Optimal. Leaf size=37 \[ \frac{\left (a+b x+c x^2\right )^2}{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^4} \]
[Out]
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Rubi [A] time = 0.039003, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{\left (a+b x+c x^2\right )^2}{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^5,x]
[Out]
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Rubi in Sympy [A] time = 8.35374, size = 32, normalized size = 0.86 \[ \frac{\left (a + b x + c x^{2}\right )^{2}}{2 d^{5} \left (b + 2 c x\right )^{4} \left (- 4 a c + b^{2}\right )} \]
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)**5,x)
[Out]
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Mathematica [A] time = 0.0265915, size = 38, normalized size = 1.03 \[ -\frac{4 c \left (a+2 c x^2\right )+b^2+8 b c x}{32 c^2 d^5 (b+2 c x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^5,x]
[Out]
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Maple [A] time = 0.008, size = 42, normalized size = 1.1 \[{\frac{1}{{d}^{5}} \left ( -{\frac{1}{16\,{c}^{2} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{4\,ac-{b}^{2}}{32\,{c}^{2} \left ( 2\,cx+b \right ) ^{4}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/(2*c*d*x+b*d)^5,x)
[Out]
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Maxima [A] time = 0.689551, size = 115, normalized size = 3.11 \[ -\frac{8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{32 \,{\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.207861, size = 115, normalized size = 3.11 \[ -\frac{8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c}{32 \,{\left (16 \, c^{6} d^{5} x^{4} + 32 \, b c^{5} d^{5} x^{3} + 24 \, b^{2} c^{4} d^{5} x^{2} + 8 \, b^{3} c^{3} d^{5} x + b^{4} c^{2} d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.96659, size = 90, normalized size = 2.43 \[ - \frac{4 a c + b^{2} + 8 b c x + 8 c^{2} x^{2}}{32 b^{4} c^{2} d^{5} + 256 b^{3} c^{3} d^{5} x + 768 b^{2} c^{4} d^{5} x^{2} + 1024 b c^{5} d^{5} x^{3} + 512 c^{6} d^{5} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.215177, size = 84, normalized size = 2.27 \[ \frac{\frac{b^{2} d}{{\left (2 \, c d x + b d\right )}^{4} c^{2}} - \frac{4 \, a d}{{\left (2 \, c d x + b d\right )}^{4} c} - \frac{2}{{\left (2 \, c d x + b d\right )}^{2} c^{2} d}}{32 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^5,x, algorithm="giac")
[Out]