3.1487 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^9} \, dx\)

Optimal. Leaf size=58 \[ \frac{b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac{(a+b x)^7}{8 (d+e x)^8 (b d-a e)} \]

[Out]

(a + b*x)^7/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a + b*x)^7)/(56*(b*d - a*e)^2*(d +
e*x)^7)

_______________________________________________________________________________________

Rubi [A]  time = 0.0549843, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{b (a+b x)^7}{56 (d+e x)^7 (b d-a e)^2}+\frac{(a+b x)^7}{8 (d+e x)^8 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^9,x]

[Out]

(a + b*x)^7/(8*(b*d - a*e)*(d + e*x)^8) + (b*(a + b*x)^7)/(56*(b*d - a*e)^2*(d +
e*x)^7)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 19.9848, size = 46, normalized size = 0.79 \[ \frac{b \left (a + b x\right )^{7}}{56 \left (d + e x\right )^{7} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{7}}{8 \left (d + e x\right )^{8} \left (a e - b d\right )} \]

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)**9,x)

[Out]

b*(a + b*x)**7/(56*(d + e*x)**7*(a*e - b*d)**2) - (a + b*x)**7/(8*(d + e*x)**8*(
a*e - b*d))

_______________________________________________________________________________________

Mathematica [B]  time = 0.184435, size = 277, normalized size = 4.78 \[ -\frac{7 a^6 e^6+6 a^5 b e^5 (d+8 e x)+5 a^4 b^2 e^4 \left (d^2+8 d e x+28 e^2 x^2\right )+4 a^3 b^3 e^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 a^2 b^4 e^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+2 a b^5 e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+b^6 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )}{56 e^7 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^9,x]

[Out]

-(7*a^6*e^6 + 6*a^5*b*e^5*(d + 8*e*x) + 5*a^4*b^2*e^4*(d^2 + 8*d*e*x + 28*e^2*x^
2) + 4*a^3*b^3*e^3*(d^3 + 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3) + 3*a^2*b^4*e^2
*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4*x^4) + 2*a*b^5*e*(d^5
+ 8*d^4*e*x + 28*d^3*e^2*x^2 + 56*d^2*e^3*x^3 + 70*d*e^4*x^4 + 56*e^5*x^5) + b^
6*(d^6 + 8*d^5*e*x + 28*d^4*e^2*x^2 + 56*d^3*e^3*x^3 + 70*d^2*e^4*x^4 + 56*d*e^5
*x^5 + 28*e^6*x^6))/(56*e^7*(d + e*x)^8)

_______________________________________________________________________________________

Maple [B]  time = 0.011, size = 357, normalized size = 6.2 \[ -{\frac{{e}^{6}{a}^{6}-6\,d{e}^{5}{a}^{5}b+15\,{d}^{2}{e}^{4}{b}^{2}{a}^{4}-20\,{d}^{3}{e}^{3}{a}^{3}{b}^{3}+15\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-6\,{d}^{5}ea{b}^{5}+{d}^{6}{b}^{6}}{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{6\,b \left ({a}^{5}{e}^{5}-5\,{a}^{4}bd{e}^{4}+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-10\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+5\,a{b}^{4}{d}^{4}e-{b}^{5}{d}^{5} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-4\,{\frac{{b}^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{5\,{b}^{2} \left ({e}^{4}{a}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{6}}}-2\,{\frac{{b}^{5} \left ( ae-bd \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{15\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{6}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}} \]

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)^9,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.70224, size = 581, normalized size = 10.02 \[ -\frac{28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \,{\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \,{\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \,{\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \,{\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \,{\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]

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)^9,x, algorithm="maxima")

[Out]

-1/56*(28*b^6*e^6*x^6 + b^6*d^6 + 2*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 + 4*a^3*b^3*
d^3*e^3 + 5*a^4*b^2*d^2*e^4 + 6*a^5*b*d*e^5 + 7*a^6*e^6 + 56*(b^6*d*e^5 + 2*a*b^
5*e^6)*x^5 + 70*(b^6*d^2*e^4 + 2*a*b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 + 56*(b^6*d^3*
e^3 + 2*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + 4*a^3*b^3*e^6)*x^3 + 28*(b^6*d^4*e^2 +
2*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 + 4*a^3*b^3*d*e^5 + 5*a^4*b^2*e^6)*x^2 + 8*
(b^6*d^5*e + 2*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3*e^3 + 4*a^3*b^3*d^2*e^4 + 5*a^4*b^2
*d*e^5 + 6*a^5*b*e^6)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^1
2*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e
^7)

_______________________________________________________________________________________

Fricas [A]  time = 0.20051, size = 581, normalized size = 10.02 \[ -\frac{28 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 2 \, a b^{5} d^{5} e + 3 \, a^{2} b^{4} d^{4} e^{2} + 4 \, a^{3} b^{3} d^{3} e^{3} + 5 \, a^{4} b^{2} d^{2} e^{4} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6} + 56 \,{\left (b^{6} d e^{5} + 2 \, a b^{5} e^{6}\right )} x^{5} + 70 \,{\left (b^{6} d^{2} e^{4} + 2 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 56 \,{\left (b^{6} d^{3} e^{3} + 2 \, a b^{5} d^{2} e^{4} + 3 \, a^{2} b^{4} d e^{5} + 4 \, a^{3} b^{3} e^{6}\right )} x^{3} + 28 \,{\left (b^{6} d^{4} e^{2} + 2 \, a b^{5} d^{3} e^{3} + 3 \, a^{2} b^{4} d^{2} e^{4} + 4 \, a^{3} b^{3} d e^{5} + 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 8 \,{\left (b^{6} d^{5} e + 2 \, a b^{5} d^{4} e^{2} + 3 \, a^{2} b^{4} d^{3} e^{3} + 4 \, a^{3} b^{3} d^{2} e^{4} + 5 \, a^{4} b^{2} d e^{5} + 6 \, a^{5} b e^{6}\right )} x}{56 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]

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)^9,x, algorithm="fricas")

[Out]

-1/56*(28*b^6*e^6*x^6 + b^6*d^6 + 2*a*b^5*d^5*e + 3*a^2*b^4*d^4*e^2 + 4*a^3*b^3*
d^3*e^3 + 5*a^4*b^2*d^2*e^4 + 6*a^5*b*d*e^5 + 7*a^6*e^6 + 56*(b^6*d*e^5 + 2*a*b^
5*e^6)*x^5 + 70*(b^6*d^2*e^4 + 2*a*b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 + 56*(b^6*d^3*
e^3 + 2*a*b^5*d^2*e^4 + 3*a^2*b^4*d*e^5 + 4*a^3*b^3*e^6)*x^3 + 28*(b^6*d^4*e^2 +
2*a*b^5*d^3*e^3 + 3*a^2*b^4*d^2*e^4 + 4*a^3*b^3*d*e^5 + 5*a^4*b^2*e^6)*x^2 + 8*
(b^6*d^5*e + 2*a*b^5*d^4*e^2 + 3*a^2*b^4*d^3*e^3 + 4*a^3*b^3*d^2*e^4 + 5*a^4*b^2
*d*e^5 + 6*a^5*b*e^6)*x)/(e^15*x^8 + 8*d*e^14*x^7 + 28*d^2*e^13*x^6 + 56*d^3*e^1
2*x^5 + 70*d^4*e^11*x^4 + 56*d^5*e^10*x^3 + 28*d^6*e^9*x^2 + 8*d^7*e^8*x + d^8*e
^7)

_______________________________________________________________________________________

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)**9,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.21156, size = 475, normalized size = 8.19 \[ -\frac{{\left (28 \, b^{6} x^{6} e^{6} + 56 \, b^{6} d x^{5} e^{5} + 70 \, b^{6} d^{2} x^{4} e^{4} + 56 \, b^{6} d^{3} x^{3} e^{3} + 28 \, b^{6} d^{4} x^{2} e^{2} + 8 \, b^{6} d^{5} x e + b^{6} d^{6} + 112 \, a b^{5} x^{5} e^{6} + 140 \, a b^{5} d x^{4} e^{5} + 112 \, a b^{5} d^{2} x^{3} e^{4} + 56 \, a b^{5} d^{3} x^{2} e^{3} + 16 \, a b^{5} d^{4} x e^{2} + 2 \, a b^{5} d^{5} e + 210 \, a^{2} b^{4} x^{4} e^{6} + 168 \, a^{2} b^{4} d x^{3} e^{5} + 84 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 24 \, a^{2} b^{4} d^{3} x e^{3} + 3 \, a^{2} b^{4} d^{4} e^{2} + 224 \, a^{3} b^{3} x^{3} e^{6} + 112 \, a^{3} b^{3} d x^{2} e^{5} + 32 \, a^{3} b^{3} d^{2} x e^{4} + 4 \, a^{3} b^{3} d^{3} e^{3} + 140 \, a^{4} b^{2} x^{2} e^{6} + 40 \, a^{4} b^{2} d x e^{5} + 5 \, a^{4} b^{2} d^{2} e^{4} + 48 \, a^{5} b x e^{6} + 6 \, a^{5} b d e^{5} + 7 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{56 \,{\left (x e + d\right )}^{8}} \]

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)^9,x, algorithm="giac")

[Out]

-1/56*(28*b^6*x^6*e^6 + 56*b^6*d*x^5*e^5 + 70*b^6*d^2*x^4*e^4 + 56*b^6*d^3*x^3*e
^3 + 28*b^6*d^4*x^2*e^2 + 8*b^6*d^5*x*e + b^6*d^6 + 112*a*b^5*x^5*e^6 + 140*a*b^
5*d*x^4*e^5 + 112*a*b^5*d^2*x^3*e^4 + 56*a*b^5*d^3*x^2*e^3 + 16*a*b^5*d^4*x*e^2
+ 2*a*b^5*d^5*e + 210*a^2*b^4*x^4*e^6 + 168*a^2*b^4*d*x^3*e^5 + 84*a^2*b^4*d^2*x
^2*e^4 + 24*a^2*b^4*d^3*x*e^3 + 3*a^2*b^4*d^4*e^2 + 224*a^3*b^3*x^3*e^6 + 112*a^
3*b^3*d*x^2*e^5 + 32*a^3*b^3*d^2*x*e^4 + 4*a^3*b^3*d^3*e^3 + 140*a^4*b^2*x^2*e^6
+ 40*a^4*b^2*d*x*e^5 + 5*a^4*b^2*d^2*e^4 + 48*a^5*b*x*e^6 + 6*a^5*b*d*e^5 + 7*a
^6*e^6)*e^(-7)/(x*e + d)^8