Optimal. Leaf size=268 \[ -\frac {\left (c d^2-b d e+a e^2\right )^3}{7 e^7 (d+e x)^7}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^7 (d+e x)^6}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{4 e^7 (d+e x)^4}-\frac {c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {c^3}{e^7 (d+e x)} \]
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Rubi [A]
time = 0.14, antiderivative size = 268, 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 {c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (d+e x)^3}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{4 e^7 (d+e x)^4}-\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^6}-\frac {\left (a e^2-b d e+c d^2\right )^3}{7 e^7 (d+e x)^7}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {c^3}{e^7 (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 )^3}{(d+e x)^8} \, dx &=\int \left (\frac {\left (c d^2-b d e+a e^2\right )^3}{e^6 (d+e x)^8}+\frac {3 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^6 (d+e x)^7}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^6 (d+e x)^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)^5}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)^4}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^3}+\frac {c^3}{e^6 (d+e x)^2}\right ) \, dx\\ &=-\frac {\left (c d^2-b d e+a e^2\right )^3}{7 e^7 (d+e x)^7}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^7 (d+e x)^6}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{5 e^7 (d+e x)^5}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{4 e^7 (d+e x)^4}-\frac {c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^3}+\frac {3 c^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {c^3}{e^7 (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 377, normalized size = 1.41 \begin {gather*} -\frac {20 c^3 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )+e^3 \left (20 a^3 e^3+10 a^2 b e^2 (d+7 e x)+4 a b^2 e \left (d^2+7 d e x+21 e^2 x^2\right )+b^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c e^2 \left (2 a^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+2 b^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+2 c^2 e \left (2 a e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 b \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{140 e^7 (d+e x)^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 461, normalized size = 1.72
method | result | size |
risch | \(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 c^{2} \left (b e +2 c d \right ) x^{5}}{2 e^{2}}-\frac {c \left (2 a c \,e^{2}+2 b^{2} e^{2}+5 b c d e +10 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+4 d \,e^{2} c^{2} a +b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{3}}{4 e^{4}}-\frac {3 \left (4 e^{4} a^{2} c +4 a \,b^{2} e^{4}+6 a b c d \,e^{3}+4 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+4 b^{2} c \,d^{2} e^{2}+10 d^{3} e b \,c^{2}+20 d^{4} c^{3}\right ) x^{2}}{20 e^{5}}-\frac {\left (10 a^{2} b \,e^{5}+4 d \,e^{4} a^{2} c +4 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+4 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+4 b^{2} c \,d^{3} e^{2}+10 b \,c^{2} d^{4} e +20 d^{5} c^{3}\right ) x}{20 e^{6}}-\frac {20 e^{6} a^{3}+10 a^{2} b d \,e^{5}+4 e^{4} d^{2} a^{2} c +4 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+4 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 d^{6} c^{3}}{140 e^{7}}}{\left (e x +d \right )^{7}}\) | \(437\) |
norman | \(\frac {-\frac {c^{3} x^{6}}{e}-\frac {3 \left (b \,c^{2} e +2 c^{3} d \right ) x^{5}}{2 e^{2}}-\frac {\left (2 e^{2} c^{2} a +2 b^{2} e^{2} c +5 d e b \,c^{2}+10 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+4 d \,e^{2} c^{2} a +b^{3} e^{3}+4 b^{2} d \,e^{2} c +10 b \,c^{2} d^{2} e +20 c^{3} d^{3}\right ) x^{3}}{4 e^{4}}-\frac {3 \left (4 e^{4} a^{2} c +4 a \,b^{2} e^{4}+6 a b c d \,e^{3}+4 d^{2} e^{2} c^{2} a +b^{3} d \,e^{3}+4 b^{2} c \,d^{2} e^{2}+10 d^{3} e b \,c^{2}+20 d^{4} c^{3}\right ) x^{2}}{20 e^{5}}-\frac {\left (10 a^{2} b \,e^{5}+4 d \,e^{4} a^{2} c +4 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+4 d^{3} e^{2} c^{2} a +b^{3} d^{2} e^{3}+4 b^{2} c \,d^{3} e^{2}+10 b \,c^{2} d^{4} e +20 d^{5} c^{3}\right ) x}{20 e^{6}}-\frac {20 e^{6} a^{3}+10 a^{2} b d \,e^{5}+4 e^{4} d^{2} a^{2} c +4 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+4 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 d^{6} c^{3}}{140 e^{7}}}{\left (e x +d \right )^{7}}\) | \(443\) |
default | \(-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} c^{2} a +3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{6 e^{7} \left (e x +d \right )^{6}}-\frac {c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{e^{7} \left (e x +d \right )^{3}}-\frac {c^{3}}{e^{7} \left (e x +d \right )}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 e^{4} d^{2} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} c^{2} a -b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {3 c^{2} \left (b e -2 c d \right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} c^{2} a -3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 d^{3} e b \,c^{2}+15 d^{4} c^{3}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {6 a b c \,e^{3}-12 d \,e^{2} c^{2} a +b^{3} e^{3}-12 b^{2} d \,e^{2} c +30 b \,c^{2} d^{2} e -20 c^{3} d^{3}}{4 e^{7} \left (e x +d \right )^{4}}\) | \(461\) |
gosper | \(-\frac {140 c^{3} e^{6} x^{6}+210 b \,c^{2} e^{6} x^{5}+420 c^{3} d \,e^{5} x^{5}+140 a \,c^{2} e^{6} x^{4}+140 b^{2} c \,e^{6} x^{4}+350 b \,c^{2} d \,e^{5} x^{4}+700 c^{3} d^{2} e^{4} x^{4}+210 a b c \,e^{6} x^{3}+140 a \,c^{2} d \,e^{5} x^{3}+35 b^{3} e^{6} x^{3}+140 b^{2} c d \,e^{5} x^{3}+350 b \,c^{2} d^{2} e^{4} x^{3}+700 c^{3} d^{3} e^{3} x^{3}+84 a^{2} c \,e^{6} x^{2}+84 a \,b^{2} e^{6} x^{2}+126 a b c d \,e^{5} x^{2}+84 a \,c^{2} d^{2} e^{4} x^{2}+21 b^{3} d \,e^{5} x^{2}+84 b^{2} c \,d^{2} e^{4} x^{2}+210 b \,c^{2} d^{3} e^{3} x^{2}+420 c^{3} d^{4} e^{2} x^{2}+70 a^{2} b \,e^{6} x +28 a^{2} c d \,e^{5} x +28 a \,b^{2} d \,e^{5} x +42 a b c \,d^{2} e^{4} x +28 a \,c^{2} d^{3} e^{3} x +7 b^{3} d^{2} e^{4} x +28 b^{2} c \,d^{3} e^{3} x +70 b \,c^{2} d^{4} e^{2} x +140 c^{3} d^{5} e x +20 e^{6} a^{3}+10 a^{2} b d \,e^{5}+4 e^{4} d^{2} a^{2} c +4 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+4 d^{4} e^{2} c^{2} a +b^{3} d^{3} e^{3}+4 b^{2} c \,d^{4} e^{2}+10 b \,c^{2} d^{5} e +20 d^{6} c^{3}}{140 \left (e x +d \right )^{7} e^{7}}\) | \(494\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 465, normalized size = 1.74 \begin {gather*} -\frac {140 \, c^{3} x^{6} e^{6} + 20 \, c^{3} d^{6} + 10 \, b c^{2} d^{5} e + 210 \, {\left (2 \, c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 4 \, {\left (b^{2} c e^{2} + a c^{2} e^{2}\right )} d^{4} + 70 \, {\left (10 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 2 \, b^{2} c e^{6} + 2 \, a c^{2} e^{6}\right )} x^{4} + 10 \, a^{2} b d e^{5} + {\left (b^{3} e^{3} + 6 \, a b c e^{3}\right )} d^{3} + 35 \, {\left (20 \, c^{3} d^{3} e^{3} + 10 \, b c^{2} d^{2} e^{4} + b^{3} e^{6} + 6 \, a b c e^{6} + 4 \, {\left (b^{2} c e^{5} + a c^{2} e^{5}\right )} d\right )} x^{3} + 20 \, a^{3} e^{6} + 4 \, {\left (a b^{2} e^{4} + a^{2} c e^{4}\right )} d^{2} + 21 \, {\left (20 \, c^{3} d^{4} e^{2} + 10 \, b c^{2} d^{3} e^{3} + 4 \, a b^{2} e^{6} + 4 \, a^{2} c e^{6} + 4 \, {\left (b^{2} c e^{4} + a c^{2} e^{4}\right )} d^{2} + {\left (b^{3} e^{5} + 6 \, a b c e^{5}\right )} d\right )} x^{2} + 7 \, {\left (20 \, c^{3} d^{5} e + 10 \, b c^{2} d^{4} e^{2} + 4 \, {\left (b^{2} c e^{3} + a c^{2} e^{3}\right )} d^{3} + 10 \, a^{2} b e^{6} + {\left (b^{3} e^{4} + 6 \, a b c e^{4}\right )} d^{2} + 4 \, {\left (a b^{2} e^{5} + a^{2} c e^{5}\right )} d\right )} x}{140 \, {\left (x^{7} e^{14} + 7 \, d x^{6} e^{13} + 21 \, d^{2} x^{5} e^{12} + 35 \, d^{3} x^{4} e^{11} + 35 \, d^{4} x^{3} e^{10} + 21 \, d^{5} x^{2} e^{9} + 7 \, d^{6} x e^{8} + d^{7} e^{7}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.68, size = 436, normalized size = 1.63 \begin {gather*} -\frac {20 \, c^{3} d^{6} + {\left (140 \, c^{3} x^{6} + 210 \, b c^{2} x^{5} + 140 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 70 \, a^{2} b x + 35 \, {\left (b^{3} + 6 \, a b c\right )} x^{3} + 20 \, a^{3} + 84 \, {\left (a b^{2} + a^{2} c\right )} x^{2}\right )} e^{6} + {\left (420 \, c^{3} d x^{5} + 350 \, b c^{2} d x^{4} + 140 \, {\left (b^{2} c + a c^{2}\right )} d x^{3} + 10 \, a^{2} b d + 21 \, {\left (b^{3} + 6 \, a b c\right )} d x^{2} + 28 \, {\left (a b^{2} + a^{2} c\right )} d x\right )} e^{5} + {\left (700 \, c^{3} d^{2} x^{4} + 350 \, b c^{2} d^{2} x^{3} + 84 \, {\left (b^{2} c + a c^{2}\right )} d^{2} x^{2} + 7 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} x + 4 \, {\left (a b^{2} + a^{2} c\right )} d^{2}\right )} e^{4} + {\left (700 \, c^{3} d^{3} x^{3} + 210 \, b c^{2} d^{3} x^{2} + 28 \, {\left (b^{2} c + a c^{2}\right )} d^{3} x + {\left (b^{3} + 6 \, a b c\right )} d^{3}\right )} e^{3} + 2 \, {\left (210 \, c^{3} d^{4} x^{2} + 35 \, b c^{2} d^{4} x + 2 \, {\left (b^{2} c + a c^{2}\right )} d^{4}\right )} e^{2} + 10 \, {\left (14 \, c^{3} d^{5} x + b c^{2} d^{5}\right )} e}{140 \, {\left (x^{7} e^{14} + 7 \, d x^{6} e^{13} + 21 \, d^{2} x^{5} e^{12} + 35 \, d^{3} x^{4} e^{11} + 35 \, d^{4} x^{3} e^{10} + 21 \, d^{5} x^{2} e^{9} + 7 \, d^{6} x e^{8} + d^{7} e^{7}\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 = 1.10, size = 458, normalized size = 1.71 \begin {gather*} -\frac {{\left (140 \, c^{3} x^{6} e^{6} + 420 \, c^{3} d x^{5} e^{5} + 700 \, c^{3} d^{2} x^{4} e^{4} + 700 \, c^{3} d^{3} x^{3} e^{3} + 420 \, c^{3} d^{4} x^{2} e^{2} + 140 \, c^{3} d^{5} x e + 20 \, c^{3} d^{6} + 210 \, b c^{2} x^{5} e^{6} + 350 \, b c^{2} d x^{4} e^{5} + 350 \, b c^{2} d^{2} x^{3} e^{4} + 210 \, b c^{2} d^{3} x^{2} e^{3} + 70 \, b c^{2} d^{4} x e^{2} + 10 \, b c^{2} d^{5} e + 140 \, b^{2} c x^{4} e^{6} + 140 \, a c^{2} x^{4} e^{6} + 140 \, b^{2} c d x^{3} e^{5} + 140 \, a c^{2} d x^{3} e^{5} + 84 \, b^{2} c d^{2} x^{2} e^{4} + 84 \, a c^{2} d^{2} x^{2} e^{4} + 28 \, b^{2} c d^{3} x e^{3} + 28 \, a c^{2} d^{3} x e^{3} + 4 \, b^{2} c d^{4} e^{2} + 4 \, a c^{2} d^{4} e^{2} + 35 \, b^{3} x^{3} e^{6} + 210 \, a b c x^{3} e^{6} + 21 \, b^{3} d x^{2} e^{5} + 126 \, a b c d x^{2} e^{5} + 7 \, b^{3} d^{2} x e^{4} + 42 \, a b c d^{2} x e^{4} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 84 \, a b^{2} x^{2} e^{6} + 84 \, a^{2} c x^{2} e^{6} + 28 \, a b^{2} d x e^{5} + 28 \, a^{2} c d x e^{5} + 4 \, a b^{2} d^{2} e^{4} + 4 \, a^{2} c d^{2} e^{4} + 70 \, a^{2} b x e^{6} + 10 \, a^{2} b d e^{5} + 20 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{140 \, {\left (x e + d\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 502, normalized size = 1.87 \begin {gather*} -\frac {\frac {20\,a^3\,e^6+10\,a^2\,b\,d\,e^5+4\,a^2\,c\,d^2\,e^4+4\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+4\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+4\,b^2\,c\,d^4\,e^2+10\,b\,c^2\,d^5\,e+20\,c^3\,d^6}{140\,e^7}+\frac {x^3\,\left (b^3\,e^3+4\,b^2\,c\,d\,e^2+10\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3+20\,c^3\,d^3+4\,a\,c^2\,d\,e^2\right )}{4\,e^4}+\frac {3\,x^2\,\left (4\,a^2\,c\,e^4+4\,a\,b^2\,e^4+6\,a\,b\,c\,d\,e^3+4\,a\,c^2\,d^2\,e^2+b^3\,d\,e^3+4\,b^2\,c\,d^2\,e^2+10\,b\,c^2\,d^3\,e+20\,c^3\,d^4\right )}{20\,e^5}+\frac {c^3\,x^6}{e}+\frac {x\,\left (10\,a^2\,b\,e^5+4\,a^2\,c\,d\,e^4+4\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+4\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+4\,b^2\,c\,d^3\,e^2+10\,b\,c^2\,d^4\,e+20\,c^3\,d^5\right )}{20\,e^6}+\frac {c\,x^4\,\left (2\,b^2\,e^2+5\,b\,c\,d\,e+10\,c^2\,d^2+2\,a\,c\,e^2\right )}{2\,e^3}+\frac {3\,c^2\,x^5\,\left (b\,e+2\,c\,d\right )}{2\,e^2}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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