Optimal. Leaf size=32 \[ \frac{d \log (a+b x)}{b^2}-\frac{b c-a d}{b^2 (a+b x)} \]
[Out]
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Rubi [A] time = 0.0623704, antiderivative size = 32, 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 \[ \frac{d \log (a+b x)}{b^2}-\frac{b c-a d}{b^2 (a+b x)} \]
Antiderivative was successfully verified.
[In] Int[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 13.1272, size = 26, normalized size = 0.81 \[ \frac{d \log{\left (a + b x \right )}}{b^{2}} + \frac{a d - b c}{b^{2} \left (a + b x\right )} \]
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)**3,x)
[Out]
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Mathematica [A] time = 0.0192003, size = 31, normalized size = 0.97 \[ \frac{a d-b c}{b^2 (a+b x)}+\frac{d \log (a+b x)}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^3,x]
[Out]
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Maple [A] time = 0.008, size = 39, normalized size = 1.2 \[{\frac{d\ln \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ad}{ \left ( bx+a \right ){b}^{2}}}-{\frac{c}{b \left ( bx+a \right ) }} \]
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)^3,x)
[Out]
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Maxima [A] time = 0.740994, size = 47, normalized size = 1.47 \[ -\frac{b c - a d}{b^{3} x + a b^{2}} + \frac{d \log \left (b x + a\right )}{b^{2}} \]
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)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206167, size = 53, normalized size = 1.66 \[ -\frac{b c - a d -{\left (b d x + a d\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]
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)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.43684, size = 27, normalized size = 0.84 \[ \frac{a d - b c}{a b^{2} + b^{3} x} + \frac{d \log{\left (a + b x \right )}}{b^{2}} \]
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)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.210462, size = 45, normalized size = 1.41 \[ \frac{d{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} - \frac{b c - a d}{{\left (b x + a\right )} b^{2}} \]
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)^3,x, algorithm="giac")
[Out]