Optimal. Leaf size=71 \[ -\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}} \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^{9/2}} \, dx &=-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}+\frac {(4 b) \int \frac {\sqrt {a-b x}}{x^{7/2}} \, dx}{7 a}\\ &=-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}+\frac {\left (8 b^2\right ) \int \frac {\sqrt {a-b x}}{x^{5/2}} \, dx}{35 a^2}\\ &=-\frac {2 (a-b x)^{3/2}}{7 a x^{7/2}}-\frac {8 b (a-b x)^{3/2}}{35 a^2 x^{5/2}}-\frac {16 b^2 (a-b x)^{3/2}}{105 a^3 x^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 52, normalized size = 0.73 \begin {gather*} -\frac {2 \sqrt {a-b x} \left (15 a^3-3 a^2 b x-4 a b^2 x^2-8 b^3 x^3\right )}{105 a^3 x^{7/2}} \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 = 45.14, size = 461, normalized size = 6.49 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b} \left (-15 a^5+33 a^4 b x-17 a^3 b^2 x^2+3 a^2 b^3 x^3-12 a b^4 x^4+8 b^5 x^5\right ) \sqrt {\frac {a-b x}{b x}}}{105 a^3 x^3 \left (a^2-2 a b x+b^2 x^2\right )},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-30 I a^5 b^{\frac {9}{2}} \sqrt {1-\frac {a}{b x}}}{105 a^5 b^4 x^3-210 a^4 b^5 x^4+105 a^3 b^6 x^5}+\frac {I 66 a^4 b^{\frac {11}{2}} x \sqrt {1-\frac {a}{b x}}}{105 a^5 b^4 x^3-210 a^4 b^5 x^4+105 a^3 b^6 x^5}-\frac {34 I a^3 b^{\frac {13}{2}} x^2 \sqrt {1-\frac {a}{b x}}}{105 a^5 b^4 x^3-210 a^4 b^5 x^4+105 a^3 b^6 x^5}+\frac {I 6 a^2 b^{\frac {15}{2}} x^3 \sqrt {1-\frac {a}{b x}}}{105 a^5 b^4 x^3-210 a^4 b^5 x^4+105 a^3 b^6 x^5}-\frac {24 I a b^{\frac {17}{2}} x^4 \sqrt {1-\frac {a}{b x}}}{105 a^5 b^4 x^3-210 a^4 b^5 x^4+105 a^3 b^6 x^5}+\frac {I 16 b^{\frac {19}{2}} x^5 \sqrt {1-\frac {a}{b x}}}{105 a^5 b^4 x^3-210 a^4 b^5 x^4+105 a^3 b^6 x^5}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.10, size = 98, normalized size = 1.38
method | result | size |
gosper | \(-\frac {2 \left (-b x +a \right )^{\frac {3}{2}} \left (8 x^{2} b^{2}+12 a b x +15 a^{2}\right )}{105 x^{\frac {7}{2}} a^{3}}\) | \(36\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (-8 b^{3} x^{3}-4 a \,b^{2} x^{2}-3 a^{2} b x +15 a^{3}\right )}{105 x^{\frac {7}{2}} a^{3}}\) | \(47\) |
default | \(-\frac {\sqrt {-b x +a}}{3 x^{\frac {7}{2}}}-\frac {a \left (-\frac {2 \sqrt {-b x +a}}{7 a \,x^{\frac {7}{2}}}+\frac {6 b \left (-\frac {2 \sqrt {-b x +a}}{5 a \,x^{\frac {5}{2}}}+\frac {4 b \left (-\frac {2 \sqrt {-b x +a}}{3 a \,x^{\frac {3}{2}}}-\frac {4 b \sqrt {-b x +a}}{3 a^{2} \sqrt {x}}\right )}{5 a}\right )}{7 a}\right )}{6}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 49, normalized size = 0.69 \begin {gather*} -\frac {2 \, {\left (\frac {35 \, {\left (-b x + a\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}} + \frac {42 \, {\left (-b x + a\right )}^{\frac {5}{2}} b}{x^{\frac {5}{2}}} + \frac {15 \, {\left (-b x + a\right )}^{\frac {7}{2}}}{x^{\frac {7}{2}}}\right )}}{105 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 46, normalized size = 0.65 \begin {gather*} \frac {2 \, {\left (8 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} + 3 \, a^{2} b x - 15 \, a^{3}\right )} \sqrt {-b x + a}}{105 \, a^{3} x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 11.98, size = 707, normalized size = 9.96 \begin {gather*} \begin {cases} - \frac {30 a^{5} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {66 a^{4} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {34 a^{3} b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {6 a^{2} b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {24 a b^{\frac {17}{2}} x^{4} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {16 b^{\frac {19}{2}} x^{5} \sqrt {\frac {a}{b x} - 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {30 i a^{5} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {66 i a^{4} b^{\frac {11}{2}} x \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {34 i a^{3} b^{\frac {13}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {6 i a^{2} b^{\frac {15}{2}} x^{3} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} - \frac {24 i a b^{\frac {17}{2}} x^{4} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} + \frac {16 i b^{\frac {19}{2}} x^{5} \sqrt {- \frac {a}{b x} + 1}}{105 a^{5} b^{4} x^{3} - 210 a^{4} b^{5} x^{4} + 105 a^{3} b^{6} x^{5}} & \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 = 158, normalized size = 2.23 \begin {gather*} \frac {2 b^{2} \left (\left (-\frac {\frac {1}{3675}\cdot 280 b^{7} \sqrt {a-b x} \sqrt {a-b x}}{a^{3}}+\frac {\frac {1}{3675}\cdot 980 b^{7} a}{a^{3}}\right ) \sqrt {a-b x} \sqrt {a-b x}-\frac {\frac {1}{3675}\cdot 1225 b^{7} a^{2}}{a^{3}}\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 )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 43, normalized size = 0.61 \begin {gather*} \frac {\sqrt {a-b\,x}\,\left (\frac {8\,b^2\,x^2}{105\,a^2}+\frac {16\,b^3\,x^3}{105\,a^3}+\frac {2\,b\,x}{35\,a}-\frac {2}{7}\right )}{x^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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