Optimal. Leaf size=104 \[ -\frac{b^2 x (-3 a d f+b c f+b d e)}{d^2 f^2}-\frac{(b c-a d)^3 \log (c+d x)}{d^3 (d e-c f)}+\frac{(b e-a f)^3 \log (e+f x)}{f^3 (d e-c f)}+\frac{b^3 x^2}{2 d f} \]
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Rubi [A] time = 0.241618, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{b^2 x (-3 a d f+b c f+b d e)}{d^2 f^2}-\frac{(b c-a d)^3 \log (c+d x)}{d^3 (d e-c f)}+\frac{(b e-a f)^3 \log (e+f x)}{f^3 (d e-c f)}+\frac{b^3 x^2}{2 d f} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^3/((c + d*x)*(e + f*x)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{3} \int x\, dx}{d f} + \frac{\left (a f - b e\right )^{3} \log{\left (e + f x \right )}}{f^{3} \left (c f - d e\right )} + \frac{\left (3 a d f - b c f - b d e\right ) \int b^{2}\, dx}{d^{2} f^{2}} - \frac{\left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{3} \left (c f - d e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**3/(d*x+c)/(f*x+e),x)
[Out]
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Mathematica [A] time = 0.131686, size = 99, normalized size = 0.95 \[ \frac{b^2 d f x (d e-c f) (6 a d f+b (-2 c f-2 d e+d f x))-2 f^3 (b c-a d)^3 \log (c+d x)+2 d^3 (b e-a f)^3 \log (e+f x)}{2 d^3 f^3 (d e-c f)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^3/((c + d*x)*(e + f*x)),x]
[Out]
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Maple [B] time = 0.013, size = 257, normalized size = 2.5 \[{\frac{{b}^{3}{x}^{2}}{2\,df}}+3\,{\frac{{b}^{2}ax}{df}}-{\frac{{b}^{3}cx}{{d}^{2}f}}-{\frac{{b}^{3}ex}{d{f}^{2}}}-{\frac{\ln \left ( dx+c \right ){a}^{3}}{cf-de}}+3\,{\frac{\ln \left ( dx+c \right ){a}^{2}cb}{d \left ( cf-de \right ) }}-3\,{\frac{\ln \left ( dx+c \right ) a{b}^{2}{c}^{2}}{{d}^{2} \left ( cf-de \right ) }}+{\frac{\ln \left ( dx+c \right ){b}^{3}{c}^{3}}{{d}^{3} \left ( cf-de \right ) }}+{\frac{\ln \left ( fx+e \right ){a}^{3}}{cf-de}}-3\,{\frac{\ln \left ( fx+e \right ){a}^{2}be}{f \left ( cf-de \right ) }}+3\,{\frac{\ln \left ( fx+e \right ) a{b}^{2}{e}^{2}}{{f}^{2} \left ( cf-de \right ) }}-{\frac{\ln \left ( fx+e \right ){b}^{3}{e}^{3}}{{f}^{3} \left ( cf-de \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^3/(d*x+c)/(f*x+e),x)
[Out]
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Maxima [A] time = 1.36075, size = 217, normalized size = 2.09 \[ -\frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (d x + c\right )}{d^{4} e - c d^{3} f} + \frac{{\left (b^{3} e^{3} - 3 \, a b^{2} e^{2} f + 3 \, a^{2} b e f^{2} - a^{3} f^{3}\right )} \log \left (f x + e\right )}{d e f^{3} - c f^{4}} + \frac{b^{3} d f x^{2} - 2 \,{\left (b^{3} d e +{\left (b^{3} c - 3 \, a b^{2} d\right )} f\right )} x}{2 \, d^{2} f^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/((d*x + c)*(f*x + e)),x, algorithm="maxima")
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Fricas [A] time = 0.336399, size = 279, normalized size = 2.68 \[ -\frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} f^{3} \log \left (d x + c\right ) -{\left (b^{3} d^{3} e f^{2} - b^{3} c d^{2} f^{3}\right )} x^{2} + 2 \,{\left (b^{3} d^{3} e^{2} f - 3 \, a b^{2} d^{3} e f^{2} -{\left (b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} f^{3}\right )} x - 2 \,{\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{3} e^{2} f + 3 \, a^{2} b d^{3} e f^{2} - a^{3} d^{3} f^{3}\right )} \log \left (f x + e\right )}{2 \,{\left (d^{4} e f^{3} - c d^{3} f^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/((d*x + c)*(f*x + e)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 26.5688, size = 614, normalized size = 5.9 \[ \frac{b^{3} x^{2}}{2 d f} + \frac{\left (a f - b e\right )^{3} \log{\left (x + \frac{a^{3} c d^{2} f^{3} + a^{3} d^{3} e f^{2} - 6 a^{2} b c d^{2} e f^{2} + 3 a b^{2} c^{2} d e f^{2} + 3 a b^{2} c d^{2} e^{2} f - b^{3} c^{3} e f^{2} - b^{3} c d^{2} e^{3} - \frac{c^{2} d^{2} f \left (a f - b e\right )^{3}}{c f - d e} + \frac{2 c d^{3} e \left (a f - b e\right )^{3}}{c f - d e} - \frac{d^{4} e^{2} \left (a f - b e\right )^{3}}{f \left (c f - d e\right )}}{2 a^{3} d^{3} f^{3} - 3 a^{2} b c d^{2} f^{3} - 3 a^{2} b d^{3} e f^{2} + 3 a b^{2} c^{2} d f^{3} + 3 a b^{2} d^{3} e^{2} f - b^{3} c^{3} f^{3} - b^{3} d^{3} e^{3}} \right )}}{f^{3} \left (c f - d e\right )} + \frac{x \left (3 a b^{2} d f - b^{3} c f - b^{3} d e\right )}{d^{2} f^{2}} - \frac{\left (a d - b c\right )^{3} \log{\left (x + \frac{a^{3} c d^{2} f^{3} + a^{3} d^{3} e f^{2} - 6 a^{2} b c d^{2} e f^{2} + 3 a b^{2} c^{2} d e f^{2} + 3 a b^{2} c d^{2} e^{2} f - b^{3} c^{3} e f^{2} - b^{3} c d^{2} e^{3} + \frac{c^{2} f^{4} \left (a d - b c\right )^{3}}{d \left (c f - d e\right )} - \frac{2 c e f^{3} \left (a d - b c\right )^{3}}{c f - d e} + \frac{d e^{2} f^{2} \left (a d - b c\right )^{3}}{c f - d e}}{2 a^{3} d^{3} f^{3} - 3 a^{2} b c d^{2} f^{3} - 3 a^{2} b d^{3} e f^{2} + 3 a b^{2} c^{2} d f^{3} + 3 a b^{2} d^{3} e^{2} f - b^{3} c^{3} f^{3} - b^{3} d^{3} e^{3}} \right )}}{d^{3} \left (c f - d e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**3/(d*x+c)/(f*x+e),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^3/((d*x + c)*(f*x + e)),x, algorithm="giac")
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