Optimal. Leaf size=161 \[ -\frac{d^2 (B d-A e) (c d-b e)^2 \log (d+e x)}{e^6}+\frac{d x (B d-A e) (c d-b e)^2}{e^5}-\frac{x^2 (B d-A e) (c d-b e)^2}{2 e^4}-\frac{x^3 \left (A c e (c d-2 b e)-B (c d-b e)^2\right )}{3 e^3}-\frac{c x^4 (-A c e-2 b B e+B c d)}{4 e^2}+\frac{B c^2 x^5}{5 e} \]
[Out]
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Rubi [A] time = 0.493956, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{d^2 (B d-A e) (c d-b e)^2 \log (d+e x)}{e^6}+\frac{d x (B d-A e) (c d-b e)^2}{e^5}-\frac{x^2 (B d-A e) (c d-b e)^2}{2 e^4}-\frac{x^3 \left (A c e (c d-2 b e)-B (c d-b e)^2\right )}{3 e^3}-\frac{c x^4 (-A c e-2 b B e+B c d)}{4 e^2}+\frac{B c^2 x^5}{5 e} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{B c^{2} x^{5}}{5 e} + \frac{c x^{4} \left (A c e + 2 B b e - B c d\right )}{4 e^{2}} + \frac{d^{2} \left (A e - B d\right ) \left (b e - c d\right )^{2} \log{\left (d + e x \right )}}{e^{6}} + \frac{x^{3} \left (2 A b c e^{2} - A c^{2} d e + B b^{2} e^{2} - 2 B b c d e + B c^{2} d^{2}\right )}{3 e^{3}} + \frac{\left (A e - B d\right ) \left (b e - c d\right )^{2} \int x\, dx}{e^{4}} - \frac{\left (A e - B d\right ) \left (b e - c d\right )^{2} \int d\, dx}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.257834, size = 156, normalized size = 0.97 \[ \frac{-60 d^2 (B d-A e) (c d-b e)^2 \log (d+e x)+15 c e^4 x^4 (A c e+2 b B e-B c d)+20 e^3 x^3 \left (A c e (2 b e-c d)+B (c d-b e)^2\right )+30 e^2 x^2 (A e-B d) (c d-b e)^2+60 d e x (B d-A e) (c d-b e)^2+12 B c^2 e^5 x^5}{60 e^6} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(b*x + c*x^2)^2)/(d + e*x),x]
[Out]
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Maple [B] time = 0.006, size = 369, normalized size = 2.3 \[ -2\,{\frac{{d}^{3}\ln \left ( ex+d \right ) Abc}{{e}^{4}}}+{\frac{B{c}^{2}{x}^{5}}{5\,e}}-{\frac{{d}^{3}\ln \left ( ex+d \right ) B{b}^{2}}{{e}^{4}}}+{\frac{{b}^{2}B{d}^{2}x}{{e}^{3}}}+{\frac{B{c}^{2}{d}^{4}x}{{e}^{5}}}-{\frac{B{c}^{2}{x}^{2}{d}^{3}}{2\,{e}^{4}}}-{\frac{{b}^{2}Adx}{{e}^{2}}}-{\frac{A{d}^{3}{c}^{2}x}{{e}^{4}}}+{\frac{A{b}^{2}{x}^{2}}{2\,e}}+{\frac{B{x}^{4}bc}{2\,e}}-{\frac{B{c}^{2}{x}^{4}d}{4\,{e}^{2}}}-{\frac{A{c}^{2}{x}^{3}d}{3\,{e}^{2}}}-{\frac{{b}^{2}B{x}^{2}d}{2\,{e}^{2}}}+{\frac{B{c}^{2}{x}^{3}{d}^{2}}{3\,{e}^{3}}}+{\frac{A{c}^{2}{x}^{2}{d}^{2}}{2\,{e}^{3}}}+{\frac{{d}^{2}\ln \left ( ex+d \right ) A{b}^{2}}{{e}^{3}}}+{\frac{{d}^{4}\ln \left ( ex+d \right ) A{c}^{2}}{{e}^{5}}}+{\frac{A{c}^{2}{x}^{4}}{4\,e}}+2\,{\frac{{d}^{4}\ln \left ( ex+d \right ) Bbc}{{e}^{5}}}-{\frac{A{x}^{2}bcd}{{e}^{2}}}+{\frac{B{x}^{2}bc{d}^{2}}{{e}^{3}}}+2\,{\frac{A{d}^{2}bcx}{{e}^{3}}}-{\frac{2\,B{x}^{3}bcd}{3\,{e}^{2}}}-2\,{\frac{Bbc{d}^{3}x}{{e}^{4}}}+{\frac{2\,Ab{x}^{3}c}{3\,e}}+{\frac{B{x}^{3}{b}^{2}}{3\,e}}-{\frac{{d}^{5}\ln \left ( ex+d \right ) B{c}^{2}}{{e}^{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^2/(e*x+d),x)
[Out]
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Maxima [A] time = 0.69126, size = 381, normalized size = 2.37 \[ \frac{12 \, B c^{2} e^{4} x^{5} - 15 \,{\left (B c^{2} d e^{3} -{\left (2 \, B b c + A c^{2}\right )} e^{4}\right )} x^{4} + 20 \,{\left (B c^{2} d^{2} e^{2} -{\left (2 \, B b c + A c^{2}\right )} d e^{3} +{\left (B b^{2} + 2 \, A b c\right )} e^{4}\right )} x^{3} - 30 \,{\left (B c^{2} d^{3} e - A b^{2} e^{4} -{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{2} +{\left (B b^{2} + 2 \, A b c\right )} d e^{3}\right )} x^{2} + 60 \,{\left (B c^{2} d^{4} - A b^{2} d e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{3} e +{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{2}\right )} x}{60 \, e^{5}} - \frac{{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275429, size = 382, normalized size = 2.37 \[ \frac{12 \, B c^{2} e^{5} x^{5} - 15 \,{\left (B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 20 \,{\left (B c^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 30 \,{\left (B c^{2} d^{3} e^{2} - A b^{2} e^{5} -{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} +{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 60 \,{\left (B c^{2} d^{4} e - A b^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} +{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x - 60 \,{\left (B c^{2} d^{5} - A b^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2}\right )} \log \left (e x + d\right )}{60 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.22853, size = 265, normalized size = 1.65 \[ \frac{B c^{2} x^{5}}{5 e} - \frac{d^{2} \left (- A e + B d\right ) \left (b e - c d\right )^{2} \log{\left (d + e x \right )}}{e^{6}} + \frac{x^{4} \left (A c^{2} e + 2 B b c e - B c^{2} d\right )}{4 e^{2}} + \frac{x^{3} \left (2 A b c e^{2} - A c^{2} d e + B b^{2} e^{2} - 2 B b c d e + B c^{2} d^{2}\right )}{3 e^{3}} - \frac{x^{2} \left (- A b^{2} e^{3} + 2 A b c d e^{2} - A c^{2} d^{2} e + B b^{2} d e^{2} - 2 B b c d^{2} e + B c^{2} d^{3}\right )}{2 e^{4}} + \frac{x \left (- A b^{2} d e^{3} + 2 A b c d^{2} e^{2} - A c^{2} d^{3} e + B b^{2} d^{2} e^{2} - 2 B b c d^{3} e + B c^{2} d^{4}\right )}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.280397, size = 435, normalized size = 2.7 \[ -{\left (B c^{2} d^{5} - 2 \, B b c d^{4} e - A c^{2} d^{4} e + B b^{2} d^{3} e^{2} + 2 \, A b c d^{3} e^{2} - A b^{2} d^{2} e^{3}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{60} \,{\left (12 \, B c^{2} x^{5} e^{4} - 15 \, B c^{2} d x^{4} e^{3} + 20 \, B c^{2} d^{2} x^{3} e^{2} - 30 \, B c^{2} d^{3} x^{2} e + 60 \, B c^{2} d^{4} x + 30 \, B b c x^{4} e^{4} + 15 \, A c^{2} x^{4} e^{4} - 40 \, B b c d x^{3} e^{3} - 20 \, A c^{2} d x^{3} e^{3} + 60 \, B b c d^{2} x^{2} e^{2} + 30 \, A c^{2} d^{2} x^{2} e^{2} - 120 \, B b c d^{3} x e - 60 \, A c^{2} d^{3} x e + 20 \, B b^{2} x^{3} e^{4} + 40 \, A b c x^{3} e^{4} - 30 \, B b^{2} d x^{2} e^{3} - 60 \, A b c d x^{2} e^{3} + 60 \, B b^{2} d^{2} x e^{2} + 120 \, A b c d^{2} x e^{2} + 30 \, A b^{2} x^{2} e^{4} - 60 \, A b^{2} d x e^{3}\right )} e^{\left (-5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d),x, algorithm="giac")
[Out]