Optimal. Leaf size=126 \[ \frac{8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{16 \sqrt{x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{64}{35 a^4 \sqrt{x} \sqrt{a x+b x^3}}-\frac{128 \sqrt{a x+b x^3}}{35 a^5 x^{3/2}}+\frac{x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.192445, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2015, 2014} \[ \frac{8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{16 \sqrt{x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{64}{35 a^4 \sqrt{x} \sqrt{a x+b x^3}}-\frac{128 \sqrt{a x+b x^3}}{35 a^5 x^{3/2}}+\frac{x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2015
Rule 2014
Rubi steps
\begin{align*} \int \frac{x^{5/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=\frac{x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{8 \int \frac{x^{3/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 a}\\ &=\frac{x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{48 \int \frac{\sqrt{x}}{\left (a x+b x^3\right )^{5/2}} \, dx}{35 a^2}\\ &=\frac{x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{16 \sqrt{x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{64 \int \frac{1}{\sqrt{x} \left (a x+b x^3\right )^{3/2}} \, dx}{35 a^3}\\ &=\frac{x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{16 \sqrt{x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{64}{35 a^4 \sqrt{x} \sqrt{a x+b x^3}}+\frac{128 \int \frac{1}{x^{3/2} \sqrt{a x+b x^3}} \, dx}{35 a^4}\\ &=\frac{x^{5/2}}{7 a \left (a x+b x^3\right )^{7/2}}+\frac{8 x^{3/2}}{35 a^2 \left (a x+b x^3\right )^{5/2}}+\frac{16 \sqrt{x}}{35 a^3 \left (a x+b x^3\right )^{3/2}}+\frac{64}{35 a^4 \sqrt{x} \sqrt{a x+b x^3}}-\frac{128 \sqrt{a x+b x^3}}{35 a^5 x^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0297992, size = 77, normalized size = 0.61 \[ -\frac{\sqrt{x \left (a+b x^2\right )} \left (560 a^2 b^2 x^4+280 a^3 b x^2+35 a^4+448 a b^3 x^6+128 b^4 x^8\right )}{35 a^5 x^{3/2} \left (a+b x^2\right )^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 70, normalized size = 0.6 \begin{align*} -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 128\,{b}^{4}{x}^{8}+448\,{b}^{3}{x}^{6}a+560\,{b}^{2}{x}^{4}{a}^{2}+280\,b{x}^{2}{a}^{3}+35\,{a}^{4} \right ) }{35\,{a}^{5}}{x}^{{\frac{7}{2}}} \left ( b{x}^{3}+ax \right ) ^{-{\frac{9}{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{x^{\frac{5}{2}}}{{\left (b x^{3} + a x\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67136, size = 239, normalized size = 1.9 \begin{align*} -\frac{{\left (128 \, b^{4} x^{8} + 448 \, a b^{3} x^{6} + 560 \, a^{2} b^{2} x^{4} + 280 \, a^{3} b x^{2} + 35 \, a^{4}\right )} \sqrt{b x^{3} + a x} \sqrt{x}}{35 \,{\left (a^{5} b^{4} x^{10} + 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} + 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.44085, size = 96, normalized size = 0.76 \begin{align*} -\frac{{\left ({\left (x^{2}{\left (\frac{93 \, b^{4} x^{2}}{a^{5}} + \frac{308 \, b^{3}}{a^{4}}\right )} + \frac{350 \, b^{2}}{a^{3}}\right )} x^{2} + \frac{140 \, b}{a^{2}}\right )} x}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}}} - \frac{\sqrt{b + \frac{a}{x^{2}}}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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