Optimal. Leaf size=102 \[ -\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b^2}-\frac {a x^{3/2} \sqrt {a-b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a-b x}+\frac {a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {52, 65, 223,
209} \begin {gather*} \frac {a^3 \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{5/2}}-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b^2}-\frac {a x^{3/2} \sqrt {a-b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a-b x} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps
\begin {align*} \int x^{3/2} \sqrt {a-b x} \, dx &=\frac {1}{3} x^{5/2} \sqrt {a-b x}+\frac {1}{6} a \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx\\ &=-\frac {a x^{3/2} \sqrt {a-b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a-b x}+\frac {a^2 \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{8 b}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b^2}-\frac {a x^{3/2} \sqrt {a-b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a-b x}+\frac {a^3 \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{16 b^2}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b^2}-\frac {a x^{3/2} \sqrt {a-b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a-b x}+\frac {a^3 \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{8 b^2}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b^2}-\frac {a x^{3/2} \sqrt {a-b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a-b x}+\frac {a^3 \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^2}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b^2}-\frac {a x^{3/2} \sqrt {a-b x}}{12 b}+\frac {1}{3} x^{5/2} \sqrt {a-b x}+\frac {a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 81, normalized size = 0.79 \begin {gather*} \frac {1}{24} \left (\frac {\sqrt {x} \sqrt {a-b x} \left (-3 a^2-2 a b x+8 b^2 x^2\right )}{b^2}-\frac {3 a^3 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{(-b)^{5/2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 112, normalized size = 1.10
method | result | size |
risch | \(-\frac {\left (-8 x^{2} b^{2}+2 a b x +3 a^{2}\right ) \sqrt {x}\, \sqrt {-b x +a}}{24 b^{2}}+\frac {a^{3} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{16 b^{\frac {5}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(91\) |
default | \(-\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {3}{2}}}{3 b}+\frac {a \left (-\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {3}{2}}}{2 b}+\frac {a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4 b}\right )}{2 b}\) | \(112\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 135, normalized size = 1.32 \begin {gather*} -\frac {a^{3} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{\frac {5}{2}}} + \frac {\frac {3 \, \sqrt {-b x + a} a^{3} b^{2}}{\sqrt {x}} - \frac {8 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{3} b}{x^{\frac {3}{2}}} - \frac {3 \, {\left (-b x + a\right )}^{\frac {5}{2}} a^{3}}{x^{\frac {5}{2}}}}{24 \, {\left (b^{5} - \frac {3 \, {\left (b x - a\right )} b^{4}}{x} + \frac {3 \, {\left (b x - a\right )}^{2} b^{3}}{x^{2}} - \frac {{\left (b x - a\right )}^{3} b^{2}}{x^{3}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 142, normalized size = 1.39 \begin {gather*} \left [-\frac {3 \, a^{3} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (8 \, b^{3} x^{2} - 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{48 \, b^{3}}, -\frac {3 \, a^{3} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (8 \, b^{3} x^{2} - 2 \, a b^{2} x - 3 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{24 \, b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 5.56, size = 260, normalized size = 2.55 \begin {gather*} \begin {cases} \frac {i a^{\frac {5}{2}} \sqrt {x}}{8 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b \sqrt {-1 + \frac {b x}{a}}} - \frac {5 i \sqrt {a} x^{\frac {5}{2}}}{12 \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{3} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} + \frac {i b x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {a^{\frac {5}{2}} \sqrt {x}}{8 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 b \sqrt {1 - \frac {b x}{a}}} + \frac {5 \sqrt {a} x^{\frac {5}{2}}}{12 \sqrt {1 - \frac {b x}{a}}} + \frac {a^{3} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{8 b^{\frac {5}{2}}} - \frac {b x^{\frac {7}{2}}}{3 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^{3/2}\,\sqrt {a-b\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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