Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac{b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac{1}{96 c^3 d^8 (b+2 c x)^3} \]
[Out]
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Rubi [A] time = 0.141904, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{\left (b^2-4 a c\right )^2}{224 c^3 d^8 (b+2 c x)^7}+\frac{b^2-4 a c}{80 c^3 d^8 (b+2 c x)^5}-\frac{1}{96 c^3 d^8 (b+2 c x)^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^8,x]
[Out]
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Rubi in Sympy [A] time = 32.5542, size = 70, normalized size = 0.96 \[ - \frac{1}{96 c^{3} d^{8} \left (b + 2 c x\right )^{3}} + \frac{- 4 a c + b^{2}}{80 c^{3} d^{8} \left (b + 2 c x\right )^{5}} - \frac{\left (- 4 a c + b^{2}\right )^{2}}{224 c^{3} d^{8} \left (b + 2 c x\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**8,x)
[Out]
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Mathematica [A] time = 0.0624556, size = 59, normalized size = 0.81 \[ \frac{42 \left (b^2-4 a c\right ) (b+2 c x)^2-15 \left (b^2-4 a c\right )^2-35 (b+2 c x)^4}{3360 c^3 d^8 (b+2 c x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^8,x]
[Out]
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Maple [A] time = 0.007, size = 74, normalized size = 1. \[{\frac{1}{{d}^{8}} \left ( -{\frac{4\,ac-{b}^{2}}{80\,{c}^{3} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{1}{96\,{c}^{3} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{224\,{c}^{3} \left ( 2\,cx+b \right ) ^{7}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^8,x)
[Out]
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Maxima [A] time = 0.711782, size = 238, normalized size = 3.26 \[ -\frac{70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \,{\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \,{\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \,{\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22354, size = 238, normalized size = 3.26 \[ -\frac{70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2} + 84 \,{\left (b^{2} c^{2} + a c^{3}\right )} x^{2} + 14 \,{\left (b^{3} c + 6 \, a b c^{2}\right )} x}{420 \,{\left (128 \, c^{10} d^{8} x^{7} + 448 \, b c^{9} d^{8} x^{6} + 672 \, b^{2} c^{8} d^{8} x^{5} + 560 \, b^{3} c^{7} d^{8} x^{4} + 280 \, b^{4} c^{6} d^{8} x^{3} + 84 \, b^{5} c^{5} d^{8} x^{2} + 14 \, b^{6} c^{4} d^{8} x + b^{7} c^{3} d^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^8,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.77141, size = 187, normalized size = 2.56 \[ - \frac{30 a^{2} c^{2} + 6 a b^{2} c + b^{4} + 140 b c^{3} x^{3} + 70 c^{4} x^{4} + x^{2} \left (84 a c^{3} + 84 b^{2} c^{2}\right ) + x \left (84 a b c^{2} + 14 b^{3} c\right )}{420 b^{7} c^{3} d^{8} + 5880 b^{6} c^{4} d^{8} x + 35280 b^{5} c^{5} d^{8} x^{2} + 117600 b^{4} c^{6} d^{8} x^{3} + 235200 b^{3} c^{7} d^{8} x^{4} + 282240 b^{2} c^{8} d^{8} x^{5} + 188160 b c^{9} d^{8} x^{6} + 53760 c^{10} d^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**8,x)
[Out]
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GIAC/XCAS [A] time = 0.213857, size = 117, normalized size = 1.6 \[ -\frac{70 \, c^{4} x^{4} + 140 \, b c^{3} x^{3} + 84 \, b^{2} c^{2} x^{2} + 84 \, a c^{3} x^{2} + 14 \, b^{3} c x + 84 \, a b c^{2} x + b^{4} + 6 \, a b^{2} c + 30 \, a^{2} c^{2}}{420 \,{\left (2 \, c x + b\right )}^{7} c^{3} d^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^8,x, algorithm="giac")
[Out]