Optimal. Leaf size=19 \[ -\frac {2 a}{3 x^{3/2}}-\frac {2 b}{\sqrt {x}} \]
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Rubi [A]
time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {45}
\begin {gather*} -\frac {2 a}{3 x^{3/2}}-\frac {2 b}{\sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {a+b x}{x^{5/2}} \, dx &=\int \left (\frac {a}{x^{5/2}}+\frac {b}{x^{3/2}}\right ) \, dx\\ &=-\frac {2 a}{3 x^{3/2}}-\frac {2 b}{\sqrt {x}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 0.79 \begin {gather*} -\frac {2 (a+3 b x)}{3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 14, normalized size = 0.74
method | result | size |
gosper | \(-\frac {2 \left (3 b x +a \right )}{3 x^{\frac {3}{2}}}\) | \(12\) |
trager | \(-\frac {2 \left (3 b x +a \right )}{3 x^{\frac {3}{2}}}\) | \(12\) |
risch | \(-\frac {2 \left (3 b x +a \right )}{3 x^{\frac {3}{2}}}\) | \(12\) |
derivativedivides | \(-\frac {2 a}{3 x^{\frac {3}{2}}}-\frac {2 b}{\sqrt {x}}\) | \(14\) |
default | \(-\frac {2 a}{3 x^{\frac {3}{2}}}-\frac {2 b}{\sqrt {x}}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 11, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 11, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 19, normalized size = 1.00 \begin {gather*} - \frac {2 a}{3 x^{\frac {3}{2}}} - \frac {2 b}{\sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 11, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left (3 \, b x + a\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 13, normalized size = 0.68 \begin {gather*} -\frac {2\,a+6\,b\,x}{3\,x^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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