Optimal. Leaf size=38 \[ -\frac{b c-a d}{4 b^2 (a+b x)^4}-\frac{d}{3 b^2 (a+b x)^3} \]
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Rubi [A] time = 0.0654448, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ -\frac{b c-a d}{4 b^2 (a+b x)^4}-\frac{d}{3 b^2 (a+b x)^3} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 13.6438, size = 31, normalized size = 0.82 \[ - \frac{d}{3 b^{2} \left (a + b x\right )^{3}} + \frac{a d - b c}{4 b^{2} \left (a + b x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*c+(a*d+b*c)*x+b*d*x**2)/(b*x+a)**6,x)
[Out]
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Mathematica [A] time = 0.0181802, size = 27, normalized size = 0.71 \[ -\frac{a d+3 b c+4 b d x}{12 b^2 (a+b x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^6,x]
[Out]
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Maple [A] time = 0.006, size = 35, normalized size = 0.9 \[ -{\frac{-ad+bc}{4\,{b}^{2} \left ( bx+a \right ) ^{4}}}-{\frac{d}{3\,{b}^{2} \left ( bx+a \right ) ^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*c+(a*d+b*c)*x+x^2*b*d)/(b*x+a)^6,x)
[Out]
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Maxima [A] time = 0.733081, size = 82, normalized size = 2.16 \[ -\frac{4 \, b d x + 3 \, b c + a d}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)/(b*x + a)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.19928, size = 82, normalized size = 2.16 \[ -\frac{4 \, b d x + 3 \, b c + a d}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)/(b*x + a)^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.34579, size = 65, normalized size = 1.71 \[ - \frac{a d + 3 b c + 4 b d x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*c+(a*d+b*c)*x+b*d*x**2)/(b*x+a)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.209844, size = 34, normalized size = 0.89 \[ -\frac{4 \, b d x + 3 \, b c + a d}{12 \,{\left (b x + a\right )}^{4} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*d*x^2 + a*c + (b*c + a*d)*x)/(b*x + a)^6,x, algorithm="giac")
[Out]