Optimal. Leaf size=67 \[ 2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+\frac {2 b \sqrt {a-b x}}{\sqrt {x}} \]
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Rubi [A] time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {47, 63, 217, 203} \begin {gather*} 2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+\frac {2 b \sqrt {a-b x}}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {(a-b x)^{3/2}}{x^{5/2}} \, dx &=-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}-b \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx\\ &=\frac {2 b \sqrt {a-b x}}{\sqrt {x}}-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+b^2 \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx\\ &=\frac {2 b \sqrt {a-b x}}{\sqrt {x}}-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )\\ &=\frac {2 b \sqrt {a-b x}}{\sqrt {x}}-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )\\ &=\frac {2 b \sqrt {a-b x}}{\sqrt {x}}-\frac {2 (a-b x)^{3/2}}{3 x^{3/2}}+2 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 49, normalized size = 0.73 \begin {gather*} -\frac {2 a \sqrt {a-b x} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {b x}{a}\right )}{3 x^{3/2} \sqrt {1-\frac {b x}{a}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 62, normalized size = 0.93 \begin {gather*} 2 \sqrt {-b} b \log \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )-\frac {2 (a-4 b x) \sqrt {a-b x}}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.62, size = 115, normalized size = 1.72 \begin {gather*} \left [\frac {3 \, \sqrt {-b} b x^{2} \log \left (-2 \, b x - 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (4 \, b x - a\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, x^{2}}, -\frac {2 \, {\left (3 \, b^{\frac {3}{2}} x^{2} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - {\left (4 \, b x - a\right )} \sqrt {-b x + a} \sqrt {x}\right )}}{3 \, x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-b x +a \right )^{\frac {3}{2}}}{x^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.86, size = 49, normalized size = 0.73 \begin {gather*} -2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + \frac {2 \, \sqrt {-b x + a} b}{\sqrt {x}} - \frac {2 \, {\left (-b x + a\right )}^{\frac {3}{2}}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a-b\,x\right )}^{3/2}}{x^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.24, size = 187, normalized size = 2.79 \begin {gather*} \begin {cases} - \frac {2 a \sqrt {b} \sqrt {\frac {a}{b x} - 1}}{3 x} + \frac {8 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}}{3} - 2 i b^{\frac {3}{2}} \log {\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + i b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )} + 2 b^{\frac {3}{2}} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 i a \sqrt {b} \sqrt {- \frac {a}{b x} + 1}}{3 x} + \frac {8 i b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}}{3} + i b^{\frac {3}{2}} \log {\left (\frac {a}{b x} \right )} - 2 i b^{\frac {3}{2}} \log {\left (\sqrt {- \frac {a}{b x} + 1} + 1 \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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