Optimal. Leaf size=67 \[ -\frac {c d^2-b d e+a e^2}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 67, 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}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)} \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)^4} \, dx &=\int \left (\frac {c d^2-b d e+a e^2}{e^2 (d+e x)^4}+\frac {-2 c d+b e}{e^2 (d+e x)^3}+\frac {c}{e^2 (d+e x)^2}\right ) \, dx\\ &=-\frac {c d^2-b d e+a e^2}{3 e^3 (d+e x)^3}+\frac {2 c d-b e}{2 e^3 (d+e x)^2}-\frac {c}{e^3 (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 0.75 \begin {gather*} -\frac {2 c \left (d^2+3 d e x+3 e^2 x^2\right )+e (2 a e+b (d+3 e x))}{6 e^3 (d+e x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 63, normalized size = 0.94
method | result | size |
gosper | \(-\frac {6 x^{2} c \,e^{2}+3 b \,e^{2} x +6 c d e x +2 e^{2} a +b d e +2 c \,d^{2}}{6 \left (e x +d \right )^{3} e^{3}}\) | \(52\) |
norman | \(\frac {-\frac {c \,x^{2}}{e}-\frac {\left (b e +2 c d \right ) x}{2 e^{2}}-\frac {2 e^{2} a +b d e +2 c \,d^{2}}{6 e^{3}}}{\left (e x +d \right )^{3}}\) | \(55\) |
risch | \(\frac {-\frac {c \,x^{2}}{e}-\frac {\left (b e +2 c d \right ) x}{2 e^{2}}-\frac {2 e^{2} a +b d e +2 c \,d^{2}}{6 e^{3}}}{\left (e x +d \right )^{3}}\) | \(55\) |
default | \(-\frac {e^{2} a -b d e +c \,d^{2}}{3 e^{3} \left (e x +d \right )^{3}}-\frac {c}{e^{3} \left (e x +d \right )}-\frac {b e -2 c d}{2 e^{3} \left (e x +d \right )^{2}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 72, normalized size = 1.07 \begin {gather*} -\frac {6 \, c x^{2} e^{2} + 2 \, c d^{2} + b d e + 3 \, {\left (2 \, c d e + b e^{2}\right )} x + 2 \, a e^{2}}{6 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.81, size = 69, normalized size = 1.03 \begin {gather*} -\frac {2 \, c d^{2} + {\left (6 \, c x^{2} + 3 \, b x + 2 \, a\right )} e^{2} + {\left (6 \, c d x + b d\right )} e}{6 \, {\left (x^{3} e^{6} + 3 \, d x^{2} e^{5} + 3 \, d^{2} x e^{4} + d^{3} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.47, size = 82, normalized size = 1.22 \begin {gather*} \frac {- 2 a e^{2} - b d e - 2 c d^{2} - 6 c e^{2} x^{2} + x \left (- 3 b e^{2} - 6 c d e\right )}{6 d^{3} e^{3} + 18 d^{2} e^{4} x + 18 d e^{5} x^{2} + 6 e^{6} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.78, size = 50, normalized size = 0.75 \begin {gather*} -\frac {{\left (6 \, c x^{2} e^{2} + 6 \, c d x e + 2 \, c d^{2} + 3 \, b x e^{2} + b d e + 2 \, a e^{2}\right )} e^{\left (-3\right )}}{6 \, {\left (x e + d\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 76, normalized size = 1.13 \begin {gather*} -\frac {\frac {2\,c\,d^2+b\,d\,e+2\,a\,e^2}{6\,e^3}+\frac {x\,\left (b\,e+2\,c\,d\right )}{2\,e^2}+\frac {c\,x^2}{e}}{d^3+3\,d^2\,e\,x+3\,d\,e^2\,x^2+e^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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