Optimal. Leaf size=120 \[ \frac{b^3 (a+b x)^7}{840 (d+e x)^7 (b d-a e)^4}+\frac{b^2 (a+b x)^7}{120 (d+e x)^8 (b d-a e)^3}+\frac{b (a+b x)^7}{30 (d+e x)^9 (b d-a e)^2}+\frac{(a+b x)^7}{10 (d+e x)^{10} (b d-a e)} \]
[Out]
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Rubi [A] time = 0.117617, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{b^3 (a+b x)^7}{840 (d+e x)^7 (b d-a e)^4}+\frac{b^2 (a+b x)^7}{120 (d+e x)^8 (b d-a e)^3}+\frac{b (a+b x)^7}{30 (d+e x)^9 (b d-a e)^2}+\frac{(a+b x)^7}{10 (d+e x)^{10} (b d-a e)} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^11,x]
[Out]
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Rubi in Sympy [A] time = 43.1681, size = 100, normalized size = 0.83 \[ \frac{b^{3} \left (a + b x\right )^{7}}{840 \left (d + e x\right )^{7} \left (a e - b d\right )^{4}} - \frac{b^{2} \left (a + b x\right )^{7}}{120 \left (d + e x\right )^{8} \left (a e - b d\right )^{3}} + \frac{b \left (a + b x\right )^{7}}{30 \left (d + e x\right )^{9} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{7}}{10 \left (d + e x\right )^{10} \left (a e - b d\right )} \]
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)**11,x)
[Out]
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Mathematica [B] time = 0.191827, size = 277, normalized size = 2.31 \[ -\frac{84 a^6 e^6+56 a^5 b e^5 (d+10 e x)+35 a^4 b^2 e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+20 a^3 b^3 e^3 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+10 a^2 b^4 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+4 a b^5 e \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+b^6 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )}{840 e^7 (d+e x)^{10}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^11,x]
[Out]
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Maple [B] time = 0.01, size = 357, normalized size = 3. \[ -{\frac{15\,{b}^{2} \left ({e}^{4}{a}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{{e}^{6}{a}^{6}-6\,d{e}^{5}{a}^{5}b+15\,{d}^{2}{e}^{4}{b}^{2}{a}^{4}-20\,{d}^{3}{e}^{3}{a}^{3}{b}^{3}+15\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-6\,{d}^{5}ea{b}^{5}+{d}^{6}{b}^{6}}{10\,{e}^{7} \left ( ex+d \right ) ^{10}}}-{\frac{20\,{b}^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{6\,{b}^{5} \left ( ae-bd \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{5\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{6}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{2\,b \left ({a}^{5}{e}^{5}-5\,{a}^{4}bd{e}^{4}+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-10\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+5\,a{b}^{4}{d}^{4}e-{b}^{5}{d}^{5} \right ) }{3\,{e}^{7} \left ( ex+d \right ) ^{9}}} \]
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)^11,x)
[Out]
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Maxima [A] time = 0.704243, size = 610, normalized size = 5.08 \[ -\frac{210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \,{\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \,{\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \,{\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \,{\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \,{\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \,{\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\right )}} \]
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)^11,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.200938, size = 610, normalized size = 5.08 \[ -\frac{210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \,{\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \,{\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \,{\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \,{\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \,{\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \,{\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\right )}} \]
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)^11,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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)**11,x)
[Out]
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GIAC/XCAS [A] time = 0.209942, size = 475, normalized size = 3.96 \[ -\frac{{\left (210 \, b^{6} x^{6} e^{6} + 252 \, b^{6} d x^{5} e^{5} + 210 \, b^{6} d^{2} x^{4} e^{4} + 120 \, b^{6} d^{3} x^{3} e^{3} + 45 \, b^{6} d^{4} x^{2} e^{2} + 10 \, b^{6} d^{5} x e + b^{6} d^{6} + 1008 \, a b^{5} x^{5} e^{6} + 840 \, a b^{5} d x^{4} e^{5} + 480 \, a b^{5} d^{2} x^{3} e^{4} + 180 \, a b^{5} d^{3} x^{2} e^{3} + 40 \, a b^{5} d^{4} x e^{2} + 4 \, a b^{5} d^{5} e + 2100 \, a^{2} b^{4} x^{4} e^{6} + 1200 \, a^{2} b^{4} d x^{3} e^{5} + 450 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 100 \, a^{2} b^{4} d^{3} x e^{3} + 10 \, a^{2} b^{4} d^{4} e^{2} + 2400 \, a^{3} b^{3} x^{3} e^{6} + 900 \, a^{3} b^{3} d x^{2} e^{5} + 200 \, a^{3} b^{3} d^{2} x e^{4} + 20 \, a^{3} b^{3} d^{3} e^{3} + 1575 \, a^{4} b^{2} x^{2} e^{6} + 350 \, a^{4} b^{2} d x e^{5} + 35 \, a^{4} b^{2} d^{2} e^{4} + 560 \, a^{5} b x e^{6} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{840 \,{\left (x e + d\right )}^{10}} \]
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)^11,x, algorithm="giac")
[Out]