Optimal. Leaf size=87 \[ -\frac{b \sqrt{c x^2}}{a^2 x (a+b x)}-\frac{2 b \sqrt{c x^2} \log (x)}{a^3 x}+\frac{2 b \sqrt{c x^2} \log (a+b x)}{a^3 x}-\frac{\sqrt{c x^2}}{a^2 x^2} \]
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Rubi [A] time = 0.0270424, antiderivative size = 87, 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} \[ -\frac{b \sqrt{c x^2}}{a^2 x (a+b x)}-\frac{2 b \sqrt{c x^2} \log (x)}{a^3 x}+\frac{2 b \sqrt{c x^2} \log (a+b x)}{a^3 x}-\frac{\sqrt{c x^2}}{a^2 x^2} \]
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)^2} \, dx &=\frac{\sqrt{c x^2} \int \frac{1}{x^2 (a+b x)^2} \, dx}{x}\\ &=\frac{\sqrt{c x^2} \int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx}{x}\\ &=-\frac{\sqrt{c x^2}}{a^2 x^2}-\frac{b \sqrt{c x^2}}{a^2 x (a+b x)}-\frac{2 b \sqrt{c x^2} \log (x)}{a^3 x}+\frac{2 b \sqrt{c x^2} \log (a+b x)}{a^3 x}\\ \end{align*}
Mathematica [A] time = 0.0268839, size = 57, normalized size = 0.66 \[ -\frac{c (a (a+2 b x)+2 b x \log (x) (a+b x)-2 b x (a+b x) \log (a+b x))}{a^3 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 74, normalized size = 0.9 \begin{align*} -{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,ab\ln \left ( x \right ) x-2\,\ln \left ( bx+a \right ) xab+2\,abx+{a}^{2}}{{x}^{2}{a}^{3} \left ( bx+a \right ) }\sqrt{c{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02047, size = 78, normalized size = 0.9 \begin{align*} -\frac{2 \, b \sqrt{c} x + a \sqrt{c}}{a^{2} b x^{2} + a^{3} x} + \frac{2 \, b \sqrt{c} \log \left (b x + a\right )}{a^{3}} - \frac{2 \, b \sqrt{c} \log \left (x\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32187, size = 123, normalized size = 1.41 \begin{align*} -\frac{{\left (2 \, a b x + a^{2} - 2 \,{\left (b^{2} x^{2} + a b x\right )} \log \left (\frac{b x + a}{x}\right )\right )} \sqrt{c x^{2}}}{a^{3} b x^{3} + a^{4} 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 )^{2}}\, 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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