Optimal. Leaf size=41 \[ \frac{1}{3} a c^2 x^2 \sqrt{c x^2}+\frac{1}{4} b c^2 x^3 \sqrt{c x^2} \]
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Rubi [A] time = 0.0100404, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ \frac{1}{3} a c^2 x^2 \sqrt{c x^2}+\frac{1}{4} b c^2 x^3 \sqrt{c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c x^2\right )^{5/2} (a+b x)}{x^3} \, dx &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int x^2 (a+b x) \, dx}{x}\\ &=\frac{\left (c^2 \sqrt{c x^2}\right ) \int \left (a x^2+b x^3\right ) \, dx}{x}\\ &=\frac{1}{3} a c^2 x^2 \sqrt{c x^2}+\frac{1}{4} b c^2 x^3 \sqrt{c x^2}\\ \end{align*}
Mathematica [A] time = 0.0026706, size = 27, normalized size = 0.66 \[ \frac{1}{12} c^2 x^2 \sqrt{c x^2} (4 a+3 b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.5 \begin{align*}{\frac{3\,bx+4\,a}{12\,{x}^{2}} \left ( c{x}^{2} \right ) ^{{\frac{5}{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 = 1.88457, size = 62, normalized size = 1.51 \begin{align*} \frac{1}{12} \,{\left (3 \, b c^{2} x^{3} + 4 \, a c^{2} 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 = 1.52712, size = 34, normalized size = 0.83 \begin{align*} \frac{a c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{3 x^{2}} + \frac{b c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{4 x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05465, size = 38, normalized size = 0.93 \begin{align*} \frac{1}{12} \,{\left (3 \, b c^{2} x^{4} \mathrm{sgn}\left (x\right ) + 4 \, a c^{2} x^{3} \mathrm{sgn}\left (x\right )\right )} \sqrt{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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