Optimal. Leaf size=44 \[ \frac {4 b \sqrt {a+b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}} \]
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Rubi [A] time = 0.00, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {45, 37} \begin {gather*} \frac {4 b \sqrt {a+b x}}{3 a^2 \sqrt {x}}-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} \sqrt {a+b x}} \, dx &=-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}-\frac {(2 b) \int \frac {1}{x^{3/2} \sqrt {a+b x}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {a+b x}}{3 a x^{3/2}}+\frac {4 b \sqrt {a+b x}}{3 a^2 \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.61 \begin {gather*} -\frac {2 (a-2 b x) \sqrt {a+b x}}{3 a^2 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 29, normalized size = 0.66 \begin {gather*} \frac {2 \sqrt {a+b x} (2 b x-a)}{3 a^2 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.15, size = 23, normalized size = 0.52 \begin {gather*} \frac {2 \, {\left (2 \, b x - a\right )} \sqrt {b x + a}}{3 \, a^{2} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.90, size = 50, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (\frac {2 \, {\left (b x + a\right )} b^{3}}{a^{2}} - \frac {3 \, b^{3}}{a}\right )} \sqrt {b x + a} b}{3 \, {\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}} {\left | b \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.50 \begin {gather*} -\frac {2 \sqrt {b x +a}\, \left (-2 b x +a \right )}{3 a^{2} x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 31, normalized size = 0.70 \begin {gather*} \frac {2 \, {\left (\frac {3 \, \sqrt {b x + a} b}{\sqrt {x}} - \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{x^{\frac {3}{2}}}\right )}}{3 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 25, normalized size = 0.57 \begin {gather*} -\frac {\left (\frac {2}{3\,a}-\frac {4\,b\,x}{3\,a^2}\right )\,\sqrt {a+b\,x}}{x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.92, size = 42, normalized size = 0.95 \begin {gather*} - \frac {2 \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 a x} + \frac {4 b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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