Optimal. Leaf size=57 \[ \frac{a^2 x^4}{3 \sqrt{c x^2}}+\frac{a b x^5}{2 \sqrt{c x^2}}+\frac{b^2 x^6}{5 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0132603, 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} \[ \frac{a^2 x^4}{3 \sqrt{c x^2}}+\frac{a b x^5}{2 \sqrt{c x^2}}+\frac{b^2 x^6}{5 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3 (a+b x)^2}{\sqrt{c x^2}} \, dx &=\frac{x \int x^2 (a+b x)^2 \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (a^2 x^2+2 a b x^3+b^2 x^4\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{a^2 x^4}{3 \sqrt{c x^2}}+\frac{a b x^5}{2 \sqrt{c x^2}}+\frac{b^2 x^6}{5 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0060105, size = 35, normalized size = 0.61 \[ \frac{x^4 \left (10 a^2+15 a b x+6 b^2 x^2\right )}{30 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.6 \begin{align*}{\frac{{x}^{4} \left ( 6\,{b}^{2}{x}^{2}+15\,abx+10\,{a}^{2} \right ) }{30}{\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 = 1.03058, size = 73, normalized size = 1.28 \begin{align*} \frac{\sqrt{c x^{2}} b^{2} x^{4}}{5 \, c} + \frac{\sqrt{c x^{2}} a b x^{3}}{2 \, c} + \frac{\sqrt{c x^{2}} a^{2} x^{2}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44338, size = 78, normalized size = 1.37 \begin{align*} \frac{{\left (6 \, b^{2} x^{4} + 15 \, a b x^{3} + 10 \, a^{2} x^{2}\right )} \sqrt{c x^{2}}}{30 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.695983, size = 60, normalized size = 1.05 \begin{align*} \frac{a^{2} x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} + \frac{a b x^{5}}{2 \sqrt{c} \sqrt{x^{2}}} + \frac{b^{2} x^{6}}{5 \sqrt{c} \sqrt{x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06773, size = 55, normalized size = 0.96 \begin{align*} \frac{1}{30} \, \sqrt{c x^{2}}{\left (3 \,{\left (\frac{2 \, b^{2} x}{c} + \frac{5 \, a b}{c}\right )} x + \frac{10 \, a^{2}}{c}\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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