2}}}{6 \sqrt {- b x + 2}} + \frac {17 x^{\frac {3}{2}}}{6 \sqrt {- b x + 2}} - \frac {\sqrt {x}}{b \sqrt {- b x + 2}} + \frac {\operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )}}{b^{\frac {3}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 245 vs.
\(2 (59) = 118\).
time = 3.38, size = 389, normalized size = 4.63 \begin {gather*} \frac {-\frac {2 b^{2} \left |b\right | \left (2 \left (\left (\frac {\frac {1}{144}\cdot 12 b^{5} \sqrt {-b x+2} \sqrt {-b x+2}}{b^{7}}-\frac {\frac {1}{144}\cdot 78 b^{5}}{b^{7}}\right ) \sqrt {-b x+2} \sqrt {-b x+2}+\frac {\frac {1}{144}\cdot 198 b^{5}}{b^{7}}\right ) \sqrt {-b x+2} \sqrt {-b \left (-b x+2\right )+2 b}-\frac {5 \ln \left |\sqrt {-b \left (-b x+2\right )+2 b}-\sqrt {-b} \sqrt {-b x+2}\right |}{2 b \sqrt {-b}}\right )}{b^{2}}-\frac {8 b \left |b\right | \left (2 \left (\frac {1}{8} \sqrt {-b x+2} \sqrt {-b x+2}-\frac {5}{8}\right ) \sqrt {-b x+2} \sqrt {-b \left (-b x+2\right )+2 b}+\frac {6 b \ln \left |\sqrt {-b \left (-b x+2\right )+2 b}-\sqrt {-b} \sqrt {-b x+2}\right |}{4 \sqrt {-b}}\right )}{b^{2} b}-\frac {8 \left |b\right | \left (\frac {1}{2} \sqrt {-b x+2} \sqrt {-b \left (-b x+2\right )+2 b}-\frac {2 b \ln \left |\sqrt {-b \left (-b x+2\right )+2 b}-\sqrt {-b} \sqrt {-b x+2}\right |}{2 \sqrt {-b}}\right )}{b^{2}}}{b} \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 \sqrt {x}\,{\left (2-b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________