Optimal. Leaf size=26 \[ -\frac{\sqrt{c x^2} (a+b x)^2}{2 a x^3} \]
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Rubi [A] time = 0.0040421, antiderivative size = 26, 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, 37} \[ -\frac{\sqrt{c x^2} (a+b x)^2}{2 a x^3} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2} (a+b x)}{x^4} \, dx &=\frac{\sqrt{c x^2} \int \frac{a+b x}{x^3} \, dx}{x}\\ &=-\frac{\sqrt{c x^2} (a+b x)^2}{2 a x^3}\\ \end{align*}
Mathematica [A] time = 0.0042912, size = 22, normalized size = 0.85 \[ -\frac{\sqrt{c x^2} (a+2 b x)}{2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 19, normalized size = 0.7 \begin{align*} -{\frac{2\,bx+a}{2\,{x}^{3}}\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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45878, size = 46, normalized size = 1.77 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )}}{2 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.482106, size = 36, normalized size = 1.38 \begin{align*} - \frac{a \sqrt{c} \sqrt{x^{2}}}{2 x^{3}} - \frac{b \sqrt{c} \sqrt{x^{2}}}{x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06677, size = 26, normalized size = 1. \begin{align*} -\frac{{\left (2 \, b x \mathrm{sgn}\left (x\right ) + a \mathrm{sgn}\left (x\right )\right )} \sqrt{c}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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