Optimal. Leaf size=100 \[ \frac{32 c^3 \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6} \]
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Rubi [A] time = 0.0407514, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {658, 650} \[ \frac{32 c^3 \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6} \]
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^6} \, dx &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}-\frac{(2 c) \int \frac{\sqrt{b x+c x^2}}{x^5} \, dx}{3 b}\\ &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}+\frac{\left (8 c^2\right ) \int \frac{\sqrt{b x+c x^2}}{x^4} \, dx}{21 b^2}\\ &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}-\frac{\left (16 c^3\right ) \int \frac{\sqrt{b x+c x^2}}{x^3} \, dx}{105 b^3}\\ &=-\frac{2 \left (b x+c x^2\right )^{3/2}}{9 b x^6}+\frac{4 c \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}-\frac{16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}+\frac{32 c^3 \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3}\\ \end{align*}
Mathematica [A] time = 0.0125828, size = 62, normalized size = 0.62 \[ \frac{2 \sqrt{x (b+c x)} \left (6 b^2 c^2 x^2-5 b^3 c x-35 b^4-8 b c^3 x^3+16 c^4 x^4\right )}{315 b^4 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 55, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -16\,{x}^{3}{c}^{3}+24\,b{x}^{2}{c}^{2}-30\,{b}^{2}xc+35\,{b}^{3} \right ) }{315\,{b}^{4}{x}^{5}}\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 = 2.08463, size = 134, normalized size = 1.34 \begin{align*} \frac{2 \,{\left (16 \, c^{4} x^{4} - 8 \, b c^{3} x^{3} + 6 \, b^{2} c^{2} x^{2} - 5 \, b^{3} c x - 35 \, b^{4}\right )} \sqrt{c x^{2} + b x}}{315 \, b^{4} 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{\sqrt{x \left (b + c x\right )}}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35586, size = 223, normalized size = 2.23 \begin{align*} \frac{2 \,{\left (630 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5} c^{\frac{5}{2}} + 1764 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{4} b c^{2} + 1995 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3} b^{2} c^{\frac{3}{2}} + 1125 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} b^{3} c + 315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b^{4} \sqrt{c} + 35 \, b^{5}\right )}}{315 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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