Optimal. Leaf size=58 \[ \frac{1}{2} a^2 c x \sqrt{c x^2}+\frac{2}{3} a b c x^2 \sqrt{c x^2}+\frac{1}{4} b^2 c x^3 \sqrt{c x^2} \]
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Rubi [A] time = 0.0131542, antiderivative size = 58, 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} \[ \frac{1}{2} a^2 c x \sqrt{c x^2}+\frac{2}{3} a b c x^2 \sqrt{c x^2}+\frac{1}{4} b^2 c x^3 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2} (a+b x)^2}{x^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x (a+b x)^2 \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{x}\\ &=\frac{1}{2} a^2 c x \sqrt{c x^2}+\frac{2}{3} a b c x^2 \sqrt{c x^2}+\frac{1}{4} b^2 c x^3 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0036646, size = 34, normalized size = 0.59 \[ \frac{1}{12} c x \sqrt{c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 32, normalized size = 0.6 \begin{align*}{\frac{3\,{b}^{2}{x}^{2}+8\,abx+6\,{a}^{2}}{12\,x} \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 = 1.41586, size = 78, normalized size = 1.34 \begin{align*} \frac{1}{12} \,{\left (3 \, b^{2} c x^{3} + 8 \, a b c x^{2} + 6 \, a^{2} c x\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.709454, size = 54, normalized size = 0.93 \begin{align*} \frac{a^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{2 x} + \frac{2 a b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} + \frac{b^{2} c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05584, size = 47, normalized size = 0.81 \begin{align*} \frac{1}{12} \,{\left (3 \, b^{2} x^{4} \mathrm{sgn}\left (x\right ) + 8 \, a b x^{3} \mathrm{sgn}\left (x\right ) + 6 \, a^{2} x^{2} \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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