Optimal. Leaf size=17 \[ -\frac {1}{6 c^3 e (d+e x)^6} \]
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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32}
\begin {gather*} -\frac {1}{6 c^3 e (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {1}{(d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^3} \, dx &=\int \frac {1}{c^3 (d+e x)^7} \, dx\\ &=\frac {\int \frac {1}{(d+e x)^7} \, dx}{c^3}\\ &=-\frac {1}{6 c^3 e (d+e x)^6}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} -\frac {1}{6 c^3 e (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 16, normalized size = 0.94
method | result | size |
default | \(-\frac {1}{6 c^{3} e \left (e x +d \right )^{6}}\) | \(16\) |
norman | \(-\frac {1}{6 c^{3} e \left (e x +d \right )^{6}}\) | \(16\) |
gosper | \(-\frac {1}{6 e \,c^{3} \left (e^{2} x^{2}+2 d x e +d^{2}\right )^{3}}\) | \(27\) |
risch | \(-\frac {1}{6 e \left (e x +d \right )^{2} c^{3} \left (e^{2} x^{2}+2 d x e +d^{2}\right )^{2}}\) | \(34\) |
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. 84 vs.
\(2 (15) = 30\).
time = 0.31, size = 84, normalized size = 4.94 \begin {gather*} -\frac {1}{6 \, {\left (c^{3} x^{6} e^{7} + 6 \, c^{3} d x^{5} e^{6} + 15 \, c^{3} d^{2} x^{4} e^{5} + 20 \, c^{3} d^{3} x^{3} e^{4} + 15 \, c^{3} d^{4} x^{2} e^{3} + 6 \, c^{3} d^{5} x e^{2} + c^{3} d^{6} 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. 84 vs.
\(2 (15) = 30\).
time = 3.31, size = 84, normalized size = 4.94 \begin {gather*} -\frac {1}{6 \, {\left (c^{3} x^{6} e^{7} + 6 \, c^{3} d x^{5} e^{6} + 15 \, c^{3} d^{2} x^{4} e^{5} + 20 \, c^{3} d^{3} x^{3} e^{4} + 15 \, c^{3} d^{4} x^{2} e^{3} + 6 \, c^{3} d^{5} x e^{2} + c^{3} d^{6} e\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. 97 vs.
\(2 (15) = 30\).
time = 0.20, size = 97, normalized size = 5.71 \begin {gather*} - \frac {1}{6 c^{3} d^{6} e + 36 c^{3} d^{5} e^{2} x + 90 c^{3} d^{4} e^{3} x^{2} + 120 c^{3} d^{3} e^{4} x^{3} + 90 c^{3} d^{2} e^{5} x^{4} + 36 c^{3} d e^{6} x^{5} + 6 c^{3} e^{7} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 15, normalized size = 0.88 \begin {gather*} -\frac {e^{\left (-1\right )}}{6 \, {\left (x e + d\right )}^{6} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.45, size = 91, normalized size = 5.35 \begin {gather*} -\frac {1}{6\,c^3\,d^6\,e+36\,c^3\,d^5\,e^2\,x+90\,c^3\,d^4\,e^3\,x^2+120\,c^3\,d^3\,e^4\,x^3+90\,c^3\,d^2\,e^5\,x^4+36\,c^3\,d\,e^6\,x^5+6\,c^3\,e^7\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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