Optimal. Leaf size=45 \[ \frac{b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac{1}{40 c^2 d^8 (b+2 c x)^5} \]
[Out]
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Rubi [A] time = 0.0815275, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac{1}{40 c^2 d^8 (b+2 c x)^5} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^8,x]
[Out]
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Rubi in Sympy [A] time = 17.7967, size = 42, normalized size = 0.93 \[ - \frac{1}{40 c^{2} d^{8} \left (b + 2 c x\right )^{5}} + \frac{- a c + \frac{b^{2}}{4}}{14 c^{2} d^{8} \left (b + 2 c x\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**8,x)
[Out]
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Mathematica [A] time = 0.0275477, size = 39, normalized size = 0.87 \[ \frac{5 \left (b^2-4 a c\right )-7 (b+2 c x)^2}{280 c^2 d^8 (b+2 c x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^8,x]
[Out]
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Maple [A] time = 0.007, size = 42, normalized size = 0.9 \[{\frac{1}{{d}^{8}} \left ( -{\frac{1}{40\,{c}^{2} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{4\,ac-{b}^{2}}{56\,{c}^{2} \left ( 2\,cx+b \right ) ^{7}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/(2*c*d*x+b*d)^8,x)
[Out]
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Maxima [A] time = 0.689621, size = 171, normalized size = 3.8 \[ -\frac{14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \,{\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.211773, size = 171, normalized size = 3.8 \[ -\frac{14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \,{\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^8,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.13545, size = 136, normalized size = 3.02 \[ - \frac{10 a c + b^{2} + 14 b c x + 14 c^{2} x^{2}}{140 b^{7} c^{2} d^{8} + 1960 b^{6} c^{3} d^{8} x + 11760 b^{5} c^{4} d^{8} x^{2} + 39200 b^{4} c^{5} d^{8} x^{3} + 78400 b^{3} c^{6} d^{8} x^{4} + 94080 b^{2} c^{7} d^{8} x^{5} + 62720 b c^{8} d^{8} x^{6} + 17920 c^{9} d^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**8,x)
[Out]
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GIAC/XCAS [A] time = 0.211122, size = 50, normalized size = 1.11 \[ -\frac{14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \,{\left (2 \, c x + b\right )}^{7} c^{2} d^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^8,x, algorithm="giac")
[Out]