Optimal. Leaf size=60 \[ \frac{1}{6} a^2 c x^5 \sqrt{c x^2}+\frac{2}{7} a b c x^6 \sqrt{c x^2}+\frac{1}{8} b^2 c x^7 \sqrt{c x^2} \]
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Rubi [A] time = 0.017045, antiderivative size = 60, 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} \[ \frac{1}{6} a^2 c x^5 \sqrt{c x^2}+\frac{2}{7} a b c x^6 \sqrt{c x^2}+\frac{1}{8} b^2 c x^7 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int x^2 \left (c x^2\right )^{3/2} (a+b x)^2 \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^5 (a+b x)^2 \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a^2 x^5+2 a b x^6+b^2 x^7\right ) \, dx}{x}\\ &=\frac{1}{6} a^2 c x^5 \sqrt{c x^2}+\frac{2}{7} a b c x^6 \sqrt{c x^2}+\frac{1}{8} b^2 c x^7 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0088444, size = 35, normalized size = 0.58 \[ \frac{1}{168} x^3 \left (c x^2\right )^{3/2} \left (28 a^2+48 a b x+21 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 0.5 \begin{align*}{\frac{{x}^{3} \left ( 21\,{b}^{2}{x}^{2}+48\,abx+28\,{a}^{2} \right ) }{168} \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 = 1.45785, size = 86, normalized size = 1.43 \begin{align*} \frac{1}{168} \,{\left (21 \, b^{2} c x^{7} + 48 \, a b c x^{6} + 28 \, a^{2} c x^{5}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.13822, size = 61, normalized size = 1.02 \begin{align*} \frac{a^{2} c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{6} + \frac{2 a b c^{\frac{3}{2}} x^{4} \left (x^{2}\right )^{\frac{3}{2}}}{7} + \frac{b^{2} c^{\frac{3}{2}} x^{5} \left (x^{2}\right )^{\frac{3}{2}}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05139, size = 47, normalized size = 0.78 \begin{align*} \frac{1}{168} \,{\left (21 \, b^{2} x^{8} \mathrm{sgn}\left (x\right ) + 48 \, a b x^{7} \mathrm{sgn}\left (x\right ) + 28 \, a^{2} x^{6} \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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