Optimal. Leaf size=38 \[ -\frac{b d-a e}{5 b^2 (a+b x)^5}-\frac{e}{4 b^2 (a+b x)^4} \]
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Rubi [A] time = 0.0617724, antiderivative size = 38, 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 \[ -\frac{b d-a e}{5 b^2 (a+b x)^5}-\frac{e}{4 b^2 (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 19.9694, size = 31, normalized size = 0.82 \[ - \frac{e}{4 b^{2} \left (a + b x\right )^{4}} + \frac{a e - b d}{5 b^{2} \left (a + b x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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Mathematica [A] time = 0.0171789, size = 27, normalized size = 0.71 \[ -\frac{a e+4 b d+5 b e x}{20 b^2 (a+b x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Maple [A] time = 0.008, size = 35, normalized size = 0.9 \[ -{\frac{-ae+bd}{5\,{b}^{2} \left ( bx+a \right ) ^{5}}}-{\frac{e}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
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Maxima [A] time = 0.690478, size = 97, normalized size = 2.55 \[ -\frac{5 \, b e x + 4 \, b d + a e}{20 \,{\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.196858, size = 97, normalized size = 2.55 \[ -\frac{5 \, b e x + 4 \, b d + a e}{20 \,{\left (b^{7} x^{5} + 5 \, a b^{6} x^{4} + 10 \, a^{2} b^{5} x^{3} + 10 \, a^{3} b^{4} x^{2} + 5 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.69749, size = 76, normalized size = 2. \[ - \frac{a e + 4 b d + 5 b e x}{20 a^{5} b^{2} + 100 a^{4} b^{3} x + 200 a^{3} b^{4} x^{2} + 200 a^{2} b^{5} x^{3} + 100 a b^{6} x^{4} + 20 b^{7} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.209069, size = 36, normalized size = 0.95 \[ -\frac{5 \, b x e + 4 \, b d + a e}{20 \,{\left (b x + a\right )}^{5} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
[Out]