Optimal. Leaf size=55 \[ 2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )-2 a \sqrt {b x-a}+\frac {2}{3} (b x-a)^{3/2} \]
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Rubi [A] time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {50, 63, 205} \begin {gather*} 2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )-2 a \sqrt {b x-a}+\frac {2}{3} (b x-a)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {(-a+b x)^{3/2}}{x} \, dx &=\frac {2}{3} (-a+b x)^{3/2}-a \int \frac {\sqrt {-a+b x}}{x} \, dx\\ &=-2 a \sqrt {-a+b x}+\frac {2}{3} (-a+b x)^{3/2}+a^2 \int \frac {1}{x \sqrt {-a+b x}} \, dx\\ &=-2 a \sqrt {-a+b x}+\frac {2}{3} (-a+b x)^{3/2}+\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{b}\\ &=-2 a \sqrt {-a+b x}+\frac {2}{3} (-a+b x)^{3/2}+2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.87 \begin {gather*} 2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )+\frac {2}{3} (b x-4 a) \sqrt {b x-a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 58, normalized size = 1.05 \begin {gather*} 2 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )-\frac {2}{3} \left (3 a \sqrt {b x-a}-(b x-a)^{3/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 93, normalized size = 1.69 \begin {gather*} \left [\sqrt {-a} a \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + \frac {2}{3} \, \sqrt {b x - a} {\left (b x - 4 \, a\right )}, 2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, \sqrt {b x - a} {\left (b x - 4 \, a\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 43, normalized size = 0.78 \begin {gather*} 2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x - a\right )}^{\frac {3}{2}} - 2 \, \sqrt {b x - a} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 44, normalized size = 0.80 \begin {gather*} 2 a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )-2 \sqrt {b x -a}\, a +\frac {2 \left (b x -a \right )^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 43, normalized size = 0.78 \begin {gather*} 2 \, a^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \frac {2}{3} \, {\left (b x - a\right )}^{\frac {3}{2}} - 2 \, \sqrt {b x - a} a \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 43, normalized size = 0.78 \begin {gather*} 2\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )-2\,a\,\sqrt {b\,x-a}+\frac {2\,{\left (b\,x-a\right )}^{3/2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.46, size = 187, normalized size = 3.40 \begin {gather*} \begin {cases} - \frac {8 a^{\frac {3}{2}} \sqrt {-1 + \frac {b x}{a}}}{3} - i a^{\frac {3}{2}} \log {\left (\frac {b x}{a} \right )} + 2 i a^{\frac {3}{2}} \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - 2 a^{\frac {3}{2}} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} + \frac {2 \sqrt {a} b x \sqrt {-1 + \frac {b x}{a}}}{3} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {8 i a^{\frac {3}{2}} \sqrt {1 - \frac {b x}{a}}}{3} - i a^{\frac {3}{2}} \log {\left (\frac {b x}{a} \right )} + 2 i a^{\frac {3}{2}} \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )} + \frac {2 i \sqrt {a} b x \sqrt {1 - \frac {b x}{a}}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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