Optimal. Leaf size=48 \[ \frac{2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac{8 (a+2 b x)}{3 a^3 \sqrt{a x+b x^2}} \]
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Rubi [A] time = 0.0111226, antiderivative size = 48, 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 = {638, 613} \[ \frac{2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac{8 (a+2 b x)}{3 a^3 \sqrt{a x+b x^2}} \]
Antiderivative was successfully verified.
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Rule 638
Rule 613
Rubi steps
\begin{align*} \int \frac{x}{\left (a x+b x^2\right )^{5/2}} \, dx &=\frac{2 x}{3 a \left (a x+b x^2\right )^{3/2}}+\frac{4 \int \frac{1}{\left (a x+b x^2\right )^{3/2}} \, dx}{3 a}\\ &=\frac{2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac{8 (a+2 b x)}{3 a^3 \sqrt{a x+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0128015, size = 38, normalized size = 0.79 \[ -\frac{2 x \left (3 a^2+12 a b x+8 b^2 x^2\right )}{3 a^3 (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 44, normalized size = 0.9 \begin{align*} -{\frac{2\,{x}^{2} \left ( bx+a \right ) \left ( 8\,{b}^{2}{x}^{2}+12\,bxa+3\,{a}^{2} \right ) }{3\,{a}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04203, size = 70, normalized size = 1.46 \begin{align*} \frac{2 \, x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} a} - \frac{16 \, b x}{3 \, \sqrt{b x^{2} + a x} a^{3}} - \frac{8}{3 \, \sqrt{b x^{2} + a x} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86979, size = 123, normalized size = 2.56 \begin{align*} -\frac{2 \,{\left (8 \, b^{2} x^{2} + 12 \, a b x + 3 \, a^{2}\right )} \sqrt{b x^{2} + a x}}{3 \,{\left (a^{3} b^{2} x^{3} + 2 \, a^{4} b x^{2} + a^{5} x\right )}} \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{x}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b x^{2} + a x\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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