Optimal. Leaf size=77 \[ -\frac {a \sqrt {x} \sqrt {a-b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a-b x}+\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, 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^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{3/2}}+\frac {1}{2} x^{3/2} \sqrt {a-b x}-\frac {a \sqrt {x} \sqrt {a-b x}}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 223
Rubi steps
\begin {align*} \int \sqrt {x} \sqrt {a-b x} \, dx &=\frac {1}{2} x^{3/2} \sqrt {a-b x}+\frac {1}{4} a \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx\\ &=-\frac {a \sqrt {x} \sqrt {a-b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a-b x}+\frac {a^2 \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{8 b}\\ &=-\frac {a \sqrt {x} \sqrt {a-b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a-b x}+\frac {a^2 \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b}\\ &=-\frac {a \sqrt {x} \sqrt {a-b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a-b x}+\frac {a^2 \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{4 b}\\ &=-\frac {a \sqrt {x} \sqrt {a-b x}}{4 b}+\frac {1}{2} x^{3/2} \sqrt {a-b x}+\frac {a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 71, normalized size = 0.92 \begin {gather*} \frac {\sqrt {x} \sqrt {a-b x} (-a+2 b x)}{4 b}+\frac {a^2 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{4 (-b)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 4.21, size = 192, normalized size = 2.49 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-a^{\frac {7}{2}} b \text {ArcCosh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {-a+b x}{a}\right )^{\frac {3}{2}}+a^2 b^{\frac {3}{2}} \sqrt {x} \left (-a+b x\right )+3 a b^{\frac {5}{2}} x^{\frac {3}{2}} \left (a-b x\right )+2 b^{\frac {7}{2}} x^{\frac {5}{2}} \left (-a+b x\right )\right )}{4 a^{\frac {3}{2}} b^{\frac {5}{2}} \left (\frac {-a+b x}{a}\right )^{\frac {3}{2}}},\text {Abs}\left [\frac {b x}{a}\right ]>1\right \}\right \},\frac {a^2 \text {ArcSin}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{4 b^{\frac {3}{2}}}-\frac {a^{\frac {3}{2}} \sqrt {x}}{4 b \sqrt {1-\frac {b x}{a}}}+\frac {3 \sqrt {a} x^{\frac {3}{2}}}{4 \sqrt {1-\frac {b x}{a}}}-\frac {b x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1-\frac {b x}{a}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.11, size = 89, normalized size = 1.16
method | result | size |
risch | \(-\frac {\left (-2 b x +a \right ) \sqrt {x}\, \sqrt {-b x +a}}{4 b}+\frac {a^{2} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{8 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(78\) |
default | \(-\frac {\sqrt {x}\, \left (-b x +a \right )^{\frac {3}{2}}}{2 b}+\frac {a \left (\sqrt {x}\, \sqrt {-b x +a}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 \sqrt {-b x +a}\, \sqrt {x}\, \sqrt {b}}\right )}{4 b}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.41, size = 95, normalized size = 1.23 \begin {gather*} -\frac {a^{2} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {3}{2}}} + \frac {\frac {\sqrt {-b x + a} a^{2} b}{\sqrt {x}} - \frac {{\left (-b x + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {3}{2}}}}{4 \, {\left (b^{3} - \frac {2 \, {\left (b x - a\right )} b^{2}}{x} + \frac {{\left (b x - a\right )}^{2} b}{x^{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 118, normalized size = 1.53 \begin {gather*} \left [-\frac {a^{2} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (2 \, b^{2} x - a b\right )} \sqrt {-b x + a} \sqrt {x}}{8 \, b^{2}}, -\frac {a^{2} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (2 \, b^{2} x - a b\right )} \sqrt {-b x + a} \sqrt {x}}{4 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.01, size = 207, normalized size = 2.69 \begin {gather*} \begin {cases} \frac {i a^{\frac {3}{2}} \sqrt {x}}{4 b \sqrt {-1 + \frac {b x}{a}}} - \frac {3 i \sqrt {a} x^{\frac {3}{2}}}{4 \sqrt {-1 + \frac {b x}{a}}} - \frac {i a^{2} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {3}{2}}} + \frac {i b x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {a^{\frac {3}{2}} \sqrt {x}}{4 b \sqrt {1 - \frac {b x}{a}}} + \frac {3 \sqrt {a} x^{\frac {3}{2}}}{4 \sqrt {1 - \frac {b x}{a}}} + \frac {a^{2} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {3}{2}}} - \frac {b x^{\frac {5}{2}}}{2 \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 [A]
time = 0.01, size = 98, normalized size = 1.27 \begin {gather*} 2 \left (2 \left (\frac {\frac {1}{32}\cdot 4 b^{2} \sqrt {x} \sqrt {x}}{b^{2}}-\frac {\frac {1}{32}\cdot 2 b a}{b^{2}}\right ) \sqrt {x} \sqrt {a-b x}-\frac {2 a^{2} \ln \left |\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right |}{16 b \sqrt {-b}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 58, normalized size = 0.75 \begin {gather*} \sqrt {x}\,\left (\frac {x}{2}-\frac {a}{4\,b}\right )\,\sqrt {a-b\,x}-\frac {a^2\,\ln \left (a-2\,b\,x+2\,\sqrt {-b}\,\sqrt {x}\,\sqrt {a-b\,x}\right )}{8\,{\left (-b\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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