Optimal. Leaf size=61 \[ -\frac {a (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^2}+\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {654, 623}
\begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}-\frac {a (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rule 654
Rubi steps
\begin {align*} \int x \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}-\frac {a \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx}{b}\\ &=-\frac {a (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^2}+\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.54 \begin {gather*} \frac {x^2 \sqrt {(a+b x)^2} (3 a+2 b x)}{6 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
2.
time = 0.14, size = 25, normalized size = 0.41
method | result | size |
default | \(-\frac {\mathrm {csgn}\left (b x +a \right ) \left (b x +a \right )^{2} \left (-2 b x +a \right )}{6 b^{2}}\) | \(25\) |
gosper | \(\frac {x^{2} \left (2 b x +3 a \right ) \sqrt {\left (b x +a \right )^{2}}}{6 b x +6 a}\) | \(30\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, a \,x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, b \,x^{3}}{3 b x +3 a}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 75, normalized size = 1.23 \begin {gather*} -\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a x}{2 \, b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.87, size = 13, normalized size = 0.21 \begin {gather*} \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 12, normalized size = 0.20 \begin {gather*} \frac {a x^{2}}{2} + \frac {b x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.69, size = 39, normalized size = 0.64 \begin {gather*} \frac {1}{3} \, b x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a x^{2} \mathrm {sgn}\left (b x + a\right ) - \frac {a^{3} \mathrm {sgn}\left (b x + a\right )}{6 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 55, normalized size = 0.90 \begin {gather*} \frac {\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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