Optimal. Leaf size=155 \[ -\frac{6 b^5 (b d-a e) \log (d+e x)}{e^7}-\frac{15 b^4 (b d-a e)^2}{e^7 (d+e x)}+\frac{10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac{5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac{3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac{(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac{b^6 x}{e^6} \]
[Out]
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Rubi [A] time = 0.379633, 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 \[ -\frac{6 b^5 (b d-a e) \log (d+e x)}{e^7}-\frac{15 b^4 (b d-a e)^2}{e^7 (d+e x)}+\frac{10 b^3 (b d-a e)^3}{e^7 (d+e x)^2}-\frac{5 b^2 (b d-a e)^4}{e^7 (d+e x)^3}+\frac{3 b (b d-a e)^5}{2 e^7 (d+e x)^4}-\frac{(b d-a e)^6}{5 e^7 (d+e x)^5}+\frac{b^6 x}{e^6} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^6,x]
[Out]
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Rubi in Sympy [A] time = 88.0701, size = 143, normalized size = 0.92 \[ \frac{b^{6} x}{e^{6}} + \frac{6 b^{5} \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{15 b^{4} \left (a e - b d\right )^{2}}{e^{7} \left (d + e x\right )} - \frac{10 b^{3} \left (a e - b d\right )^{3}}{e^{7} \left (d + e x\right )^{2}} - \frac{5 b^{2} \left (a e - b d\right )^{4}}{e^{7} \left (d + e x\right )^{3}} - \frac{3 b \left (a e - b d\right )^{5}}{2 e^{7} \left (d + e x\right )^{4}} - \frac{\left (a e - b d\right )^{6}}{5 e^{7} \left (d + e x\right )^{5}} \]
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)**6,x)
[Out]
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Mathematica [A] time = 0.234189, size = 297, normalized size = 1.92 \[ -\frac{2 a^6 e^6+3 a^5 b e^5 (d+5 e x)+5 a^4 b^2 e^4 \left (d^2+5 d e x+10 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+30 a^2 b^4 e^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )-a b^5 d e \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )+60 b^5 (d+e x)^5 (b d-a e) \log (d+e x)+b^6 \left (87 d^6+375 d^5 e x+600 d^4 e^2 x^2+400 d^3 e^3 x^3+50 d^2 e^4 x^4-50 d e^5 x^5-10 e^6 x^6\right )}{10 e^7 (d+e x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^6,x]
[Out]
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Maple [B] time = 0.018, size = 508, normalized size = 3.3 \[ -15\,{\frac{{b}^{6}{d}^{2}}{{e}^{7} \left ( ex+d \right ) }}-{\frac{{a}^{6}}{5\,e \left ( ex+d \right ) ^{5}}}+{\frac{15\,d{b}^{2}{a}^{4}}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}-15\,{\frac{{d}^{2}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{4}}}+15\,{\frac{{d}^{3}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{6\,{d}^{5}a{b}^{5}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}+{\frac{6\,d{a}^{5}b}{5\,{e}^{2} \left ( ex+d \right ) ^{5}}}-3\,{\frac{{d}^{2}{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) ^{5}}}+30\,{\frac{d{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-30\,{\frac{{d}^{2}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+30\,{\frac{da{b}^{5}}{{e}^{6} \left ( ex+d \right ) }}+20\,{\frac{d{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{3}}}-30\,{\frac{{b}^{4}{d}^{2}{a}^{2}}{{e}^{5} \left ( ex+d \right ) ^{3}}}+20\,{\frac{{d}^{3}a{b}^{5}}{{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{15\,{d}^{4}a{b}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}+4\,{\frac{{d}^{3}{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{5}}}-3\,{\frac{{d}^{4}{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) ^{5}}}+{\frac{3\,{d}^{5}{b}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{4}}}-5\,{\frac{{d}^{4}{b}^{6}}{{e}^{7} \left ( ex+d \right ) ^{3}}}+6\,{\frac{{b}^{5}\ln \left ( ex+d \right ) a}{{e}^{6}}}-6\,{\frac{{b}^{6}\ln \left ( ex+d \right ) d}{{e}^{7}}}-{\frac{3\,{a}^{5}b}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{{d}^{6}{b}^{6}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-5\,{\frac{{b}^{2}{a}^{4}}{{e}^{3} \left ( ex+d \right ) ^{3}}}-10\,{\frac{{a}^{3}{b}^{3}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+10\,{\frac{{d}^{3}{b}^{6}}{{e}^{7} \left ( ex+d \right ) ^{2}}}-15\,{\frac{{a}^{2}{b}^{4}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{{b}^{6}x}{{e}^{6}}} \]
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)^6,x)
[Out]
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Maxima [A] time = 0.708666, size = 536, normalized size = 3.46 \[ \frac{b^{6} x}{e^{6}} - \frac{87 \, b^{6} d^{6} - 137 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + 2 \, a^{6} e^{6} + 150 \,{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \,{\left (5 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 50 \,{\left (13 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 5 \,{\left (77 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} - \frac{6 \,{\left (b^{6} d - a b^{5} e\right )} \log \left (e x + d\right )}{e^{7}} \]
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)^6,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.206431, size = 732, normalized size = 4.72 \[ \frac{10 \, b^{6} e^{6} x^{6} + 50 \, b^{6} d e^{5} x^{5} - 87 \, b^{6} d^{6} + 137 \, a b^{5} d^{5} e - 30 \, a^{2} b^{4} d^{4} e^{2} - 10 \, a^{3} b^{3} d^{3} e^{3} - 5 \, a^{4} b^{2} d^{2} e^{4} - 3 \, a^{5} b d e^{5} - 2 \, a^{6} e^{6} - 50 \,{\left (b^{6} d^{2} e^{4} - 6 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} - 100 \,{\left (4 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} - 50 \,{\left (12 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} - 5 \,{\left (75 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x - 60 \,{\left (b^{6} d^{6} - a b^{5} d^{5} e +{\left (b^{6} d e^{5} - a b^{5} e^{6}\right )} x^{5} + 5 \,{\left (b^{6} d^{2} e^{4} - a b^{5} d e^{5}\right )} x^{4} + 10 \,{\left (b^{6} d^{3} e^{3} - a b^{5} d^{2} e^{4}\right )} x^{3} + 10 \,{\left (b^{6} d^{4} e^{2} - a b^{5} d^{3} e^{3}\right )} x^{2} + 5 \,{\left (b^{6} d^{5} e - a b^{5} d^{4} e^{2}\right )} x\right )} \log \left (e x + d\right )}{10 \,{\left (e^{12} x^{5} + 5 \, d e^{11} x^{4} + 10 \, d^{2} e^{10} x^{3} + 10 \, d^{3} e^{9} x^{2} + 5 \, d^{4} e^{8} x + d^{5} e^{7}\right )}} \]
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)^6,x, algorithm="fricas")
[Out]
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Sympy [A] time = 90.9358, size = 420, normalized size = 2.71 \[ \frac{b^{6} x}{e^{6}} + \frac{6 b^{5} \left (a e - b d\right ) \log{\left (d + e x \right )}}{e^{7}} - \frac{2 a^{6} e^{6} + 3 a^{5} b d e^{5} + 5 a^{4} b^{2} d^{2} e^{4} + 10 a^{3} b^{3} d^{3} e^{3} + 30 a^{2} b^{4} d^{4} e^{2} - 137 a b^{5} d^{5} e + 87 b^{6} d^{6} + x^{4} \left (150 a^{2} b^{4} e^{6} - 300 a b^{5} d e^{5} + 150 b^{6} d^{2} e^{4}\right ) + x^{3} \left (100 a^{3} b^{3} e^{6} + 300 a^{2} b^{4} d e^{5} - 900 a b^{5} d^{2} e^{4} + 500 b^{6} d^{3} e^{3}\right ) + x^{2} \left (50 a^{4} b^{2} e^{6} + 100 a^{3} b^{3} d e^{5} + 300 a^{2} b^{4} d^{2} e^{4} - 1100 a b^{5} d^{3} e^{3} + 650 b^{6} d^{4} e^{2}\right ) + x \left (15 a^{5} b e^{6} + 25 a^{4} b^{2} d e^{5} + 50 a^{3} b^{3} d^{2} e^{4} + 150 a^{2} b^{4} d^{3} e^{3} - 625 a b^{5} d^{4} e^{2} + 385 b^{6} d^{5} e\right )}{10 d^{5} e^{7} + 50 d^{4} e^{8} x + 100 d^{3} e^{9} x^{2} + 100 d^{2} e^{10} x^{3} + 50 d e^{11} x^{4} + 10 e^{12} x^{5}} \]
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)**6,x)
[Out]
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GIAC/XCAS [A] time = 0.213389, size = 447, normalized size = 2.88 \[ b^{6} x e^{\left (-6\right )} - 6 \,{\left (b^{6} d - a b^{5} e\right )} e^{\left (-7\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (87 \, b^{6} d^{6} - 137 \, a b^{5} d^{5} e + 30 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 3 \, a^{5} b d e^{5} + 2 \, a^{6} e^{6} + 150 \,{\left (b^{6} d^{2} e^{4} - 2 \, a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 100 \,{\left (5 \, b^{6} d^{3} e^{3} - 9 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 50 \,{\left (13 \, b^{6} d^{4} e^{2} - 22 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 2 \, a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 5 \,{\left (77 \, b^{6} d^{5} e - 125 \, a b^{5} d^{4} e^{2} + 30 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 3 \, a^{5} b e^{6}\right )} x\right )} e^{\left (-7\right )}}{10 \,{\left (x e + d\right )}^{5}} \]
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)^6,x, algorithm="giac")
[Out]