Optimal. Leaf size=63 \[ -\frac{b x \log (x)}{a^2 c \sqrt{c x^2}}+\frac{b x \log (a+b x)}{a^2 c \sqrt{c x^2}}-\frac{1}{a c \sqrt{c x^2}} \]
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Rubi [A] time = 0.0169767, antiderivative size = 63, 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, 44} \[ -\frac{b x \log (x)}{a^2 c \sqrt{c x^2}}+\frac{b x \log (a+b x)}{a^2 c \sqrt{c x^2}}-\frac{1}{a c \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{x}{\left (c x^2\right )^{3/2} (a+b x)} \, dx &=\frac{x \int \frac{1}{x^2 (a+b x)} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{1}{a c \sqrt{c x^2}}-\frac{b x \log (x)}{a^2 c \sqrt{c x^2}}+\frac{b x \log (a+b x)}{a^2 c \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0059688, size = 35, normalized size = 0.56 \[ \frac{x^2 (b x \log (a+b x)-a-b x \log (x))}{a^2 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 33, normalized size = 0.5 \begin{align*} -{\frac{{x}^{2} \left ( b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a \right ) }{{a}^{2}} \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 = 1.63943, size = 73, normalized size = 1.16 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x \log \left (\frac{b x + a}{x}\right ) - a\right )}}{a^{2} c^{2} x^{2}} \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}{\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 = 1.08624, size = 123, normalized size = 1.95 \begin{align*} -\frac{\frac{b \log \left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{a^{2} c} - \frac{b \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{a^{2} c} - \frac{2}{{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} a \sqrt{c}}}{\sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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