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 = {45}
\begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 45
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 = 0.02, size = 39, normalized size = 0.87 \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 36, normalized size = 0.80
method | result | size |
gosper | \(-\frac {2 \left (-b^{3} x^{3}-5 a \,b^{2} x^{2}-15 a^{2} b x +5 a^{3}\right )}{5 \sqrt {x}}\) | \(36\) |
derivativedivides | \(2 a \,b^{2} x^{\frac {3}{2}}+\frac {2 b^{3} x^{\frac {5}{2}}}{5}-\frac {2 a^{3}}{\sqrt {x}}+6 a^{2} b \sqrt {x}\) | \(36\) |
default | \(2 a \,b^{2} x^{\frac {3}{2}}+\frac {2 b^{3} x^{\frac {5}{2}}}{5}-\frac {2 a^{3}}{\sqrt {x}}+6 a^{2} b \sqrt {x}\) | \(36\) |
trager | \(-\frac {2 \left (-b^{3} x^{3}-5 a \,b^{2} x^{2}-15 a^{2} b x +5 a^{3}\right )}{5 \sqrt {x}}\) | \(36\) |
risch | \(-\frac {2 \left (-b^{3} x^{3}-5 a \,b^{2} x^{2}-15 a^{2} b x +5 a^{3}\right )}{5 \sqrt {x}}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 35, normalized size = 0.78 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.69, size = 34, normalized size = 0.76 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 44, normalized size = 0.98 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 35, normalized size = 0.78 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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