Optimal. Leaf size=23 \[ \frac {2 (c+d (a+b x))^{3/2}}{3 b d} \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {33, 32}
\begin {gather*} \frac {2 (d (a+b x)+c)^{3/2}}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 33
Rubi steps
\begin {align*} \int \sqrt {c+d (a+b x)} \, dx &=\frac {\text {Subst}\left (\int \sqrt {c+d x} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 (c+d (a+b x))^{3/2}}{3 b d}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 (c+a d+b d x)^{3/2}}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 1.80, size = 84, normalized size = 3.65 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\sqrt {c} x,d\text {==}0\text {\&\&}b\text {==}0\text {$\vert $$\vert $}d\text {==}0\right \},\left \{x \sqrt {a d+c},b\text {==}0\right \}\right \},\frac {2 a \sqrt {a d+b d x+c}}{3 b}+\frac {2 c \sqrt {a d+b d x+c}}{3 b d}+\frac {2 x \sqrt {a d+b d x+c}}{3}\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 20, normalized size = 0.87
method | result | size |
gosper | \(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 b d}\) | \(20\) |
derivativedivides | \(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 b d}\) | \(20\) |
default | \(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 b d}\) | \(20\) |
trager | \(\frac {2 \left (b d x +a d +c \right )^{\frac {3}{2}}}{3 b d}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left ({\left (b x + a\right )} d + c\right )}^{\frac {3}{2}}}{3 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, {\left (b d x + a d + c\right )}^{\frac {3}{2}}}{3 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 82, normalized size = 3.57 \begin {gather*} \begin {cases} \sqrt {c} x & \text {for}\: b = 0 \wedge d = 0 \\x \sqrt {a d + c} & \text {for}\: b = 0 \\\sqrt {c} x & \text {for}\: d = 0 \\\frac {2 a \sqrt {a d + b d x + c}}{3 b} + \frac {2 x \sqrt {a d + b d x + c}}{3} + \frac {2 c \sqrt {a d + b d x + c}}{3 b d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 30, normalized size = 1.30 \begin {gather*} \frac {\sqrt {a d+b d x+c} \left (a d+b d x+c\right )}{\frac {3}{2} b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 19, normalized size = 0.83 \begin {gather*} \frac {2\,{\left (c+d\,\left (a+b\,x\right )\right )}^{3/2}}{3\,b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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