Optimal. Leaf size=142 \[ \frac{b (a+b x)}{a^2 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 \log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
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Rubi [A] time = 0.0447885, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {646, 44} \[ \frac{b (a+b x)}{a^2 x \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{a+b x}{2 a x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 \log (x) (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 \sqrt{a^2+2 a b x+b^2 x^2}} \, dx &=\frac{\left (a b+b^2 x\right ) \int \frac{1}{x^3 \left (a b+b^2 x\right )} \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=\frac{\left (a b+b^2 x\right ) \int \left (\frac{1}{a b x^3}-\frac{1}{a^2 x^2}+\frac{b}{a^3 x}-\frac{b^2}{a^3 (a+b x)}\right ) \, dx}{\sqrt{a^2+2 a b x+b^2 x^2}}\\ &=-\frac{a+b x}{2 a x^2 \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b (a+b x)}{a^2 x \sqrt{a^2+2 a b x+b^2 x^2}}+\frac{b^2 (a+b x) \log (x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}-\frac{b^2 (a+b x) \log (a+b x)}{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0179001, size = 59, normalized size = 0.42 \[ -\frac{(a+b x) \left (2 b^2 x^2 \log (a+b x)+a (a-2 b x)-2 b^2 x^2 \log (x)\right )}{2 a^3 x^2 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 58, normalized size = 0.4 \begin{align*}{\frac{ \left ( bx+a \right ) \left ( 2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2} \right ) }{2\,{x}^{2}{a}^{3}}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{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 = 1.76156, size = 103, normalized size = 0.73 \begin{align*} -\frac{2 \, b^{2} x^{2} \log \left (b x + a\right ) - 2 \, b^{2} x^{2} \log \left (x\right ) - 2 \, a b x + a^{2}}{2 \, a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.911211, size = 31, normalized size = 0.22 \begin{align*} \frac{- a + 2 b x}{2 a^{2} x^{2}} + \frac{b^{2} \left (\log{\left (x \right )} - \log{\left (\frac{a}{b} + x \right )}\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3977, size = 73, normalized size = 0.51 \begin{align*} -\frac{1}{2} \,{\left (\frac{2 \, b^{2} \log \left ({\left | b x + a \right |}\right )}{a^{3}} - \frac{2 \, b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{2 \, a b x - a^{2}}{a^{3} x^{2}}\right )} \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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