Optimal. Leaf size=143 \[ \frac{3 a^2 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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Rubi [A] time = 0.0341437, antiderivative size = 143, 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} \[ \frac{3 a^2 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^3 \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x} \]
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} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \frac{\left (a b+b^2 x\right )^3}{x} \, 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^2 b^4+\frac{a^3 b^3}{x}+3 a b^5 x+b^6 x^2\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{3 a^2 b x \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}+\frac{3 a b^2 x^2 \sqrt{a^2+2 a b x+b^2 x^2}}{2 (a+b x)}+\frac{b^3 x^3 \sqrt{a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac{a^3 \sqrt{a^2+2 a b x+b^2 x^2} \log (x)}{a+b x}\\ \end{align*}
Mathematica [A] time = 0.0159683, size = 52, normalized size = 0.36 \[ \frac{\sqrt{(a+b x)^2} \left (b x \left (18 a^2+9 a b x+2 b^2 x^2\right )+6 a^3 \log (x)\right )}{6 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.219, size = 51, normalized size = 0.4 \begin{align*}{\frac{2\,{b}^{3}{x}^{3}+9\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( x \right ) +18\,b{a}^{2}x}{6\, \left ( bx+a \right ) ^{3}} \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 = 1.96813, size = 73, normalized size = 0.51 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\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{\left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13416, size = 76, normalized size = 0.53 \begin{align*} \frac{1}{3} \, b^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a b^{2} x^{2} \mathrm{sgn}\left (b x + a\right ) + 3 \, a^{2} b x \mathrm{sgn}\left (b x + a\right ) + a^{3} \log \left ({\left | x \right |}\right ) \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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