Optimal. Leaf size=39 \[ -\frac {a-\frac {c d^2}{e^2}}{4 (d+e x)^4}-\frac {c d}{3 e^2 (d+e x)^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 45}
\begin {gather*} -\frac {a-\frac {c d^2}{e^2}}{4 (d+e x)^4}-\frac {c d}{3 e^2 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 24
Rule 45
Rubi steps
\begin {align*} \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^6} \, dx &=\frac {\int \frac {a e^3+c d e^2 x}{(d+e x)^5} \, dx}{e^2}\\ &=\frac {\int \left (\frac {-c d^2 e+a e^3}{(d+e x)^5}+\frac {c d e}{(d+e x)^4}\right ) \, dx}{e^2}\\ &=-\frac {a-\frac {c d^2}{e^2}}{4 (d+e x)^4}-\frac {c d}{3 e^2 (d+e x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 30, normalized size = 0.77 \begin {gather*} -\frac {3 a e^2+c d (d+4 e x)}{12 e^2 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 40, normalized size = 1.03
method | result | size |
gosper | \(-\frac {4 c d e x +3 e^{2} a +c \,d^{2}}{12 e^{2} \left (e x +d \right )^{4}}\) | \(31\) |
risch | \(\frac {-\frac {c d x}{3 e}-\frac {3 e^{2} a +c \,d^{2}}{12 e^{2}}}{\left (e x +d \right )^{4}}\) | \(35\) |
default | \(-\frac {e^{2} a -c \,d^{2}}{4 e^{2} \left (e x +d \right )^{4}}-\frac {c d}{3 e^{2} \left (e x +d \right )^{3}}\) | \(40\) |
norman | \(\frac {-\frac {d \left (3 a \,e^{5}+c \,d^{2} e^{3}\right )}{12 e^{5}}-\frac {\left (3 a \,e^{5}+5 c \,d^{2} e^{3}\right ) x}{12 e^{4}}-\frac {c d \,x^{2}}{3}}{\left (e x +d \right )^{5}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 61, normalized size = 1.56 \begin {gather*} -\frac {4 \, c d x e + c d^{2} + 3 \, a e^{2}}{12 \, {\left (x^{4} e^{6} + 4 \, d x^{3} e^{5} + 6 \, d^{2} x^{2} e^{4} + 4 \, d^{3} x e^{3} + d^{4} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.13, size = 61, normalized size = 1.56 \begin {gather*} -\frac {4 \, c d x e + c d^{2} + 3 \, a e^{2}}{12 \, {\left (x^{4} e^{6} + 4 \, d x^{3} e^{5} + 6 \, d^{2} x^{2} e^{4} + 4 \, d^{3} x e^{3} + d^{4} e^{2}\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. 70 vs.
\(2 (34) = 68\).
time = 0.24, size = 70, normalized size = 1.79 \begin {gather*} \frac {- 3 a e^{2} - c d^{2} - 4 c d e x}{12 d^{4} e^{2} + 48 d^{3} e^{3} x + 72 d^{2} e^{4} x^{2} + 48 d e^{5} x^{3} + 12 e^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.25, size = 30, normalized size = 0.77 \begin {gather*} -\frac {{\left (4 \, c d x e + c d^{2} + 3 \, a e^{2}\right )} e^{\left (-2\right )}}{12 \, {\left (x e + d\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.58, size = 68, normalized size = 1.74 \begin {gather*} -\frac {\frac {c\,d^2+3\,a\,e^2}{12\,e^2}+\frac {c\,d\,x}{3\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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