Optimal. Leaf size=57 \[ -\frac{a^2}{4 x^3 \sqrt{c x^2}}-\frac{2 a b}{3 x^2 \sqrt{c x^2}}-\frac{b^2}{2 x \sqrt{c x^2}} \]
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Rubi [A] time = 0.0128204, antiderivative size = 57, 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} \[ -\frac{a^2}{4 x^3 \sqrt{c x^2}}-\frac{2 a b}{3 x^2 \sqrt{c x^2}}-\frac{b^2}{2 x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x^4 \sqrt{c x^2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^5} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a^2}{x^5}+\frac{2 a b}{x^4}+\frac{b^2}{x^3}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{a^2}{4 x^3 \sqrt{c x^2}}-\frac{2 a b}{3 x^2 \sqrt{c x^2}}-\frac{b^2}{2 x \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0080514, size = 35, normalized size = 0.61 \[ \frac{-3 a^2-8 a b x-6 b^2 x^2}{12 x^3 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.6 \begin{align*} -{\frac{6\,{b}^{2}{x}^{2}+8\,abx+3\,{a}^{2}}{12\,{x}^{3}}{\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 = 1.08714, size = 45, normalized size = 0.79 \begin{align*} -\frac{b^{2}}{2 \, \sqrt{c} x^{2}} - \frac{2 \, a b}{3 \, \sqrt{c} x^{3}} - \frac{a^{2}}{4 \, \sqrt{c} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60211, size = 77, normalized size = 1.35 \begin{align*} -\frac{{\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} \sqrt{c x^{2}}}{12 \, c x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.75472, size = 61, normalized size = 1.07 \begin{align*} - \frac{a^{2}}{4 \sqrt{c} x^{3} \sqrt{x^{2}}} - \frac{2 a b}{3 \sqrt{c} x^{2} \sqrt{x^{2}}} - \frac{b^{2}}{2 \sqrt{c} x \sqrt{x^{2}}} \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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