Optimal. Leaf size=32 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b} \]
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Rubi [A] time = 0.0055318, 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} \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b} \]
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 )^{5/2} \, dx &=\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b}\\ \end{align*}
Mathematica [A] time = 0.0128349, size = 23, normalized size = 0.72 \[ \frac{(a+b x) \left ((a+b x)^2\right )^{5/2}}{6 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 71, normalized size = 2.2 \begin{align*}{\frac{x \left ({b}^{5}{x}^{5}+6\,a{b}^{4}{x}^{4}+15\,{a}^{2}{b}^{3}{x}^{3}+20\,{a}^{3}{b}^{2}{x}^{2}+15\,{a}^{4}bx+6\,{a}^{5} \right ) }{6\, \left ( bx+a \right ) ^{5}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{5}{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 = 1.53539, size = 116, normalized size = 3.62 \begin{align*} \frac{1}{6} \, b^{5} x^{6} + a b^{4} x^{5} + \frac{5}{2} \, a^{2} b^{3} x^{4} + \frac{10}{3} \, a^{3} b^{2} x^{3} + \frac{5}{2} \, a^{4} b x^{2} + a^{5} 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{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17674, size = 139, normalized size = 4.34 \begin{align*} \frac{1}{6} \, b^{5} x^{6} \mathrm{sgn}\left (b x + a\right ) + a b^{4} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{2} b^{3} x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{10}{3} \, a^{3} b^{2} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{5}{2} \, a^{4} b x^{2} \mathrm{sgn}\left (b x + a\right ) + a^{5} x \mathrm{sgn}\left (b x + a\right ) + \frac{a^{6} \mathrm{sgn}\left (b x + a\right )}{6 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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