Optimal. Leaf size=46 \[ -\frac {2 (a-b x)^{3/2}}{5 a x^{5/2}}-\frac {4 b (a-b x)^{3/2}}{15 a^2 x^{3/2}} \]
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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 (a-b x)^{3/2}}{15 a^2 x^{3/2}}-\frac {2 (a-b x)^{3/2}}{5 a x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx &=-\frac {2 (a-b x)^{3/2}}{5 a x^{5/2}}+\frac {(2 b) \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx}{5 a}\\ &=-\frac {2 (a-b x)^{3/2}}{5 a x^{5/2}}-\frac {4 b (a-b x)^{3/2}}{15 a^2 x^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 41, normalized size = 0.89 \begin {gather*} -\frac {2 \sqrt {a-b x} \left (3 a^2-a b x-2 b^2 x^2\right )}{15 a^2 x^{5/2}} \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 = 4.53, size = 229, normalized size = 4.98 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b} \left (-3 a^2+a b x+2 b^2 x^2\right ) \sqrt {\frac {a-b x}{b x}}}{15 a^2 x^2},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-6 I a^3 b^{\frac {3}{2}} \sqrt {1-\frac {a}{b x}}}{x \left (15 a^3 b x-15 a^2 b^2 x^2\right )}+\frac {I 8 a^2 b^{\frac {5}{2}} \sqrt {1-\frac {a}{b x}}}{15 a^3 b x-15 a^2 b^2 x^2}+\frac {I 2 a b^{\frac {7}{2}} x \sqrt {1-\frac {a}{b x}}}{15 a^3 b x-15 a^2 b^2 x^2}-\frac {4 I b^{\frac {9}{2}} x^2 \sqrt {1-\frac {a}{b x}}}{15 a^3 b x-15 a^2 b^2 x^2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs.
\(2(34)=68\).
time = 0.12, size = 75, normalized size = 1.63
method | result | size |
gosper | \(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}} \left (2 b x +3 a \right )}{15 x^{\frac {5}{2}} a^{2}}\) | \(25\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (-2 x^{2} b^{2}-a b x +3 a^{2}\right )}{15 x^{\frac {5}{2}} a^{2}}\) | \(36\) |
default | \(-\frac {\sqrt {-b x +a}}{2 x^{\frac {5}{2}}}-\frac {a \left (-\frac {2 \sqrt {-b x +a}}{5 a \,x^{\frac {5}{2}}}+\frac {4 b \left (-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\right )}{5 a}\right )}{4}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 33, normalized size = 0.72 \begin {gather*} -\frac {2 \, {\left (\frac {5 \, {\left (-b x + a\right )}^{\frac {3}{2}} b}{x^{\frac {3}{2}}} + \frac {3 \, {\left (-b x + a\right )}^{\frac {5}{2}}}{x^{\frac {5}{2}}}\right )}}{15 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 34, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (2 \, b^{2} x^{2} + a b x - 3 \, a^{2}\right )} \sqrt {-b x + a}}{15 \, a^{2} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.00, size = 241, normalized size = 5.24 \begin {gather*} \begin {cases} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{5 x^{2}} + \frac {2 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{15 a x} + \frac {4 b^{\frac {5}{2}} \sqrt {\frac {a}{b x} - 1}}{15 a^{2}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {6 i a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{x \left (- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}\right )} - \frac {8 i a^{2} b^{\frac {5}{2}} \sqrt {- \frac {a}{b x} + 1}}{- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}} - \frac {2 i a b^{\frac {7}{2}} x \sqrt {- \frac {a}{b x} + 1}}{- 15 a^{3} b x + 15 a^{2} b^{2} x^{2}} + \frac {4 i b^{\frac {9}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{- 15 a^{3} b x + 15 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 = 120, normalized size = 2.61 \begin {gather*} \frac {2 b^{2} \left (\frac {\frac {1}{225}\cdot 30 b^{5} \sqrt {a-b x} \sqrt {a-b x}}{a^{2}}-\frac {\frac {1}{225}\cdot 75 b^{5} a}{a^{2}}\right ) \sqrt {a-b x} \sqrt {a-b x} \sqrt {a-b x} \sqrt {a b-b \left (a-b x\right )}}{\left |b\right | b \left (a b-b \left (a-b x\right )\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 32, normalized size = 0.70 \begin {gather*} \frac {\sqrt {a-b\,x}\,\left (\frac {4\,b^2\,x^2}{15\,a^2}+\frac {2\,b\,x}{15\,a}-\frac {2}{5}\right )}{x^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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