Optimal. Leaf size=34 \[ \frac{2 (a+b x)^{7/2}}{7 b^2}-\frac{2 a (a+b x)^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.0077795, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2 (a+b x)^{7/2}}{7 b^2}-\frac{2 a (a+b x)^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x (a+b x)^{3/2} \, dx &=\int \left (-\frac{a (a+b x)^{3/2}}{b}+\frac{(a+b x)^{5/2}}{b}\right ) \, dx\\ &=-\frac{2 a (a+b x)^{5/2}}{5 b^2}+\frac{2 (a+b x)^{7/2}}{7 b^2}\\ \end{align*}
Mathematica [A] time = 0.0248003, size = 24, normalized size = 0.71 \[ \frac{2 (a+b x)^{5/2} (5 b x-2 a)}{35 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{-10\,bx+4\,a}{35\,{b}^{2}} \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 = 1.0367, size = 35, normalized size = 1.03 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}}}{7 \, b^{2}} - \frac{2 \,{\left (b x + a\right )}^{\frac{5}{2}} a}{5 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55714, size = 92, normalized size = 2.71 \begin{align*} \frac{2 \,{\left (5 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} + a^{2} b x - 2 \, a^{3}\right )} \sqrt{b x + a}}{35 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.843876, size = 80, normalized size = 2.35 \begin{align*} \begin{cases} - \frac{4 a^{3} \sqrt{a + b x}}{35 b^{2}} + \frac{2 a^{2} x \sqrt{a + b x}}{35 b} + \frac{16 a x^{2} \sqrt{a + b x}}{35} + \frac{2 b x^{3} \sqrt{a + b x}}{7} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{2}} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17375, size = 92, normalized size = 2.71 \begin{align*} \frac{2 \,{\left (\frac{7 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} a}{b} + \frac{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}}{b}\right )}}{105 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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