Optimal. Leaf size=71 \[ \frac {a x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {b x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
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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^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {a x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 660
Rubi steps
\begin {align*} \int x^3 \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^3 \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^3+b^2 x^4\right ) \, dx}{a b+b^2 x}\\ &=\frac {a x^4 \sqrt {a^2+2 a b x+b^2 x^2}}{4 (a+b x)}+\frac {b x^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.46 \begin {gather*} \frac {x^4 \sqrt {(a+b x)^2} (5 a+4 b x)}{20 (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 = 47, normalized size = 0.66
method | result | size |
gosper | \(\frac {x^{4} \left (4 b x +5 a \right ) \sqrt {\left (b x +a \right )^{2}}}{20 b x +20 a}\) | \(30\) |
risch | \(\frac {a \,x^{4} \sqrt {\left (b x +a \right )^{2}}}{4 b x +4 a}+\frac {b \,x^{5} \sqrt {\left (b x +a \right )^{2}}}{5 b x +5 a}\) | \(46\) |
default | \(-\frac {\mathrm {csgn}\left (b x +a \right ) \left (b x +a \right )^{2} \left (-4 b^{3} x^{3}+3 a \,b^{2} x^{2}-2 a^{2} b x +a^{3}\right )}{20 b^{4}}\) | \(47\) |
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. 131 vs.
\(2 (45) = 90\).
time = 0.30, size = 131, normalized size = 1.85 \begin {gather*} -\frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{3} x}{2 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x^{2}}{5 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{4}}{2 \, b^{4}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a x}{20 \, b^{3}} + \frac {9 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a^{2}}{20 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.59, size = 13, normalized size = 0.18 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} \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^{4}}{4} + \frac {b x^{5}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 39, normalized size = 0.55 \begin {gather*} \frac {1}{5} \, b x^{5} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{4} \, a x^{4} \mathrm {sgn}\left (b x + a\right ) - \frac {a^{5} \mathrm {sgn}\left (b x + a\right )}{20 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 92, normalized size = 1.30 \begin {gather*} \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (4\,b^2\,x^2\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )-a^4+9\,a^2\,b^2\,x^2+8\,a^3\,b\,x-7\,a\,b\,x\,\left (a^2+2\,a\,b\,x+b^2\,x^2\right )\right )}{20\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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