Optimal. Leaf size=32 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]
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Rubi [A] time = 0.0048205, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {609} \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin{align*} \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b}\\ \end{align*}
Mathematica [A] time = 0.0101818, size = 23, normalized size = 0.72 \[ \frac{(a+b x) \left ((a+b x)^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 49, normalized size = 1.5 \begin{align*}{\frac{x \left ({b}^{3}{x}^{3}+4\,a{b}^{2}{x}^{2}+6\,{a}^{2}bx+4\,{a}^{3} \right ) }{4\, \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 = 1.92746, size = 66, normalized size = 2.06 \begin{align*} \frac{1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac{3}{2} \, a^{2} b x^{2} + a^{3} 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 \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20249, size = 93, normalized size = 2.91 \begin{align*} \frac{1}{4} \, b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + a b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b x^{2} \mathrm{sgn}\left (b x + a\right ) + a^{3} x \mathrm{sgn}\left (b x + a\right ) + \frac{a^{4} \mathrm{sgn}\left (b x + a\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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