Optimal. Leaf size=28 \[ -\frac{(d+e x)^4}{4 (a+b x)^4 (b d-a e)} \]
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Rubi [A] time = 0.0199561, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{(d+e x)^4}{4 (a+b x)^4 (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x)^3)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
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Rubi in Sympy [A] time = 15.9114, size = 20, normalized size = 0.71 \[ \frac{\left (d + e x\right )^{4}}{4 \left (a + b x\right )^{4} \left (a e - b d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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Mathematica [B] time = 0.0560978, size = 91, normalized size = 3.25 \[ -\frac{a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )}{4 b^4 (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x)^3)/(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
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Maple [B] time = 0.009, size = 122, normalized size = 4.4 \[ -{\frac{-{a}^{3}{e}^{3}+3\,{a}^{2}bd{e}^{2}-3\,a{b}^{2}{d}^{2}e+{b}^{3}{d}^{3}}{4\,{b}^{4} \left ( bx+a \right ) ^{4}}}-{\frac{e \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{{b}^{4} \left ( bx+a \right ) ^{3}}}+{\frac{3\,{e}^{2} \left ( ae-bd \right ) }{2\,{b}^{4} \left ( bx+a \right ) ^{2}}}-{\frac{{e}^{3}}{{b}^{4} \left ( bx+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^3/(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
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Maxima [A] time = 0.719161, size = 193, normalized size = 6.89 \[ -\frac{4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.279278, size = 193, normalized size = 6.89 \[ -\frac{4 \, b^{3} e^{3} x^{3} + b^{3} d^{3} + a b^{2} d^{2} e + a^{2} b d e^{2} + a^{3} e^{3} + 6 \,{\left (b^{3} d e^{2} + a b^{2} e^{3}\right )} x^{2} + 4 \,{\left (b^{3} d^{2} e + a b^{2} d e^{2} + a^{2} b e^{3}\right )} x}{4 \,{\left (b^{8} x^{4} + 4 \, a b^{7} x^{3} + 6 \, a^{2} b^{6} x^{2} + 4 \, a^{3} b^{5} x + a^{4} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.0674, size = 153, normalized size = 5.46 \[ - \frac{a^{3} e^{3} + a^{2} b d e^{2} + a b^{2} d^{2} e + b^{3} d^{3} + 4 b^{3} e^{3} x^{3} + x^{2} \left (6 a b^{2} e^{3} + 6 b^{3} d e^{2}\right ) + x \left (4 a^{2} b e^{3} + 4 a b^{2} d e^{2} + 4 b^{3} d^{2} e\right )}{4 a^{4} b^{4} + 16 a^{3} b^{5} x + 24 a^{2} b^{6} x^{2} + 16 a b^{7} x^{3} + 4 b^{8} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**3/(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.282091, size = 143, normalized size = 5.11 \[ -\frac{4 \, b^{3} x^{3} e^{3} + 6 \, b^{3} d x^{2} e^{2} + 4 \, b^{3} d^{2} x e + b^{3} d^{3} + 6 \, a b^{2} x^{2} e^{3} + 4 \, a b^{2} d x e^{2} + a b^{2} d^{2} e + 4 \, a^{2} b x e^{3} + a^{2} b d e^{2} + a^{3} e^{3}}{4 \,{\left (b x + a\right )}^{4} b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^3/(b^2*x^2 + 2*a*b*x + a^2)^3,x, algorithm="giac")
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