Optimal. Leaf size=155 \[ -\frac{b^5 x (5 b d-6 a e)}{e^6}+\frac{15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac{20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac{15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac{2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac{(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac{b^6 x^2}{2 e^5} \]
[Out]
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Rubi [A] time = 0.37691, antiderivative size = 155, 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 \[ -\frac{b^5 x (5 b d-6 a e)}{e^6}+\frac{15 b^4 (b d-a e)^2 \log (d+e x)}{e^7}+\frac{20 b^3 (b d-a e)^3}{e^7 (d+e x)}-\frac{15 b^2 (b d-a e)^4}{2 e^7 (d+e x)^2}+\frac{2 b (b d-a e)^5}{e^7 (d+e x)^3}-\frac{(b d-a e)^6}{4 e^7 (d+e x)^4}+\frac{b^6 x^2}{2 e^5} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^5,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b^{6} \int x\, dx}{e^{5}} + \frac{15 b^{4} \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{7}} - \frac{20 b^{3} \left (a e - b d\right )^{3}}{e^{7} \left (d + e x\right )} - \frac{15 b^{2} \left (a e - b d\right )^{4}}{2 e^{7} \left (d + e x\right )^{2}} - \frac{2 b \left (a e - b d\right )^{5}}{e^{7} \left (d + e x\right )^{3}} - \frac{\left (a e - b d\right )^{6}}{4 e^{7} \left (d + e x\right )^{4}} + \frac{\left (6 a e - 5 b d\right ) \int b^{11}\, dx}{b^{6} e^{6}} \]
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)**5,x)
[Out]
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Mathematica [A] time = 0.23702, size = 301, normalized size = 1.94 \[ -\frac{a^6 e^6+2 a^5 b e^5 (d+4 e x)+5 a^4 b^2 e^4 \left (d^2+4 d e x+6 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a^2 b^4 d e^2 \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+2 a b^5 e \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )-60 b^4 (d+e x)^4 (b d-a e)^2 \log (d+e x)+b^6 \left (-\left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )}{4 e^7 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^5,x]
[Out]
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Maple [B] time = 0.016, size = 498, normalized size = 3.2 \[ -{\frac{{a}^{6}}{4\,e \left ( ex+d \right ) ^{4}}}-30\,{\frac{{b}^{5}\ln \left ( ex+d \right ) da}{{e}^{6}}}+{\frac{3\,d{a}^{5}b}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}+20\,{\frac{{d}^{3}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+60\,{\frac{d{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}-60\,{\frac{{d}^{2}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}-10\,{\frac{{d}^{4}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{3}}}+5\,{\frac{{d}^{3}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{d}^{4}{a}^{2}{b}^{4}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{{b}^{6}{x}^{2}}{2\,{e}^{5}}}+{\frac{3\,{d}^{5}a{b}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}+30\,{\frac{d{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-45\,{\frac{{d}^{2}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+30\,{\frac{{d}^{3}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{{d}^{6}{b}^{6}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{15\,{b}^{2}{a}^{4}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{15\,{d}^{4}{b}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}+6\,{\frac{{b}^{5}xa}{{e}^{5}}}-5\,{\frac{{b}^{6}xd}{{e}^{6}}}-2\,{\frac{{a}^{5}b}{{e}^{2} \left ( ex+d \right ) ^{3}}}+2\,{\frac{{d}^{5}{b}^{6}}{{e}^{7} \left ( ex+d \right ) ^{3}}}+15\,{\frac{{b}^{4}\ln \left ( ex+d \right ){a}^{2}}{{e}^{5}}}+15\,{\frac{{b}^{6}\ln \left ( ex+d \right ){d}^{2}}{{e}^{7}}}-20\,{\frac{{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) }}+20\,{\frac{{d}^{3}{b}^{6}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{15\,{d}^{2}{b}^{2}{a}^{4}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+10\,{\frac{d{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-20\,{\frac{{d}^{2}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{3}}} \]
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)^5,x)
[Out]
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Maxima [A] time = 0.707129, size = 522, normalized size = 3.37 \[ \frac{57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} + 80 \,{\left (b^{6} d^{3} e^{3} - 3 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} - a^{3} b^{3} e^{6}\right )} x^{3} + 30 \,{\left (7 \, b^{6} d^{4} e^{2} - 20 \, a b^{5} d^{3} e^{3} + 18 \, a^{2} b^{4} d^{2} e^{4} - 4 \, a^{3} b^{3} d e^{5} - a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (47 \, b^{6} d^{5} e - 130 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac{b^{6} e x^{2} - 2 \,{\left (5 \, b^{6} d - 6 \, a b^{5} e\right )} x}{2 \, e^{6}} + \frac{15 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \]
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)^5,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203162, size = 771, normalized size = 4.97 \[ \frac{2 \, b^{6} e^{6} x^{6} + 57 \, b^{6} d^{6} - 154 \, a b^{5} d^{5} e + 125 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 2 \, a^{5} b d e^{5} - a^{6} e^{6} - 12 \,{\left (b^{6} d e^{5} - 2 \, a b^{5} e^{6}\right )} x^{5} - 4 \,{\left (17 \, b^{6} d^{2} e^{4} - 24 \, a b^{5} d e^{5}\right )} x^{4} - 16 \,{\left (2 \, b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} - 15 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 6 \,{\left (22 \, b^{6} d^{4} e^{2} - 84 \, a b^{5} d^{3} e^{3} + 90 \, a^{2} b^{4} d^{2} e^{4} - 20 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 4 \,{\left (42 \, b^{6} d^{5} e - 124 \, a b^{5} d^{4} e^{2} + 110 \, a^{2} b^{4} d^{3} e^{3} - 20 \, a^{3} b^{3} d^{2} e^{4} - 5 \, a^{4} b^{2} d e^{5} - 2 \, a^{5} b e^{6}\right )} x + 60 \,{\left (b^{6} d^{6} - 2 \, a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} +{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 4 \,{\left (b^{6} d^{3} e^{3} - 2 \, a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5}\right )} x^{3} + 6 \,{\left (b^{6} d^{4} e^{2} - 2 \, a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (b^{6} d^{5} e - 2 \, a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \,{\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \]
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)^5,x, algorithm="fricas")
[Out]
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Sympy [A] time = 37.1173, size = 393, normalized size = 2.54 \[ \frac{b^{6} x^{2}}{2 e^{5}} + \frac{15 b^{4} \left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{7}} - \frac{a^{6} e^{6} + 2 a^{5} b d e^{5} + 5 a^{4} b^{2} d^{2} e^{4} + 20 a^{3} b^{3} d^{3} e^{3} - 125 a^{2} b^{4} d^{4} e^{2} + 154 a b^{5} d^{5} e - 57 b^{6} d^{6} + x^{3} \left (80 a^{3} b^{3} e^{6} - 240 a^{2} b^{4} d e^{5} + 240 a b^{5} d^{2} e^{4} - 80 b^{6} d^{3} e^{3}\right ) + x^{2} \left (30 a^{4} b^{2} e^{6} + 120 a^{3} b^{3} d e^{5} - 540 a^{2} b^{4} d^{2} e^{4} + 600 a b^{5} d^{3} e^{3} - 210 b^{6} d^{4} e^{2}\right ) + x \left (8 a^{5} b e^{6} + 20 a^{4} b^{2} d e^{5} + 80 a^{3} b^{3} d^{2} e^{4} - 440 a^{2} b^{4} d^{3} e^{3} + 520 a b^{5} d^{4} e^{2} - 188 b^{6} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} + \frac{x \left (6 a b^{5} e - 5 b^{6} d\right )}{e^{6}} \]
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)**5,x)
[Out]
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GIAC/XCAS [A] time = 0.217456, size = 694, normalized size = 4.48 \[ \frac{1}{2} \,{\left (b^{6} - \frac{12 \,{\left (b^{6} d e - a b^{5} e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-7\right )} - 15 \,{\left (b^{6} d^{2} - 2 \, a b^{5} d e + a^{2} b^{4} e^{2}\right )} e^{\left (-7\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) + \frac{1}{4} \,{\left (\frac{80 \, b^{6} d^{3} e^{29}}{x e + d} - \frac{30 \, b^{6} d^{4} e^{29}}{{\left (x e + d\right )}^{2}} + \frac{8 \, b^{6} d^{5} e^{29}}{{\left (x e + d\right )}^{3}} - \frac{b^{6} d^{6} e^{29}}{{\left (x e + d\right )}^{4}} - \frac{240 \, a b^{5} d^{2} e^{30}}{x e + d} + \frac{120 \, a b^{5} d^{3} e^{30}}{{\left (x e + d\right )}^{2}} - \frac{40 \, a b^{5} d^{4} e^{30}}{{\left (x e + d\right )}^{3}} + \frac{6 \, a b^{5} d^{5} e^{30}}{{\left (x e + d\right )}^{4}} + \frac{240 \, a^{2} b^{4} d e^{31}}{x e + d} - \frac{180 \, a^{2} b^{4} d^{2} e^{31}}{{\left (x e + d\right )}^{2}} + \frac{80 \, a^{2} b^{4} d^{3} e^{31}}{{\left (x e + d\right )}^{3}} - \frac{15 \, a^{2} b^{4} d^{4} e^{31}}{{\left (x e + d\right )}^{4}} - \frac{80 \, a^{3} b^{3} e^{32}}{x e + d} + \frac{120 \, a^{3} b^{3} d e^{32}}{{\left (x e + d\right )}^{2}} - \frac{80 \, a^{3} b^{3} d^{2} e^{32}}{{\left (x e + d\right )}^{3}} + \frac{20 \, a^{3} b^{3} d^{3} e^{32}}{{\left (x e + d\right )}^{4}} - \frac{30 \, a^{4} b^{2} e^{33}}{{\left (x e + d\right )}^{2}} + \frac{40 \, a^{4} b^{2} d e^{33}}{{\left (x e + d\right )}^{3}} - \frac{15 \, a^{4} b^{2} d^{2} e^{33}}{{\left (x e + d\right )}^{4}} - \frac{8 \, a^{5} b e^{34}}{{\left (x e + d\right )}^{3}} + \frac{6 \, a^{5} b d e^{34}}{{\left (x e + d\right )}^{4}} - \frac{a^{6} e^{35}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-36\right )} \]
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)^5,x, algorithm="giac")
[Out]