Optimal. Leaf size=66 \[ \frac {2}{a x^{3/2} \sqrt {a-b x}}-\frac {8 \sqrt {a-b x}}{3 a^2 x^{3/2}}-\frac {16 b \sqrt {a-b x}}{3 a^3 \sqrt {x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 66, 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 \sqrt {a-b x}}{3 a^3 \sqrt {x}}-\frac {8 \sqrt {a-b x}}{3 a^2 x^{3/2}}+\frac {2}{a x^{3/2} \sqrt {a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} (a-b x)^{3/2}} \, dx &=\frac {2}{a x^{3/2} \sqrt {a-b x}}+\frac {4 \int \frac {1}{x^{5/2} \sqrt {a-b x}} \, dx}{a}\\ &=\frac {2}{a x^{3/2} \sqrt {a-b x}}-\frac {8 \sqrt {a-b x}}{3 a^2 x^{3/2}}+\frac {(8 b) \int \frac {1}{x^{3/2} \sqrt {a-b x}} \, dx}{3 a^2}\\ &=\frac {2}{a x^{3/2} \sqrt {a-b x}}-\frac {8 \sqrt {a-b x}}{3 a^2 x^{3/2}}-\frac {16 b \sqrt {a-b x}}{3 a^3 \sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 39, normalized size = 0.59 \begin {gather*} -\frac {2 \left (a^2+4 a b x-8 b^2 x^2\right )}{3 a^3 x^{3/2} \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 = 8.56, size = 311, normalized size = 4.71 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \sqrt {b} \left (-a^3-3 a^2 b x+12 a b^2 x^2-8 b^3 x^3\right ) \sqrt {\frac {a-b x}{b x}}}{3 a^3 x \left (a^2-2 a b x+b^2 x^2\right )},\text {Abs}\left [\frac {a}{b x}\right ]>1\right \}\right \},\frac {-2 I a^3 b^{\frac {9}{2}} \sqrt {1-\frac {a}{b x}}}{3 a^5 b^4 x-6 a^4 b^5 x^2+3 a^3 b^6 x^3}-\frac {6 I a^2 b^{\frac {11}{2}} x \sqrt {1-\frac {a}{b x}}}{3 a^5 b^4 x-6 a^4 b^5 x^2+3 a^3 b^6 x^3}+\frac {I 24 a b^{\frac {13}{2}} x^2 \sqrt {1-\frac {a}{b x}}}{3 a^5 b^4 x-6 a^4 b^5 x^2+3 a^3 b^6 x^3}-\frac {16 I b^{\frac {15}{2}} x^3 \sqrt {1-\frac {a}{b x}}}{3 a^5 b^4 x-6 a^4 b^5 x^2+3 a^3 b^6 x^3}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 58, normalized size = 0.88
method | result | size |
gosper | \(-\frac {2 \left (-8 x^{2} b^{2}+4 a b x +a^{2}\right )}{3 x^{\frac {3}{2}} \sqrt {-b x +a}\, a^{3}}\) | \(34\) |
risch | \(-\frac {2 \sqrt {-b x +a}\, \left (5 b x +a \right )}{3 a^{3} x^{\frac {3}{2}}}+\frac {2 b^{2} \sqrt {x}}{a^{3} \sqrt {-b x +a}}\) | \(43\) |
default | \(-\frac {2}{3 a \,x^{\frac {3}{2}} \sqrt {-b x +a}}+\frac {4 b \left (-\frac {2}{a \sqrt {x}\, \sqrt {-b x +a}}+\frac {4 b \sqrt {x}}{a^{2} \sqrt {-b x +a}}\right )}{3 a}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 52, normalized size = 0.79 \begin {gather*} \frac {2 \, b^{2} \sqrt {x}}{\sqrt {-b x + a} a^{3}} - \frac {2 \, {\left (\frac {6 \, \sqrt {-b x + a} b}{\sqrt {x}} + \frac {{\left (-b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 51, normalized size = 0.77 \begin {gather*} -\frac {2 \, {\left (8 \, b^{2} x^{2} - 4 \, a b x - a^{2}\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (a^{3} b x^{3} - a^{4} x^{2}\right )}} \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 = 2.67, size = 452, normalized size = 6.85 \begin {gather*} \begin {cases} - \frac {2 a^{3} b^{\frac {9}{2}} \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac {6 a^{2} b^{\frac {11}{2}} x \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {24 a b^{\frac {13}{2}} x^{2} \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac {16 b^{\frac {15}{2}} x^{3} \sqrt {\frac {a}{b x} - 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i a^{3} b^{\frac {9}{2}} \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac {6 i a^{2} b^{\frac {11}{2}} x \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} + \frac {24 i a b^{\frac {13}{2}} x^{2} \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} - \frac {16 i b^{\frac {15}{2}} x^{3} \sqrt {- \frac {a}{b x} + 1}}{3 a^{5} b^{4} x - 6 a^{4} b^{5} x^{2} + 3 a^{3} b^{6} x^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs.
\(2 (50) = 100\).
time = 0.01, size = 162, normalized size = 2.45 \begin {gather*} 2 \left (\frac {\frac {1}{2}\cdot 2 b^{2} \sqrt {x} \sqrt {a-b x}}{a^{3} \left (a-b x\right )}+\frac {2 \left (3 b \sqrt {-b} \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{4}-12 b \sqrt {-b} \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2} a+5 b \sqrt {-b} a^{2}\right )}{3 a^{2} \left (\left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )^{2}-a\right )^{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.43, size = 48, normalized size = 0.73 \begin {gather*} \frac {\sqrt {a-b\,x}\,\left (\frac {8\,x}{3\,a^2}+\frac {2}{3\,a\,b}-\frac {16\,b\,x^2}{3\,a^3}\right )}{x^{5/2}-\frac {a\,x^{3/2}}{b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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