Optimal. Leaf size=29 \[ a c \sqrt{c x^2}+\frac{1}{2} b c x \sqrt{c x^2} \]
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Rubi [A] time = 0.0043358, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {15} \[ a c \sqrt{c x^2}+\frac{1}{2} b c x \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{3/2} (a+b x)}{x^3} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int (a+b x) \, dx}{x}\\ &=a c \sqrt{c x^2}+\frac{1}{2} b c x \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0028447, size = 21, normalized size = 0.72 \[ \frac{1}{2} c \sqrt{c x^2} (2 a+b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 20, normalized size = 0.7 \begin{align*}{\frac{bx+2\,a}{2\,{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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49662, size = 45, normalized size = 1.55 \begin{align*} \frac{1}{2} \,{\left (b c x + 2 \, a c\right )} \sqrt{c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.695981, size = 32, normalized size = 1.1 \begin{align*} \frac{a c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x^{2}} + \frac{b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{2 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06579, size = 23, normalized size = 0.79 \begin{align*} \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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