Optimal. Leaf size=62 \[ -\frac {3 b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3 b}{a^2 \sqrt {b x-a}}+\frac {1}{a x \sqrt {b x-a}} \]
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Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.05, 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 = {51, 63, 205} \[ -\frac {3 \sqrt {b x-a}}{a^2 x}-\frac {3 b \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {2}{a x \sqrt {b x-a}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x^2 (-a+b x)^{3/2}} \, dx &=-\frac {2}{a x \sqrt {-a+b x}}-\frac {3 \int \frac {1}{x^2 \sqrt {-a+b x}} \, dx}{a}\\ &=-\frac {2}{a x \sqrt {-a+b x}}-\frac {3 \sqrt {-a+b x}}{a^2 x}-\frac {(3 b) \int \frac {1}{x \sqrt {-a+b x}} \, dx}{2 a^2}\\ &=-\frac {2}{a x \sqrt {-a+b x}}-\frac {3 \sqrt {-a+b x}}{a^2 x}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{a^2}\\ &=-\frac {2}{a x \sqrt {-a+b x}}-\frac {3 \sqrt {-a+b x}}{a^2 x}-\frac {3 b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.55 \[ -\frac {2 b \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};1-\frac {b x}{a}\right )}{a^2 \sqrt {b x-a}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 164, normalized size = 2.65 \[ \left [-\frac {3 \, {\left (b^{2} x^{2} - a b x\right )} \sqrt {-a} \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, {\left (3 \, a b x - a^{2}\right )} \sqrt {b x - a}}{2 \, {\left (a^{3} b x^{2} - a^{4} x\right )}}, -\frac {3 \, {\left (b^{2} x^{2} - a b x\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (3 \, a b x - a^{2}\right )} \sqrt {b x - a}}{a^{3} b x^{2} - a^{4} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 64, normalized size = 1.03 \[ -\frac {3 \, b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} - \frac {3 \, {\left (b x - a\right )} b + 2 \, a b}{{\left ({\left (b x - a\right )}^{\frac {3}{2}} + \sqrt {b x - a} a\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 54, normalized size = 0.87 \[ -\frac {3 b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}-\frac {2 b}{\sqrt {b x -a}\, a^{2}}-\frac {\sqrt {b x -a}}{a^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 67, normalized size = 1.08 \[ -\frac {3 \, {\left (b x - a\right )} b + 2 \, a b}{{\left (b x - a\right )}^{\frac {3}{2}} a^{2} + \sqrt {b x - a} a^{3}} - \frac {3 \, b \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 52, normalized size = 0.84 \[ \frac {1}{a\,x\,\sqrt {b\,x-a}}-\frac {3\,b}{a^2\,\sqrt {b\,x-a}}-\frac {3\,b\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.50, size = 156, normalized size = 2.52 \[ \begin {cases} - \frac {i}{a \sqrt {b} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}} + \frac {3 i \sqrt {b}}{a^{2} \sqrt {x} \sqrt {\frac {a}{b x} - 1}} - \frac {3 i b \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {5}{2}}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {1}{a \sqrt {b} x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} - \frac {3 \sqrt {b}}{a^{2} \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} + \frac {3 b \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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