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 = {45, 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 45
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 = 0.01, 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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IntegrateAlgebraic [A] time = 0.10, size = 37, normalized size = 0.90 \begin {gather*} -\frac {2 \sqrt {a-b x} (2 b x-a)}{a^2 \sqrt {x} (b x-a)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, 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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giac [B] time = 1.44, size = 94, normalized size = 2.29 \begin {gather*} -\frac {4 \, \sqrt {-b} b^{2}}{{\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} a {\left | b \right |}} - \frac {2 \, \sqrt {-b x + a} b^{2}}{\sqrt {{\left (b x - a\right )} b + a b} a^{2} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.56 \begin {gather*} -\frac {2 \left (-2 b x +a \right )}{\sqrt {-b x +a}\, a^{2} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, 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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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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sympy [A] time = 1.68, 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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