Optimal. Leaf size=27 \[ \frac{c \sqrt{c x^2} (a+b x)^3}{3 b x} \]
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Rubi [A] time = 0.0044034, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 32} \[ \frac{c \sqrt{c x^2} (a+b x)^3}{3 b x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2} (a+b x)^2}{x^3} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int (a+b x)^2 \, dx}{x}\\ &=\frac{c \sqrt{c x^2} (a+b x)^3}{3 b x}\\ \end{align*}
Mathematica [A] time = 0.0052405, size = 26, normalized size = 0.96 \[ \frac{\left (c x^2\right )^{3/2} (a+b x)^3}{3 b x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 31, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}{x}^{2}+3\,abx+3\,{a}^{2}}{3\,{x}^{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 [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 = 1.43776, size = 69, normalized size = 2.56 \begin{align*} \frac{1}{3} \,{\left (b^{2} c x^{2} + 3 \, a b c x + 3 \, a^{2} c\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.854937, size = 51, normalized size = 1.89 \begin{align*} \frac{a^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x^{2}} + \frac{a b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x} + \frac{b^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06025, size = 39, normalized size = 1.44 \begin{align*} \frac{1}{3} \,{\left (\frac{{\left (b x + a\right )}^{3} \mathrm{sgn}\left (x\right )}{b} - \frac{a^{3} \mathrm{sgn}\left (x\right )}{b}\right )} c^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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