Optimal. Leaf size=54 \[ -\frac{a^2 \sqrt{c x^2}}{2 x^3}-\frac{2 a b \sqrt{c x^2}}{x^2}+\frac{b^2 \sqrt{c x^2} \log (x)}{x} \]
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Rubi [A] time = 0.0115592, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ -\frac{a^2 \sqrt{c x^2}}{2 x^3}-\frac{2 a b \sqrt{c x^2}}{x^2}+\frac{b^2 \sqrt{c x^2} \log (x)}{x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2} (a+b x)^2}{x^4} \, dx &=\frac{\sqrt{c x^2} \int \frac{(a+b x)^2}{x^3} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{a^2}{x^3}+\frac{2 a b}{x^2}+\frac{b^2}{x}\right ) \, dx}{x}\\ &=-\frac{a^2 \sqrt{c x^2}}{2 x^3}-\frac{2 a b \sqrt{c x^2}}{x^2}+\frac{b^2 \sqrt{c x^2} \log (x)}{x}\\ \end{align*}
Mathematica [A] time = 0.0092095, size = 36, normalized size = 0.67 \[ \frac{\sqrt{c x^2} \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 34, normalized size = 0.6 \begin{align*}{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-4\,abx-{a}^{2}}{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 = 1.60008, size = 76, normalized size = 1.41 \begin{align*} \frac{{\left (2 \, b^{2} x^{2} \log \left (x\right ) - 4 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, x^{3}} \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{c x^{2}} \left (a + b x\right )^{2}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06497, size = 47, normalized size = 0.87 \begin{align*} \frac{1}{2} \,{\left (2 \, b^{2} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right ) - \frac{4 \, a b x \mathrm{sgn}\left (x\right ) + a^{2} \mathrm{sgn}\left (x\right )}{x^{2}}\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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