Optimal. Leaf size=49 \[ \frac{a^2 \sqrt{c x^2} \log (x)}{x}+2 a b \sqrt{c x^2}+\frac{1}{2} b^2 x \sqrt{c x^2} \]
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Rubi [A] time = 0.009421, 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} \[ \frac{a^2 \sqrt{c x^2} \log (x)}{x}+2 a b \sqrt{c x^2}+\frac{1}{2} b^2 x \sqrt{c x^2} \]
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^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{(a+b x)^2}{x} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx}{x}\\ &=2 a b \sqrt{c x^2}+\frac{1}{2} b^2 x \sqrt{c x^2}+\frac{a^2 \sqrt{c x^2} \log (x)}{x}\\ \end{align*}
Mathematica [A] time = 0.0097211, size = 33, normalized size = 0.67 \[ \frac{c x \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 33, normalized size = 0.7 \begin{align*}{\frac{{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( x \right ) +4\,abx}{2\,x}\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 = 1.58105, size = 73, normalized size = 1.49 \begin{align*} \frac{{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right )\right )} \sqrt{c x^{2}}}{2 \, x} \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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0843, size = 43, normalized size = 0.88 \begin{align*} \frac{1}{2} \,{\left (b^{2} x^{2} \mathrm{sgn}\left (x\right ) + 4 \, a b x \mathrm{sgn}\left (x\right ) + 2 \, a^{2} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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