Optimal. Leaf size=35 \[ -\frac {(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 37}
\begin {gather*} -\frac {(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {d+e x}{(a e+c d x)^3} \, dx\\ &=-\frac {(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} -\frac {a e^2+c d (d+2 e x)}{2 c^2 d^2 (a e+c d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.70, size = 51, normalized size = 1.46
method | result | size |
gosper | \(-\frac {2 c d e x +e^{2} a +c \,d^{2}}{2 \left (c d x +a e \right )^{2} c^{2} d^{2}}\) | \(36\) |
risch | \(\frac {-\frac {e x}{c d}-\frac {e^{2} a +c \,d^{2}}{2 c^{2} d^{2}}}{\left (c d x +a e \right )^{2}}\) | \(42\) |
default | \(-\frac {e}{c^{2} d^{2} \left (c d x +a e \right )}-\frac {-e^{2} a +c \,d^{2}}{2 c^{2} d^{2} \left (c d x +a e \right )^{2}}\) | \(51\) |
norman | \(\frac {-\frac {e^{3} x^{3}}{c d}+\frac {\left (-e^{4} a -2 d^{2} e^{2} c \right ) x}{c^{2} d e}+\frac {-e^{2} a -c \,d^{2}}{2 c^{2}}+\frac {\left (-a \,e^{6}-5 c \,d^{2} e^{4}\right ) x^{2}}{2 d^{2} e^{2} c^{2}}}{\left (c d x +a e \right )^{2} \left (e x +d \right )^{2}}\) | \(109\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 56, normalized size = 1.60 \begin {gather*} -\frac {2 \, c d x e + c d^{2} + a e^{2}}{2 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} x e + a^{2} c^{2} d^{2} e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.05, size = 56, normalized size = 1.60 \begin {gather*} -\frac {2 \, c d x e + c d^{2} + a e^{2}}{2 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} x e + a^{2} c^{2} d^{2} 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. 60 vs.
\(2 (29) = 58\).
time = 0.18, size = 60, normalized size = 1.71 \begin {gather*} \frac {- a e^{2} - c d^{2} - 2 c d e x}{2 a^{2} c^{2} d^{2} e^{2} + 4 a c^{3} d^{3} e x + 2 c^{4} d^{4} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.48, size = 36, normalized size = 1.03 \begin {gather*} -\frac {2 \, c d x e + c d^{2} + a e^{2}}{2 \, {\left (c d x + a e\right )}^{2} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 43, normalized size = 1.23 \begin {gather*} -\frac {\frac {1}{2\,c}-\frac {x^2}{2\,a}}{a^2\,e^2+2\,a\,c\,d\,e\,x+c^2\,d^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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