Optimal. Leaf size=46 \[ -\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}-\frac {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 46, 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 {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx &=-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}+\frac {(2 b) \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {a-b x}}{3 a x^{3/2}}-\frac {4 b \sqrt {a-b x}}{3 a^2 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 28, normalized size = 0.61 \begin {gather*} -\frac {2 \sqrt {a-b x} (a+2 b x)}{3 a^2 x^{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 2 in
optimal.
time = 2.94, size = 174, normalized size = 3.78 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b} \left (-a-2 b x\right ) \sqrt {\frac {a-b x}{b x}}}{3 a^2 x},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-2 I a^2 b^{\frac {3}{2}} \sqrt {1-\frac {a}{b x}}}{3 a^3 b x-3 a^2 b^2 x^2}-\frac {2 I a b^{\frac {5}{2}} x \sqrt {1-\frac {a}{b x}}}{3 a^3 b x-3 a^2 b^2 x^2}+\frac {I 4 b^{\frac {7}{2}} x^2 \sqrt {1-\frac {a}{b x}}}{3 a^3 b x-3 a^2 b^2 x^2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 35, normalized size = 0.76
method | result | size |
gosper | \(-\frac {2 \sqrt {-b x +a}\, \left (2 b x +a \right )}{3 x^{\frac {3}{2}} a^{2}}\) | \(23\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (2 b x +a \right )}{3 x^{\frac {3}{2}} a^{2}}\) | \(23\) |
default | \(-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 32, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (\frac {3 \, \sqrt {-b x + a} b}{\sqrt {x}} + \frac {{\left (-b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 22, normalized size = 0.48 \begin {gather*} -\frac {2 \, {\left (2 \, b x + a\right )} \sqrt {-b x + a}}{3 \, a^{2} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.14, size = 177, normalized size = 3.85 \begin {gather*} \begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{3 a x} - \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{3 a^{2}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {2 i a^{2} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} + \frac {2 i a b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} - \frac {4 i b^{\frac {7}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} & \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 = 76, normalized size = 1.65 \begin {gather*} \frac {32 \sqrt {-b} b \left (-3 \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2}+a\right )}{2\cdot 6 \left (\left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2}-a\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 26, normalized size = 0.57 \begin {gather*} -\frac {\left (\frac {2}{3\,a}+\frac {4\,b\,x}{3\,a^2}\right )\,\sqrt {a-b\,x}}{x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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