Optimal. Leaf size=56 \[ \frac{4 b \sqrt{a x^2+b x^3}}{3 a^2 x^{3/2}}-\frac{2 \sqrt{a x^2+b x^3}}{3 a x^{5/2}} \]
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Rubi [A] time = 0.0766247, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2016, 2014} \[ \frac{4 b \sqrt{a x^2+b x^3}}{3 a^2 x^{3/2}}-\frac{2 \sqrt{a x^2+b x^3}}{3 a x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{x^{3/2} \sqrt{a x^2+b x^3}} \, dx &=-\frac{2 \sqrt{a x^2+b x^3}}{3 a x^{5/2}}-\frac{(2 b) \int \frac{1}{\sqrt{x} \sqrt{a x^2+b x^3}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{a x^2+b x^3}}{3 a x^{5/2}}+\frac{4 b \sqrt{a x^2+b x^3}}{3 a^2 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0150395, size = 31, normalized size = 0.55 \[ -\frac{2 (a-2 b x) \sqrt{x^2 (a+b x)}}{3 a^2 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 33, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,bx+2\,a \right ) \left ( -2\,bx+a \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{3} + a x^{2}} x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.890541, size = 69, normalized size = 1.23 \begin{align*} \frac{2 \, \sqrt{b x^{3} + a x^{2}}{\left (2 \, b x - a\right )}}{3 \, a^{2} x^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{3}{2}} \sqrt{x^{2} \left (a + b x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20807, size = 74, normalized size = 1.32 \begin{align*} \frac{8 \,{\left (3 \,{\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a\right )} b^{\frac{3}{2}}}{3 \,{\left ({\left (\sqrt{b} \sqrt{x} - \sqrt{b x + a}\right )}^{2} - a\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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