Optimal. Leaf size=49 \[ -\frac{a^2 \sqrt{c x^2}}{x^2}+\frac{2 a b \sqrt{c x^2} \log (x)}{x}+b^2 \sqrt{c x^2} \]
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Rubi [A] time = 0.0113116, antiderivative size = 49, 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} \[ -\frac{a^2 \sqrt{c x^2}}{x^2}+\frac{2 a b \sqrt{c x^2} \log (x)}{x}+b^2 \sqrt{c x^2} \]
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^3} \, dx &=\frac{\sqrt{c x^2} \int \frac{(a+b x)^2}{x^2} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (b^2+\frac{a^2}{x^2}+\frac{2 a b}{x}\right ) \, dx}{x}\\ &=b^2 \sqrt{c x^2}-\frac{a^2 \sqrt{c x^2}}{x^2}+\frac{2 a b \sqrt{c x^2} \log (x)}{x}\\ \end{align*}
Mathematica [A] time = 0.0119556, size = 31, normalized size = 0.63 \[ \frac{c \left (-a^2+2 a b x \log (x)+b^2 x^2\right )}{\sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 32, normalized size = 0.7 \begin{align*}{\frac{2\,ab\ln \left ( x \right ) x+{b}^{2}{x}^{2}-{a}^{2}}{{x}^{2}}\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 = 1.73218, size = 68, normalized size = 1.39 \begin{align*} \frac{{\left (b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}\right )} \sqrt{c x^{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{\sqrt{c x^{2}} \left (a + b x\right )^{2}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06002, size = 42, normalized size = 0.86 \begin{align*}{\left (b^{2} x \mathrm{sgn}\left (x\right ) + 2 \, a b \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right ) - \frac{a^{2} \mathrm{sgn}\left (x\right )}{x}\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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