Optimal. Leaf size=48 \[ \frac{4 c \sqrt{b x+c x^2}}{3 b^2 x}-\frac{2 \sqrt{b x+c x^2}}{3 b x^2} \]
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Rubi [A] time = 0.0163138, 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 = {658, 650} \[ \frac{4 c \sqrt{b x+c x^2}}{3 b^2 x}-\frac{2 \sqrt{b x+c x^2}}{3 b x^2} \]
Antiderivative was successfully verified.
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Rule 658
Rule 650
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{b x+c x^2}} \, dx &=-\frac{2 \sqrt{b x+c x^2}}{3 b x^2}-\frac{(2 c) \int \frac{1}{x \sqrt{b x+c x^2}} \, dx}{3 b}\\ &=-\frac{2 \sqrt{b x+c x^2}}{3 b x^2}+\frac{4 c \sqrt{b x+c x^2}}{3 b^2 x}\\ \end{align*}
Mathematica [A] time = 0.0130494, size = 29, normalized size = 0.6 \[ \frac{2 \sqrt{x (b+c x)} (2 c x-b)}{3 b^2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 31, normalized size = 0.7 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -2\,cx+b \right ) }{3\,{b}^{2}x}{\frac{1}{\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 = 2.01151, size = 61, normalized size = 1.27 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x - b\right )}}{3 \, b^{2} x^{2}} \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^{2} \sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22839, size = 66, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} + b\right )}}{3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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