Optimal. Leaf size=60 \[ \frac{1}{3} a^2 c x^2 \sqrt{c x^2}+\frac{1}{2} a b c x^3 \sqrt{c x^2}+\frac{1}{5} b^2 c x^4 \sqrt{c x^2} \]
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Rubi [A] time = 0.0146076, 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} \[ \frac{1}{3} a^2 c x^2 \sqrt{c x^2}+\frac{1}{2} a b c x^3 \sqrt{c x^2}+\frac{1}{5} b^2 c x^4 \sqrt{c x^2} \]
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} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int x^2 (a+b x)^2 \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (a^2 x^2+2 a b x^3+b^2 x^4\right ) \, dx}{x}\\ &=\frac{1}{3} a^2 c x^2 \sqrt{c x^2}+\frac{1}{2} a b c x^3 \sqrt{c x^2}+\frac{1}{5} b^2 c x^4 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0039654, size = 36, normalized size = 0.6 \[ \frac{1}{30} c x^2 \sqrt{c x^2} \left (10 a^2+15 a b x+6 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 0.5 \begin{align*}{\frac{6\,{b}^{2}{x}^{2}+15\,abx+10\,{a}^{2}}{30} \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 = 1.51166, size = 84, normalized size = 1.4 \begin{align*} \frac{1}{30} \,{\left (6 \, b^{2} c x^{4} + 15 \, a b c x^{3} + 10 \, a^{2} c x^{2}\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.687988, size = 54, normalized size = 0.9 \begin{align*} \frac{a^{2} c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} + \frac{a b c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}}{2} + \frac{b^{2} c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08133, size = 47, normalized size = 0.78 \begin{align*} \frac{1}{30} \,{\left (6 \, b^{2} x^{5} \mathrm{sgn}\left (x\right ) + 15 \, a b x^{4} \mathrm{sgn}\left (x\right ) + 10 \, a^{2} x^{3} \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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