Optimal. Leaf size=69 \[ -\frac {c d^2-b d e+a e^2}{4 e^3 (d+e x)^4}+\frac {2 c d-b e}{3 e^3 (d+e x)^3}-\frac {c}{2 e^3 (d+e x)^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {712}
\begin {gather*} -\frac {a e^2-b d e+c d^2}{4 e^3 (d+e x)^4}+\frac {2 c d-b e}{3 e^3 (d+e x)^3}-\frac {c}{2 e^3 (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {a+b x+c x^2}{(d+e x)^5} \, dx &=\int \left (\frac {c d^2-b d e+a e^2}{e^2 (d+e x)^5}+\frac {-2 c d+b e}{e^2 (d+e x)^4}+\frac {c}{e^2 (d+e x)^3}\right ) \, dx\\ &=-\frac {c d^2-b d e+a e^2}{4 e^3 (d+e x)^4}+\frac {2 c d-b e}{3 e^3 (d+e x)^3}-\frac {c}{2 e^3 (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 0.71 \begin {gather*} -\frac {c \left (d^2+4 d e x+6 e^2 x^2\right )+e (3 a e+b (d+4 e x))}{12 e^3 (d+e x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 63, normalized size = 0.91
method | result | size |
gosper | \(-\frac {6 x^{2} c \,e^{2}+4 b \,e^{2} x +4 c d e x +3 e^{2} a +b d e +c \,d^{2}}{12 e^{3} \left (e x +d \right )^{4}}\) | \(51\) |
risch | \(\frac {-\frac {c \,x^{2}}{2 e}-\frac {\left (b e +c d \right ) x}{3 e^{2}}-\frac {3 e^{2} a +b d e +c \,d^{2}}{12 e^{3}}}{\left (e x +d \right )^{4}}\) | \(53\) |
norman | \(\frac {-\frac {c \,x^{2}}{2 e}-\frac {\left (b \,e^{2}+c d e \right ) x}{3 e^{3}}-\frac {3 a \,e^{3}+b d \,e^{2}+c \,d^{2} e}{12 e^{4}}}{\left (e x +d \right )^{4}}\) | \(59\) |
default | \(-\frac {b e -2 c d}{3 e^{3} \left (e x +d \right )^{3}}-\frac {c}{2 e^{3} \left (e x +d \right )^{2}}-\frac {e^{2} a -b d e +c \,d^{2}}{4 e^{3} \left (e x +d \right )^{4}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 80, normalized size = 1.16 \begin {gather*} -\frac {6 \, c x^{2} e^{2} + c d^{2} + b d e + 4 \, {\left (c d e + b e^{2}\right )} x + 3 \, a e^{2}}{12 \, {\left (x^{4} e^{7} + 4 \, d x^{3} e^{6} + 6 \, d^{2} x^{2} e^{5} + 4 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.84, size = 78, normalized size = 1.13 \begin {gather*} -\frac {c d^{2} + {\left (6 \, c x^{2} + 4 \, b x + 3 \, a\right )} e^{2} + {\left (4 \, c d x + b d\right )} e}{12 \, {\left (x^{4} e^{7} + 4 \, d x^{3} e^{6} + 6 \, d^{2} x^{2} e^{5} + 4 \, d^{3} x e^{4} + d^{4} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.72, size = 92, normalized size = 1.33 \begin {gather*} \frac {- 3 a e^{2} - b d e - c d^{2} - 6 c e^{2} x^{2} + x \left (- 4 b e^{2} - 4 c d e\right )}{12 d^{4} e^{3} + 48 d^{3} e^{4} x + 72 d^{2} e^{5} x^{2} + 48 d e^{6} x^{3} + 12 e^{7} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.96, size = 86, normalized size = 1.25 \begin {gather*} -\frac {1}{12} \, {\left (\frac {6 \, c e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {8 \, c d e^{\left (-2\right )}}{{\left (x e + d\right )}^{3}} + \frac {3 \, c d^{2} e^{\left (-2\right )}}{{\left (x e + d\right )}^{4}} + \frac {4 \, b e^{\left (-1\right )}}{{\left (x e + d\right )}^{3}} - \frac {3 \, b d e^{\left (-1\right )}}{{\left (x e + d\right )}^{4}} + \frac {3 \, a}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.65, size = 86, normalized size = 1.25 \begin {gather*} -\frac {\frac {c\,d^2+b\,d\,e+3\,a\,e^2}{12\,e^3}+\frac {x\,\left (b\,e+c\,d\right )}{3\,e^2}+\frac {c\,x^2}{2\,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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