Optimal. Leaf size=47 \[ 2 a^2 b x^{3/2}+2 a^3 \sqrt{x}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{7} b^3 x^{7/2} \]
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Rubi [A] time = 0.0114421, 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} \[ 2 a^2 b x^{3/2}+2 a^3 \sqrt{x}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{7} b^3 x^{7/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^3}{\sqrt{x}} \, dx &=\int \left (\frac{a^3}{\sqrt{x}}+3 a^2 b \sqrt{x}+3 a b^2 x^{3/2}+b^3 x^{5/2}\right ) \, dx\\ &=2 a^3 \sqrt{x}+2 a^2 b x^{3/2}+\frac{6}{5} a b^2 x^{5/2}+\frac{2}{7} b^3 x^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0102448, size = 39, normalized size = 0.83 \[ \frac{2}{35} \sqrt{x} \left (35 a^2 b x+35 a^3+21 a b^2 x^2+5 b^3 x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.8 \begin{align*}{\frac{10\,{b}^{3}{x}^{3}+42\,a{b}^{2}{x}^{2}+70\,{a}^{2}bx+70\,{a}^{3}}{35}\sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02586, size = 47, normalized size = 1. \begin{align*} \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} + 2 \, a^{3} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44814, size = 85, normalized size = 1.81 \begin{align*} \frac{2}{35} \,{\left (5 \, b^{3} x^{3} + 21 \, a b^{2} x^{2} + 35 \, a^{2} b x + 35 \, a^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.644315, size = 46, normalized size = 0.98 \begin{align*} 2 a^{3} \sqrt{x} + 2 a^{2} b x^{\frac{3}{2}} + \frac{6 a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19342, size = 47, normalized size = 1. \begin{align*} \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + \frac{6}{5} \, a b^{2} x^{\frac{5}{2}} + 2 \, a^{2} b x^{\frac{3}{2}} + 2 \, a^{3} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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