Optimal. Leaf size=34 \[ -\frac {2 a (a+b x)^{3/2}}{3 b^2}+\frac {2 (a+b x)^{5/2}}{5 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, 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)^{5/2}}{5 b^2}-\frac {2 a (a+b x)^{3/2}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int x \sqrt {a+b x} \, dx &=\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx\\ &=-\frac {2 a (a+b x)^{3/2}}{3 b^2}+\frac {2 (a+b x)^{5/2}}{5 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a+b x} \left (-2 a^2+a b x+3 b^2 x^2\right )}{15 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. \(108\) vs. \(2(34)=68\).
time = 3.80, size = 96, normalized size = 2.82 \begin {gather*} \frac {2 \sqrt {a} \left (2 a^3 \left (1-\sqrt {\frac {a+b x}{a}}\right )+a^2 b x \left (2-\sqrt {\frac {a+b x}{a}}\right )+4 a b^2 x^2 \sqrt {\frac {a+b x}{a}}+3 b^3 x^3 \sqrt {\frac {a+b x}{a}}\right )}{15 b^2 \left (a+b x\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.04, size = 26, normalized size = 0.76
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-3 b x +2 a \right )}{15 b^{2}}\) | \(21\) |
derivativedivides | \(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{2}}\) | \(26\) |
default | \(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{2}}\) | \(26\) |
trager | \(-\frac {2 \left (-3 x^{2} b^{2}-a b x +2 a^{2}\right ) \sqrt {b x +a}}{15 b^{2}}\) | \(32\) |
risch | \(-\frac {2 \left (-3 x^{2} b^{2}-a b x +2 a^{2}\right ) \sqrt {b x +a}}{15 b^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 26, normalized size = 0.76 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}}}{5 \, b^{2}} - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} a}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 30, normalized size = 0.88 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} x^{2} + a b x - 2 \, a^{2}\right )} \sqrt {b x + a}}{15 \, 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. 202 vs.
\(2 (31) = 62\)
time = 0.56, size = 202, normalized size = 5.94 \begin {gather*} - \frac {4 a^{\frac {9}{2}} \sqrt {1 + \frac {b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac {4 a^{\frac {9}{2}}}{15 a^{2} b^{2} + 15 a b^{3} x} - \frac {2 a^{\frac {7}{2}} b x \sqrt {1 + \frac {b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac {4 a^{\frac {7}{2}} b x}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac {8 a^{\frac {5}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} + \frac {6 a^{\frac {3}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{15 a^{2} b^{2} + 15 a b^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (26) = 52\).
time = 0.00, size = 99, normalized size = 2.91 \begin {gather*} \frac {\frac {2 b \left (\frac {1}{5} \sqrt {a+b x} \left (a+b x\right )^{2}-\frac {2}{3} \sqrt {a+b x} \left (a+b x\right ) a+\sqrt {a+b x} a^{2}\right )}{b^{2}}+\frac {2 a \left (\frac {1}{3} \sqrt {a+b x} \left (a+b x\right )-a \sqrt {a+b x}\right )}{b}}{b} \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 = 0.74 \begin {gather*} -\frac {10\,a\,{\left (a+b\,x\right )}^{3/2}-6\,{\left (a+b\,x\right )}^{5/2}}{15\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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