Optimal. Leaf size=54 \[ -\frac {c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac {e}{2 c^2 d^2 (a e+c d x)^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 45}
\begin {gather*} -\frac {c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac {e}{2 c^2 d^2 (a e+c d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 640
Rubi steps
\begin {align*} \int \frac {(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac {d+e x}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac {c d^2-a e^2}{c d (a e+c d x)^4}+\frac {e}{c d (a e+c d x)^3}\right ) \, dx\\ &=-\frac {c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac {e}{2 c^2 d^2 (a e+c d x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 0.69 \begin {gather*} -\frac {a e^2+c d (2 d+3 e x)}{6 c^2 d^2 (a e+c d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.73, size = 51, normalized size = 0.94
method | result | size |
gosper | \(-\frac {3 c d e x +e^{2} a +2 c \,d^{2}}{6 c^{2} d^{2} \left (c d x +a e \right )^{3}}\) | \(37\) |
risch | \(\frac {-\frac {e x}{2 c d}-\frac {e^{2} a +2 c \,d^{2}}{6 c^{2} d^{2}}}{\left (c d x +a e \right )^{3}}\) | \(43\) |
default | \(-\frac {-e^{2} a +c \,d^{2}}{3 c^{2} d^{2} \left (c d x +a e \right )^{3}}-\frac {e}{2 c^{2} d^{2} \left (c d x +a e \right )^{2}}\) | \(51\) |
norman | \(\frac {\frac {-a d \,e^{2} c -2 c^{2} d^{3}}{6 c^{3}}+\frac {\left (-a c d \,e^{4}-3 c^{2} d^{3} e^{2}\right ) x}{2 c^{3} d e}-\frac {e^{4} x^{4}}{2 c d}+\frac {\left (-a c d \,e^{6}-5 e^{4} d^{3} c^{2}\right ) x^{2}}{2 c^{3} e^{2} d^{2}}+\frac {\left (-a c d \,e^{8}-11 e^{6} d^{3} c^{2}\right ) x^{3}}{6 c^{3} d^{3} e^{3}}}{\left (c d x +a e \right )^{3} \left (e x +d \right )^{3}}\) | \(156\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 73, normalized size = 1.35 \begin {gather*} -\frac {3 \, c d x e + 2 \, c d^{2} + a e^{2}}{6 \, {\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} x^{2} e + 3 \, a^{2} c^{3} d^{3} x e^{2} + a^{3} c^{2} d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.23, size = 73, normalized size = 1.35 \begin {gather*} -\frac {3 \, c d x e + 2 \, c d^{2} + a e^{2}}{6 \, {\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} x^{2} e + 3 \, a^{2} c^{3} d^{3} x e^{2} + a^{3} c^{2} d^{2} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 80, normalized size = 1.48 \begin {gather*} \frac {- a e^{2} - 2 c d^{2} - 3 c d e x}{6 a^{3} c^{2} d^{2} e^{3} + 18 a^{2} c^{3} d^{3} e^{2} x + 18 a c^{4} d^{4} e x^{2} + 6 c^{5} d^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 37, normalized size = 0.69 \begin {gather*} -\frac {3 \, c d x e + 2 \, c d^{2} + a e^{2}}{6 \, {\left (c d x + a e\right )}^{3} c^{2} d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.57, size = 77, normalized size = 1.43 \begin {gather*} -\frac {\frac {2\,c\,d^2+a\,e^2}{6\,c^2\,d^2}+\frac {e\,x}{2\,c\,d}}{a^3\,e^3+3\,a^2\,c\,d\,e^2\,x+3\,a\,c^2\,d^2\,e\,x^2+c^3\,d^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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