Optimal. Leaf size=43 \[ \frac{a x}{b^2 \sqrt{c x^2} (a+b x)}+\frac{x \log (a+b x)}{b^2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0131761, antiderivative size = 43, 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} \[ \frac{a x}{b^2 \sqrt{c x^2} (a+b x)}+\frac{x \log (a+b x)}{b^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{c x^2} (a+b x)^2} \, dx &=\frac{x \int \frac{x}{(a+b x)^2} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{a x}{b^2 \sqrt{c x^2} (a+b x)}+\frac{x \log (a+b x)}{b^2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0090779, size = 35, normalized size = 0.81 \[ \frac{x ((a+b x) \log (a+b x)+a)}{b^2 \sqrt{c x^2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 39, normalized size = 0.9 \begin{align*}{\frac{x \left ( b\ln \left ( bx+a \right ) x+a\ln \left ( bx+a \right ) +a \right ) }{{b}^{2} \left ( bx+a \right ) }{\frac{1}{\sqrt{c{x}^{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 = 1.29448, size = 89, normalized size = 2.07 \begin{align*} \frac{\sqrt{c x^{2}}{\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )}}{b^{3} c x^{2} + a b^{2} c 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{x^{2}}{\sqrt{c x^{2}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{c x^{2}}{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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