Optimal. Leaf size=62 \[ -\frac {3 b}{a^2 \sqrt {-a+b x}}+\frac {1}{a x \sqrt {-a+b x}}-\frac {3 b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {44, 53, 65, 211}
\begin {gather*} -\frac {3 b \text {ArcTan}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 211
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 \text {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 [A]
time = 0.06, size = 51, normalized size = 0.82 \begin {gather*} \frac {a-3 b x}{a^2 x \sqrt {-a+b x}}-\frac {3 b \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 61, normalized size = 0.98
method | result | size |
risch | \(\frac {-b x +a}{a^{2} x \sqrt {b x -a}}-\frac {2 b}{a^{2} \sqrt {b x -a}}-\frac {3 b \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}\) | \(59\) |
derivativedivides | \(2 b \left (-\frac {\frac {\sqrt {b x -a}}{2 b x}+\frac {3 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a^{2}}-\frac {1}{a^{2} \sqrt {b x -a}}\right )\) | \(61\) |
default | \(2 b \left (-\frac {\frac {\sqrt {b x -a}}{2 b x}+\frac {3 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{2 \sqrt {a}}}{a^{2}}-\frac {1}{a^{2} \sqrt {b x -a}}\right )\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 67, normalized size = 1.08 \begin {gather*} -\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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.54, size = 164, normalized size = 2.65 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.72, size = 156, normalized size = 2.52 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 64, normalized size = 1.03 \begin {gather*} -\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}} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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