Optimal. Leaf size=34 \[ \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \]
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Rubi [A]
time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {657, 643}
\begin {gather*} \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rule 657
Rubi steps
\begin {align*} \int (d+e x)^3 \left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2} \, dx &=\frac {\int (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{7/2} \, dx}{c}\\ &=\frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{9/2}}{9 c^2 e}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 27, normalized size = 0.79 \begin {gather*} \frac {(d+e x)^4 \left (c (d+e x)^2\right )^{5/2}}{9 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 35, normalized size = 1.03
method | result | size |
risch | \(\frac {c^{2} \left (e x +d \right )^{8} \sqrt {\left (e x +d \right )^{2} c}}{9 e}\) | \(27\) |
default | \(\frac {\left (e x +d \right )^{4} \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{9 e}\) | \(35\) |
gosper | \(\frac {x \left (e^{8} x^{8}+9 d \,e^{7} x^{7}+36 d^{2} e^{6} x^{6}+84 d^{3} e^{5} x^{5}+126 d^{4} e^{4} x^{4}+126 d^{5} e^{3} x^{3}+84 d^{6} e^{2} x^{2}+36 d^{7} e x +9 d^{8}\right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{9 \left (e x +d \right )^{5}}\) | \(117\) |
trager | \(\frac {c^{2} x \left (e^{8} x^{8}+9 d \,e^{7} x^{7}+36 d^{2} e^{6} x^{6}+84 d^{3} e^{5} x^{5}+126 d^{4} e^{4} x^{4}+126 d^{5} e^{3} x^{3}+84 d^{6} e^{2} x^{2}+36 d^{7} e x +9 d^{8}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{9 e x +9 d}\) | \(120\) |
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. 94 vs.
\(2 (29) = 58\).
time = 0.29, size = 94, normalized size = 2.76 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {7}{2}} x^{2} e}{9 \, c} + \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {7}{2}} d^{2} e^{\left (-1\right )}}{9 \, c} + \frac {2 \, {\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {7}{2}} d x}{9 \, 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. 140 vs.
\(2 (29) = 58\).
time = 4.06, size = 140, normalized size = 4.12 \begin {gather*} \frac {{\left (c^{2} x^{9} e^{8} + 9 \, c^{2} d x^{8} e^{7} + 36 \, c^{2} d^{2} x^{7} e^{6} + 84 \, c^{2} d^{3} x^{6} e^{5} + 126 \, c^{2} d^{4} x^{5} e^{4} + 126 \, c^{2} d^{5} x^{4} e^{3} + 84 \, c^{2} d^{6} x^{3} e^{2} + 36 \, c^{2} d^{7} x^{2} e + 9 \, c^{2} d^{8} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{9 \, {\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. 374 vs.
\(2 (31) = 62\).
time = 0.54, size = 374, normalized size = 11.00 \begin {gather*} \begin {cases} \frac {c^{2} d^{8} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9 e} + \frac {8 c^{2} d^{7} x \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {28 c^{2} d^{6} e x^{2} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {56 c^{2} d^{5} e^{2} x^{3} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {70 c^{2} d^{4} e^{3} x^{4} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {56 c^{2} d^{3} e^{4} x^{5} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {28 c^{2} d^{2} e^{5} x^{6} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {8 c^{2} d e^{6} x^{7} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} + \frac {c^{2} e^{7} x^{8} \sqrt {c d^{2} + 2 c d e x + c e^{2} x^{2}}}{9} & \text {for}\: e \neq 0 \\d^{3} x \left (c d^{2}\right )^{\frac {5}{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. 176 vs.
\(2 (29) = 58\).
time = 1.07, size = 176, normalized size = 5.18 \begin {gather*} \frac {1}{9} \, {\left (c^{2} x^{9} e^{8} \mathrm {sgn}\left (x e + d\right ) + 9 \, c^{2} d x^{8} e^{7} \mathrm {sgn}\left (x e + d\right ) + 36 \, c^{2} d^{2} x^{7} e^{6} \mathrm {sgn}\left (x e + d\right ) + 84 \, c^{2} d^{3} x^{6} e^{5} \mathrm {sgn}\left (x e + d\right ) + 126 \, c^{2} d^{4} x^{5} e^{4} \mathrm {sgn}\left (x e + d\right ) + 126 \, c^{2} d^{5} x^{4} e^{3} \mathrm {sgn}\left (x e + d\right ) + 84 \, c^{2} d^{6} x^{3} e^{2} \mathrm {sgn}\left (x e + d\right ) + 36 \, c^{2} d^{7} x^{2} e \mathrm {sgn}\left (x e + d\right ) + 9 \, c^{2} d^{8} 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 [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (d+e\,x\right )}^3\,{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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