3.1490 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=170 \[ \frac{b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac{15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac{5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac{5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac{3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac{(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac{b^6}{5 e^7 (d+e x)^5} \]

[Out]

-(b*d - a*e)^6/(11*e^7*(d + e*x)^11) + (3*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^10)
- (5*b^2*(b*d - a*e)^4)/(3*e^7*(d + e*x)^9) + (5*b^3*(b*d - a*e)^3)/(2*e^7*(d +
e*x)^8) - (15*b^4*(b*d - a*e)^2)/(7*e^7*(d + e*x)^7) + (b^5*(b*d - a*e))/(e^7*(d
+ e*x)^6) - b^6/(5*e^7*(d + e*x)^5)

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Rubi [A]  time = 0.377384, antiderivative size = 170, 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 \[ \frac{b^5 (b d-a e)}{e^7 (d+e x)^6}-\frac{15 b^4 (b d-a e)^2}{7 e^7 (d+e x)^7}+\frac{5 b^3 (b d-a e)^3}{2 e^7 (d+e x)^8}-\frac{5 b^2 (b d-a e)^4}{3 e^7 (d+e x)^9}+\frac{3 b (b d-a e)^5}{5 e^7 (d+e x)^{10}}-\frac{(b d-a e)^6}{11 e^7 (d+e x)^{11}}-\frac{b^6}{5 e^7 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^12,x]

[Out]

-(b*d - a*e)^6/(11*e^7*(d + e*x)^11) + (3*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^10)
- (5*b^2*(b*d - a*e)^4)/(3*e^7*(d + e*x)^9) + (5*b^3*(b*d - a*e)^3)/(2*e^7*(d +
e*x)^8) - (15*b^4*(b*d - a*e)^2)/(7*e^7*(d + e*x)^7) + (b^5*(b*d - a*e))/(e^7*(d
+ e*x)^6) - b^6/(5*e^7*(d + e*x)^5)

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Rubi in Sympy [A]  time = 94.7138, size = 156, normalized size = 0.92 \[ - \frac{b^{6}}{5 e^{7} \left (d + e x\right )^{5}} - \frac{b^{5} \left (a e - b d\right )}{e^{7} \left (d + e x\right )^{6}} - \frac{15 b^{4} \left (a e - b d\right )^{2}}{7 e^{7} \left (d + e x\right )^{7}} - \frac{5 b^{3} \left (a e - b d\right )^{3}}{2 e^{7} \left (d + e x\right )^{8}} - \frac{5 b^{2} \left (a e - b d\right )^{4}}{3 e^{7} \left (d + e x\right )^{9}} - \frac{3 b \left (a e - b d\right )^{5}}{5 e^{7} \left (d + e x\right )^{10}} - \frac{\left (a e - b d\right )^{6}}{11 e^{7} \left (d + e x\right )^{11}} \]

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)**12,x)

[Out]

-b**6/(5*e**7*(d + e*x)**5) - b**5*(a*e - b*d)/(e**7*(d + e*x)**6) - 15*b**4*(a*
e - b*d)**2/(7*e**7*(d + e*x)**7) - 5*b**3*(a*e - b*d)**3/(2*e**7*(d + e*x)**8)
- 5*b**2*(a*e - b*d)**4/(3*e**7*(d + e*x)**9) - 3*b*(a*e - b*d)**5/(5*e**7*(d +
e*x)**10) - (a*e - b*d)**6/(11*e**7*(d + e*x)**11)

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Mathematica [A]  time = 0.194537, size = 277, normalized size = 1.63 \[ -\frac{210 a^6 e^6+126 a^5 b e^5 (d+11 e x)+70 a^4 b^2 e^4 \left (d^2+11 d e x+55 e^2 x^2\right )+35 a^3 b^3 e^3 \left (d^3+11 d^2 e x+55 d e^2 x^2+165 e^3 x^3\right )+15 a^2 b^4 e^2 \left (d^4+11 d^3 e x+55 d^2 e^2 x^2+165 d e^3 x^3+330 e^4 x^4\right )+5 a b^5 e \left (d^5+11 d^4 e x+55 d^3 e^2 x^2+165 d^2 e^3 x^3+330 d e^4 x^4+462 e^5 x^5\right )+b^6 \left (d^6+11 d^5 e x+55 d^4 e^2 x^2+165 d^3 e^3 x^3+330 d^2 e^4 x^4+462 d e^5 x^5+462 e^6 x^6\right )}{2310 e^7 (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^12,x]

[Out]

-(210*a^6*e^6 + 126*a^5*b*e^5*(d + 11*e*x) + 70*a^4*b^2*e^4*(d^2 + 11*d*e*x + 55
*e^2*x^2) + 35*a^3*b^3*e^3*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 15*
a^2*b^4*e^2*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4) +
5*a*b^5*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 +
462*e^5*x^5) + b^6*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d
^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6))/(2310*e^7*(d + e*x)^11)

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Maple [B]  time = 0.011, size = 357, normalized size = 2.1 \[ -{\frac{5\,{b}^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{3\,b \left ({a}^{5}{e}^{5}-5\,{a}^{4}bd{e}^{4}+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-10\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+5\,a{b}^{4}{d}^{4}e-{b}^{5}{d}^{5} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{10}}}-{\frac{15\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{{b}^{6}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{5} \left ( ae-bd \right ) }{{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{e}^{6}{a}^{6}-6\,d{e}^{5}{a}^{5}b+15\,{d}^{2}{e}^{4}{b}^{2}{a}^{4}-20\,{d}^{3}{e}^{3}{a}^{3}{b}^{3}+15\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-6\,{d}^{5}ea{b}^{5}+{d}^{6}{b}^{6}}{11\,{e}^{7} \left ( ex+d \right ) ^{11}}}-{\frac{5\,{b}^{2} \left ({e}^{4}{a}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{3\,{e}^{7} \left ( ex+d \right ) ^{9}}} \]

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)^12,x)

[Out]

-5/2*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)^8-3/5*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)^10-15/7*b^4*(a^2*e^2-2*a*b*d*e+b^2*d^2)/e^7/(e*x+d)^7-1/5*b^6/e^7/(e*
x+d)^5-b^5*(a*e-b*d)/e^7/(e*x+d)^6-1/11*(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)^11-5/
3*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
)^9

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Maxima [A]  time = 0.703081, size = 625, normalized size = 3.68 \[ -\frac{462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \,{\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \,{\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \,{\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \,{\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \,{\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \,{\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \]

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)^12,x, algorithm="maxima")

[Out]

-1/2310*(462*b^6*e^6*x^6 + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 35*a^3
*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 210*a^6*e^6 + 462*(b^6*d*e
^5 + 5*a*b^5*e^6)*x^5 + 330*(b^6*d^2*e^4 + 5*a*b^5*d*e^5 + 15*a^2*b^4*e^6)*x^4 +
165*(b^6*d^3*e^3 + 5*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 5
5*(b^6*d^4*e^2 + 5*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2*e^4 + 35*a^3*b^3*d*e^5 + 70*a^
4*b^2*e^6)*x^2 + 11*(b^6*d^5*e + 5*a*b^5*d^4*e^2 + 15*a^2*b^4*d^3*e^3 + 35*a^3*b
^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 126*a^5*b*e^6)*x)/(e^18*x^11 + 11*d*e^17*x^10 +
55*d^2*e^16*x^9 + 165*d^3*e^15*x^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d
^6*e^12*x^5 + 330*d^7*e^11*x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8
*x + d^11*e^7)

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Fricas [A]  time = 0.222984, size = 625, normalized size = 3.68 \[ -\frac{462 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 5 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} + 35 \, a^{3} b^{3} d^{3} e^{3} + 70 \, a^{4} b^{2} d^{2} e^{4} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6} + 462 \,{\left (b^{6} d e^{5} + 5 \, a b^{5} e^{6}\right )} x^{5} + 330 \,{\left (b^{6} d^{2} e^{4} + 5 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 165 \,{\left (b^{6} d^{3} e^{3} + 5 \, a b^{5} d^{2} e^{4} + 15 \, a^{2} b^{4} d e^{5} + 35 \, a^{3} b^{3} e^{6}\right )} x^{3} + 55 \,{\left (b^{6} d^{4} e^{2} + 5 \, a b^{5} d^{3} e^{3} + 15 \, a^{2} b^{4} d^{2} e^{4} + 35 \, a^{3} b^{3} d e^{5} + 70 \, a^{4} b^{2} e^{6}\right )} x^{2} + 11 \,{\left (b^{6} d^{5} e + 5 \, a b^{5} d^{4} e^{2} + 15 \, a^{2} b^{4} d^{3} e^{3} + 35 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 126 \, a^{5} b e^{6}\right )} x}{2310 \,{\left (e^{18} x^{11} + 11 \, d e^{17} x^{10} + 55 \, d^{2} e^{16} x^{9} + 165 \, d^{3} e^{15} x^{8} + 330 \, d^{4} e^{14} x^{7} + 462 \, d^{5} e^{13} x^{6} + 462 \, d^{6} e^{12} x^{5} + 330 \, d^{7} e^{11} x^{4} + 165 \, d^{8} e^{10} x^{3} + 55 \, d^{9} e^{9} x^{2} + 11 \, d^{10} e^{8} x + d^{11} e^{7}\right )}} \]

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)^12,x, algorithm="fricas")

[Out]

-1/2310*(462*b^6*e^6*x^6 + b^6*d^6 + 5*a*b^5*d^5*e + 15*a^2*b^4*d^4*e^2 + 35*a^3
*b^3*d^3*e^3 + 70*a^4*b^2*d^2*e^4 + 126*a^5*b*d*e^5 + 210*a^6*e^6 + 462*(b^6*d*e
^5 + 5*a*b^5*e^6)*x^5 + 330*(b^6*d^2*e^4 + 5*a*b^5*d*e^5 + 15*a^2*b^4*e^6)*x^4 +
165*(b^6*d^3*e^3 + 5*a*b^5*d^2*e^4 + 15*a^2*b^4*d*e^5 + 35*a^3*b^3*e^6)*x^3 + 5
5*(b^6*d^4*e^2 + 5*a*b^5*d^3*e^3 + 15*a^2*b^4*d^2*e^4 + 35*a^3*b^3*d*e^5 + 70*a^
4*b^2*e^6)*x^2 + 11*(b^6*d^5*e + 5*a*b^5*d^4*e^2 + 15*a^2*b^4*d^3*e^3 + 35*a^3*b
^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 126*a^5*b*e^6)*x)/(e^18*x^11 + 11*d*e^17*x^10 +
55*d^2*e^16*x^9 + 165*d^3*e^15*x^8 + 330*d^4*e^14*x^7 + 462*d^5*e^13*x^6 + 462*d
^6*e^12*x^5 + 330*d^7*e^11*x^4 + 165*d^8*e^10*x^3 + 55*d^9*e^9*x^2 + 11*d^10*e^8
*x + d^11*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)**12,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.211115, size = 475, normalized size = 2.79 \[ -\frac{{\left (462 \, b^{6} x^{6} e^{6} + 462 \, b^{6} d x^{5} e^{5} + 330 \, b^{6} d^{2} x^{4} e^{4} + 165 \, b^{6} d^{3} x^{3} e^{3} + 55 \, b^{6} d^{4} x^{2} e^{2} + 11 \, b^{6} d^{5} x e + b^{6} d^{6} + 2310 \, a b^{5} x^{5} e^{6} + 1650 \, a b^{5} d x^{4} e^{5} + 825 \, a b^{5} d^{2} x^{3} e^{4} + 275 \, a b^{5} d^{3} x^{2} e^{3} + 55 \, a b^{5} d^{4} x e^{2} + 5 \, a b^{5} d^{5} e + 4950 \, a^{2} b^{4} x^{4} e^{6} + 2475 \, a^{2} b^{4} d x^{3} e^{5} + 825 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 165 \, a^{2} b^{4} d^{3} x e^{3} + 15 \, a^{2} b^{4} d^{4} e^{2} + 5775 \, a^{3} b^{3} x^{3} e^{6} + 1925 \, a^{3} b^{3} d x^{2} e^{5} + 385 \, a^{3} b^{3} d^{2} x e^{4} + 35 \, a^{3} b^{3} d^{3} e^{3} + 3850 \, a^{4} b^{2} x^{2} e^{6} + 770 \, a^{4} b^{2} d x e^{5} + 70 \, a^{4} b^{2} d^{2} e^{4} + 1386 \, a^{5} b x e^{6} + 126 \, a^{5} b d e^{5} + 210 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{2310 \,{\left (x e + d\right )}^{11}} \]

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)^12,x, algorithm="giac")

[Out]

-1/2310*(462*b^6*x^6*e^6 + 462*b^6*d*x^5*e^5 + 330*b^6*d^2*x^4*e^4 + 165*b^6*d^3
*x^3*e^3 + 55*b^6*d^4*x^2*e^2 + 11*b^6*d^5*x*e + b^6*d^6 + 2310*a*b^5*x^5*e^6 +
1650*a*b^5*d*x^4*e^5 + 825*a*b^5*d^2*x^3*e^4 + 275*a*b^5*d^3*x^2*e^3 + 55*a*b^5*
d^4*x*e^2 + 5*a*b^5*d^5*e + 4950*a^2*b^4*x^4*e^6 + 2475*a^2*b^4*d*x^3*e^5 + 825*
a^2*b^4*d^2*x^2*e^4 + 165*a^2*b^4*d^3*x*e^3 + 15*a^2*b^4*d^4*e^2 + 5775*a^3*b^3*
x^3*e^6 + 1925*a^3*b^3*d*x^2*e^5 + 385*a^3*b^3*d^2*x*e^4 + 35*a^3*b^3*d^3*e^3 +
3850*a^4*b^2*x^2*e^6 + 770*a^4*b^2*d*x*e^5 + 70*a^4*b^2*d^2*e^4 + 1386*a^5*b*x*e
^6 + 126*a^5*b*d*e^5 + 210*a^6*e^6)*e^(-7)/(x*e + d)^11