3.1489 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx\)

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]

(a + b*x)^7/(10*(b*d - a*e)*(d + e*x)^10) + (b*(a + b*x)^7)/(30*(b*d - a*e)^2*(d
+ e*x)^9) + (b^2*(a + b*x)^7)/(120*(b*d - a*e)^3*(d + e*x)^8) + (b^3*(a + b*x)^
7)/(840*(b*d - a*e)^4*(d + e*x)^7)

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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]

(a + b*x)^7/(10*(b*d - a*e)*(d + e*x)^10) + (b*(a + b*x)^7)/(30*(b*d - a*e)^2*(d
+ e*x)^9) + (b^2*(a + b*x)^7)/(120*(b*d - a*e)^3*(d + e*x)^8) + (b^3*(a + b*x)^
7)/(840*(b*d - a*e)^4*(d + e*x)^7)

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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]

b**3*(a + b*x)**7/(840*(d + e*x)**7*(a*e - b*d)**4) - b**2*(a + b*x)**7/(120*(d
+ e*x)**8*(a*e - b*d)**3) + b*(a + b*x)**7/(30*(d + e*x)**9*(a*e - b*d)**2) - (a
+ b*x)**7/(10*(d + e*x)**10*(a*e - b*d))

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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]

-(84*a^6*e^6 + 56*a^5*b*e^5*(d + 10*e*x) + 35*a^4*b^2*e^4*(d^2 + 10*d*e*x + 45*e
^2*x^2) + 20*a^3*b^3*e^3*(d^3 + 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 10*a^
2*b^4*e^2*(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) + 4*
a*b^5*e*(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 + 2
52*e^5*x^5) + b^6*(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))/(840*e^7*(d + e*x)^10)

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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]

-15/8*b^2*(a^4*e^4-4*a^3*b*d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/e^7/(e
*x+d)^8-1/10*(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)/e^7/(e*x+d)^10-20/7*b^3*(a^3*e^3-3*a^2*b*d*e
^2+3*a*b^2*d^2*e-b^3*d^3)/e^7/(e*x+d)^7-6/5*b^5*(a*e-b*d)/e^7/(e*x+d)^5-5/2*b^4*
(a^2*e^2-2*a*b*d*e+b^2*d^2)/e^7/(e*x+d)^6-1/4*b^6/e^7/(e*x+d)^4-2/3*b*(a^5*e^5-5
*a^4*b*d*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)/e^7/(e
*x+d)^9

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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]

-1/840*(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*(b^6*d*e^5
+ 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a^2*b^4*e^6)*x^4 + 12
0*(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)*x^3 + 45*(
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)*x^2 + 10*(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)*x)/(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)

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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]

-1/840*(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*(b^6*d*e^5
+ 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a^2*b^4*e^6)*x^4 + 12
0*(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)*x^3 + 45*(
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)*x^2 + 10*(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)*x)/(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)

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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]

Timed 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]

-1/840*(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 + 8
40*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 + 157
5*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)*e^(-7)/(x*e + d)^10