Optimal. Leaf size=60 \[ \frac{1}{5} a^2 c x^4 \sqrt{c x^2}+\frac{1}{3} a b c x^5 \sqrt{c x^2}+\frac{1}{7} b^2 c x^6 \sqrt{c x^2} \]
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Rubi [A] time = 0.0161078, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{1}{5} a^2 c x^4 \sqrt{c x^2}+\frac{1}{3} a b c x^5 \sqrt{c x^2}+\frac{1}{7} b^2 c x^6 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int x \left (c x^2\right )^{3/2} (a+b x)^2 \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^4 (a+b x)^2 \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a^2 x^4+2 a b x^5+b^2 x^6\right ) \, dx}{x}\\ &=\frac{1}{5} a^2 c x^4 \sqrt{c x^2}+\frac{1}{3} a b c x^5 \sqrt{c x^2}+\frac{1}{7} b^2 c x^6 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0075748, size = 35, normalized size = 0.58 \[ \frac{1}{105} x^2 \left (c x^2\right )^{3/2} \left (21 a^2+35 a b x+15 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 32, normalized size = 0.5 \begin{align*}{\frac{{x}^{2} \left ( 15\,{b}^{2}{x}^{2}+35\,abx+21\,{a}^{2} \right ) }{105} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42549, size = 86, normalized size = 1.43 \begin{align*} \frac{1}{105} \,{\left (15 \, b^{2} c x^{6} + 35 \, a b c x^{5} + 21 \, a^{2} c x^{4}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.877928, size = 60, normalized size = 1. \begin{align*} \frac{a^{2} c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} + \frac{a b c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{3} + \frac{b^{2} c^{\frac{3}{2}} x^{4} \left (x^{2}\right )^{\frac{3}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0684, size = 47, normalized size = 0.78 \begin{align*} \frac{1}{105} \,{\left (15 \, b^{2} x^{7} \mathrm{sgn}\left (x\right ) + 35 \, a b x^{6} \mathrm{sgn}\left (x\right ) + 21 \, a^{2} x^{5} \mathrm{sgn}\left (x\right )\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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