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 = 0.0046125, antiderivative size = 43, normalized size of antiderivative = 1., 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} \[ \frac{4 \sqrt{x}}{3 a^2 \sqrt{a+b x}}+\frac{2 \sqrt{x}}{3 a (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
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*}
Mathematica [A] time = 0.0117225, size = 29, normalized size = 0.67 \[ \frac{2 \sqrt{x} (3 a+2 b x)}{3 a^2 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 24, normalized size = 0.6 \begin{align*}{\frac{4\,bx+6\,a}{3\,{a}^{2}}\sqrt{x} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10958, size = 36, normalized size = 0.84 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16803, size = 99, normalized size = 2.3 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.69002, size = 92, normalized size = 2.14 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15429, size = 109, normalized size = 2.53 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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