Optimal. Leaf size=80 \[ \frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b} \]
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Rubi [A] time = 0.02, antiderivative size = 80, 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 = {50, 63, 217, 203} \[ \frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{5/2}}-\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 217
Rubi steps
\begin {align*} \int \frac {x^{3/2}}{\sqrt {a-b x}} \, dx &=-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {(3 a) \int \frac {\sqrt {x}}{\sqrt {a-b x}} \, dx}{4 b}\\ &=-\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {\left (3 a^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx}{8 b^2}\\ &=-\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right )}{4 b^2}\\ &=-\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^2}\\ &=-\frac {3 a \sqrt {x} \sqrt {a-b x}}{4 b^2}-\frac {x^{3/2} \sqrt {a-b x}}{2 b}+\frac {3 a^2 \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 86, normalized size = 1.08 \[ \frac {3 a^{5/2} \sqrt {1-\frac {b x}{a}} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )+\sqrt {b} \sqrt {x} \left (-3 a^2+a b x+2 b^2 x^2\right )}{4 b^{5/2} \sqrt {a-b x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 119, normalized size = 1.49 \[ \left [-\frac {3 \, a^{2} \sqrt {-b} \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) + 2 \, {\left (2 \, b^{2} x + 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{8 \, b^{3}}, -\frac {3 \, a^{2} \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) + {\left (2 \, b^{2} x + 3 \, a b\right )} \sqrt {-b x + a} \sqrt {x}}{4 \, b^{3}}\right ] \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 89, normalized size = 1.11 \[ -\frac {\sqrt {-b x +a}\, x^{\frac {3}{2}}}{2 b}+\frac {3 \sqrt {\left (-b x +a \right ) x}\, a^{2} \arctan \left (\frac {\left (x -\frac {a}{2 b}\right ) \sqrt {b}}{\sqrt {-b \,x^{2}+a x}}\right )}{8 \sqrt {-b x +a}\, b^{\frac {5}{2}} \sqrt {x}}-\frac {3 \sqrt {-b x +a}\, a \sqrt {x}}{4 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 98, normalized size = 1.22 \[ -\frac {3 \, a^{2} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right )}{4 \, b^{\frac {5}{2}}} - \frac {\frac {5 \, \sqrt {-b x + a} a^{2} b}{\sqrt {x}} + \frac {3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{2}}{x^{\frac {3}{2}}}}{4 \, {\left (b^{4} - \frac {2 \, {\left (b x - a\right )} b^{3}}{x} + \frac {{\left (b x - a\right )}^{2} b^{2}}{x^{2}}\right )}} \]
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 \[ \int \frac {x^{3/2}}{\sqrt {a-b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.30, size = 214, normalized size = 2.68 \[ \begin {cases} \frac {3 i a^{\frac {3}{2}} \sqrt {x}}{4 b^{2} \sqrt {-1 + \frac {b x}{a}}} - \frac {i \sqrt {a} x^{\frac {3}{2}}}{4 b \sqrt {-1 + \frac {b x}{a}}} - \frac {3 i a^{2} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}}} - \frac {i x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {3 a^{\frac {3}{2}} \sqrt {x}}{4 b^{2} \sqrt {1 - \frac {b x}{a}}} + \frac {\sqrt {a} x^{\frac {3}{2}}}{4 b \sqrt {1 - \frac {b x}{a}}} + \frac {3 a^{2} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{4 b^{\frac {5}{2}}} + \frac {x^{\frac {5}{2}}}{2 \sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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