Optimal. Leaf size=156 \[ -\frac{3 b^5 (d+e x)^4 (b d-a e)}{2 e^7}+\frac{5 b^4 (d+e x)^3 (b d-a e)^2}{e^7}-\frac{10 b^3 (d+e x)^2 (b d-a e)^3}{e^7}+\frac{15 b^2 x (b d-a e)^4}{e^6}-\frac{(b d-a e)^6}{e^7 (d+e x)}-\frac{6 b (b d-a e)^5 \log (d+e x)}{e^7}+\frac{b^6 (d+e x)^5}{5 e^7} \]
[Out]
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Rubi [A] time = 0.449748, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{3 b^5 (d+e x)^4 (b d-a e)}{2 e^7}+\frac{5 b^4 (d+e x)^3 (b d-a e)^2}{e^7}-\frac{10 b^3 (d+e x)^2 (b d-a e)^3}{e^7}+\frac{15 b^2 x (b d-a e)^4}{e^6}-\frac{(b d-a e)^6}{e^7 (d+e x)}-\frac{6 b (b d-a e)^5 \log (d+e x)}{e^7}+\frac{b^6 (d+e x)^5}{5 e^7} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^2,x]
[Out]
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Rubi in Sympy [A] time = 97.8553, size = 143, normalized size = 0.92 \[ \frac{b^{6} \left (d + e x\right )^{5}}{5 e^{7}} + \frac{3 b^{5} \left (d + e x\right )^{4} \left (a e - b d\right )}{2 e^{7}} + \frac{5 b^{4} \left (d + e x\right )^{3} \left (a e - b d\right )^{2}}{e^{7}} + \frac{10 b^{3} \left (d + e x\right )^{2} \left (a e - b d\right )^{3}}{e^{7}} + \frac{15 b^{2} x \left (a e - b d\right )^{4}}{e^{6}} + \frac{6 b \left (a e - b d\right )^{5} \log{\left (d + e x \right )}}{e^{7}} - \frac{\left (a e - b d\right )^{6}}{e^{7} \left (d + e x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**2,x)
[Out]
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Mathematica [A] time = 0.19485, size = 302, normalized size = 1.94 \[ \frac{-10 a^6 e^6+60 a^5 b d e^5+150 a^4 b^2 e^4 \left (-d^2+d e x+e^2 x^2\right )+100 a^3 b^3 e^3 \left (2 d^3-4 d^2 e x-3 d e^2 x^2+e^3 x^3\right )+50 a^2 b^4 e^2 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+5 a b^5 e \left (12 d^5-48 d^4 e x-30 d^3 e^2 x^2+10 d^2 e^3 x^3-5 d e^4 x^4+3 e^5 x^5\right )-60 b (d+e x) (b d-a e)^5 \log (d+e x)+b^6 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )}{10 e^7 (d+e x)} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^2,x]
[Out]
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Maple [B] time = 0.016, size = 440, normalized size = 2.8 \[ -15\,{\frac{{d}^{2}{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) }}+20\,{\frac{{d}^{3}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}-15\,{\frac{{d}^{4}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}+6\,{\frac{{d}^{5}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}-15\,{\frac{{b}^{4}{x}^{2}{a}^{2}d}{{e}^{3}}}-30\,{\frac{{b}^{2}\ln \left ( ex+d \right ){a}^{4}d}{{e}^{3}}}-4\,{\frac{{b}^{5}{x}^{3}ad}{{e}^{3}}}+6\,{\frac{d{a}^{5}b}{{e}^{2} \left ( ex+d \right ) }}+{\frac{{b}^{6}{x}^{5}}{5\,{e}^{2}}}-{\frac{{a}^{6}}{e \left ( ex+d \right ) }}-{\frac{{b}^{6}{x}^{4}d}{2\,{e}^{3}}}+5\,{\frac{{b}^{4}{x}^{3}{a}^{2}}{{e}^{2}}}+{\frac{{b}^{6}{x}^{3}{d}^{2}}{{e}^{4}}}+10\,{\frac{{b}^{3}{x}^{2}{a}^{3}}{{e}^{2}}}-2\,{\frac{{b}^{6}{x}^{2}{d}^{3}}{{e}^{5}}}+15\,{\frac{{b}^{2}{a}^{4}x}{{e}^{2}}}+5\,{\frac{{d}^{4}{b}^{6}x}{{e}^{6}}}+6\,{\frac{b\ln \left ( ex+d \right ){a}^{5}}{{e}^{2}}}-6\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{5}}{{e}^{7}}}-{\frac{{d}^{6}{b}^{6}}{{e}^{7} \left ( ex+d \right ) }}+{\frac{3\,{b}^{5}{x}^{4}a}{2\,{e}^{2}}}+60\,{\frac{{b}^{3}\ln \left ( ex+d \right ){a}^{3}{d}^{2}}{{e}^{4}}}-60\,{\frac{{b}^{4}\ln \left ( ex+d \right ){a}^{2}{d}^{3}}{{e}^{5}}}+30\,{\frac{{b}^{5}\ln \left ( ex+d \right ) a{d}^{4}}{{e}^{6}}}+9\,{\frac{{b}^{5}{x}^{2}a{d}^{2}}{{e}^{4}}}-40\,{\frac{d{a}^{3}{b}^{3}x}{{e}^{3}}}+45\,{\frac{{d}^{2}{a}^{2}{b}^{4}x}{{e}^{4}}}-24\,{\frac{{d}^{3}a{b}^{5}x}{{e}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^2,x)
[Out]
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Maxima [A] time = 0.690343, size = 482, normalized size = 3.09 \[ -\frac{b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}}{e^{8} x + d e^{7}} + \frac{2 \, b^{6} e^{4} x^{5} - 5 \,{\left (b^{6} d e^{3} - 3 \, a b^{5} e^{4}\right )} x^{4} + 10 \,{\left (b^{6} d^{2} e^{2} - 4 \, a b^{5} d e^{3} + 5 \, a^{2} b^{4} e^{4}\right )} x^{3} - 10 \,{\left (2 \, b^{6} d^{3} e - 9 \, a b^{5} d^{2} e^{2} + 15 \, a^{2} b^{4} d e^{3} - 10 \, a^{3} b^{3} e^{4}\right )} x^{2} + 10 \,{\left (5 \, b^{6} d^{4} - 24 \, a b^{5} d^{3} e + 45 \, a^{2} b^{4} d^{2} e^{2} - 40 \, a^{3} b^{3} d e^{3} + 15 \, a^{4} b^{2} e^{4}\right )} x}{10 \, e^{6}} - \frac{6 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} \log \left (e x + d\right )}{e^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.202707, size = 670, normalized size = 4.29 \[ \frac{2 \, b^{6} e^{6} x^{6} - 10 \, b^{6} d^{6} + 60 \, a b^{5} d^{5} e - 150 \, a^{2} b^{4} d^{4} e^{2} + 200 \, a^{3} b^{3} d^{3} e^{3} - 150 \, a^{4} b^{2} d^{2} e^{4} + 60 \, a^{5} b d e^{5} - 10 \, a^{6} e^{6} - 3 \,{\left (b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - 5 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} - 10 \,{\left (b^{6} d^{3} e^{3} - 5 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} - 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 30 \,{\left (b^{6} d^{4} e^{2} - 5 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} - 10 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \,{\left (5 \, b^{6} d^{5} e - 24 \, a b^{5} d^{4} e^{2} + 45 \, a^{2} b^{4} d^{3} e^{3} - 40 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5}\right )} x - 60 \,{\left (b^{6} d^{6} - 5 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} - a^{5} b d e^{5} +{\left (b^{6} d^{5} e - 5 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} - 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} - a^{5} b e^{6}\right )} x\right )} \log \left (e x + d\right )}{10 \,{\left (e^{8} x + d e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.84298, size = 303, normalized size = 1.94 \[ \frac{b^{6} x^{5}}{5 e^{2}} + \frac{6 b \left (a e - b d\right )^{5} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} - 6 a^{5} b d e^{5} + 15 a^{4} b^{2} d^{2} e^{4} - 20 a^{3} b^{3} d^{3} e^{3} + 15 a^{2} b^{4} d^{4} e^{2} - 6 a b^{5} d^{5} e + b^{6} d^{6}}{d e^{7} + e^{8} x} + \frac{x^{4} \left (3 a b^{5} e - b^{6} d\right )}{2 e^{3}} + \frac{x^{3} \left (5 a^{2} b^{4} e^{2} - 4 a b^{5} d e + b^{6} d^{2}\right )}{e^{4}} + \frac{x^{2} \left (10 a^{3} b^{3} e^{3} - 15 a^{2} b^{4} d e^{2} + 9 a b^{5} d^{2} e - 2 b^{6} d^{3}\right )}{e^{5}} + \frac{x \left (15 a^{4} b^{2} e^{4} - 40 a^{3} b^{3} d e^{3} + 45 a^{2} b^{4} d^{2} e^{2} - 24 a b^{5} d^{3} e + 5 b^{6} d^{4}\right )}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.214755, size = 579, normalized size = 3.71 \[ \frac{1}{10} \,{\left (2 \, b^{6} - \frac{15 \,{\left (b^{6} d e - a b^{5} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{50 \,{\left (b^{6} d^{2} e^{2} - 2 \, a b^{5} d e^{3} + a^{2} b^{4} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{100 \,{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac{150 \,{\left (b^{6} d^{4} e^{4} - 4 \, a b^{5} d^{3} e^{5} + 6 \, a^{2} b^{4} d^{2} e^{6} - 4 \, a^{3} b^{3} d e^{7} + a^{4} b^{2} e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )}{\left (x e + d\right )}^{5} e^{\left (-7\right )} + 6 \,{\left (b^{6} d^{5} - 5 \, a b^{5} d^{4} e + 10 \, a^{2} b^{4} d^{3} e^{2} - 10 \, a^{3} b^{3} d^{2} e^{3} + 5 \, a^{4} b^{2} d e^{4} - a^{5} b e^{5}\right )} e^{\left (-7\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{b^{6} d^{6} e^{5}}{x e + d} - \frac{6 \, a b^{5} d^{5} e^{6}}{x e + d} + \frac{15 \, a^{2} b^{4} d^{4} e^{7}}{x e + d} - \frac{20 \, a^{3} b^{3} d^{3} e^{8}}{x e + d} + \frac{15 \, a^{4} b^{2} d^{2} e^{9}}{x e + d} - \frac{6 \, a^{5} b d e^{10}}{x e + d} + \frac{a^{6} e^{11}}{x e + d}\right )} e^{\left (-12\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^2,x, algorithm="giac")
[Out]