Optimal. Leaf size=49 \[ \frac{a c \sqrt{c x^2}}{b^2 x (a+b x)}+\frac{c \sqrt{c x^2} \log (a+b x)}{b^2 x} \]
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Rubi [A] time = 0.0145969, antiderivative size = 49, 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} \[ \frac{a c \sqrt{c x^2}}{b^2 x (a+b x)}+\frac{c \sqrt{c x^2} \log (a+b x)}{b^2 x} \]
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}}{x^2 (a+b x)^2} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{x}{(a+b x)^2} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx}{x}\\ &=\frac{a c \sqrt{c x^2}}{b^2 x (a+b x)}+\frac{c \sqrt{c x^2} \log (a+b x)}{b^2 x}\\ \end{align*}
Mathematica [A] time = 0.0088697, size = 38, normalized size = 0.78 \[ \frac{c^2 x ((a+b x) \log (a+b x)+a)}{b^2 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 41, normalized size = 0.8 \begin{align*}{\frac{b\ln \left ( bx+a \right ) x+a\ln \left ( bx+a \right ) +a}{{x}^{3}{b}^{2} \left ( bx+a \right ) } \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 = 1.21269, size = 92, normalized size = 1.88 \begin{align*} \frac{\sqrt{c x^{2}}{\left (a c +{\left (b c x + a c\right )} \log \left (b x + a\right )\right )}}{b^{3} x^{2} + a b^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x^{2}\right )^{\frac{3}{2}}}{x^{2} \left (a + b x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06821, size = 62, normalized size = 1.27 \begin{align*} -c^{\frac{3}{2}}{\left (\frac{{\left (\log \left ({\left | a \right |}\right ) + 1\right )} \mathrm{sgn}\left (x\right )}{b^{2}} - \frac{\log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{2}} - \frac{a \mathrm{sgn}\left (x\right )}{{\left (b x + a\right )} b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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