Optimal. Leaf size=61 \[ -\frac{b \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b \sqrt{c x^2} \log (a+b x)}{a^2 x}-\frac{\sqrt{c x^2}}{a x^2} \]
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Rubi [A] time = 0.017243, 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, 44} \[ -\frac{b \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b \sqrt{c x^2} \log (a+b x)}{a^2 x}-\frac{\sqrt{c x^2}}{a x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{\sqrt{c x^2}}{x^3 (a+b x)} \, dx &=\frac{\sqrt{c x^2} \int \frac{1}{x^2 (a+b x)} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{1}{a x^2}-\frac{b}{a^2 x}+\frac{b^2}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{\sqrt{c x^2}}{a x^2}-\frac{b \sqrt{c x^2} \log (x)}{a^2 x}+\frac{b \sqrt{c x^2} \log (a+b x)}{a^2 x}\\ \end{align*}
Mathematica [A] time = 0.0115274, size = 32, normalized size = 0.52 \[ -\frac{c (-b x \log (a+b x)+a+b x \log (x))}{a^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 33, normalized size = 0.5 \begin{align*} -{\frac{b\ln \left ( x \right ) x-b\ln \left ( bx+a \right ) x+a}{{a}^{2}{x}^{2}}\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07367, size = 50, normalized size = 0.82 \begin{align*} \frac{b \sqrt{c} \log \left (b x + a\right )}{a^{2}} - \frac{b \sqrt{c} \log \left (x\right )}{a^{2}} - \frac{\sqrt{c}}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57438, size = 68, normalized size = 1.11 \begin{align*} \frac{\sqrt{c x^{2}}{\left (b x \log \left (\frac{b x + a}{x}\right ) - a\right )}}{a^{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{\sqrt{c x^{2}}}{x^{3} \left (a + b x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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