Optimal. Leaf size=136 \[ \frac{6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac{6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}+\frac{(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac{d^3 (3 c d-4 b e)}{b^4 x}+\frac{(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}-\frac{d^4}{2 b^3 x^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.338598, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{6 d^2 \log (x) (c d-b e)^2}{b^5}-\frac{6 d^2 (c d-b e)^2 \log (b+c x)}{b^5}+\frac{(c d-b e)^3 (b e+3 c d)}{b^4 c^2 (b+c x)}+\frac{d^3 (3 c d-4 b e)}{b^4 x}+\frac{(c d-b e)^4}{2 b^3 c^2 (b+c x)^2}-\frac{d^4}{2 b^3 x^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^4/(b*x + c*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 44.1403, size = 126, normalized size = 0.93 \[ - \frac{d^{4}}{2 b^{3} x^{2}} + \frac{\left (b e - c d\right )^{4}}{2 b^{3} c^{2} \left (b + c x\right )^{2}} - \frac{d^{3} \left (4 b e - 3 c d\right )}{b^{4} x} - \frac{\left (b e - c d\right )^{3} \left (b e + 3 c d\right )}{b^{4} c^{2} \left (b + c x\right )} + \frac{6 d^{2} \left (b e - c d\right )^{2} \log{\left (x \right )}}{b^{5}} - \frac{6 d^{2} \left (b e - c d\right )^{2} \log{\left (b + c x \right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**4/(c*x**2+b*x)**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.135743, size = 130, normalized size = 0.96 \[ -\frac{-\frac{b^2 (c d-b e)^4}{c^2 (b+c x)^2}+\frac{b^2 d^4}{x^2}+\frac{2 b (b e-c d)^3 (b e+3 c d)}{c^2 (b+c x)}+\frac{2 b d^3 (4 b e-3 c d)}{x}-12 d^2 \log (x) (c d-b e)^2+12 d^2 (c d-b e)^2 \log (b+c x)}{2 b^5} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^4/(b*x + c*x^2)^3,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.018, size = 278, normalized size = 2. \[ -{\frac{{d}^{4}}{2\,{b}^{3}{x}^{2}}}-4\,{\frac{{d}^{3}e}{{b}^{3}x}}+3\,{\frac{{d}^{4}c}{{b}^{4}x}}+6\,{\frac{{d}^{2}\ln \left ( x \right ){e}^{2}}{{b}^{3}}}-12\,{\frac{{d}^{3}\ln \left ( x \right ) ce}{{b}^{4}}}+6\,{\frac{{d}^{4}\ln \left ( x \right ){c}^{2}}{{b}^{5}}}-{\frac{{e}^{4}}{{c}^{2} \left ( cx+b \right ) }}+6\,{\frac{{d}^{2}{e}^{2}}{{b}^{2} \left ( cx+b \right ) }}-8\,{\frac{{d}^{3}ec}{{b}^{3} \left ( cx+b \right ) }}+3\,{\frac{{c}^{2}{d}^{4}}{{b}^{4} \left ( cx+b \right ) }}+{\frac{b{e}^{4}}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}}-2\,{\frac{d{e}^{3}}{c \left ( cx+b \right ) ^{2}}}+3\,{\frac{{d}^{2}{e}^{2}}{b \left ( cx+b \right ) ^{2}}}-2\,{\frac{{d}^{3}ec}{{b}^{2} \left ( cx+b \right ) ^{2}}}+{\frac{{c}^{2}{d}^{4}}{2\,{b}^{3} \left ( cx+b \right ) ^{2}}}-6\,{\frac{{d}^{2}\ln \left ( cx+b \right ){e}^{2}}{{b}^{3}}}+12\,{\frac{{d}^{3}\ln \left ( cx+b \right ) ce}{{b}^{4}}}-6\,{\frac{{d}^{4}\ln \left ( cx+b \right ){c}^{2}}{{b}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^4/(c*x^2+b*x)^3,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.699215, size = 338, normalized size = 2.49 \[ -\frac{b^{3} c^{2} d^{4} - 2 \,{\left (6 \, c^{5} d^{4} - 12 \, b c^{4} d^{3} e + 6 \, b^{2} c^{3} d^{2} e^{2} - b^{4} c e^{4}\right )} x^{3} -{\left (18 \, b c^{4} d^{4} - 36 \, b^{2} c^{3} d^{3} e + 18 \, b^{3} c^{2} d^{2} e^{2} - 4 \, b^{4} c d e^{3} - b^{5} e^{4}\right )} x^{2} - 4 \,{\left (b^{2} c^{3} d^{4} - 2 \, b^{3} c^{2} d^{3} e\right )} x}{2 \,{\left (b^{4} c^{4} x^{4} + 2 \, b^{5} c^{3} x^{3} + b^{6} c^{2} x^{2}\right )}} - \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (c x + b\right )}{b^{5}} + \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (x\right )}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(c*x^2 + b*x)^3,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.227725, size = 575, normalized size = 4.23 \[ -\frac{b^{4} c^{2} d^{4} - 2 \,{\left (6 \, b c^{5} d^{4} - 12 \, b^{2} c^{4} d^{3} e + 6 \, b^{3} c^{3} d^{2} e^{2} - b^{5} c e^{4}\right )} x^{3} -{\left (18 \, b^{2} c^{4} d^{4} - 36 \, b^{3} c^{3} d^{3} e + 18 \, b^{4} c^{2} d^{2} e^{2} - 4 \, b^{5} c d e^{3} - b^{6} e^{4}\right )} x^{2} - 4 \,{\left (b^{3} c^{3} d^{4} - 2 \, b^{4} c^{2} d^{3} e\right )} x + 12 \,{\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} +{\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (c x + b\right ) - 12 \,{\left ({\left (c^{6} d^{4} - 2 \, b c^{5} d^{3} e + b^{2} c^{4} d^{2} e^{2}\right )} x^{4} + 2 \,{\left (b c^{5} d^{4} - 2 \, b^{2} c^{4} d^{3} e + b^{3} c^{3} d^{2} e^{2}\right )} x^{3} +{\left (b^{2} c^{4} d^{4} - 2 \, b^{3} c^{3} d^{3} e + b^{4} c^{2} d^{2} e^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{5} c^{4} x^{4} + 2 \, b^{6} c^{3} x^{3} + b^{7} c^{2} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(c*x^2 + b*x)^3,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 11.5754, size = 389, normalized size = 2.86 \[ - \frac{b^{3} c^{2} d^{4} + x^{3} \left (2 b^{4} c e^{4} - 12 b^{2} c^{3} d^{2} e^{2} + 24 b c^{4} d^{3} e - 12 c^{5} d^{4}\right ) + x^{2} \left (b^{5} e^{4} + 4 b^{4} c d e^{3} - 18 b^{3} c^{2} d^{2} e^{2} + 36 b^{2} c^{3} d^{3} e - 18 b c^{4} d^{4}\right ) + x \left (8 b^{3} c^{2} d^{3} e - 4 b^{2} c^{3} d^{4}\right )}{2 b^{6} c^{2} x^{2} + 4 b^{5} c^{3} x^{3} + 2 b^{4} c^{4} x^{4}} + \frac{6 d^{2} \left (b e - c d\right )^{2} \log{\left (x + \frac{6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} - 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} - \frac{6 d^{2} \left (b e - c d\right )^{2} \log{\left (x + \frac{6 b^{3} d^{2} e^{2} - 12 b^{2} c d^{3} e + 6 b c^{2} d^{4} + 6 b d^{2} \left (b e - c d\right )^{2}}{12 b^{2} c d^{2} e^{2} - 24 b c^{2} d^{3} e + 12 c^{3} d^{4}} \right )}}{b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**4/(c*x**2+b*x)**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.209429, size = 343, normalized size = 2.52 \[ \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{5}} - \frac{6 \,{\left (c^{3} d^{4} - 2 \, b c^{2} d^{3} e + b^{2} c d^{2} e^{2}\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{5} c} + \frac{12 \, c^{5} d^{4} x^{3} - 24 \, b c^{4} d^{3} x^{3} e + 18 \, b c^{4} d^{4} x^{2} + 12 \, b^{2} c^{3} d^{2} x^{3} e^{2} - 36 \, b^{2} c^{3} d^{3} x^{2} e + 4 \, b^{2} c^{3} d^{4} x + 18 \, b^{3} c^{2} d^{2} x^{2} e^{2} - 8 \, b^{3} c^{2} d^{3} x e - b^{3} c^{2} d^{4} - 2 \, b^{4} c x^{3} e^{4} - 4 \, b^{4} c d x^{2} e^{3} - b^{5} x^{2} e^{4}}{2 \,{\left (c x^{2} + b x\right )}^{2} b^{4} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(c*x^2 + b*x)^3,x, algorithm="giac")
[Out]