Optimal. Leaf size=31 \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]
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Rubi [A] time = 0.0698133, antiderivative size = 31, 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 \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 14.1453, size = 26, normalized size = 0.84 \[ \frac{b \log{\left (c + d x \right )}}{d^{2}} - \frac{a d - b c}{d^{2} \left (c + d x\right )} \]
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)**2,x)
[Out]
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Mathematica [A] time = 0.0195983, size = 31, normalized size = 1. \[ \frac{b c-a d}{d^2 (c+d x)}+\frac{b \log (c+d x)}{d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Maple [A] time = 0.007, size = 39, normalized size = 1.3 \[ -{\frac{a}{d \left ( dx+c \right ) }}+{\frac{bc}{{d}^{2} \left ( dx+c \right ) }}+{\frac{b\ln \left ( dx+c \right ) }{{d}^{2}}} \]
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)^2,x)
[Out]
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Maxima [A] time = 0.730316, size = 46, normalized size = 1.48 \[ \frac{b c - a d}{d^{3} x + c d^{2}} + \frac{b \log \left (d x + c\right )}{d^{2}} \]
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)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237424, size = 50, normalized size = 1.61 \[ \frac{b c - a d +{\left (b d x + b c\right )} \log \left (d x + c\right )}{d^{3} x + c d^{2}} \]
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)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.49113, size = 27, normalized size = 0.87 \[ \frac{b \log{\left (c + d x \right )}}{d^{2}} - \frac{a d - b c}{c d^{2} + d^{3} x} \]
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)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.213626, size = 43, normalized size = 1.39 \[ \frac{b{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{2}} + \frac{b c - a d}{{\left (d x + c\right )} d^{2}} \]
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)^2,x, algorithm="giac")
[Out]