Optimal. Leaf size=131 \[ \frac{x \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )}{e^4}-\frac{\left (a e^2-b d e+c d^2\right )^2}{e^5 (d+e x)}-\frac{2 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac{c x^2 (c d-b e)}{e^3}+\frac{c^2 x^3}{3 e^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.351437, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{x \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )}{e^4}-\frac{\left (a e^2-b d e+c d^2\right )^2}{e^5 (d+e x)}-\frac{2 (2 c d-b e) \log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac{c x^2 (c d-b e)}{e^3}+\frac{c^2 x^3}{3 e^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^2/(d + e*x)^2,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{c^{2} x^{3}}{3 e^{2}} + \frac{2 c \left (b e - c d\right ) \int x\, dx}{e^{3}} + \left (2 a c e^{2} + b^{2} e^{2} - 4 b c d e + 3 c^{2} d^{2}\right ) \int \frac{1}{e^{4}}\, dx + \frac{2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{5} \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**2/(e*x+d)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.242201, size = 127, normalized size = 0.97 \[ \frac{3 e x \left (2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right )-\frac{3 \left (e (a e-b d)+c d^2\right )^2}{d+e x}-6 (2 c d-b e) \log (d+e x) \left (e (a e-b d)+c d^2\right )+3 c e^2 x^2 (b e-c d)+c^2 e^3 x^3}{3 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^2/(d + e*x)^2,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.012, size = 246, normalized size = 1.9 \[{\frac{{c}^{2}{x}^{3}}{3\,{e}^{2}}}+{\frac{b{x}^{2}c}{{e}^{2}}}-{\frac{{c}^{2}d{x}^{2}}{{e}^{3}}}+2\,{\frac{acx}{{e}^{2}}}+{\frac{{b}^{2}x}{{e}^{2}}}-4\,{\frac{bcdx}{{e}^{3}}}+3\,{\frac{{c}^{2}{d}^{2}x}{{e}^{4}}}+2\,{\frac{\ln \left ( ex+d \right ) ab}{{e}^{2}}}-4\,{\frac{\ln \left ( ex+d \right ) adc}{{e}^{3}}}-2\,{\frac{\ln \left ( ex+d \right ){b}^{2}d}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ) bc{d}^{2}}{{e}^{4}}}-4\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{3}}{{e}^{5}}}-{\frac{{a}^{2}}{e \left ( ex+d \right ) }}+2\,{\frac{bda}{{e}^{2} \left ( ex+d \right ) }}-2\,{\frac{ac{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{{b}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{{d}^{3}bc}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^2/(e*x+d)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.802628, size = 236, normalized size = 1.8 \[ -\frac{c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}}{e^{6} x + d e^{5}} + \frac{c^{2} e^{2} x^{3} - 3 \,{\left (c^{2} d e - b c e^{2}\right )} x^{2} + 3 \,{\left (3 \, c^{2} d^{2} - 4 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} x}{3 \, e^{4}} - \frac{2 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^2,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.213457, size = 350, normalized size = 2.67 \[ \frac{c^{2} e^{4} x^{4} - 3 \, c^{2} d^{4} + 6 \, b c d^{3} e + 6 \, a b d e^{3} - 3 \, a^{2} e^{4} - 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} -{\left (2 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} + 3 \,{\left (2 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 3 \,{\left (3 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{4} - 3 \, b c d^{3} e - a b d e^{3} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} +{\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} - a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{6} x + d e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^2,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.89274, size = 167, normalized size = 1.27 \[ \frac{c^{2} x^{3}}{3 e^{2}} - \frac{a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}}{d e^{5} + e^{6} x} + \frac{x^{2} \left (b c e - c^{2} d\right )}{e^{3}} + \frac{x \left (2 a c e^{2} + b^{2} e^{2} - 4 b c d e + 3 c^{2} d^{2}\right )}{e^{4}} + \frac{2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**2/(e*x+d)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.206985, size = 336, normalized size = 2.56 \[ \frac{1}{3} \,{\left (c^{2} - \frac{3 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-5\right )} + 2 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2} + 2 \, a c d e^{2} - a b e^{3}\right )} e^{\left (-5\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{2} d^{4} e^{3}}{x e + d} - \frac{2 \, b c d^{3} e^{4}}{x e + d} + \frac{b^{2} d^{2} e^{5}}{x e + d} + \frac{2 \, a c d^{2} e^{5}}{x e + d} - \frac{2 \, a b d e^{6}}{x e + d} + \frac{a^{2} e^{7}}{x e + d}\right )} e^{\left (-8\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^2,x, algorithm="giac")
[Out]