Optimal. Leaf size=60 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {2}{a^2 \sqrt {b x-a}}-\frac {2}{3 a (b x-a)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 60, 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 = {51, 63, 205} \begin {gather*} \frac {2}{a^2 \sqrt {b x-a}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {2}{3 a (b x-a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 205
Rubi steps
\begin {align*} \int \frac {1}{x (-a+b x)^{5/2}} \, dx &=-\frac {2}{3 a (-a+b x)^{3/2}}-\frac {\int \frac {1}{x (-a+b x)^{3/2}} \, dx}{a}\\ &=-\frac {2}{3 a (-a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {-a+b x}}+\frac {\int \frac {1}{x \sqrt {-a+b x}} \, dx}{a^2}\\ &=-\frac {2}{3 a (-a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {-a+b x}}+\frac {2 \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{a^2 b}\\ &=-\frac {2}{3 a (-a+b x)^{3/2}}+\frac {2}{a^2 \sqrt {-a+b x}}+\frac {2 \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 = 35, normalized size = 0.58 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};1-\frac {b x}{a}\right )}{3 a (b x-a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 55, normalized size = 0.92 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {2 (a-3 (b x-a))}{3 a^2 (b x-a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 182, normalized size = 3.03 \begin {gather*} \left [-\frac {3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\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 - 4 \, a^{2}\right )} \sqrt {b x - a}}{3 \, {\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (3 \, a b x - 4 \, a^{2}\right )} \sqrt {b x - a}\right )}}{3 \, {\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.06, size = 42, normalized size = 0.70 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x - 4 \, a\right )}}{3 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 49, normalized size = 0.82 \begin {gather*} -\frac {2}{3 \left (b x -a \right )^{\frac {3}{2}} a}+\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}}+\frac {2}{\sqrt {b x -a}\, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.94, size = 42, normalized size = 0.70 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, b x - 4 \, a\right )}}{3 \, {\left (b x - a\right )}^{\frac {3}{2}} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 48, normalized size = 0.80 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2\,\left (a-b\,x\right )}{a^2}+\frac {2}{3\,a}}{{\left (b\,x-a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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