Optimal. Leaf size=44 \[ \frac{x \log (x)}{a c \sqrt{c x^2}}-\frac{x \log (a+b x)}{a c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0080417, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {15, 36, 29, 31} \[ \frac{x \log (x)}{a c \sqrt{c x^2}}-\frac{x \log (a+b x)}{a c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac{x \int \frac{1}{x (a+b x)} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \frac{1}{x} \, dx}{a c \sqrt{c x^2}}-\frac{(b x) \int \frac{1}{a+b x} \, dx}{a c \sqrt{c x^2}}\\ &=\frac{x \log (x)}{a c \sqrt{c x^2}}-\frac{x \log (a+b x)}{a c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0071789, size = 27, normalized size = 0.61 \[ \frac{x^3 (\log (x)-\log (a+b x))}{a \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 26, normalized size = 0.6 \begin{align*}{\frac{{x}^{3} \left ( \ln \left ( x \right ) -\ln \left ( bx+a \right ) \right ) }{a} \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 = 1.54608, size = 153, normalized size = 3.48 \begin{align*} \left [\frac{\sqrt{c x^{2}} \log \left (\frac{x}{b x + a}\right )}{a c^{2} x}, \frac{2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )} \sqrt{-c}}{a c x}\right )}{a c^{2}}\right ] \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{x^{2}}{\left (c x^{2}\right )^{\frac{3}{2}} \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 = 1.08139, size = 85, normalized size = 1.93 \begin{align*} \frac{\frac{\log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{a \sqrt{c}} - \frac{\log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{a \sqrt{c}}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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