Optimal. Leaf size=41 \[ \frac {2}{a \sqrt {x} \sqrt {a-b x}}-\frac {4 \sqrt {a-b x}}{a^2 \sqrt {x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37}
\begin {gather*} \frac {2}{a \sqrt {x} \sqrt {a-b x}}-\frac {4 \sqrt {a-b x}}{a^2 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} (a-b x)^{3/2}} \, dx &=\frac {2}{a \sqrt {x} \sqrt {a-b x}}+\frac {2 \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx}{a}\\ &=\frac {2}{a \sqrt {x} \sqrt {a-b x}}-\frac {4 \sqrt {a-b x}}{a^2 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 26, normalized size = 0.63 \begin {gather*} -\frac {2 (a-2 b x)}{a^2 \sqrt {x} \sqrt {a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 2.73, size = 126, normalized size = 3.07 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b} \left (-a+2 b x\right ) \sqrt {\frac {a-b x}{b x}}}{a^2 \left (a-b x\right )},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-2 I a b^{\frac {3}{2}} \sqrt {1-\frac {a}{b x}}}{a^3 b-a^2 b^2 x}+\frac {I 4 b^{\frac {5}{2}} x \sqrt {1-\frac {a}{b x}}}{a^3 b-a^2 b^2 x}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.13, size = 35, normalized size = 0.85
method | result | size |
gosper | \(-\frac {2 \left (-2 b x +a \right )}{\sqrt {x}\, \sqrt {-b x +a}\, a^{2}}\) | \(23\) |
default | \(-\frac {2}{a \sqrt {x}\, \sqrt {-b x +a}}+\frac {4 b \sqrt {x}}{a^{2} \sqrt {-b x +a}}\) | \(35\) |
risch | \(-\frac {2 \sqrt {-b x +a}}{a^{2} \sqrt {x}}+\frac {2 b \sqrt {x}}{a^{2} \sqrt {-b x +a}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 34, normalized size = 0.83 \begin {gather*} \frac {2 \, b \sqrt {x}}{\sqrt {-b x + a} a^{2}} - \frac {2 \, \sqrt {-b x + a}}{a^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 38, normalized size = 0.93 \begin {gather*} -\frac {2 \, {\left (2 \, b x - a\right )} \sqrt {-b x + a} \sqrt {x}}{a^{2} b x^{2} - a^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.92, size = 112, normalized size = 2.73 \begin {gather*} \begin {cases} - \frac {2}{a \sqrt {b} x \sqrt {\frac {a}{b x} - 1}} + \frac {4 \sqrt {b}}{a^{2} \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {2 i a b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{- a^{3} b + a^{2} b^{2} x} - \frac {4 i b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- a^{3} b + a^{2} b^{2} x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 77, normalized size = 1.88 \begin {gather*} 2 \left (\frac {\frac {1}{2}\cdot 2 b \sqrt {x} \sqrt {a-b x}}{a^{2} \left (a-b x\right )}+\frac {4 \sqrt {-b}}{2 a \left (\left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2}-a\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 42, normalized size = 1.02 \begin {gather*} -\frac {2\,a\,\sqrt {a-b\,x}-4\,b\,x\,\sqrt {a-b\,x}}{\sqrt {x}\,\left (a^3-a^2\,b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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