Optimal. Leaf size=57 \[ \frac{a^2 x^3}{2 \sqrt{c x^2}}+\frac{2 a b x^4}{3 \sqrt{c x^2}}+\frac{b^2 x^5}{4 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0121125, 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} \[ \frac{a^2 x^3}{2 \sqrt{c x^2}}+\frac{2 a b x^4}{3 \sqrt{c x^2}}+\frac{b^2 x^5}{4 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2 (a+b x)^2}{\sqrt{c x^2}} \, dx &=\frac{x \int x (a+b x)^2 \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{a^2 x^3}{2 \sqrt{c x^2}}+\frac{2 a b x^4}{3 \sqrt{c x^2}}+\frac{b^2 x^5}{4 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0048516, size = 35, normalized size = 0.61 \[ \frac{x^3 \left (6 a^2+8 a b x+3 b^2 x^2\right )}{12 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ( 3\,{b}^{2}{x}^{2}+8\,abx+6\,{a}^{2} \right ) }{12}{\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 = 1.04988, size = 63, normalized size = 1.11 \begin{align*} \frac{\sqrt{c x^{2}} b^{2} x^{3}}{4 \, c} + \frac{2 \, \sqrt{c x^{2}} a b x^{2}}{3 \, c} + \frac{a^{2} x^{2}}{2 \, \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52525, size = 73, normalized size = 1.28 \begin{align*} \frac{{\left (3 \, b^{2} x^{3} + 8 \, a b x^{2} + 6 \, a^{2} x\right )} \sqrt{c x^{2}}}{12 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.59411, size = 61, normalized size = 1.07 \begin{align*} \frac{a^{2} x^{3}}{2 \sqrt{c} \sqrt{x^{2}}} + \frac{2 a b x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} + \frac{b^{2} x^{5}}{4 \sqrt{c} \sqrt{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08845, size = 51, normalized size = 0.89 \begin{align*} \frac{1}{12} \, \sqrt{c x^{2}}{\left ({\left (\frac{3 \, b^{2} x}{c} + \frac{8 \, a b}{c}\right )} x + \frac{6 \, a^{2}}{c}\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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