Optimal. Leaf size=43 \[ \frac {4 \sqrt {x}}{3 a^2 \sqrt {a+b x}}+\frac {2 \sqrt {x}}{3 a (a+b x)^{3/2}} \]
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Rubi [A] time = 0.00, antiderivative size = 43, 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 \sqrt {x}}{3 a^2 \sqrt {a+b x}}+\frac {2 \sqrt {x}}{3 a (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x} (a+b x)^{5/2}} \, dx &=\frac {2 \sqrt {x}}{3 a (a+b x)^{3/2}}+\frac {2 \int \frac {1}{\sqrt {x} (a+b x)^{3/2}} \, dx}{3 a}\\ &=\frac {2 \sqrt {x}}{3 a (a+b x)^{3/2}}+\frac {4 \sqrt {x}}{3 a^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {x} (3 a+2 b x)}{3 a^2 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 29, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {x} (3 a+2 b x)}{3 a^2 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.49, size = 43, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (2 \, b x + 3 \, a\right )} \sqrt {b x + a} \sqrt {x}}{3 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.50, size = 81, normalized size = 1.88 \begin {gather*} \frac {8 \, {\left (3 \, {\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )} b^{\frac {5}{2}}}{3 \, {\left ({\left (\sqrt {b x + a} \sqrt {b} - \sqrt {{\left (b x + a\right )} b - a b}\right )}^{2} + a b\right )}^{3} {\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 = 24, normalized size = 0.56 \begin {gather*} \frac {2 \left (2 b x +3 a \right ) \sqrt {x}}{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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maxima [A] time = 1.34, size = 27, normalized size = 0.63 \begin {gather*} -\frac {2 \, {\left (b - \frac {3 \, {\left (b x + a\right )}}{x}\right )} x^{\frac {3}{2}}}{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 = 0.40, size = 54, normalized size = 1.26 \begin {gather*} \frac {6\,a\,\sqrt {x}\,\sqrt {a+b\,x}+4\,b\,x^{3/2}\,\sqrt {a+b\,x}}{3\,a^4+6\,a^3\,b\,x+3\,a^2\,b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.90, size = 92, normalized size = 2.14 \begin {gather*} \frac {6 a}{3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} + 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} + 1}} + \frac {4 b x}{3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} + 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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