Optimal. Leaf size=58 \[ -\frac{a^2}{2 c x \sqrt{c x^2}}-\frac{2 a b}{c \sqrt{c x^2}}+\frac{b^2 x \log (x)}{c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0122137, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {15, 43} \[ -\frac{a^2}{2 c x \sqrt{c x^2}}-\frac{2 a b}{c \sqrt{c x^2}}+\frac{b^2 x \log (x)}{c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^3} \, dx}{c \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}{c \sqrt{c x^2}}\\ &=-\frac{2 a b}{c \sqrt{c x^2}}-\frac{a^2}{2 c x \sqrt{c x^2}}+\frac{b^2 x \log (x)}{c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0041649, size = 34, normalized size = 0.59 \[ \frac{x \left (2 b^2 x^2 \log (x)-a (a+4 b x)\right )}{2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 32, normalized size = 0.6 \begin{align*}{\frac{x \left ( 2\,{b}^{2}\ln \left ( x \right ){x}^{2}-4\,abx-{a}^{2} \right ) }{2} \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 = 1.05825, size = 47, normalized size = 0.81 \begin{align*} \frac{b^{2} \log \left (x\right )}{c^{\frac{3}{2}}} - \frac{2 \, a b}{\sqrt{c x^{2}} c} - \frac{a^{2}}{2 \, c^{\frac{3}{2}} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58561, size = 84, normalized size = 1.45 \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^{2} 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}}{\left (c x^{2}\right )^{\frac{3}{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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