Optimal. Leaf size=61 \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}-\frac{a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^2} \]
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Rubi [A] time = 0.014963, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {640, 609} \[ \frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}-\frac{a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 640
Rule 609
Rubi steps
\begin{align*} \int x \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}-\frac{a \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx}{b}\\ &=-\frac{a (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^2}+\frac{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0140692, size = 55, normalized size = 0.9 \[ \frac{x^2 \sqrt{(a+b x)^2} \left (20 a^2 b x+10 a^3+15 a b^2 x^2+4 b^3 x^3\right )}{20 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.171, size = 52, normalized size = 0.9 \begin{align*}{\frac{{x}^{2} \left ( 4\,{x}^{3}{b}^{3}+15\,a{b}^{2}{x}^{2}+20\,{a}^{2}bx+10\,{a}^{3} \right ) }{20\, \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 = 1.96812, size = 74, normalized size = 1.21 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac{1}{2} \, a^{3} 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 x \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19716, size = 97, normalized size = 1.59 \begin{align*} \frac{1}{5} \, b^{3} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, a b^{2} x^{4} \mathrm{sgn}\left (b x + a\right ) + a^{2} b x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, a^{3} x^{2} \mathrm{sgn}\left (b x + a\right ) - \frac{a^{5} \mathrm{sgn}\left (b x + a\right )}{20 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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