Optimal. Leaf size=69 \[ \frac {(b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^2}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {654, 623}
\begin {gather*} \frac {(a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{2 b^2}+\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rule 654
Rubi steps
\begin {align*} \int (d+e x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {e \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b^2}+\frac {\left (2 b^2 d-2 a b e\right ) \int \sqrt {a^2+2 a b x+b^2 x^2} \, dx}{2 b^2}\\ &=\frac {(b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}{2 b^2}+\frac {e \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 = 45, normalized size = 0.65 \begin {gather*} \frac {x \sqrt {(a+b x)^2} (3 a (2 d+e x)+b x (3 d+2 e 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 = 0.19, size = 32, normalized size = 0.46
method | result | size |
default | \(-\frac {\mathrm {csgn}\left (b x +a \right ) \left (b x +a \right )^{2} \left (-2 b e x +a e -3 b d \right )}{6 b^{2}}\) | \(32\) |
gosper | \(\frac {x \left (2 b e \,x^{2}+3 a e x +3 x b d +6 a d \right ) \sqrt {\left (b x +a \right )^{2}}}{6 b x +6 a}\) | \(42\) |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, b e \,x^{3}}{3 b x +3 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (a e +b d \right ) x^{2}}{2 b x +2 a}+\frac {\sqrt {\left (b x +a \right )^{2}}\, a d x}{b x +a}\) | \(73\) |
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. 128 vs.
\(2 (54) = 108\).
time = 0.30, size = 128, normalized size = 1.86 \begin {gather*} \frac {1}{2} \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} d x - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a x e}{2 \, b} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a d}{2 \, b} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2}} a^{2} e}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} e}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.16, size = 29, normalized size = 0.42 \begin {gather*} \frac {1}{2} \, b d x^{2} + a d x + \frac {1}{6} \, {\left (2 \, b x^{3} + 3 \, a x^{2}\right )} e \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.01, size = 26, normalized size = 0.38 \begin {gather*} a d x + \frac {b e x^{3}}{3} + x^{2} \left (\frac {a e}{2} + \frac {b d}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.73, size = 52, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, b x^{3} e \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, b d x^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {1}{2} \, a x^{2} e \mathrm {sgn}\left (b x + a\right ) + a d x \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 77, normalized size = 1.12 \begin {gather*} \frac {e\,\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}+\frac {d\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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