Optimal. Leaf size=23 \[ -\frac{2 \left (b x+c x^2\right )^{5/2}}{5 b x^5} \]
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Rubi [A] time = 0.0075107, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {650} \[ -\frac{2 \left (b x+c x^2\right )^{5/2}}{5 b x^5} \]
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^{3/2}}{x^5} \, dx &=-\frac{2 \left (b x+c x^2\right )^{5/2}}{5 b x^5}\\ \end{align*}
Mathematica [A] time = 0.0120201, size = 21, normalized size = 0.91 \[ -\frac{2 (x (b+c x))^{5/2}}{5 b x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 25, normalized size = 1.1 \begin{align*} -{\frac{2\,cx+2\,b}{5\,b{x}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98778, size = 78, normalized size = 3.39 \begin{align*} -\frac{2 \,{\left (c^{2} x^{2} + 2 \, b c x + b^{2}\right )} \sqrt{c x^{2} + b x}}{5 \, b x^{3}} \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{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25282, size = 181, normalized size = 7.87 \begin{align*} \frac{2 \,{\left (5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} c^{2} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c^{\frac{3}{2}} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} c + 5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} \sqrt{c} + b^{4}\right )}}{5 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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