Optimal. Leaf size=53 \[ \frac{2 a^2 (a+b x)^{5/2}}{5 b^3}+\frac{2 (a+b x)^{9/2}}{9 b^3}-\frac{4 a (a+b x)^{7/2}}{7 b^3} \]
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Rubi [A] time = 0.0128408, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ \frac{2 a^2 (a+b x)^{5/2}}{5 b^3}+\frac{2 (a+b x)^{9/2}}{9 b^3}-\frac{4 a (a+b x)^{7/2}}{7 b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x^2 (a+b x)^{3/2} \, dx &=\int \left (\frac{a^2 (a+b x)^{3/2}}{b^2}-\frac{2 a (a+b x)^{5/2}}{b^2}+\frac{(a+b x)^{7/2}}{b^2}\right ) \, dx\\ &=\frac{2 a^2 (a+b x)^{5/2}}{5 b^3}-\frac{4 a (a+b x)^{7/2}}{7 b^3}+\frac{2 (a+b x)^{9/2}}{9 b^3}\\ \end{align*}
Mathematica [A] time = 0.025302, size = 35, normalized size = 0.66 \[ \frac{2 (a+b x)^{5/2} \left (8 a^2-20 a b x+35 b^2 x^2\right )}{315 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 32, normalized size = 0.6 \begin{align*}{\frac{70\,{b}^{2}{x}^{2}-40\,abx+16\,{a}^{2}}{315\,{b}^{3}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07241, size = 55, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{9}{2}}}{9 \, b^{3}} - \frac{4 \,{\left (b x + a\right )}^{\frac{7}{2}} a}{7 \, b^{3}} + \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2}}{5 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51169, size = 120, normalized size = 2.26 \begin{align*} \frac{2 \,{\left (35 \, b^{4} x^{4} + 50 \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} - 4 \, a^{3} b x + 8 \, a^{4}\right )} \sqrt{b x + a}}{315 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.51985, size = 733, normalized size = 13.83 \begin{align*} \frac{16 a^{\frac{25}{2}} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{25}{2}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{40 a^{\frac{23}{2}} b x \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{23}{2}} b x}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{30 a^{\frac{21}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac{48 a^{\frac{21}{2}} b^{2} x^{2}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{110 a^{\frac{19}{2}} b^{3} x^{3} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} - \frac{16 a^{\frac{19}{2}} b^{3} x^{3}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{380 a^{\frac{17}{2}} b^{4} x^{4} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{516 a^{\frac{15}{2}} b^{5} x^{5} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{310 a^{\frac{13}{2}} b^{6} x^{6} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} + \frac{70 a^{\frac{11}{2}} b^{7} x^{7} \sqrt{1 + \frac{b x}{a}}}{315 a^{8} b^{3} + 945 a^{7} b^{4} x + 945 a^{6} b^{5} x^{2} + 315 a^{5} b^{6} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17771, size = 124, normalized size = 2.34 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} a}{b^{2}} + \frac{35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}}{b^{2}}\right )}}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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