Optimal. Leaf size=80 \[ \frac{16 a^2 \sqrt{a x^2+b x^5}}{45 b^3 x}-\frac{8 a x^2 \sqrt{a x^2+b x^5}}{45 b^2}+\frac{2 x^5 \sqrt{a x^2+b x^5}}{15 b} \]
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Rubi [A] time = 0.114813, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2016, 1588} \[ \frac{16 a^2 \sqrt{a x^2+b x^5}}{45 b^3 x}-\frac{8 a x^2 \sqrt{a x^2+b x^5}}{45 b^2}+\frac{2 x^5 \sqrt{a x^2+b x^5}}{15 b} \]
Antiderivative was successfully verified.
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Rule 2016
Rule 1588
Rubi steps
\begin{align*} \int \frac{x^9}{\sqrt{a x^2+b x^5}} \, dx &=\frac{2 x^5 \sqrt{a x^2+b x^5}}{15 b}-\frac{(4 a) \int \frac{x^6}{\sqrt{a x^2+b x^5}} \, dx}{5 b}\\ &=-\frac{8 a x^2 \sqrt{a x^2+b x^5}}{45 b^2}+\frac{2 x^5 \sqrt{a x^2+b x^5}}{15 b}+\frac{\left (8 a^2\right ) \int \frac{x^3}{\sqrt{a x^2+b x^5}} \, dx}{15 b^2}\\ &=\frac{16 a^2 \sqrt{a x^2+b x^5}}{45 b^3 x}-\frac{8 a x^2 \sqrt{a x^2+b x^5}}{45 b^2}+\frac{2 x^5 \sqrt{a x^2+b x^5}}{15 b}\\ \end{align*}
Mathematica [A] time = 0.0246324, size = 46, normalized size = 0.57 \[ \frac{2 \sqrt{x^2 \left (a+b x^3\right )} \left (8 a^2-4 a b x^3+3 b^2 x^6\right )}{45 b^3 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 48, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,b{x}^{3}+2\,a \right ) \left ( 3\,{b}^{2}{x}^{6}-4\,ab{x}^{3}+8\,{a}^{2} \right ) x}{45\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0347, size = 62, normalized size = 0.78 \begin{align*} \frac{2 \,{\left (3 \, b^{3} x^{9} - a b^{2} x^{6} + 4 \, a^{2} b x^{3} + 8 \, a^{3}\right )}}{45 \, \sqrt{b x^{3} + a} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.65072, size = 89, normalized size = 1.11 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{6} - 4 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{b x^{5} + a x^{2}}}{45 \, b^{3} x} \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^{9}}{\sqrt{x^{2} \left (a + b x^{3}\right )}}\, 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^{9}}{\sqrt{b x^{5} + a x^{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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