Optimal. Leaf size=45 \[ \frac{b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac{1}{32 c^2 d^7 (b+2 c x)^4} \]
[Out]
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Rubi [A] time = 0.0843846, 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 \[ \frac{b^2-4 a c}{48 c^2 d^7 (b+2 c x)^6}-\frac{1}{32 c^2 d^7 (b+2 c x)^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^7,x]
[Out]
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Rubi in Sympy [A] time = 17.1447, size = 42, normalized size = 0.93 \[ - \frac{1}{32 c^{2} d^{7} \left (b + 2 c x\right )^{4}} + \frac{- a c + \frac{b^{2}}{4}}{12 c^{2} d^{7} \left (b + 2 c x\right )^{6}} \]
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)**7,x)
[Out]
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Mathematica [A] time = 0.0249318, size = 43, normalized size = 0.96 \[ \frac{\frac{b^2-4 a c}{48 c^2 (b+2 c x)^6}-\frac{1}{32 c^2 (b+2 c x)^4}}{d^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^7,x]
[Out]
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Maple [A] time = 0.007, size = 42, normalized size = 0.9 \[{\frac{1}{{d}^{7}} \left ( -{\frac{1}{32\,{c}^{2} \left ( 2\,cx+b \right ) ^{4}}}-{\frac{4\,ac-{b}^{2}}{48\,{c}^{2} \left ( 2\,cx+b \right ) ^{6}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/(2*c*d*x+b*d)^7,x)
[Out]
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Maxima [A] time = 0.68909, size = 153, normalized size = 3.4 \[ -\frac{12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \,{\left (64 \, c^{8} d^{7} x^{6} + 192 \, b c^{7} d^{7} x^{5} + 240 \, b^{2} c^{6} d^{7} x^{4} + 160 \, b^{3} c^{5} d^{7} x^{3} + 60 \, b^{4} c^{4} d^{7} x^{2} + 12 \, b^{5} c^{3} d^{7} x + b^{6} c^{2} d^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205423, size = 153, normalized size = 3.4 \[ -\frac{12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \,{\left (64 \, c^{8} d^{7} x^{6} + 192 \, b c^{7} d^{7} x^{5} + 240 \, b^{2} c^{6} d^{7} x^{4} + 160 \, b^{3} c^{5} d^{7} x^{3} + 60 \, b^{4} c^{4} d^{7} x^{2} + 12 \, b^{5} c^{3} d^{7} x + b^{6} c^{2} d^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.06013, size = 121, normalized size = 2.69 \[ - \frac{8 a c + b^{2} + 12 b c x + 12 c^{2} x^{2}}{96 b^{6} c^{2} d^{7} + 1152 b^{5} c^{3} d^{7} x + 5760 b^{4} c^{4} d^{7} x^{2} + 15360 b^{3} c^{5} d^{7} x^{3} + 23040 b^{2} c^{6} d^{7} x^{4} + 18432 b c^{7} d^{7} x^{5} + 6144 c^{8} d^{7} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**7,x)
[Out]
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GIAC/XCAS [A] time = 0.212065, size = 50, normalized size = 1.11 \[ -\frac{12 \, c^{2} x^{2} + 12 \, b c x + b^{2} + 8 \, a c}{96 \,{\left (2 \, c x + b\right )}^{6} c^{2} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^7,x, algorithm="giac")
[Out]