Optimal. Leaf size=71 \[ \frac {a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {660, 45}
\begin {gather*} \frac {b x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int x^2 \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^2 \left (a b+b^2 x\right ) \, dx}{a b+b^2 x}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b x^2+b^2 x^3\right ) \, dx}{a b+b^2 x}\\ &=\frac {a x^3 \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {b x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.46 \begin {gather*} \frac {x^3 \sqrt {(a+b x)^2} (4 a+3 b x)}{12 (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 = 0.13, size = 36, normalized size = 0.51
method | result | size |
gosper | \(\frac {x^{3} \left (3 b x +4 a \right ) \sqrt {\left (b x +a \right )^{2}}}{12 b x +12 a}\) | \(30\) |
default | \(\frac {\mathrm {csgn}\left (b x +a \right ) \left (b x +a \right )^{2} \left (3 b^{2} x^{2}-2 a b x +a^{2}\right )}{12 b^{3}}\) | \(36\) |
risch | \(\frac {a \,x^{3} \sqrt {\left (b x +a \right )^{2}}}{3 b x +3 a}+\frac {b \,x^{4} \sqrt {\left (b x +a \right )^{2}}}{4 b x +4 a}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (45) = 90\).
time = 0.29, size = 102, normalized size = 1.44 \begin {gather*} \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} x}{2 \, b^{2}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3}}{2 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x}{4 \, b^{2}} - \frac {5 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a}{12 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.76, size = 13, normalized size = 0.18 \begin {gather*} \frac {1}{4} \, b x^{4} + \frac {1}{3} \, a x^{3} \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 = 0.17 \begin {gather*} \frac {a x^{3}}{3} + \frac {b x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.98, size = 39, normalized size = 0.55 \begin {gather*} \frac {1}{4} \, b x^{4} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{3} \, a x^{3} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{4} \mathrm {sgn}\left (b x + a\right )}{12 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 63, normalized size = 0.89 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (a^3-5\,a\,b^2\,x^2+3\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-4\,a^2\,b\,x\right )}{12\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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