Optimal. Leaf size=61 \[ \frac{a^2 x \log (x)}{c \sqrt{c x^2}}+\frac{2 a b x^2}{c \sqrt{c x^2}}+\frac{b^2 x^3}{2 c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0109616, antiderivative size = 61, 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} \[ \frac{a^2 x \log (x)}{c \sqrt{c x^2}}+\frac{2 a b x^2}{c \sqrt{c x^2}}+\frac{b^2 x^3}{2 c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 (a+b x)^2}{\left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx}{c \sqrt{c x^2}}\\ &=\frac{2 a b x^2}{c \sqrt{c x^2}}+\frac{b^2 x^3}{2 c \sqrt{c x^2}}+\frac{a^2 x \log (x)}{c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0072139, size = 34, normalized size = 0.56 \[ \frac{x^3 \left (2 a^2 \log (x)+b x (4 a+b x)\right )}{2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 33, normalized size = 0.5 \begin{align*}{\frac{{x}^{3} \left ({b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( x \right ) +4\,abx \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 = 1.04843, size = 61, normalized size = 1. \begin{align*} \frac{b^{2} x^{3}}{2 \, \sqrt{c x^{2}} c} + \frac{2 \, a b x^{2}}{\sqrt{c x^{2}} c} + \frac{a^{2} \log \left (x\right )}{c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54526, size = 81, normalized size = 1.33 \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 \, c^{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{x^{2} \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 [A] time = 1.07647, size = 74, normalized size = 1.21 \begin{align*} -\frac{\frac{2 \, a^{2} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{\sqrt{c}} - \sqrt{c x^{2}}{\left (\frac{b^{2} x}{c} + \frac{4 \, a b}{c}\right )}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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