Optimal. Leaf size=103 \[ \frac{b^2 x}{a^3 \sqrt{c x^2} (a+b x)}+\frac{3 b^2 x \log (x)}{a^4 \sqrt{c x^2}}-\frac{3 b^2 x \log (a+b x)}{a^4 \sqrt{c x^2}}+\frac{2 b}{a^3 \sqrt{c x^2}}-\frac{1}{2 a^2 x \sqrt{c x^2}} \]
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Rubi [A] time = 0.0316542, antiderivative size = 103, 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} \[ \frac{b^2 x}{a^3 \sqrt{c x^2} (a+b x)}+\frac{3 b^2 x \log (x)}{a^4 \sqrt{c x^2}}-\frac{3 b^2 x \log (a+b x)}{a^4 \sqrt{c x^2}}+\frac{2 b}{a^3 \sqrt{c x^2}}-\frac{1}{2 a^2 x \sqrt{c x^2}} \]
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)^2} \, dx &=\frac{x \int \frac{1}{x^3 (a+b x)^2} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{1}{a^2 x^3}-\frac{2 b}{a^3 x^2}+\frac{3 b^2}{a^4 x}-\frac{b^3}{a^3 (a+b x)^2}-\frac{3 b^3}{a^4 (a+b x)}\right ) \, dx}{\sqrt{c x^2}}\\ &=\frac{2 b}{a^3 \sqrt{c x^2}}-\frac{1}{2 a^2 x \sqrt{c x^2}}+\frac{b^2 x}{a^3 \sqrt{c x^2} (a+b x)}+\frac{3 b^2 x \log (x)}{a^4 \sqrt{c x^2}}-\frac{3 b^2 x \log (a+b x)}{a^4 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0170383, size = 81, normalized size = 0.79 \[ \frac{c x \left (a \left (-a^2+3 a b x+6 b^2 x^2\right )+6 b^2 x^2 \log (x) (a+b x)-6 b^2 x^2 (a+b x) \log (a+b x)\right )}{2 a^4 \left (c x^2\right )^{3/2} (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 95, normalized size = 0.9 \begin{align*}{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,\ln \left ( x \right ){x}^{2}a{b}^{2}-6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}+6\,a{b}^{2}{x}^{2}+3\,{a}^{2}bx-{a}^{3}}{2\,x{a}^{4} \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 [A] time = 1.05652, size = 103, normalized size = 1. \begin{align*} \frac{6 \, b^{2} x^{2} + 3 \, a b x - a^{2}}{2 \,{\left (a^{3} b \sqrt{c} x^{3} + a^{4} \sqrt{c} x^{2}\right )}} - \frac{3 \, b^{2} \log \left (b x + a\right )}{a^{4} \sqrt{c}} + \frac{3 \, b^{2} \log \left (x\right )}{a^{4} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.3217, size = 159, normalized size = 1.54 \begin{align*} \frac{{\left (6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} + 6 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (\frac{x}{b x + a}\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (a^{4} b c x^{4} + a^{5} c x^{3}\right )}} \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 )^{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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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