Optimal. Leaf size=71 \[ -\frac {3 a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}+\frac {2 x^{3/2}}{b \sqrt {a-b x}} \]
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Rubi [A] time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {47, 50, 63, 217, 203} \begin {gather*} \frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {3 a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}}+\frac {2 x^{3/2}}{b \sqrt {a-b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{(a-b x)^{3/2}} \, dx &=\frac {2 x^{3/2}}{b \sqrt {a-b x}}-\frac {3 \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{b}\\ &=\frac {2 x^{3/2}}{b \sqrt {a-b x}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {(3 a) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{2 b^2}\\ &=\frac {2 x^{3/2}}{b \sqrt {a-b x}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{b^2}\\ &=\frac {2 x^{3/2}}{b \sqrt {a-b x}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{b^2}\\ &=\frac {2 x^{3/2}}{b \sqrt {a-b x}}+\frac {3 \sqrt {x} \sqrt {a-b x}}{b^2}-\frac {3 a \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 51, normalized size = 0.72 \begin {gather*} \frac {2 x^{5/2} \sqrt {1-\frac {b x}{a}} \, _2F_1\left (\frac {3}{2},\frac {5}{2};\frac {7}{2};\frac {b x}{a}\right )}{5 a \sqrt {a-b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 81, normalized size = 1.14 \begin {gather*} -\frac {3 a \sqrt {-b} \log \left (\sqrt {a-b x}-\sqrt {-b} \sqrt {x}\right )}{b^3}-\frac {\sqrt {a-b x} \left (3 a \sqrt {x}-b x^{3/2}\right )}{b^2 (b x-a)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.30, size = 152, normalized size = 2.14 \begin {gather*} \left [-\frac {3 \, {\left (a b x - a^{2}\right )} \sqrt {-b} \log \left (-2 \, b x - 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, {\left (b^{2} x - 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{2 \, {\left (b^{4} x - a b^{3}\right )}}, \frac {3 \, {\left (a b x - a^{2}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (b^{2} x - 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{b^{4} x - a b^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 111.13, size = 130, normalized size = 1.83 \begin {gather*} -\frac {{\left (\frac {8 \, a^{2} \sqrt {-b}}{{\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b} + \frac {3 \, a \log \left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2}\right )}{\sqrt {-b}} - \frac {2 \, \sqrt {{\left (b x - a\right )} b + a b} \sqrt {-b x + a}}{b}\right )} {\left | b \right |}}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 114, normalized size = 1.61 \begin {gather*} \frac {\left (-\frac {3 a \arctan \left (\frac {\left (x -\frac {a}{2 b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+a x}}\right )}{2 b^{\frac {5}{2}}}-\frac {2 \sqrt {-\left (x -\frac {a}{b}\right ) a -\left (x -\frac {a}{b}\right )^{2} b}\, a}{\left (x -\frac {a}{b}\right ) b^{3}}\right ) \sqrt {\left (-b x +a \right ) x}}{\sqrt {-b x +a}\, \sqrt {x}}+\frac {\sqrt {-b x +a}\, \sqrt {x}}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.95, size = 75, normalized size = 1.06 \begin {gather*} \frac {2 \, a b - \frac {3 \, {\left (b x - a\right )} a}{x}}{\frac {\sqrt {-b x + a} b^{3}}{\sqrt {x}} + \frac {{\left (-b x + a\right )}^{\frac {3}{2}} b^{2}}{x^{\frac {3}{2}}}} + \frac {3 \, a \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{b^{\frac {5}{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 {x^{3/2}}{{\left (a-b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.70, size = 155, normalized size = 2.18 \begin {gather*} \begin {cases} - \frac {3 i \sqrt {a} \sqrt {x}}{b^{2} \sqrt {-1 + \frac {b x}{a}}} + \frac {3 i a \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {5}{2}}} + \frac {i x^{\frac {3}{2}}}{\sqrt {a} b \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\\frac {3 \sqrt {a} \sqrt {x}}{b^{2} \sqrt {1 - \frac {b x}{a}}} - \frac {3 a \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{b^{\frac {5}{2}}} - \frac {x^{\frac {3}{2}}}{\sqrt {a} b \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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