Optimal. Leaf size=99 \[ -\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 99, 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 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{3/2}}-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2} \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} (a-b x)^{3/2} \, dx &=\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {1}{2} a \int \sqrt {x} \sqrt {a-b x} \, dx\\ &=\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {1}{8} a^2 \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx\\ &=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{16 b}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{8 b}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\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}\\ &=-\frac {a^2 \sqrt {x} \sqrt {a-b x}}{8 b}+\frac {1}{4} a x^{3/2} \sqrt {a-b x}+\frac {1}{3} x^{3/2} (a-b x)^{3/2}+\frac {a^3 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{8 b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 82, normalized size = 0.83 \begin {gather*} -\frac {\sqrt {x} \sqrt {a-b x} \left (3 a^2-14 a b x+8 b^2 x^2\right )}{24 b}+\frac {a^3 \log \left (-\sqrt {-b} \sqrt {x}+\sqrt {a-b x}\right )}{8 (-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 = 6.07, size = 222, normalized size = 2.24 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-3 a^{\frac {9}{2}} b \text {ArcCosh}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ] \left (\frac {-a+b x}{a}\right )^{\frac {3}{2}}+3 a^3 b^{\frac {3}{2}} \sqrt {x} \left (-a+b x\right )+a b^{\frac {5}{2}} x^{\frac {3}{2}} \left (-17 a+22 b x\right ) \left (-a+b x\right )+8 b^{\frac {9}{2}} x^{\frac {7}{2}} \left (a-b x\right )\right )}{24 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^3 \text {ArcSin}\left [\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right ]}{8 b^{\frac {3}{2}}}-\frac {a^{\frac {5}{2}} \sqrt {x}}{8 b \sqrt {1-\frac {b x}{a}}}+\frac {17 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {1-\frac {b x}{a}}}-\frac {11 \sqrt {a} b x^{\frac {5}{2}}}{12 \sqrt {1-\frac {b x}{a}}}+\frac {b^2 x^{\frac {7}{2}}}{3 \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 = 104, normalized size = 1.05
method | result | size |
risch | \(-\frac {\left (8 x^{2} b^{2}-14 a b x +3 a^{2}\right ) \sqrt {x}\, \sqrt {-b x +a}}{24 b}+\frac {a^{3} \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right ) \sqrt {x \left (-b x +a \right )}}{16 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\) | \(91\) |
default | \(\frac {x^{\frac {3}{2}} \left (-b x +a \right )^{\frac {3}{2}}}{3}+\frac {a \left (\frac {x^{\frac {3}{2}} \sqrt {-b x +a}}{2}+\frac {a \left (-\frac {\sqrt {x}\, \sqrt {-b x +a}}{b}+\frac {a \sqrt {x \left (-b x +a \right )}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-x^{2} b +a x}}\right )}{2 b^{\frac {3}{2}} \sqrt {x}\, \sqrt {-b x +a}}\right )}{4}\right )}{2}\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 133, normalized size = 1.34 \begin {gather*} -\frac {a^{3} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{8 \, b^{\frac {3}{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^{4} - \frac {3 \, {\left (b x - a\right )} b^{3}}{x} + \frac {3 \, {\left (b x - a\right )}^{2} b^{2}}{x^{2}} - \frac {{\left (b x - a\right )}^{3} b}{x^{3}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 141, normalized size = 1.42 \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} - 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{48 \, b^{2}}, -\frac {3 \, a^{3} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (8 \, b^{3} x^{2} - 14 \, a b^{2} x + 3 \, a^{2} b\right )} \sqrt {-b x + a} \sqrt {x}}{24 \, b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.00, size = 264, normalized size = 2.67 \begin {gather*} \begin {cases} \frac {i a^{\frac {5}{2}} \sqrt {x}}{8 b \sqrt {-1 + \frac {b x}{a}}} - \frac {17 i a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {-1 + \frac {b x}{a}}} + \frac {11 i \sqrt {a} b 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 {3}{2}}} - \frac {i b^{2} 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 \sqrt {1 - \frac {b x}{a}}} + \frac {17 a^{\frac {3}{2}} x^{\frac {3}{2}}}{24 \sqrt {1 - \frac {b x}{a}}} - \frac {11 \sqrt {a} b 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 {3}{2}}} + \frac {b^{2} 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 281 vs.
\(2 (71) = 142\).
time = 31.18, size = 406, normalized size = 4.10 \begin {gather*} \frac {-\frac {2 b^{2} \left |b\right | \left (2 \left (\left (\frac {\frac {1}{2304}\cdot 192 b^{5} \sqrt {a-b x} \sqrt {a-b x}}{b^{7}}-\frac {\frac {1}{2304}\cdot 624 b^{5} a}{b^{7}}\right ) \sqrt {a-b x} \sqrt {a-b x}+\frac {\frac {1}{2304}\cdot 792 b^{5} a^{2}}{b^{7}}\right ) \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}-\frac {10 a^{3} \ln \left |\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right |}{32 b \sqrt {-b}}\right )}{b^{2}}-\frac {4 a b \left |b\right | \left (2 \left (\frac {1}{8} \sqrt {a-b x} \sqrt {a-b x}-\frac {10}{32} a\right ) \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}+\frac {6 a^{2} b \ln \left |\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right |}{16 \sqrt {-b}}\right )}{b^{2} b}-\frac {2 a^{2} \left |b\right | \left (\frac {1}{2} \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}-\frac {2 a b \ln \left |\sqrt {a b-b \left (a-b x\right )}-\sqrt {-b} \sqrt {a-b x}\right |}{4 \sqrt {-b}}\right )}{b^{2}}}{b} \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 \sqrt {x}\,{\left (a-b\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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