Optimal. Leaf size=74 \[ -\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^3}+\frac{8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}-\frac{2 \left (b x+c x^2\right )^{3/2}}{7 b x^5} \]
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Rubi [A] time = 0.0282151, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ -\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^3}+\frac{8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}-\frac{2 \left (b x+c x^2\right )^{3/2}}{7 b x^5} \]
Antiderivative was successfully verified.
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Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{\sqrt{b x+c x^2}}{x^5} \, dx &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}-\frac{(4 c) \int \frac{\sqrt{b x+c x^2}}{x^4} \, dx}{7 b}\\ &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}+\frac{8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}+\frac{\left (8 c^2\right ) \int \frac{\sqrt{b x+c x^2}}{x^3} \, dx}{35 b^2}\\ &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}+\frac{8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0108996, size = 51, normalized size = 0.69 \[ -\frac{2 \sqrt{x (b+c x)} \left (3 b^2 c x+15 b^3-4 b c^2 x^2+8 c^3 x^3\right )}{105 b^3 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 44, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 8\,{c}^{2}{x}^{2}-12\,bcx+15\,{b}^{2} \right ) }{105\,{b}^{3}{x}^{4}}\sqrt{c{x}^{2}+bx}} \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 = 1.99353, size = 112, normalized size = 1.51 \begin{align*} -\frac{2 \,{\left (8 \, c^{3} x^{3} - 4 \, b c^{2} x^{2} + 3 \, b^{2} c x + 15 \, b^{3}\right )} \sqrt{c x^{2} + b x}}{105 \, b^{3} x^{4}} \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{\sqrt{x \left (b + c x\right )}}{x^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34038, size = 184, normalized size = 2.49 \begin{align*} \frac{2 \,{\left (140 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} c^{2} + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b c^{\frac{3}{2}} + 273 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{2} c + 105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{3} \sqrt{c} + 15 \, b^{4}\right )}}{105 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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