Optimal. Leaf size=39 \[ -\frac {c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, 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, 621}
\begin {gather*} -\frac {c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 621
Rule 656
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^4 \sqrt {c d^2+2 c d e x+c e^2 x^2}} \, dx &=c^2 \int \frac {1}{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}} \, dx\\ &=-\frac {c}{4 e (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.69 \begin {gather*} -\frac {1}{4 e (d+e x)^3 \sqrt {c (d+e x)^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 35, normalized size = 0.90
method | result | size |
risch | \(-\frac {1}{4 \left (e x +d \right )^{3} \sqrt {\left (e x +d \right )^{2} c}\, e}\) | \(24\) |
gosper | \(-\frac {1}{4 \left (e x +d \right )^{3} e \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\) | \(35\) |
default | \(-\frac {1}{4 \left (e x +d \right )^{3} e \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}\) | \(35\) |
trager | \(\frac {\left (e^{3} x^{3}+4 d \,e^{2} x^{2}+6 d^{2} e x +4 d^{3}\right ) x \sqrt {x^{2} c \,e^{2}+2 c d e x +c \,d^{2}}}{4 d^{4} c \left (e x +d \right )^{5}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 58, normalized size = 1.49 \begin {gather*} -\frac {1}{4 \, {\left (\sqrt {c} x^{4} e^{5} + 4 \, \sqrt {c} d x^{3} e^{4} + 6 \, \sqrt {c} d^{2} x^{2} e^{3} + 4 \, \sqrt {c} d^{3} x e^{2} + \sqrt {c} d^{4} 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. 81 vs.
\(2 (35) = 70\).
time = 1.83, size = 81, normalized size = 2.08 \begin {gather*} -\frac {\sqrt {c x^{2} e^{2} + 2 \, c d x e + c d^{2}}}{4 \, {\left (c x^{5} e^{6} + 5 \, c d x^{4} e^{5} + 10 \, c d^{2} x^{3} e^{4} + 10 \, c d^{3} x^{2} e^{3} + 5 \, c d^{4} x e^{2} + c d^{5} e\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 {1}{\sqrt {c \left (d + e x\right )^{2}} \left (d + e x\right )^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.73, size = 24, normalized size = 0.62 \begin {gather*} -\frac {e^{\left (-1\right )}}{4 \, {\left (x e + d\right )}^{4} \sqrt {c} \mathrm {sgn}\left (x e + d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.46, size = 37, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {c\,d^2+2\,c\,d\,e\,x+c\,e^2\,x^2}}{4\,c\,e\,{\left (d+e\,x\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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