Optimal. Leaf size=65 \[ \frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac{b^2}{4 e^3 (d+e x)^4} \]
[Out]
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Rubi [A] time = 0.09123, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 b (b d-a e)}{5 e^3 (d+e x)^5}-\frac{(b d-a e)^2}{6 e^3 (d+e x)^6}-\frac{b^2}{4 e^3 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^7,x]
[Out]
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Rubi in Sympy [A] time = 31.723, size = 58, normalized size = 0.89 \[ - \frac{b^{2}}{4 e^{3} \left (d + e x\right )^{4}} - \frac{2 b \left (a e - b d\right )}{5 e^{3} \left (d + e x\right )^{5}} - \frac{\left (a e - b d\right )^{2}}{6 e^{3} \left (d + e x\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**7,x)
[Out]
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Mathematica [A] time = 0.0395, size = 55, normalized size = 0.85 \[ -\frac{10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )}{60 e^3 (d+e x)^6} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x)^7,x]
[Out]
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Maple [A] time = 0.008, size = 71, normalized size = 1.1 \[ -{\frac{2\,b \left ( ae-bd \right ) }{5\,{e}^{3} \left ( ex+d \right ) ^{5}}}-{\frac{{a}^{2}{e}^{2}-2\,deab+{b}^{2}{d}^{2}}{6\,{e}^{3} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)/(e*x+d)^7,x)
[Out]
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Maxima [A] time = 0.697357, size = 162, normalized size = 2.49 \[ -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d)^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.196676, size = 162, normalized size = 2.49 \[ -\frac{15 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 4 \, a b d e + 10 \, a^{2} e^{2} + 6 \,{\left (b^{2} d e + 4 \, a b e^{2}\right )} x}{60 \,{\left (e^{9} x^{6} + 6 \, d e^{8} x^{5} + 15 \, d^{2} e^{7} x^{4} + 20 \, d^{3} e^{6} x^{3} + 15 \, d^{4} e^{5} x^{2} + 6 \, d^{5} e^{4} x + d^{6} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d)^7,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.28379, size = 128, normalized size = 1.97 \[ - \frac{10 a^{2} e^{2} + 4 a b d e + b^{2} d^{2} + 15 b^{2} e^{2} x^{2} + x \left (24 a b e^{2} + 6 b^{2} d e\right )}{60 d^{6} e^{3} + 360 d^{5} e^{4} x + 900 d^{4} e^{5} x^{2} + 1200 d^{3} e^{6} x^{3} + 900 d^{2} e^{7} x^{4} + 360 d e^{8} x^{5} + 60 e^{9} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d)**7,x)
[Out]
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GIAC/XCAS [A] time = 0.209703, size = 81, normalized size = 1.25 \[ -\frac{{\left (15 \, b^{2} x^{2} e^{2} + 6 \, b^{2} d x e + b^{2} d^{2} + 24 \, a b x e^{2} + 4 \, a b d e + 10 \, a^{2} e^{2}\right )} e^{\left (-3\right )}}{60 \,{\left (x e + d\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d)^7,x, algorithm="giac")
[Out]