Optimal. Leaf size=22 \[ -\frac{x}{b \sqrt{c x^2} (a+b x)} \]
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Rubi [A] time = 0.0033827, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 32} \[ -\frac{x}{b \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{c x^2} (a+b x)^2} \, dx &=\frac{x \int \frac{1}{(a+b x)^2} \, dx}{\sqrt{c x^2}}\\ &=-\frac{x}{b \sqrt{c x^2} (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0028723, size = 22, normalized size = 1. \[ -\frac{x}{b \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 21, normalized size = 1. \begin{align*} -{\frac{x}{b \left ( bx+a \right ) }{\frac{1}{\sqrt{c{x}^{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 = 1.33952, size = 49, normalized size = 2.23 \begin{align*} -\frac{\sqrt{c x^{2}}}{b^{2} c x^{2} + a b c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.11411, size = 85, normalized size = 3.86 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{c} \sqrt{x^{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{b^{2} \sqrt{c} \sqrt{x^{2}}} & \text{for}\: a = 0 \\\frac{\tilde{\infty } x^{2}}{\sqrt{c} \sqrt{x^{2}}} & \text{for}\: b = - \frac{a}{x} \\\frac{x^{2}}{a^{2} \sqrt{c} \sqrt{x^{2}} + a b \sqrt{c} x \sqrt{x^{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{c x^{2}}{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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