Optimal. Leaf size=37 \[ \frac {c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e} \]
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Rubi [A]
time = 0.01, antiderivative size = 37, 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 = {656, 623}
\begin {gather*} \frac {c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Rule 623
Rule 656
Rubi steps
\begin {align*} \int \frac {\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=c \int \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2} \, dx\\ &=\frac {c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{4 e}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 26, normalized size = 0.70 \begin {gather*} \frac {c (d+e x) \left (c (d+e x)^2\right )^{3/2}}{4 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 35, normalized size = 0.95
method | result | size |
risch | \(\frac {c^{2} \left (e x +d \right )^{3} \sqrt {\left (e x +d \right )^{2} c}}{4 e}\) | \(27\) |
default | \(\frac {\left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{4 \left (e x +d \right ) e}\) | \(35\) |
gosper | \(\frac {x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \left (x^{2} c \,e^{2}+2 c d e x +c \,d^{2}\right )^{\frac {5}{2}}}{4 \left (e x +d \right )^{5}}\) | \(62\) |
trager | \(\frac {c^{2} x \left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{4 e x +4 d}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 35, normalized size = 0.95 \begin {gather*} \frac {{\left (c x^{2} e^{2} + 2 \, c d x e + c d^{2}\right )}^{\frac {5}{2}}}{4 \, {\left (x e^{2} + d e\right )}} \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. 75 vs.
\(2 (33) = 66\).
time = 3.11, size = 75, normalized size = 2.03 \begin {gather*} \frac {{\left (c^{2} x^{4} e^{3} + 4 \, c^{2} d x^{3} e^{2} + 6 \, c^{2} d^{2} x^{2} e + 4 \, c^{2} d^{3} x\right )} \sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{4 \, {\left (x e + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c \left (d + e x\right )^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{2}}\, dx \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. 76 vs.
\(2 (33) = 66\).
time = 1.04, size = 76, normalized size = 2.05 \begin {gather*} \frac {1}{4} \, {\left (c^{2} x^{4} e^{3} \mathrm {sgn}\left (x e + d\right ) + 4 \, c^{2} d x^{3} e^{2} \mathrm {sgn}\left (x e + d\right ) + 6 \, c^{2} d^{2} x^{2} e \mathrm {sgn}\left (x e + d\right ) + 4 \, c^{2} d^{3} 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 \frac {{\left (c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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