Optimal. Leaf size=48 \[ \frac{4 b \sqrt{a x+b x^4}}{9 a^2 x^2}-\frac{2 \sqrt{a x+b x^4}}{9 a x^5} \]
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Rubi [A] time = 0.0664935, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2016, 2014} \[ \frac{4 b \sqrt{a x+b x^4}}{9 a^2 x^2}-\frac{2 \sqrt{a x+b x^4}}{9 a x^5} \]
Antiderivative was successfully verified.
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Rule 2016
Rule 2014
Rubi steps
\begin{align*} \int \frac{1}{x^5 \sqrt{a x+b x^4}} \, dx &=-\frac{2 \sqrt{a x+b x^4}}{9 a x^5}-\frac{(2 b) \int \frac{1}{x^2 \sqrt{a x+b x^4}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{a x+b x^4}}{9 a x^5}+\frac{4 b \sqrt{a x+b x^4}}{9 a^2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0137281, size = 31, normalized size = 0.65 \[ -\frac{2 \left (a-2 b x^3\right ) \sqrt{x \left (a+b x^3\right )}}{9 a^2 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 35, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,b{x}^{3}+2\,a \right ) \left ( -2\,b{x}^{3}+a \right ) }{9\,{a}^{2}{x}^{4}}{\frac{1}{\sqrt{b{x}^{4}+ax}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01426, size = 51, normalized size = 1.06 \begin{align*} \frac{2 \,{\left (2 \, b^{2} x^{7} + a b x^{4} - a^{2} x\right )}}{9 \, \sqrt{b x^{3} + a} a^{2} x^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19889, size = 63, normalized size = 1.31 \begin{align*} \frac{2 \, \sqrt{b x^{4} + a x}{\left (2 \, b x^{3} - a\right )}}{9 \, a^{2} x^{5}} \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^{5} \sqrt{x \left (a + b x^{3}\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19973, size = 36, normalized size = 0.75 \begin{align*} -\frac{2 \,{\left ({\left (b + \frac{a}{x^{3}}\right )}^{\frac{3}{2}} - 3 \, \sqrt{b + \frac{a}{x^{3}}} b\right )}}{9 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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