Optimal. Leaf size=151 \[ -\frac {\left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{2 e^5 (d+e x)^4}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{3 e^5 (d+e x)^3}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)} \]
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Rubi [A]
time = 0.08, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {712}
\begin {gather*} -\frac {-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^6}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^5}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^4}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^3}+\frac {c^2}{e^4 (d+e x)^2}\right ) \, dx\\ &=-\frac {\left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^5}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{2 e^5 (d+e x)^4}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{3 e^5 (d+e x)^3}+\frac {c (2 c d-b e)}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 160, normalized size = 1.06 \begin {gather*} -\frac {6 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )+e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )}{30 e^5 (d+e x)^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.78, size = 195, normalized size = 1.29
method | result | size |
risch | \(\frac {-\frac {c^{2} x^{4}}{e}-\frac {c \left (b e +2 c d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x^{2}}{3 e^{3}}-\frac {\left (3 a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}+3 d^{2} e b c +6 c^{2} d^{3}\right ) x}{6 e^{4}}-\frac {6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{5}}\) | \(179\) |
norman | \(\frac {-\frac {c^{2} x^{4}}{e}-\frac {\left (b c e +2 c^{2} d \right ) x^{3}}{e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+3 b c d e +6 c^{2} d^{2}\right ) x^{2}}{3 e^{3}}-\frac {\left (3 a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}+3 d^{2} e b c +6 c^{2} d^{3}\right ) x}{6 e^{4}}-\frac {6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{5}}\) | \(181\) |
gosper | \(-\frac {30 c^{2} x^{4} e^{4}+30 b c \,e^{4} x^{3}+60 c^{2} d \,e^{3} x^{3}+20 a c \,e^{4} x^{2}+10 b^{2} e^{4} x^{2}+30 b c d \,e^{3} x^{2}+60 c^{2} d^{2} x^{2} e^{2}+15 a b \,e^{4} x +10 a c d \,e^{3} x +5 b^{2} d \,e^{3} x +15 b c \,d^{2} e^{2} x +30 c^{2} d^{3} e x +6 a^{2} e^{4}+3 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+3 d^{3} e b c +6 c^{2} d^{4}}{30 \left (e x +d \right )^{5} e^{5}}\) | \(193\) |
default | \(-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{e^{5} \left (e x +d \right )}-\frac {c \left (b e -2 c d \right )}{e^{5} \left (e x +d \right )^{2}}-\frac {2 a b \,e^{3}-4 a d \,e^{2} c -2 b^{2} d \,e^{2}+6 d^{2} e b c -4 c^{2} d^{3}}{4 e^{5} \left (e x +d \right )^{4}}\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 210, normalized size = 1.39 \begin {gather*} -\frac {30 \, c^{2} x^{4} e^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 30 \, {\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 3 \, a b d e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d^{2} + 10 \, {\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} x^{2} + 6 \, a^{2} e^{4} + 5 \, {\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} + {\left (b^{2} e^{3} + 2 \, a c e^{3}\right )} d\right )} x}{30 \, {\left (x^{5} e^{10} + 5 \, d x^{4} e^{9} + 10 \, d^{2} x^{3} e^{8} + 10 \, d^{3} x^{2} e^{7} + 5 \, d^{4} x e^{6} + d^{5} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.28, size = 197, normalized size = 1.30 \begin {gather*} -\frac {6 \, c^{2} d^{4} + {\left (30 \, c^{2} x^{4} + 30 \, b c x^{3} + 15 \, a b x + 10 \, {\left (b^{2} + 2 \, a c\right )} x^{2} + 6 \, a^{2}\right )} e^{4} + {\left (60 \, c^{2} d x^{3} + 30 \, b c d x^{2} + 3 \, a b d + 5 \, {\left (b^{2} + 2 \, a c\right )} d x\right )} e^{3} + {\left (60 \, c^{2} d^{2} x^{2} + 15 \, b c d^{2} x + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} e^{2} + 3 \, {\left (10 \, c^{2} d^{3} x + b c d^{3}\right )} e}{30 \, {\left (x^{5} e^{10} + 5 \, d x^{4} e^{9} + 10 \, d^{2} x^{3} e^{8} + 10 \, d^{3} x^{2} e^{7} + 5 \, d^{4} x e^{6} + d^{5} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 207.79, size = 253, normalized size = 1.68 \begin {gather*} \frac {- 6 a^{2} e^{4} - 3 a b d e^{3} - 2 a c d^{2} e^{2} - b^{2} d^{2} e^{2} - 3 b c d^{3} e - 6 c^{2} d^{4} - 30 c^{2} e^{4} x^{4} + x^{3} \left (- 30 b c e^{4} - 60 c^{2} d e^{3}\right ) + x^{2} \left (- 20 a c e^{4} - 10 b^{2} e^{4} - 30 b c d e^{3} - 60 c^{2} d^{2} e^{2}\right ) + x \left (- 15 a b e^{4} - 10 a c d e^{3} - 5 b^{2} d e^{3} - 15 b c d^{2} e^{2} - 30 c^{2} d^{3} e\right )}{30 d^{5} e^{5} + 150 d^{4} e^{6} x + 300 d^{3} e^{7} x^{2} + 300 d^{2} e^{8} x^{3} + 150 d e^{9} x^{4} + 30 e^{10} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 179, normalized size = 1.19 \begin {gather*} -\frac {{\left (30 \, c^{2} x^{4} e^{4} + 60 \, c^{2} d x^{3} e^{3} + 60 \, c^{2} d^{2} x^{2} e^{2} + 30 \, c^{2} d^{3} x e + 6 \, c^{2} d^{4} + 30 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 15 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 10 \, b^{2} x^{2} e^{4} + 20 \, a c x^{2} e^{4} + 5 \, b^{2} d x e^{3} + 10 \, a c d x e^{3} + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 15 \, a b x e^{4} + 3 \, a b d e^{3} + 6 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{30 \, {\left (x e + d\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 221, normalized size = 1.46 \begin {gather*} -\frac {\frac {6\,a^2\,e^4+3\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2+3\,b\,c\,d^3\,e+6\,c^2\,d^4}{30\,e^5}+\frac {x\,\left (b^2\,d\,e^2+3\,b\,c\,d^2\,e+3\,a\,b\,e^3+6\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{6\,e^4}+\frac {c^2\,x^4}{e}+\frac {x^2\,\left (b^2\,e^2+3\,b\,c\,d\,e+6\,c^2\,d^2+2\,a\,c\,e^2\right )}{3\,e^3}+\frac {c\,x^3\,\left (b\,e+2\,c\,d\right )}{e^2}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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