Optimal. Leaf size=25 \[ -\frac{c \sqrt{c x^2}}{b x (a+b x)} \]
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Rubi [A] time = 0.004015, antiderivative size = 25, 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} \[ -\frac{c \sqrt{c x^2}}{b x (a+b x)} \]
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}}{x^3 (a+b x)^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{1}{(a+b x)^2} \, dx}{x}\\ &=-\frac{c \sqrt{c x^2}}{b x (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0055979, size = 24, normalized size = 0.96 \[ -\frac{\left (c x^2\right )^{3/2}}{b x^3 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 23, normalized size = 0.9 \begin{align*} -{\frac{1}{ \left ( bx+a \right ) b{x}^{3}} \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 [A] time = 1.04378, size = 22, normalized size = 0.88 \begin{align*} -\frac{c^{\frac{3}{2}}}{b^{2} x + a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25167, size = 46, normalized size = 1.84 \begin{align*} -\frac{\sqrt{c x^{2}} c}{b^{2} x^{2} + a b x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.19917, size = 44, normalized size = 1.76 \begin{align*} \begin{cases} - \frac{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{a b x^{3} + b^{2} x^{4}} & \text{for}\: b \neq 0 \\\frac{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{a^{2} x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0644, size = 39, normalized size = 1.56 \begin{align*} -c^{\frac{3}{2}}{\left (\frac{\mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b} - \frac{\mathrm{sgn}\left (x\right )}{a b}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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