Optimal. Leaf size=34 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \]
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Rubi [A]
time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {643}
\begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rubi steps
\begin {align*} \int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx &=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{5 c e}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.68 \begin {gather*} \frac {\left (c (d+e x)^2\right )^{5/2}}{5 c e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 35, normalized size = 1.03
method | result | size |
risch | \(\frac {c \left (e x +d \right )^{4} \sqrt {\left (e x +d \right )^{2} c}}{5 e}\) | \(25\) |
default | \(\frac {\left (e x +d \right )^{2} \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{5 e}\) | \(35\) |
gosper | \(\frac {x \left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {3}{2}}}{5 \left (e x +d \right )^{3}}\) | \(73\) |
trager | \(\frac {c x \left (e^{4} x^{4}+5 d \,e^{3} x^{3}+10 d^{2} e^{2} x^{2}+10 d^{3} e x +5 d^{4}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{5 e x +5 d}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 29, normalized size = 0.85 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {5}{2}} e^{\left (-1\right )}}{5 \, c} \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. 78 vs.
\(2 (29) = 58\).
time = 3.40, size = 78, normalized size = 2.29 \begin {gather*} \frac {{\left (c x^{5} e^{4} + 5 \, c d x^{4} e^{3} + 10 \, c d^{2} x^{3} e^{2} + 10 \, c d^{3} x^{2} e + 5 \, c d^{4} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{5 \, {\left (x e + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs.
\(2 (29) = 58\).
time = 0.16, size = 194, normalized size = 5.71 \begin {gather*} \begin {cases} \frac {c d^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5 e} + \frac {4 c d^{3} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac {6 c d^{2} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac {4 c d e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} + \frac {c e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{5} & \text {for}\: e \neq 0 \\d x \left (c d^{2}\right )^{\frac {3}{2}} & \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. 86 vs.
\(2 (29) = 58\).
time = 0.95, size = 86, normalized size = 2.53 \begin {gather*} \frac {1}{5} \, {\left (c x^{5} e^{4} \mathrm {sgn}\left (x e + d\right ) + 5 \, c d x^{4} e^{3} \mathrm {sgn}\left (x e + d\right ) + 10 \, c d^{2} x^{3} e^{2} \mathrm {sgn}\left (x e + d\right ) + 10 \, c d^{3} x^{2} e \mathrm {sgn}\left (x e + d\right ) + 5 \, c d^{4} x \mathrm {sgn}\left (x e + d\right )\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.56, size = 34, normalized size = 1.00 \begin {gather*} \frac {{\left (d+e\,x\right )}^2\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{3/2}}{5\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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