3.1688 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=135 \[ \frac{b (a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{210 e (d+e x)^5 (b d-a e)^3}+\frac{(a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{42 e (d+e x)^6 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{7 e (d+e x)^7 (b d-a e)} \]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(7*e*(b*d - a*e)*(d + e*x)^7) + ((5*b*B*d + 2*A*b*e -
7*a*B*e)*(a + b*x)^5)/(42*e*(b*d - a*e)^2*(d + e*x)^6) + (b*(5*b*B*d + 2*A*b*e
- 7*a*B*e)*(a + b*x)^5)/(210*e*(b*d - a*e)^3*(d + e*x)^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.174011, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{b (a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{210 e (d+e x)^5 (b d-a e)^3}+\frac{(a+b x)^5 (-7 a B e+2 A b e+5 b B d)}{42 e (d+e x)^6 (b d-a e)^2}-\frac{(a+b x)^5 (B d-A e)}{7 e (d+e x)^7 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^8,x]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(7*e*(b*d - a*e)*(d + e*x)^7) + ((5*b*B*d + 2*A*b*e -
7*a*B*e)*(a + b*x)^5)/(42*e*(b*d - a*e)^2*(d + e*x)^6) + (b*(5*b*B*d + 2*A*b*e
- 7*a*B*e)*(a + b*x)^5)/(210*e*(b*d - a*e)^3*(d + e*x)^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 51.4073, size = 122, normalized size = 0.9 \[ - \frac{b \left (a + b x\right )^{5} \left (2 A b e - 7 B a e + 5 B b d\right )}{210 e \left (d + e x\right )^{5} \left (a e - b d\right )^{3}} + \frac{\left (a + b x\right )^{5} \left (2 A b e - 7 B a e + 5 B b d\right )}{42 e \left (d + e x\right )^{6} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{5} \left (A e - B d\right )}{7 e \left (d + e x\right )^{7} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**8,x)

[Out]

-b*(a + b*x)**5*(2*A*b*e - 7*B*a*e + 5*B*b*d)/(210*e*(d + e*x)**5*(a*e - b*d)**3
) + (a + b*x)**5*(2*A*b*e - 7*B*a*e + 5*B*b*d)/(42*e*(d + e*x)**6*(a*e - b*d)**2
) - (a + b*x)**5*(A*e - B*d)/(7*e*(d + e*x)**7*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [B]  time = 0.299479, size = 323, normalized size = 2.39 \[ -\frac{5 a^4 e^4 (6 A e+B (d+7 e x))+4 a^3 b e^3 \left (5 A e (d+7 e x)+2 B \left (d^2+7 d e x+21 e^2 x^2\right )\right )+3 a^2 b^2 e^2 \left (4 A e \left (d^2+7 d e x+21 e^2 x^2\right )+3 B \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 a b^3 e \left (3 A e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+4 B \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )\right )+b^4 \left (2 A e \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+5 B \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )\right )}{210 e^6 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^8,x]

[Out]

-(5*a^4*e^4*(6*A*e + B*(d + 7*e*x)) + 4*a^3*b*e^3*(5*A*e*(d + 7*e*x) + 2*B*(d^2
+ 7*d*e*x + 21*e^2*x^2)) + 3*a^2*b^2*e^2*(4*A*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3
*B*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3)) + 2*a*b^3*e*(3*A*e*(d^3 + 7*d^
2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*B*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*
d*e^3*x^3 + 35*e^4*x^4)) + b^4*(2*A*e*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e
^3*x^3 + 35*e^4*x^4) + 5*B*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 +
35*d*e^4*x^4 + 21*e^5*x^5)))/(210*e^6*(d + e*x)^7)

_______________________________________________________________________________________

Maple [B]  time = 0.011, size = 430, normalized size = 3.2 \[ -{\frac{2\,b \left ( 3\,A{a}^{2}b{e}^{3}-6\,Aa{b}^{2}d{e}^{2}+3\,A{b}^{3}{d}^{2}e+2\,B{e}^{3}{a}^{3}-9\,B{a}^{2}bd{e}^{2}+12\,Ba{b}^{2}{d}^{2}e-5\,B{b}^{3}{d}^{3} \right ) }{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{{b}^{2} \left ( 2\,Aab{e}^{2}-2\,Ad{b}^{2}e+3\,{a}^{2}B{e}^{2}-8\,Bdabe+5\,{b}^{2}B{d}^{2} \right ) }{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{3} \left ( Abe+4\,aBe-5\,Bbd \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{4}B}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{A{a}^{4}{e}^{5}-4\,Ad{a}^{3}b{e}^{4}+6\,A{d}^{2}{a}^{2}{b}^{2}{e}^{3}-4\,A{d}^{3}a{b}^{3}{e}^{2}+A{d}^{4}{b}^{4}e-B{a}^{4}d{e}^{4}+4\,B{d}^{2}{a}^{3}b{e}^{3}-6\,B{d}^{3}{a}^{2}{b}^{2}{e}^{2}+4\,B{d}^{4}a{b}^{3}e-{b}^{4}B{d}^{5}}{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{4\,A{a}^{3}b{e}^{4}-12\,Ad{a}^{2}{b}^{2}{e}^{3}+12\,A{d}^{2}a{b}^{3}{e}^{2}-4\,A{d}^{3}{b}^{4}e+B{e}^{4}{a}^{4}-8\,Bd{a}^{3}b{e}^{3}+18\,B{d}^{2}{a}^{2}{b}^{2}{e}^{2}-16\,B{d}^{3}a{b}^{3}e+5\,{b}^{4}B{d}^{4}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^8,x)

[Out]

-2/5*b*(3*A*a^2*b*e^3-6*A*a*b^2*d*e^2+3*A*b^3*d^2*e+2*B*a^3*e^3-9*B*a^2*b*d*e^2+
12*B*a*b^2*d^2*e-5*B*b^3*d^3)/e^6/(e*x+d)^5-1/2*b^2*(2*A*a*b*e^2-2*A*b^2*d*e+3*B
*a^2*e^2-8*B*a*b*d*e+5*B*b^2*d^2)/e^6/(e*x+d)^4-1/3*b^3*(A*b*e+4*B*a*e-5*B*b*d)/
e^6/(e*x+d)^3-1/2*b^4*B/e^6/(e*x+d)^2-1/7*(A*a^4*e^5-4*A*a^3*b*d*e^4+6*A*a^2*b^2
*d^2*e^3-4*A*a*b^3*d^3*e^2+A*b^4*d^4*e-B*a^4*d*e^4+4*B*a^3*b*d^2*e^3-6*B*a^2*b^2
*d^3*e^2+4*B*a*b^3*d^4*e-B*b^4*d^5)/e^6/(e*x+d)^7-1/6*(4*A*a^3*b*e^4-12*A*a^2*b^
2*d*e^3+12*A*a*b^3*d^2*e^2-4*A*b^4*d^3*e+B*a^4*e^4-8*B*a^3*b*d*e^3+18*B*a^2*b^2*
d^2*e^2-16*B*a*b^3*d^3*e+5*B*b^4*d^4)/e^6/(e*x+d)^6

_______________________________________________________________________________________

Maxima [A]  time = 0.717175, size = 645, normalized size = 4.78 \[ -\frac{105 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 30 \, A a^{4} e^{5} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 35 \,{\left (5 \, B b^{4} d e^{4} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 35 \,{\left (5 \, B b^{4} d^{2} e^{3} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 21 \,{\left (5 \, B b^{4} d^{3} e^{2} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 7 \,{\left (5 \, B b^{4} d^{4} e + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 5 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{210 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^8,x, algorithm="maxima")

[Out]

-1/210*(105*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 30*A*a^4*e^5 + 2*(4*B*a*b^3 + A*b^4)*d
^4*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3
+ 5*(B*a^4 + 4*A*a^3*b)*d*e^4 + 35*(5*B*b^4*d*e^4 + 2*(4*B*a*b^3 + A*b^4)*e^5)*
x^4 + 35*(5*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a
*b^3)*e^5)*x^3 + 21*(5*B*b^4*d^3*e^2 + 2*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^
2*b^2 + 2*A*a*b^3)*d*e^4 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 7*(5*B*b^4*d^4
*e + 2*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 4*(2*
B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 5*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^13*x^7 + 7*d*e^1
2*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*
d^6*e^7*x + d^7*e^6)

_______________________________________________________________________________________

Fricas [A]  time = 0.278677, size = 645, normalized size = 4.78 \[ -\frac{105 \, B b^{4} e^{5} x^{5} + 5 \, B b^{4} d^{5} + 30 \, A a^{4} e^{5} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + 5 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 35 \,{\left (5 \, B b^{4} d e^{4} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 35 \,{\left (5 \, B b^{4} d^{2} e^{3} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 21 \,{\left (5 \, B b^{4} d^{3} e^{2} + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 7 \,{\left (5 \, B b^{4} d^{4} e + 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} + 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} + 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} + 5 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{210 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^8,x, algorithm="fricas")

[Out]

-1/210*(105*B*b^4*e^5*x^5 + 5*B*b^4*d^5 + 30*A*a^4*e^5 + 2*(4*B*a*b^3 + A*b^4)*d
^4*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3
+ 5*(B*a^4 + 4*A*a^3*b)*d*e^4 + 35*(5*B*b^4*d*e^4 + 2*(4*B*a*b^3 + A*b^4)*e^5)*
x^4 + 35*(5*B*b^4*d^2*e^3 + 2*(4*B*a*b^3 + A*b^4)*d*e^4 + 3*(3*B*a^2*b^2 + 2*A*a
*b^3)*e^5)*x^3 + 21*(5*B*b^4*d^3*e^2 + 2*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 3*(3*B*a^
2*b^2 + 2*A*a*b^3)*d*e^4 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 7*(5*B*b^4*d^4
*e + 2*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^3 + 4*(2*
B*a^3*b + 3*A*a^2*b^2)*d*e^4 + 5*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^13*x^7 + 7*d*e^1
2*x^6 + 21*d^2*e^11*x^5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*
d^6*e^7*x + d^7*e^6)

_______________________________________________________________________________________

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*x+A)*(b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.286993, size = 594, normalized size = 4.4 \[ -\frac{{\left (105 \, B b^{4} x^{5} e^{5} + 175 \, B b^{4} d x^{4} e^{4} + 175 \, B b^{4} d^{2} x^{3} e^{3} + 105 \, B b^{4} d^{3} x^{2} e^{2} + 35 \, B b^{4} d^{4} x e + 5 \, B b^{4} d^{5} + 280 \, B a b^{3} x^{4} e^{5} + 70 \, A b^{4} x^{4} e^{5} + 280 \, B a b^{3} d x^{3} e^{4} + 70 \, A b^{4} d x^{3} e^{4} + 168 \, B a b^{3} d^{2} x^{2} e^{3} + 42 \, A b^{4} d^{2} x^{2} e^{3} + 56 \, B a b^{3} d^{3} x e^{2} + 14 \, A b^{4} d^{3} x e^{2} + 8 \, B a b^{3} d^{4} e + 2 \, A b^{4} d^{4} e + 315 \, B a^{2} b^{2} x^{3} e^{5} + 210 \, A a b^{3} x^{3} e^{5} + 189 \, B a^{2} b^{2} d x^{2} e^{4} + 126 \, A a b^{3} d x^{2} e^{4} + 63 \, B a^{2} b^{2} d^{2} x e^{3} + 42 \, A a b^{3} d^{2} x e^{3} + 9 \, B a^{2} b^{2} d^{3} e^{2} + 6 \, A a b^{3} d^{3} e^{2} + 168 \, B a^{3} b x^{2} e^{5} + 252 \, A a^{2} b^{2} x^{2} e^{5} + 56 \, B a^{3} b d x e^{4} + 84 \, A a^{2} b^{2} d x e^{4} + 8 \, B a^{3} b d^{2} e^{3} + 12 \, A a^{2} b^{2} d^{2} e^{3} + 35 \, B a^{4} x e^{5} + 140 \, A a^{3} b x e^{5} + 5 \, B a^{4} d e^{4} + 20 \, A a^{3} b d e^{4} + 30 \, A a^{4} e^{5}\right )} e^{\left (-6\right )}}{210 \,{\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)/(e*x + d)^8,x, algorithm="giac")

[Out]

-1/210*(105*B*b^4*x^5*e^5 + 175*B*b^4*d*x^4*e^4 + 175*B*b^4*d^2*x^3*e^3 + 105*B*
b^4*d^3*x^2*e^2 + 35*B*b^4*d^4*x*e + 5*B*b^4*d^5 + 280*B*a*b^3*x^4*e^5 + 70*A*b^
4*x^4*e^5 + 280*B*a*b^3*d*x^3*e^4 + 70*A*b^4*d*x^3*e^4 + 168*B*a*b^3*d^2*x^2*e^3
+ 42*A*b^4*d^2*x^2*e^3 + 56*B*a*b^3*d^3*x*e^2 + 14*A*b^4*d^3*x*e^2 + 8*B*a*b^3*
d^4*e + 2*A*b^4*d^4*e + 315*B*a^2*b^2*x^3*e^5 + 210*A*a*b^3*x^3*e^5 + 189*B*a^2*
b^2*d*x^2*e^4 + 126*A*a*b^3*d*x^2*e^4 + 63*B*a^2*b^2*d^2*x*e^3 + 42*A*a*b^3*d^2*
x*e^3 + 9*B*a^2*b^2*d^3*e^2 + 6*A*a*b^3*d^3*e^2 + 168*B*a^3*b*x^2*e^5 + 252*A*a^
2*b^2*x^2*e^5 + 56*B*a^3*b*d*x*e^4 + 84*A*a^2*b^2*d*x*e^4 + 8*B*a^3*b*d^2*e^3 +
12*A*a^2*b^2*d^2*e^3 + 35*B*a^4*x*e^5 + 140*A*a^3*b*x*e^5 + 5*B*a^4*d*e^4 + 20*A
*a^3*b*d*e^4 + 30*A*a^4*e^5)*e^(-6)/(x*e + d)^7