Optimal. Leaf size=49 \[ -\frac{a^2}{2 x \sqrt{c x^2}}-\frac{2 a b}{\sqrt{c x^2}}+\frac{b^2 x \log (x)}{\sqrt{c x^2}} \]
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Rubi [A] time = 0.0113112, 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} \[ -\frac{a^2}{2 x \sqrt{c x^2}}-\frac{2 a b}{\sqrt{c x^2}}+\frac{b^2 x \log (x)}{\sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x^2 \sqrt{c x^2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^3} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a^2}{x^3}+\frac{2 a b}{x^2}+\frac{b^2}{x}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{2 a b}{\sqrt{c x^2}}-\frac{a^2}{2 x \sqrt{c x^2}}+\frac{b^2 x \log (x)}{\sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0080317, size = 35, normalized size = 0.71 \[ \frac{c x \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 34, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-4\,abx-{a}^{2}}{2\,x}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08382, size = 42, normalized size = 0.86 \begin{align*} \frac{b^{2} \log \left (x\right )}{\sqrt{c}} - \frac{2 \, a b}{\sqrt{c} x} - \frac{a^{2}}{2 \, \sqrt{c} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46495, size = 81, normalized size = 1.65 \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 \, c 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{\left (a + b x\right )^{2}}{x^{2} \sqrt{c x^{2}}}\, dx \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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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