Optimal. Leaf size=23 \[ \frac {2 (c+d (a+b x))^{5/2}}{5 b d} \]
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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)^{5/2}}{5 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 33
Rubi steps
\begin {align*} \int (c+d (a+b x))^{3/2} \, dx &=\frac {\text {Subst}\left (\int (c+d x)^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac {2 (c+d (a+b x))^{5/2}}{5 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)^{5/2}}{5 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 = 2.01, size = 142, normalized size = 6.17 \begin {gather*} \text {Piecewise}\left [\left \{\left \{c^{\frac {3}{2}} x,d\text {==}0\text {\&\&}b\text {==}0\text {$\vert $$\vert $}d\text {==}0\right \},\left \{x \left (a d+c\right )^{\frac {3}{2}},b\text {==}0\right \}\right \},\frac {2 a^2 d \sqrt {a d+b d x+c}}{5 b}+\frac {4 a c \sqrt {a d+b d x+c}}{5 b}+\frac {4 a d x \sqrt {a d+b d x+c}}{5}+\frac {2 b d x^2 \sqrt {a d+b d x+c}}{5}+\frac {2 c^2 \sqrt {a d+b d x+c}}{5 b d}+\frac {4 c x \sqrt {a d+b d x+c}}{5}\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 20, normalized size = 0.87
method | result | size |
gosper | \(\frac {2 \left (b d x +a d +c \right )^{\frac {5}{2}}}{5 b d}\) | \(20\) |
derivativedivides | \(\frac {2 \left (b d x +a d +c \right )^{\frac {5}{2}}}{5 b d}\) | \(20\) |
default | \(\frac {2 \left (b d x +a d +c \right )^{\frac {5}{2}}}{5 b d}\) | \(20\) |
trager | \(\frac {2 \left (b^{2} d^{2} x^{2}+2 a b \,d^{2} x +a^{2} d^{2}+2 b c d x +2 a c d +c^{2}\right ) \sqrt {b d x +a d +c}}{5 b d}\) | \(60\) |
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 {5}{2}}}{5 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs.
\(2 (19) = 38\).
time = 0.31, size = 59, normalized size = 2.57 \begin {gather*} \frac {2 \, {\left (b^{2} d^{2} x^{2} + a^{2} d^{2} + 2 \, a c d + c^{2} + 2 \, {\left (a b d^{2} + b c d\right )} x\right )} \sqrt {b d x + a d + c}}{5 \, b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 156, normalized size = 6.78 \begin {gather*} \begin {cases} c^{\frac {3}{2}} x & \text {for}\: b = 0 \wedge d = 0 \\x \left (a d + c\right )^{\frac {3}{2}} & \text {for}\: b = 0 \\c^{\frac {3}{2}} x & \text {for}\: d = 0 \\\frac {2 a^{2} d \sqrt {a d + b d x + c}}{5 b} + \frac {4 a d x \sqrt {a d + b d x + c}}{5} + \frac {4 a c \sqrt {a d + b d x + c}}{5 b} + \frac {2 b d x^{2} \sqrt {a d + b d x + c}}{5} + \frac {4 c x \sqrt {a d + b d x + c}}{5} + \frac {2 c^{2} \sqrt {a d + b d x + c}}{5 b d} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 195 vs.
\(2 (19) = 38\).
time = 0.00, size = 357, normalized size = 15.52 \begin {gather*} \frac {\frac {2 b^{2} d^{2} \left (\frac {1}{5} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )^{2}-\frac {2}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right ) a d-\frac {2}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right ) c+\sqrt {a d+b d x+c} a^{2} d^{2}+2 \sqrt {a d+b d x+c} a d c+\sqrt {a d+b d x+c} c^{2}\right )}{d^{2} b^{2}}+2 a^{2} d^{2} \sqrt {a d+b d x+c}+\frac {4 a b d^{2} \left (\frac {1}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )-a d \sqrt {a d+b d x+c}-c \sqrt {a d+b d x+c}\right )}{b d}+4 a c d \sqrt {a d+b d x+c}+\frac {4 b c d \left (\frac {1}{3} \sqrt {a d+b d x+c} \left (a d+b d x+c\right )-a d \sqrt {a d+b d x+c}-c \sqrt {a d+b d x+c}\right )}{b d}+2 c^{2} \sqrt {a d+b d x+c}}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 45, normalized size = 1.96 \begin {gather*} \sqrt {c+d\,\left (a+b\,x\right )}\,\left (x\,\left (\frac {4\,c}{5}+\frac {4\,a\,d}{5}\right )+\frac {2\,{\left (c+a\,d\right )}^2}{5\,b\,d}+\frac {2\,b\,d\,x^2}{5}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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