Optimal. Leaf size=32 \[ \frac {2 a}{3 b^2 (a+b x)^{3/2}}-\frac {2}{b^2 \sqrt {a+b x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, 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 b^2 (a+b x)^{3/2}}-\frac {2}{b^2 \sqrt {a+b x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x}{(a+b x)^{5/2}} \, dx &=\int \left (-\frac {a}{b (a+b x)^{5/2}}+\frac {1}{b (a+b x)^{3/2}}\right ) \, dx\\ &=\frac {2 a}{3 b^2 (a+b x)^{3/2}}-\frac {2}{b^2 \sqrt {a+b x}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.75 \begin {gather*} -\frac {2 (2 a+3 b x)}{3 b^2 (a+b x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 26, normalized size = 0.81
method | result | size |
gosper | \(-\frac {2 \left (3 b x +2 a \right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{2}}\) | \(21\) |
trager | \(-\frac {2 \left (3 b x +2 a \right )}{3 \left (b x +a \right )^{\frac {3}{2}} b^{2}}\) | \(21\) |
derivativedivides | \(\frac {-\frac {2}{\sqrt {b x +a}}+\frac {2 a}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{2}}\) | \(26\) |
default | \(\frac {-\frac {2}{\sqrt {b x +a}}+\frac {2 a}{3 \left (b x +a \right )^{\frac {3}{2}}}}{b^{2}}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 26, normalized size = 0.81 \begin {gather*} -\frac {2}{\sqrt {b x + a} b^{2}} + \frac {2 \, a}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 41, normalized size = 1.28 \begin {gather*} -\frac {2 \, {\left (3 \, b x + 2 \, a\right )} \sqrt {b x + a}}{3 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 80 vs.
\(2 (29) = 58\).
time = 0.35, size = 80, normalized size = 2.50 \begin {gather*} \begin {cases} - \frac {4 a}{3 a b^{2} \sqrt {a + b x} + 3 b^{3} x \sqrt {a + b x}} - \frac {6 b x}{3 a b^{2} \sqrt {a + b x} + 3 b^{3} x \sqrt {a + b x}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.66, size = 20, normalized size = 0.62 \begin {gather*} -\frac {2 \, {\left (3 \, b x + 2 \, a\right )}}{3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 20, normalized size = 0.62 \begin {gather*} -\frac {4\,a+6\,b\,x}{3\,b^2\,{\left (a+b\,x\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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