Optimal. Leaf size=32 \[ \frac {2 (a+b x)^{3/2}}{3 b^2}-\frac {2 a \sqrt {a+b x}}{b^2} \]
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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 = {43} \[ \frac {2 (a+b x)^{3/2}}{3 b^2}-\frac {2 a \sqrt {a+b x}}{b^2} \]
Antiderivative was successfully verified.
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Rule 43
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 \[ \frac {2 (b x-2 a) \sqrt {a+b x}}{3 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 19, normalized size = 0.59 \[ \frac {2 \, \sqrt {b x + a} {\left (b x - 2 \, a\right )}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.21, size = 23, normalized size = 0.72 \[ \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.66 \[ -\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 26, normalized size = 0.81 \[ \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, b^{2}} - \frac {2 \, \sqrt {b x + a} a}{b^{2}} \]
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 \[ -\frac {6\,a\,\sqrt {a+b\,x}-2\,{\left (a+b\,x\right )}^{3/2}}{3\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.11, size = 162, normalized size = 5.06 \[ - \frac {4 a^{\frac {7}{2}} \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {4 a^{\frac {7}{2}}}{3 a^{2} b^{2} + 3 a b^{3} x} - \frac {2 a^{\frac {5}{2}} b x \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {4 a^{\frac {5}{2}} b x}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {2 a^{\frac {3}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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