Optimal. Leaf size=156 \[ -\frac {\left (c d^2-b d e+a e^2\right )^2}{6 e^5 (d+e x)^6}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{5 e^5 (d+e x)^5}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 156, 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}{4 e^5 (d+e x)^4}+\frac {2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^2}{6 e^5 (d+e x)^6}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2} \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)^7} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^7}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^6}+\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^5}-\frac {2 c (2 c d-b e)}{e^4 (d+e x)^4}+\frac {c^2}{e^4 (d+e x)^3}\right ) \, dx\\ &=-\frac {\left (c d^2-b d e+a e^2\right )^2}{6 e^5 (d+e x)^6}+\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{5 e^5 (d+e x)^5}-\frac {6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{4 e^5 (d+e x)^4}+\frac {2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 159, normalized size = 1.02 \begin {gather*} -\frac {2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+e^2 \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+6 d e x+15 e^2 x^2\right )+b \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )}{60 e^5 (d+e x)^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 195, normalized size = 1.25
method | result | size |
risch | \(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 c \left (b e +c d \right ) x^{3}}{3 e^{2}}-\frac {\left (2 a c \,e^{2}+b^{2} e^{2}+2 b c d e +2 c^{2} d^{2}\right ) x^{2}}{4 e^{3}}-\frac {\left (4 a b \,e^{3}+2 a d \,e^{2} c +b^{2} d \,e^{2}+2 d^{2} e b c +2 c^{2} d^{3}\right ) x}{10 e^{4}}-\frac {10 a^{2} e^{4}+4 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+2 d^{3} e b c +2 c^{2} d^{4}}{60 e^{5}}}{\left (e x +d \right )^{6}}\) | \(178\) |
norman | \(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 \left (b c \,e^{2}+c^{2} d e \right ) x^{3}}{3 e^{3}}-\frac {\left (2 a c \,e^{3}+b^{2} e^{3}+2 d \,e^{2} b c +2 d^{2} e \,c^{2}\right ) x^{2}}{4 e^{4}}-\frac {\left (4 e^{4} a b +2 a c d \,e^{3}+b^{2} d \,e^{3}+2 d^{2} e^{2} b c +2 c^{2} d^{3} e \right ) x}{10 e^{5}}-\frac {10 a^{2} e^{5}+4 a b d \,e^{4}+2 a c \,e^{3} d^{2}+b^{2} d^{2} e^{3}+2 b c \,d^{3} e^{2}+2 c^{2} d^{4} e}{60 e^{6}}}{\left (e x +d \right )^{6}}\) | \(192\) |
gosper | \(-\frac {30 c^{2} x^{4} e^{4}+40 b c \,e^{4} x^{3}+40 c^{2} d \,e^{3} x^{3}+30 a c \,e^{4} x^{2}+15 b^{2} e^{4} x^{2}+30 b c d \,e^{3} x^{2}+30 c^{2} d^{2} x^{2} e^{2}+24 a b \,e^{4} x +12 a c d \,e^{3} x +6 b^{2} d \,e^{3} x +12 b c \,d^{2} e^{2} x +12 c^{2} d^{3} e x +10 a^{2} e^{4}+4 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}+2 d^{3} e b c +2 c^{2} d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\) | \(193\) |
default | \(-\frac {a^{2} e^{4}-2 d \,e^{3} a b +2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 d^{3} e b c +c^{2} d^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {2 c \left (b e -2 c d \right )}{3 e^{5} \left (e x +d \right )^{3}}-\frac {2 a b \,e^{3}-4 a d \,e^{2} c -2 b^{2} d \,e^{2}+6 d^{2} e b c -4 c^{2} d^{3}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {c^{2}}{2 e^{5} \left (e x +d \right )^{2}}-\frac {2 a c \,e^{2}+b^{2} e^{2}-6 b c d e +6 c^{2} d^{2}}{4 e^{5} \left (e x +d \right )^{4}}\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 219, normalized size = 1.40 \begin {gather*} -\frac {30 \, c^{2} x^{4} e^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + 40 \, {\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 4 \, a b d e^{3} + {\left (b^{2} e^{2} + 2 \, a c e^{2}\right )} d^{2} + 15 \, {\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} x^{2} + 10 \, a^{2} e^{4} + 6 \, {\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + 4 \, a b e^{4} + {\left (b^{2} e^{3} + 2 \, a c e^{3}\right )} d\right )} x}{60 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.18, size = 208, normalized size = 1.33 \begin {gather*} -\frac {2 \, c^{2} d^{4} + {\left (30 \, c^{2} x^{4} + 40 \, b c x^{3} + 24 \, a b x + 15 \, {\left (b^{2} + 2 \, a c\right )} x^{2} + 10 \, a^{2}\right )} e^{4} + 2 \, {\left (20 \, c^{2} d x^{3} + 15 \, b c d x^{2} + 2 \, a b d + 3 \, {\left (b^{2} + 2 \, a c\right )} d x\right )} e^{3} + {\left (30 \, c^{2} d^{2} x^{2} + 12 \, b c d^{2} x + {\left (b^{2} + 2 \, a c\right )} d^{2}\right )} e^{2} + 2 \, {\left (6 \, c^{2} d^{3} x + b c d^{3}\right )} e}{60 \, {\left (x^{6} e^{11} + 6 \, d x^{5} e^{10} + 15 \, d^{2} x^{4} e^{9} + 20 \, d^{3} x^{3} e^{8} + 15 \, d^{4} x^{2} e^{7} + 6 \, d^{5} x e^{6} + d^{6} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.19, size = 179, normalized size = 1.15 \begin {gather*} -\frac {{\left (30 \, c^{2} x^{4} e^{4} + 40 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 12 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 40 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 12 \, b c d^{2} x e^{2} + 2 \, b c d^{3} e + 15 \, b^{2} x^{2} e^{4} + 30 \, a c x^{2} e^{4} + 6 \, b^{2} d x e^{3} + 12 \, a c d x e^{3} + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 24 \, a b x e^{4} + 4 \, a b d e^{3} + 10 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{60 \, {\left (x e + d\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.71, size = 233, normalized size = 1.49 \begin {gather*} -\frac {\frac {10\,a^2\,e^4+4\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2+2\,b\,c\,d^3\,e+2\,c^2\,d^4}{60\,e^5}+\frac {x\,\left (b^2\,d\,e^2+2\,b\,c\,d^2\,e+4\,a\,b\,e^3+2\,c^2\,d^3+2\,a\,c\,d\,e^2\right )}{10\,e^4}+\frac {c^2\,x^4}{2\,e}+\frac {x^2\,\left (b^2\,e^2+2\,b\,c\,d\,e+2\,c^2\,d^2+2\,a\,c\,e^2\right )}{4\,e^3}+\frac {2\,c\,x^3\,\left (b\,e+c\,d\right )}{3\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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