Optimal. Leaf size=32 \[ -\frac {2 a \sqrt {a+b x}}{b^2}+\frac {2 (a+b x)^{3/2}}{3 b^2} \]
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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+b x)^{3/2}}{3 b^2}-\frac {2 a \sqrt {a+b x}}{b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {a+b x}} \, dx &=\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx\\ &=-\frac {2 a \sqrt {a+b x}}{b^2}+\frac {2 (a+b x)^{3/2}}{3 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.72 \begin {gather*} \frac {2 (-2 a+b x) \sqrt {a+b x}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(83\) vs. \(2(32)=64\).
time = 3.31, size = 73, normalized size = 2.28 \begin {gather*} \frac {2 \sqrt {a} \left (2 a^2 \left (1-\sqrt {\frac {a+b x}{a}}\right )+a b x \left (2-\sqrt {\frac {a+b x}{a}}\right )+b^2 x^2 \sqrt {\frac {a+b x}{a}}\right )}{3 b^2 \left (a+b x\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.08, size = 26, normalized size = 0.81
method | result | size |
gosper | \(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\) | \(21\) |
trager | \(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\) | \(21\) |
risch | \(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \sqrt {b x +a}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \sqrt {b x +a}}{b^{2}}\) | \(26\) |
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 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, b^{2}} - \frac {2 \, \sqrt {b x + a} a}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 19, normalized size = 0.59 \begin {gather*} \frac {2 \, \sqrt {b x + a} {\left (b x - 2 \, a\right )}}{3 \, b^{2}} \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. 162 vs.
\(2 (29) = 58\).
time = 0.56, size = 162, normalized size = 5.06 \begin {gather*} - \frac {4 a^{\frac {7}{2}} \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {4 a^{\frac {7}{2}}}{3 a^{2} b^{2} + 3 a b^{3} x} - \frac {2 a^{\frac {5}{2}} b x \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {4 a^{\frac {5}{2}} b x}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {2 a^{\frac {3}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 36, normalized size = 1.12 \begin {gather*} \frac {2 \left (\frac {1}{3} \sqrt {a+b x} \left (a+b x\right )-a \sqrt {a+b x}\right )}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 25, normalized size = 0.78 \begin {gather*} -\frac {6\,a\,\sqrt {a+b\,x}-2\,{\left (a+b\,x\right )}^{3/2}}{3\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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