Optimal. Leaf size=50 \[ \frac{(b c-a d)^2 \log (c+d x)}{d^3}-\frac{b x (b c-a d)}{d^2}+\frac{(a+b x)^2}{2 d} \]
[Out]
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Rubi [A] time = 0.0735782, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{(b c-a d)^2 \log (c+d x)}{d^3}-\frac{b x (b c-a d)}{d^2}+\frac{(a+b x)^2}{2 d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\left (a + b x\right )^{2}}{2 d} + \frac{\left (a d - b c\right ) \int b\, dx}{d^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**3/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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Mathematica [A] time = 0.0297085, size = 43, normalized size = 0.86 \[ \frac{b d x (4 a d-2 b c+b d x)+2 (b c-a d)^2 \log (c+d x)}{2 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2),x]
[Out]
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Maple [A] time = 0.004, size = 74, normalized size = 1.5 \[{\frac{{b}^{2}{x}^{2}}{2\,d}}+2\,{\frac{abx}{d}}-{\frac{x{b}^{2}c}{{d}^{2}}}+{\frac{\ln \left ( dx+c \right ){a}^{2}}{d}}-2\,{\frac{\ln \left ( dx+c \right ) cab}{{d}^{2}}}+{\frac{\ln \left ( dx+c \right ){b}^{2}{c}^{2}}{{d}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3/(a*c+(a*d+b*c)*x+x^2*b*d),x)
[Out]
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Maxima [A] time = 0.738803, size = 81, normalized size = 1.62 \[ \frac{b^{2} d x^{2} - 2 \,{\left (b^{2} c - 2 \, a b d\right )} x}{2 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202919, size = 84, normalized size = 1.68 \[ \frac{b^{2} d^{2} x^{2} - 2 \,{\left (b^{2} c d - 2 \, a b d^{2}\right )} x + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x + c\right )}{2 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.52897, size = 44, normalized size = 0.88 \[ \frac{b^{2} x^{2}}{2 d} + \frac{x \left (2 a b d - b^{2} c\right )}{d^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**3/(a*c+(a*d+b*c)*x+b*d*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.212231, size = 81, normalized size = 1.62 \[ \frac{b^{2} d x^{2} - 2 \, b^{2} c x + 4 \, a b d x}{2 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/(b*d*x^2 + a*c + (b*c + a*d)*x),x, algorithm="giac")
[Out]