Optimal. Leaf size=96 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{6 b^3}-\frac{2 a \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^3}+\frac{a^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^3} \]
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Rubi [A] time = 0.0310885, antiderivative size = 107, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {645} \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^5}{6 b^3}-\frac{2 a \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^4}{5 b^3}+\frac{a^2 \sqrt{a^2+2 a b x+b^2 x^2} (a+b x)^3}{4 b^3} \]
Antiderivative was successfully verified.
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Rule 645
Rubi steps
\begin{align*} \int x^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (\frac{a^2 \left (a b+b^2 x\right )^3}{b^2}-\frac{2 a \left (a b+b^2 x\right )^4}{b^3}+\frac{\left (a b+b^2 x\right )^5}{b^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{a^2 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2}}{4 b^3}-\frac{2 a (a+b x)^4 \sqrt{a^2+2 a b x+b^2 x^2}}{5 b^3}+\frac{(a+b x)^5 \sqrt{a^2+2 a b x+b^2 x^2}}{6 b^3}\\ \end{align*}
Mathematica [A] time = 0.0137966, size = 55, normalized size = 0.57 \[ \frac{x^3 \sqrt{(a+b x)^2} \left (45 a^2 b x+20 a^3+36 a b^2 x^2+10 b^3 x^3\right )}{60 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 52, normalized size = 0.5 \begin{align*}{\frac{{x}^{3} \left ( 10\,{b}^{3}{x}^{3}+36\,a{b}^{2}{x}^{2}+45\,{a}^{2}bx+20\,{a}^{3} \right ) }{60\, \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 = 1.965, size = 80, normalized size = 0.83 \begin{align*} \frac{1}{6} \, b^{3} x^{6} + \frac{3}{5} \, a b^{2} x^{5} + \frac{3}{4} \, a^{2} b x^{4} + \frac{1}{3} \, a^{3} x^{3} \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^{2} \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 = 1.30003, size = 99, normalized size = 1.03 \begin{align*} \frac{1}{6} \, b^{3} x^{6} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{5} \, a b^{2} x^{5} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, a^{2} b x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{3} \, a^{3} x^{3} \mathrm{sgn}\left (b x + a\right ) + \frac{a^{6} \mathrm{sgn}\left (b x + a\right )}{60 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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