Optimal. Leaf size=66 \[ -\frac{a^2}{4 c x^3 \sqrt{c x^2}}-\frac{2 a b}{3 c x^2 \sqrt{c x^2}}-\frac{b^2}{2 c x \sqrt{c x^2}} \]
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Rubi [A] time = 0.0128758, antiderivative size = 66, 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} \[ -\frac{a^2}{4 c x^3 \sqrt{c x^2}}-\frac{2 a b}{3 c x^2 \sqrt{c x^2}}-\frac{b^2}{2 c x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x^2 \left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^5} \, dx}{c \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}{c \sqrt{c x^2}}\\ &=-\frac{a^2}{4 c x^3 \sqrt{c x^2}}-\frac{2 a b}{3 c x^2 \sqrt{c x^2}}-\frac{b^2}{2 c x \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0095725, size = 38, normalized size = 0.58 \[ -\frac{\sqrt{c x^2} \left (3 a^2+8 a b x+6 b^2 x^2\right )}{12 c^2 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 32, normalized size = 0.5 \begin{align*} -{\frac{6\,{b}^{2}{x}^{2}+8\,abx+3\,{a}^{2}}{12\,x} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04647, size = 45, normalized size = 0.68 \begin{align*} -\frac{b^{2}}{2 \, c^{\frac{3}{2}} x^{2}} - \frac{2 \, a b}{3 \, c^{\frac{3}{2}} x^{3}} - \frac{a^{2}}{4 \, c^{\frac{3}{2}} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51039, size = 80, normalized size = 1.21 \begin{align*} -\frac{{\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} \sqrt{c x^{2}}}{12 \, c^{2} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.733367, size = 56, normalized size = 0.85 \begin{align*} - \frac{a^{2}}{4 c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}} - \frac{2 a b}{3 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b^{2} x}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{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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