Optimal. Leaf size=45 \[ -\frac {2 a^3}{\sqrt {x}}+6 a^2 b \sqrt {x}+2 a b^2 x^{3/2}+\frac {2}{5} b^3 x^{5/2} \]
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Rubi [A] time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, 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} \[ 6 a^2 b \sqrt {x}-\frac {2 a^3}{\sqrt {x}}+2 a b^2 x^{3/2}+\frac {2}{5} b^3 x^{5/2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^3}{x^{3/2}} \, dx &=\int \left (\frac {a^3}{x^{3/2}}+\frac {3 a^2 b}{\sqrt {x}}+3 a b^2 \sqrt {x}+b^3 x^{3/2}\right ) \, dx\\ &=-\frac {2 a^3}{\sqrt {x}}+6 a^2 b \sqrt {x}+2 a b^2 x^{3/2}+\frac {2}{5} b^3 x^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.84 \[ \frac {2 \left (-5 a^3+15 a^2 b x+5 a b^2 x^2+b^3 x^3\right )}{5 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 34, normalized size = 0.76 \[ \frac {2 \, {\left (b^{3} x^{3} + 5 \, a b^{2} x^{2} + 15 \, a^{2} b x - 5 \, a^{3}\right )}}{5 \, \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 35, normalized size = 0.78 \[ \frac {2}{5} \, b^{3} x^{\frac {5}{2}} + 2 \, a b^{2} x^{\frac {3}{2}} + 6 \, a^{2} b \sqrt {x} - \frac {2 \, a^{3}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.80 \[ -\frac {2 \left (-b^{3} x^{3}-5 a \,b^{2} x^{2}-15 a^{2} b x +5 a^{3}\right )}{5 \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 35, normalized size = 0.78 \[ \frac {2}{5} \, b^{3} x^{\frac {5}{2}} + 2 \, a b^{2} x^{\frac {3}{2}} + 6 \, a^{2} b \sqrt {x} - \frac {2 \, a^{3}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 35, normalized size = 0.78 \[ \frac {2\,b^3\,x^{5/2}}{5}-\frac {2\,a^3}{\sqrt {x}}+6\,a^2\,b\,\sqrt {x}+2\,a\,b^2\,x^{3/2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 44, normalized size = 0.98 \[ - \frac {2 a^{3}}{\sqrt {x}} + 6 a^{2} b \sqrt {x} + 2 a b^{2} x^{\frac {3}{2}} + \frac {2 b^{3} x^{\frac {5}{2}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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