Optimal. Leaf size=51 \[ -\frac{2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
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Rubi [A] time = 0.0736555, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2015, 2014} \[ -\frac{2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}-\frac{x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2015
Rule 2014
Rubi steps
\begin{align*} \int \frac{x^{15/2}}{\left (a x+b x^3\right )^{9/2}} \, dx &=-\frac{x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}}+\frac{2 \int \frac{x^{9/2}}{\left (a x+b x^3\right )^{7/2}} \, dx}{7 b}\\ &=-\frac{x^{11/2}}{7 b \left (a x+b x^3\right )^{7/2}}-\frac{2 x^{5/2}}{35 b^2 \left (a x+b x^3\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0195845, size = 44, normalized size = 0.86 \[ -\frac{\sqrt{x} \left (2 a+7 b x^2\right )}{35 b^2 \left (a+b x^2\right )^3 \sqrt{x \left (a+b x^2\right )}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 37, normalized size = 0.7 \begin{align*} -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 7\,b{x}^{2}+2\,a \right ) }{35\,{b}^{2}}{x}^{{\frac{9}{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{15}{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 = 1.59191, size = 158, normalized size = 3.1 \begin{align*} -\frac{\sqrt{b x^{3} + a x}{\left (7 \, b x^{2} + 2 \, a\right )} \sqrt{x}}{35 \,{\left (b^{6} x^{9} + 4 \, a b^{5} x^{7} + 6 \, a^{2} b^{4} x^{5} + 4 \, a^{3} b^{3} x^{3} + a^{4} b^{2} x\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 = 1.26349, size = 45, normalized size = 0.88 \begin{align*} -\frac{7 \, b x^{2} + 2 \, a}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} + \frac{2}{35 \, a^{\frac{5}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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