Optimal. Leaf size=141 \[ -\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.0355548, antiderivative size = 141, 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} \[ -\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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^3} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3}{x^3} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (b^6+\frac{a^3 b^3}{x^3}+\frac{3 a^2 b^4}{x^2}+\frac{3 a b^5}{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}}{2 x^2 (a+b x)}-\frac{3 a^2 b \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)}+\frac{b^3 x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0155769, size = 55, normalized size = 0.39 \[ -\frac{\sqrt{(a+b x)^2} \left (6 a^2 b x+a^3-6 a b^2 x^2 \log (x)-2 b^3 x^3\right )}{2 x^2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.209, size = 54, normalized size = 0.4 \begin{align*}{\frac{6\,{b}^{2}a\ln \left ( x \right ){x}^{2}+2\,{b}^{3}{x}^{3}-6\,b{a}^{2}x-{a}^{3}}{2\, \left ( bx+a \right ) ^{3}{x}^{2}} \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 = 1.99319, size = 81, normalized size = 0.57 \begin{align*} \frac{2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} \log \left (x\right ) - 6 \, a^{2} b x - a^{3}}{2 \, x^{2}} \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^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24739, size = 76, normalized size = 0.54 \begin{align*} b^{3} x \mathrm{sgn}\left (b x + a\right ) + 3 \, a b^{2} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) - \frac{6 \, a^{2} b x \mathrm{sgn}\left (b x + a\right ) + a^{3} \mathrm{sgn}\left (b x + a\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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