Optimal. Leaf size=142 \[ -\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{3 a b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{3 a^2 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.0344887, 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, 43} \[ -\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{3 a b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{3 a^2 b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3}{x^2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (3 a b^5+\frac{a^3 b^3}{x^2}+\frac{3 a^2 b^4}{x}+b^6 x\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{3 a b^2 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{b^3 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0171186, size = 56, normalized size = 0.39 \[ \frac{\sqrt{(a+b x)^2} \left (6 a^2 b x \log (x)-2 a^3+6 a b^2 x^2+b^3 x^3\right )}{2 x (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.222, size = 53, normalized size = 0.4 \begin{align*}{\frac{{b}^{3}{x}^{3}+6\,b{a}^{2}\ln \left ( x \right ) x+6\,a{b}^{2}{x}^{2}-2\,{a}^{3}}{2\, \left ( bx+a \right ) ^{3}x} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{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 = 1.88336, size = 78, normalized size = 0.55 \begin{align*} \frac{b^{3} x^{3} + 6 \, a b^{2} x^{2} + 6 \, a^{2} b x \log \left (x\right ) - 2 \, a^{3}}{2 \, 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{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28843, size = 77, normalized size = 0.54 \begin{align*} \frac{1}{2} \, b^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, a b^{2} x \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) - \frac{a^{3} \mathrm{sgn}\left (b x + a\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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