Optimal. Leaf size=61 \[ \frac{a^2 c \sqrt{c x^2} \log (a+b x)}{b^3 x}-\frac{a c \sqrt{c x^2}}{b^2}+\frac{c x \sqrt{c x^2}}{2 b} \]
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Rubi [A] time = 0.0185233, antiderivative size = 61, 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} \[ \frac{a^2 c \sqrt{c x^2} \log (a+b x)}{b^3 x}-\frac{a c \sqrt{c x^2}}{b^2}+\frac{c x \sqrt{c x^2}}{2 b} \]
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 (a+b x)} \, dx &=\frac{\left (c \sqrt{c x^2}\right ) \int \frac{x^2}{a+b x} \, dx}{x}\\ &=\frac{\left (c \sqrt{c x^2}\right ) \int \left (-\frac{a}{b^2}+\frac{x}{b}+\frac{a^2}{b^2 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{a c \sqrt{c x^2}}{b^2}+\frac{c x \sqrt{c x^2}}{2 b}+\frac{a^2 c \sqrt{c x^2} \log (a+b x)}{b^3 x}\\ \end{align*}
Mathematica [A] time = 0.0048889, size = 42, normalized size = 0.69 \[ \frac{c^2 x \left (2 a^2 \log (a+b x)+b x (b x-2 a)\right )}{2 b^3 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 40, normalized size = 0.7 \begin{align*}{\frac{{b}^{2}{x}^{2}+2\,{a}^{2}\ln \left ( bx+a \right ) -2\,abx}{2\,{b}^{3}{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 [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 = 1.51577, size = 97, normalized size = 1.59 \begin{align*} \frac{{\left (b^{2} c x^{2} - 2 \, a b c x + 2 \, a^{2} c \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{2 \, b^{3} 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 \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06519, size = 73, normalized size = 1.2 \begin{align*} \frac{1}{2} \, c^{\frac{3}{2}}{\left (\frac{2 \, a^{2} \log \left ({\left | b x + a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{3}} - \frac{2 \, a^{2} \log \left ({\left | a \right |}\right ) \mathrm{sgn}\left (x\right )}{b^{3}} + \frac{b x^{2} \mathrm{sgn}\left (x\right ) - 2 \, a x \mathrm{sgn}\left (x\right )}{b^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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