Optimal. Leaf size=129 \[ \frac{x^2 \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{2 e^3}-\frac{x (c d-b e) \left (c d^2-e (b d-2 a e)\right )}{e^4}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{c^2 x^4}{4 e} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.354214, antiderivative size = 129, 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 \[ \frac{x^2 \left (-2 c e (b d-a e)+b^2 e^2+c^2 d^2\right )}{2 e^3}-\frac{x (c d-b e) \left (c d^2-e (b d-2 a e)\right )}{e^4}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^5}-\frac{c x^3 (c d-2 b e)}{3 e^2}+\frac{c^2 x^4}{4 e} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^2/(d + e*x),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{c^{2} x^{4}}{4 e} + \frac{c x^{3} \left (2 b e - c d\right )}{3 e^{2}} + \left (b e - c d\right ) \left (2 a e^{2} - b d e + c d^{2}\right ) \int \frac{1}{e^{4}}\, dx + \frac{\left (2 a c e^{2} + b^{2} e^{2} - 2 b c d e + c^{2} d^{2}\right ) \int x\, dx}{e^{3}} + \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**2/(e*x+d),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.118154, size = 128, normalized size = 0.99 \[ \frac{e x \left (4 c e \left (3 a e (e x-2 d)+b \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+6 b e^2 (4 a e-2 b d+b e x)+c^2 \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 \log (d+e x) \left (e (a e-b d)+c d^2\right )^2}{12 e^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^2/(d + e*x),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 221, normalized size = 1.7 \[{\frac{{c}^{2}{x}^{4}}{4\,e}}+{\frac{2\,bc{x}^{3}}{3\,e}}-{\frac{{c}^{2}d{x}^{3}}{3\,{e}^{2}}}+{\frac{ac{x}^{2}}{e}}+{\frac{{b}^{2}{x}^{2}}{2\,e}}-{\frac{b{x}^{2}cd}{{e}^{2}}}+{\frac{{x}^{2}{c}^{2}{d}^{2}}{2\,{e}^{3}}}+2\,{\frac{abx}{e}}-2\,{\frac{cdax}{{e}^{2}}}-{\frac{{b}^{2}dx}{{e}^{2}}}+2\,{\frac{c{d}^{2}bx}{{e}^{3}}}-{\frac{{c}^{2}{d}^{3}x}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){a}^{2}}{e}}-2\,{\frac{\ln \left ( ex+d \right ) dab}{{e}^{2}}}+2\,{\frac{\ln \left ( ex+d \right ) ac{d}^{2}}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ){b}^{2}{d}^{2}}{{e}^{3}}}-2\,{\frac{\ln \left ( ex+d \right ){d}^{3}bc}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{4}}{{e}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^2/(e*x+d),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.770653, size = 227, normalized size = 1.76 \[ \frac{3 \, c^{2} e^{3} x^{4} - 4 \,{\left (c^{2} d e^{2} - 2 \, b c e^{3}\right )} x^{3} + 6 \,{\left (c^{2} d^{2} e - 2 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x^{2} - 12 \,{\left (c^{2} d^{3} - 2 \, b c d^{2} e - 2 \, a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x}{12 \, e^{4}} + \frac{{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.204017, size = 230, normalized size = 1.78 \[ \frac{3 \, c^{2} e^{4} x^{4} - 4 \,{\left (c^{2} d e^{3} - 2 \, b c e^{4}\right )} x^{3} + 6 \,{\left (c^{2} d^{2} e^{2} - 2 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 12 \,{\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} - 2 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x + 12 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.45923, size = 144, normalized size = 1.12 \[ \frac{c^{2} x^{4}}{4 e} + \frac{x^{3} \left (2 b c e - c^{2} d\right )}{3 e^{2}} + \frac{x^{2} \left (2 a c e^{2} + b^{2} e^{2} - 2 b c d e + c^{2} d^{2}\right )}{2 e^{3}} + \frac{x \left (2 a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} + 2 b c d^{2} e - c^{2} d^{3}\right )}{e^{4}} + \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**2/(e*x+d),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.204381, size = 243, normalized size = 1.88 \[{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, c^{2} x^{4} e^{3} - 4 \, c^{2} d x^{3} e^{2} + 6 \, c^{2} d^{2} x^{2} e - 12 \, c^{2} d^{3} x + 8 \, b c x^{3} e^{3} - 12 \, b c d x^{2} e^{2} + 24 \, b c d^{2} x e + 6 \, b^{2} x^{2} e^{3} + 12 \, a c x^{2} e^{3} - 12 \, b^{2} d x e^{2} - 24 \, a c d x e^{2} + 24 \, a b x e^{3}\right )} e^{\left (-4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d),x, algorithm="giac")
[Out]