Optimal. Leaf size=74 \[ \frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {3 b \sqrt {b x-a}}{4 a^2 x}+\frac {\sqrt {b x-a}}{2 a x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 74, 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 {3 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {3 b \sqrt {b x-a}}{4 a^2 x}+\frac {\sqrt {b x-a}}{2 a x^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^3 \sqrt {-a+b x}} \, dx &=\frac {\sqrt {-a+b x}}{2 a x^2}+\frac {(3 b) \int \frac {1}{x^2 \sqrt {-a+b x}} \, dx}{4 a}\\ &=\frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {\left (3 b^2\right ) \int \frac {1}{x \sqrt {-a+b x}} \, dx}{8 a^2}\\ &=\frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {-a+b x}\right )}{4 a^2}\\ &=\frac {\sqrt {-a+b x}}{2 a x^2}+\frac {3 b \sqrt {-a+b x}}{4 a^2 x}+\frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{4 a^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 0.49 \begin {gather*} \frac {2 b^2 \sqrt {b x-a} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};1-\frac {b x}{a}\right )}{a^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 69, normalized size = 0.93 \begin {gather*} \frac {3 b^2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{4 a^{5/2}}+\frac {3 (b x-a)^{3/2}+5 a \sqrt {b x-a}}{4 a^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.36, size = 128, normalized size = 1.73 \begin {gather*} \left [-\frac {3 \, \sqrt {-a} b^{2} x^{2} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) - 2 \, {\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt {b x - a}}{8 \, a^{3} x^{2}}, \frac {3 \, \sqrt {a} b^{2} x^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + {\left (3 \, a b x + 2 \, a^{2}\right )} \sqrt {b x - a}}{4 \, a^{3} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 68, normalized size = 0.92 \begin {gather*} \frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {3 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{3} + 5 \, \sqrt {b x - a} a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 59, normalized size = 0.80 \begin {gather*} \frac {3 b^{2} \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{4 a^{\frac {5}{2}}}+\frac {3 \sqrt {b x -a}\, b}{4 a^{2} x}+\frac {\sqrt {b x -a}}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.96, size = 86, normalized size = 1.16 \begin {gather*} \frac {3 \, b^{2} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{4 \, a^{\frac {5}{2}}} + \frac {3 \, {\left (b x - a\right )}^{\frac {3}{2}} b^{2} + 5 \, \sqrt {b x - a} a b^{2}}{4 \, {\left ({\left (b x - a\right )}^{2} a^{2} + 2 \, {\left (b x - a\right )} a^{3} + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 57, normalized size = 0.77 \begin {gather*} \frac {3\,b^2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{4\,a^{5/2}}+\frac {5\,\sqrt {b\,x-a}}{4\,a\,x^2}+\frac {3\,{\left (b\,x-a\right )}^{3/2}}{4\,a^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.20, size = 216, normalized size = 2.92 \begin {gather*} \begin {cases} \frac {i}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} - 1}} + \frac {i \sqrt {b}}{4 a x^{\frac {3}{2}} \sqrt {\frac {a}{b x} - 1}} - \frac {3 i b^{\frac {3}{2}}}{4 a^{2} \sqrt {x} \sqrt {\frac {a}{b x} - 1}} + \frac {3 i b^{2} \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {5}{2}}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {1}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {- \frac {a}{b x} + 1}} - \frac {\sqrt {b}}{4 a x^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1}} + \frac {3 b^{\frac {3}{2}}}{4 a^{2} \sqrt {x} \sqrt {- \frac {a}{b x} + 1}} - \frac {3 b^{2} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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