Optimal. Leaf size=34 \[ \frac{2 (a+b x)^{9/2}}{9 b^2}-\frac{2 a (a+b x)^{7/2}}{7 b^2} \]
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Rubi [A] time = 0.0086827, 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} \[ \frac{2 (a+b x)^{9/2}}{9 b^2}-\frac{2 a (a+b x)^{7/2}}{7 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x (a+b x)^{5/2} \, dx &=\int \left (-\frac{a (a+b x)^{5/2}}{b}+\frac{(a+b x)^{7/2}}{b}\right ) \, dx\\ &=-\frac{2 a (a+b x)^{7/2}}{7 b^2}+\frac{2 (a+b x)^{9/2}}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.0257153, size = 24, normalized size = 0.71 \[ \frac{2 (a+b x)^{7/2} (7 b x-2 a)}{63 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 21, normalized size = 0.6 \begin{align*} -{\frac{-14\,bx+4\,a}{63\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12851, size = 35, normalized size = 1.03 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{9}{2}}}{9 \, b^{2}} - \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}} a}{7 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47407, size = 116, normalized size = 3.41 \begin{align*} \frac{2 \,{\left (7 \, b^{4} x^{4} + 19 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + a^{3} b x - 2 \, a^{4}\right )} \sqrt{b x + a}}{63 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.0582, size = 102, normalized size = 3. \begin{align*} \begin{cases} - \frac{4 a^{4} \sqrt{a + b x}}{63 b^{2}} + \frac{2 a^{3} x \sqrt{a + b x}}{63 b} + \frac{10 a^{2} x^{2} \sqrt{a + b x}}{21} + \frac{38 a b x^{3} \sqrt{a + b x}}{63} + \frac{2 b^{2} x^{4} \sqrt{a + b x}}{9} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{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 = 1.21709, size = 162, normalized size = 4.76 \begin{align*} \frac{2 \,{\left (\frac{21 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} a^{2}}{b} + \frac{6 \,{\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} + \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}\right )}}{315 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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