Optimal. Leaf size=47 \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{2 a^3}{3 x^{3/2}}+6 a b^2 \sqrt{x}+\frac{2}{3} b^3 x^{3/2} \]
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Rubi [A] time = 0.0107336, antiderivative size = 47, 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} \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{2 a^3}{3 x^{3/2}}+6 a b^2 \sqrt{x}+\frac{2}{3} b^3 x^{3/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{x^{5/2}} \, dx &=\int \left (\frac{a^3}{x^{5/2}}+\frac{3 a^2 b}{x^{3/2}}+\frac{3 a b^2}{\sqrt{x}}+b^3 \sqrt{x}\right ) \, dx\\ &=-\frac{2 a^3}{3 x^{3/2}}-\frac{6 a^2 b}{\sqrt{x}}+6 a b^2 \sqrt{x}+\frac{2}{3} b^3 x^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0145666, size = 38, normalized size = 0.81 \[ \frac{2 \left (-9 a^2 b x-a^3+9 a b^2 x^2+b^3 x^3\right )}{3 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 34, normalized size = 0.7 \begin{align*} -{\frac{-2\,{b}^{3}{x}^{3}-18\,a{b}^{2}{x}^{2}+18\,{a}^{2}bx+2\,{a}^{3}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10933, size = 46, normalized size = 0.98 \begin{align*} \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} \sqrt{x} - \frac{2 \,{\left (9 \, a^{2} b x + a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54299, size = 74, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (b^{3} x^{3} + 9 \, a b^{2} x^{2} - 9 \, a^{2} b x - a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.19895, size = 46, normalized size = 0.98 \begin{align*} - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} - \frac{6 a^{2} b}{\sqrt{x}} + 6 a b^{2} \sqrt{x} + \frac{2 b^{3} x^{\frac{3}{2}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20573, size = 46, normalized size = 0.98 \begin{align*} \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} \sqrt{x} - \frac{2 \,{\left (9 \, a^{2} b x + a^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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