Optimal. Leaf size=139 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]
[Out]
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Rubi [A] time = 0.321256, antiderivative size = 139, 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 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{e^5 (d+e x)}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{2 c (2 c d-b e) \log (d+e x)}{e^5}+\frac{c^2 x}{e^4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^2/(d + e*x)^4,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{5}} + \frac{\int c^{2}\, dx}{e^{4}} - \frac{2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{e^{5} \left (d + e x\right )} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{5} \left (d + e x\right )^{2}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{5} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**2/(e*x+d)**4,x)
[Out]
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Mathematica [A] time = 0.17584, size = 176, normalized size = 1.27 \[ \frac{-e^2 \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )+c e \left (b d \left (11 d^2+27 d e x+18 e^2 x^2\right )-2 a e \left (d^2+3 d e x+3 e^2 x^2\right )\right )-6 c (d+e x)^3 (2 c d-b e) \log (d+e x)+c^2 \left (-13 d^4-27 d^3 e x-9 d^2 e^2 x^2+9 d e^3 x^3+3 e^4 x^4\right )}{3 e^5 (d+e x)^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^2/(d + e*x)^4,x]
[Out]
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Maple [B] time = 0.013, size = 279, normalized size = 2. \[{\frac{{c}^{2}x}{{e}^{4}}}-{\frac{ab}{{e}^{2} \left ( ex+d \right ) ^{2}}}+2\,{\frac{acd}{{e}^{3} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{2}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-3\,{\frac{c{d}^{2}b}{{e}^{4} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{2}}{3\,e \left ( ex+d \right ) ^{3}}}+{\frac{2\,bda}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{2\,ac{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{2}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{d}^{3}bc}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+2\,{\frac{c\ln \left ( ex+d \right ) b}{{e}^{4}}}-4\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) }{{e}^{5}}}-2\,{\frac{ac}{{e}^{3} \left ( ex+d \right ) }}-{\frac{{b}^{2}}{{e}^{3} \left ( ex+d \right ) }}+6\,{\frac{bcd}{{e}^{4} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^2/(e*x+d)^4,x)
[Out]
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Maxima [A] time = 0.821291, size = 262, normalized size = 1.88 \[ -\frac{13 \, c^{2} d^{4} - 11 \, b c d^{3} e + a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac{c^{2} x}{e^{4}} - \frac{2 \,{\left (2 \, c^{2} d - b c e\right )} \log \left (e x + d\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21621, size = 381, normalized size = 2.74 \[ \frac{3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} + 11 \, b c d^{3} e - a b d e^{3} - a^{2} e^{4} -{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 3 \,{\left (3 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 3 \,{\left (9 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x - 6 \,{\left (2 \, c^{2} d^{4} - b c d^{3} e +{\left (2 \, c^{2} d e^{3} - b c e^{4}\right )} x^{3} + 3 \,{\left (2 \, c^{2} d^{2} e^{2} - b c d e^{3}\right )} x^{2} + 3 \,{\left (2 \, c^{2} d^{3} e - b c d^{2} e^{2}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 26.8745, size = 218, normalized size = 1.57 \[ \frac{c^{2} x}{e^{4}} + \frac{2 c \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{a^{2} e^{4} + a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 11 b c d^{3} e + 13 c^{2} d^{4} + x^{2} \left (6 a c e^{4} + 3 b^{2} e^{4} - 18 b c d e^{3} + 18 c^{2} d^{2} e^{2}\right ) + x \left (3 a b e^{4} + 6 a c d e^{3} + 3 b^{2} d e^{3} - 27 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)**2/(e*x+d)**4,x)
[Out]
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GIAC/XCAS [A] time = 0.204778, size = 230, normalized size = 1.65 \[ c^{2} x e^{\left (-4\right )} - 2 \,{\left (2 \, c^{2} d - b c e\right )} e^{\left (-5\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (13 \, c^{2} d^{4} - 11 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + a b d e^{3} + 3 \,{\left (6 \, c^{2} d^{2} e^{2} - 6 \, b c d e^{3} + b^{2} e^{4} + 2 \, a c e^{4}\right )} x^{2} + a^{2} e^{4} + 3 \,{\left (10 \, c^{2} d^{3} e - 9 \, b c d^{2} e^{2} + b^{2} d e^{3} + 2 \, a c d e^{3} + a b e^{4}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^2/(e*x + d)^4,x, algorithm="giac")
[Out]