Optimal. Leaf size=77 \[ \frac{b^2 x \log (x)}{a^3 \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 \sqrt{c x^2}}+\frac{b}{a^2 \sqrt{c x^2}}-\frac{1}{2 a x \sqrt{c x^2}} \]
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Rubi [A] time = 0.0285021, antiderivative size = 77, 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} \[ \frac{b^2 x \log (x)}{a^3 \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 \sqrt{c x^2}}+\frac{b}{a^2 \sqrt{c x^2}}-\frac{1}{2 a x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^2 \sqrt{c x^2} (a+b x)} \, dx &=\frac{x \int \frac{1}{x^3 (a+b x)} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{b}{a^2 \sqrt{c x^2}}-\frac{1}{2 a x \sqrt{c x^2}}+\frac{b^2 x \log (x)}{a^3 \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0108957, size = 52, normalized size = 0.68 \[ \frac{c x \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 51, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2}}{2\,x{a}^{3}}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02626, size = 74, normalized size = 0.96 \begin{align*} -\frac{b^{2} \log \left (b x + a\right )}{a^{3} \sqrt{c}} + \frac{b^{2} \log \left (x\right )}{a^{3} \sqrt{c}} + \frac{2 \, b \sqrt{c} x - a \sqrt{c}}{2 \, a^{2} c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58499, size = 100, normalized size = 1.3 \begin{align*} \frac{{\left (2 \, b^{2} x^{2} \log \left (\frac{x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, a^{3} c x^{3}} \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{1}{x^{2} \sqrt{c x^{2}} \left (a + b x\right )}\, 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*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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