3.1686 \(\int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^6} \, dx\)

Optimal. Leaf size=155 \[ -\frac{(a+b x)^5 (B d-A e)}{5 e (d+e x)^5 (b d-a e)}+\frac{4 b^3 B (b d-a e)}{e^6 (d+e x)}-\frac{3 b^2 B (b d-a e)^2}{e^6 (d+e x)^2}+\frac{4 b B (b d-a e)^3}{3 e^6 (d+e x)^3}-\frac{B (b d-a e)^4}{4 e^6 (d+e x)^4}+\frac{b^4 B \log (d+e x)}{e^6} \]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(5*e*(b*d - a*e)*(d + e*x)^5) - (B*(b*d - a*e)^4)/(4*
e^6*(d + e*x)^4) + (4*b*B*(b*d - a*e)^3)/(3*e^6*(d + e*x)^3) - (3*b^2*B*(b*d - a
*e)^2)/(e^6*(d + e*x)^2) + (4*b^3*B*(b*d - a*e))/(e^6*(d + e*x)) + (b^4*B*Log[d
+ e*x])/e^6

_______________________________________________________________________________________

Rubi [A]  time = 0.342913, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ -\frac{(a+b x)^5 (B d-A e)}{5 e (d+e x)^5 (b d-a e)}+\frac{4 b^3 B (b d-a e)}{e^6 (d+e x)}-\frac{3 b^2 B (b d-a e)^2}{e^6 (d+e x)^2}+\frac{4 b B (b d-a e)^3}{3 e^6 (d+e x)^3}-\frac{B (b d-a e)^4}{4 e^6 (d+e x)^4}+\frac{b^4 B \log (d+e x)}{e^6} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^6,x]

[Out]

-((B*d - A*e)*(a + b*x)^5)/(5*e*(b*d - a*e)*(d + e*x)^5) - (B*(b*d - a*e)^4)/(4*
e^6*(d + e*x)^4) + (4*b*B*(b*d - a*e)^3)/(3*e^6*(d + e*x)^3) - (3*b^2*B*(b*d - a
*e)^2)/(e^6*(d + e*x)^2) + (4*b^3*B*(b*d - a*e))/(e^6*(d + e*x)) + (b^4*B*Log[d
+ e*x])/e^6

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 78.2395, size = 139, normalized size = 0.9 \[ \frac{B b^{4} \log{\left (d + e x \right )}}{e^{6}} - \frac{4 B b^{3} \left (a e - b d\right )}{e^{6} \left (d + e x\right )} - \frac{3 B b^{2} \left (a e - b d\right )^{2}}{e^{6} \left (d + e x\right )^{2}} - \frac{4 B b \left (a e - b d\right )^{3}}{3 e^{6} \left (d + e x\right )^{3}} - \frac{B \left (a e - b d\right )^{4}}{4 e^{6} \left (d + e x\right )^{4}} - \frac{\left (a + b x\right )^{5} \left (A e - B d\right )}{5 e \left (d + e x\right )^{5} \left (a e - b d\right )} \]

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)**6,x)

[Out]

B*b**4*log(d + e*x)/e**6 - 4*B*b**3*(a*e - b*d)/(e**6*(d + e*x)) - 3*B*b**2*(a*e
- b*d)**2/(e**6*(d + e*x)**2) - 4*B*b*(a*e - b*d)**3/(3*e**6*(d + e*x)**3) - B*
(a*e - b*d)**4/(4*e**6*(d + e*x)**4) - (a + b*x)**5*(A*e - B*d)/(5*e*(d + e*x)**
5*(a*e - b*d))

_______________________________________________________________________________________

Mathematica [B]  time = 0.290079, size = 332, normalized size = 2.14 \[ \frac{-3 a^4 e^4 (4 A e+B (d+5 e x))-4 a^3 b e^3 \left (3 A e (d+5 e x)+2 B \left (d^2+5 d e x+10 e^2 x^2\right )\right )-6 a^2 b^2 e^2 \left (2 A e \left (d^2+5 d e x+10 e^2 x^2\right )+3 B \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )\right )-12 a b^3 e \left (A e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+4 B \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+b^4 \left (B d \left (137 d^4+625 d^3 e x+1100 d^2 e^2 x^2+900 d e^3 x^3+300 e^4 x^4\right )-12 A e \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )\right )+60 b^4 B (d+e x)^5 \log (d+e x)}{60 e^6 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a^2 + 2*a*b*x + b^2*x^2)^2)/(d + e*x)^6,x]

[Out]

(-3*a^4*e^4*(4*A*e + B*(d + 5*e*x)) - 4*a^3*b*e^3*(3*A*e*(d + 5*e*x) + 2*B*(d^2
+ 5*d*e*x + 10*e^2*x^2)) - 6*a^2*b^2*e^2*(2*A*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3
*B*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) - 12*a*b^3*e*(A*e*(d^3 + 5*d^2
*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*B*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d
*e^3*x^3 + 5*e^4*x^4)) + b^4*(-12*A*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e
^3*x^3 + 5*e^4*x^4) + B*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*e^3*
x^3 + 300*e^4*x^4)) + 60*b^4*B*(d + e*x)^5*Log[d + e*x])/(60*e^6*(d + e*x)^5)

_______________________________________________________________________________________

Maple [B]  time = 0.014, size = 651, normalized size = 4.2 \[ \text{result too large to display} \]

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)^6,x)

[Out]

b^4*B*ln(e*x+d)/e^6-5/4/e^6/(e*x+d)^4*b^4*B*d^4+1/e^5/(e*x+d)^4*A*d^3*b^4-8*b^3/
e^5/(e*x+d)^3*B*a*d^2+8*b^3/e^5/(e*x+d)^2*B*d*a+3/e^3/(e*x+d)^4*A*d*a^2*b^2-3/e^
4/(e*x+d)^4*A*d^2*a*b^3-2*b^2/e^3/(e*x+d)^3*A*a^2-2*b^4/e^5/(e*x+d)^3*A*d^2+2/e^
3/(e*x+d)^4*B*d*a^3*b-9/2/e^4/(e*x+d)^4*B*d^2*a^2*b^2+4/e^5/(e*x+d)^4*B*d^3*a*b^
3+6*b^2/e^4/(e*x+d)^3*B*a^2*d+4/5/e^2/(e*x+d)^5*A*d*a^3*b-6/5/e^3/(e*x+d)^5*A*d^
2*a^2*b^2+4/5/e^4/(e*x+d)^5*A*d^3*a*b^3-4/5/e^3/(e*x+d)^5*B*d^2*a^3*b+6/5/e^4/(e
*x+d)^5*B*d^3*a^2*b^2-4/5/e^5/(e*x+d)^5*B*d^4*a*b^3-4*b^3/e^5/(e*x+d)*a*B-3*b^2/
e^4/(e*x+d)^2*a^2*B-4/3*b/e^3/(e*x+d)^3*B*a^3+10/3*b^4/e^6/(e*x+d)^3*B*d^3+5*b^4
/e^6/(e*x+d)*B*d-2*b^3/e^4/(e*x+d)^2*A*a+2*b^4/e^5/(e*x+d)^2*A*d-5*b^4/e^6/(e*x+
d)^2*B*d^2-1/5/e^5/(e*x+d)^5*A*d^4*b^4+1/5/e^2/(e*x+d)^5*B*a^4*d+1/5/e^6/(e*x+d)
^5*B*b^4*d^5-1/e^2/(e*x+d)^4*A*a^3*b+4*b^3/e^4/(e*x+d)^3*A*a*d-b^4/e^5/(e*x+d)*A
-1/5/e/(e*x+d)^5*A*a^4-1/4/e^2/(e*x+d)^4*B*a^4

_______________________________________________________________________________________

Maxima [A]  time = 0.70742, size = 620, normalized size = 4. \[ \frac{137 \, B b^{4} d^{5} - 12 \, A a^{4} e^{5} - 12 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 60 \,{\left (5 \, B b^{4} d e^{4} -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 60 \,{\left (15 \, B b^{4} d^{2} e^{3} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} -{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 20 \,{\left (55 \, B b^{4} d^{3} e^{2} - 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} - 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 5 \,{\left (125 \, B b^{4} d^{4} e - 12 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} - 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} - 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} + \frac{B b^{4} \log \left (e x + d\right )}{e^{6}} \]

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)^6,x, algorithm="maxima")

[Out]

1/60*(137*B*b^4*d^5 - 12*A*a^4*e^5 - 12*(4*B*a*b^3 + A*b^4)*d^4*e - 6*(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 - 3*(B*a^4 + 4*A*a
^3*b)*d*e^4 + 60*(5*B*b^4*d*e^4 - (4*B*a*b^3 + A*b^4)*e^5)*x^4 + 60*(15*B*b^4*d^
2*e^3 - 2*(4*B*a*b^3 + A*b^4)*d*e^4 - (3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 20*(5
5*B*b^4*d^3*e^2 - 6*(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 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 5*(125*B*b^4*d^4*e - 12*(4*B*a*b^3
+ A*b^4)*d^3*e^2 - 6*(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 - 3*(B*a^4 + 4*A*a^3*b)*e^5)*x)/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9
*x^3 + 10*d^3*e^8*x^2 + 5*d^4*e^7*x + d^5*e^6) + B*b^4*log(e*x + d)/e^6

_______________________________________________________________________________________

Fricas [A]  time = 0.275454, size = 710, normalized size = 4.58 \[ \frac{137 \, B b^{4} d^{5} - 12 \, A a^{4} e^{5} - 12 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e - 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} - 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4} + 60 \,{\left (5 \, B b^{4} d e^{4} -{\left (4 \, B a b^{3} + A b^{4}\right )} e^{5}\right )} x^{4} + 60 \,{\left (15 \, B b^{4} d^{2} e^{3} - 2 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d e^{4} -{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{5}\right )} x^{3} + 20 \,{\left (55 \, B b^{4} d^{3} e^{2} - 6 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{3} - 3 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{4} - 2 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{5}\right )} x^{2} + 5 \,{\left (125 \, B b^{4} d^{4} e - 12 \,{\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{2} - 6 \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{3} - 4 \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{4} - 3 \,{\left (B a^{4} + 4 \, A a^{3} b\right )} e^{5}\right )} x + 60 \,{\left (B b^{4} e^{5} x^{5} + 5 \, B b^{4} d e^{4} x^{4} + 10 \, B b^{4} d^{2} e^{3} x^{3} + 10 \, B b^{4} d^{3} e^{2} x^{2} + 5 \, B b^{4} d^{4} e x + B b^{4} d^{5}\right )} \log \left (e x + d\right )}{60 \,{\left (e^{11} x^{5} + 5 \, d e^{10} x^{4} + 10 \, d^{2} e^{9} x^{3} + 10 \, d^{3} e^{8} x^{2} + 5 \, d^{4} e^{7} x + d^{5} e^{6}\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

1/60*(137*B*b^4*d^5 - 12*A*a^4*e^5 - 12*(4*B*a*b^3 + A*b^4)*d^4*e - 6*(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 - 3*(B*a^4 + 4*A*a
^3*b)*d*e^4 + 60*(5*B*b^4*d*e^4 - (4*B*a*b^3 + A*b^4)*e^5)*x^4 + 60*(15*B*b^4*d^
2*e^3 - 2*(4*B*a*b^3 + A*b^4)*d*e^4 - (3*B*a^2*b^2 + 2*A*a*b^3)*e^5)*x^3 + 20*(5
5*B*b^4*d^3*e^2 - 6*(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 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^2 + 5*(125*B*b^4*d^4*e - 12*(4*B*a*b^3
+ A*b^4)*d^3*e^2 - 6*(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 - 3*(B*a^4 + 4*A*a^3*b)*e^5)*x + 60*(B*b^4*e^5*x^5 + 5*B*b^4*d*e^4*x^
4 + 10*B*b^4*d^2*e^3*x^3 + 10*B*b^4*d^3*e^2*x^2 + 5*B*b^4*d^4*e*x + B*b^4*d^5)*l
og(e*x + d))/(e^11*x^5 + 5*d*e^10*x^4 + 10*d^2*e^9*x^3 + 10*d^3*e^8*x^2 + 5*d^4*
e^7*x + d^5*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)**6,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.284544, size = 560, normalized size = 3.61 \[ B b^{4} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (60 \,{\left (5 \, B b^{4} d e^{3} - 4 \, B a b^{3} e^{4} - A b^{4} e^{4}\right )} x^{4} + 60 \,{\left (15 \, B b^{4} d^{2} e^{2} - 8 \, B a b^{3} d e^{3} - 2 \, A b^{4} d e^{3} - 3 \, B a^{2} b^{2} e^{4} - 2 \, A a b^{3} e^{4}\right )} x^{3} + 20 \,{\left (55 \, B b^{4} d^{3} e - 24 \, B a b^{3} d^{2} e^{2} - 6 \, A b^{4} d^{2} e^{2} - 9 \, B a^{2} b^{2} d e^{3} - 6 \, A a b^{3} d e^{3} - 4 \, B a^{3} b e^{4} - 6 \, A a^{2} b^{2} e^{4}\right )} x^{2} + 5 \,{\left (125 \, B b^{4} d^{4} - 48 \, B a b^{3} d^{3} e - 12 \, A b^{4} d^{3} e - 18 \, B a^{2} b^{2} d^{2} e^{2} - 12 \, A a b^{3} d^{2} e^{2} - 8 \, B a^{3} b d e^{3} - 12 \, A a^{2} b^{2} d e^{3} - 3 \, B a^{4} e^{4} - 12 \, A a^{3} b e^{4}\right )} x +{\left (137 \, B b^{4} d^{5} - 48 \, B a b^{3} d^{4} e - 12 \, A b^{4} d^{4} e - 18 \, B a^{2} b^{2} d^{3} e^{2} - 12 \, A a b^{3} d^{3} e^{2} - 8 \, B a^{3} b d^{2} e^{3} - 12 \, A a^{2} b^{2} d^{2} e^{3} - 3 \, B a^{4} d e^{4} - 12 \, A a^{3} b d e^{4} - 12 \, A a^{4} e^{5}\right )} e^{\left (-1\right )}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{5}} \]

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)^6,x, algorithm="giac")

[Out]

B*b^4*e^(-6)*ln(abs(x*e + d)) + 1/60*(60*(5*B*b^4*d*e^3 - 4*B*a*b^3*e^4 - A*b^4*
e^4)*x^4 + 60*(15*B*b^4*d^2*e^2 - 8*B*a*b^3*d*e^3 - 2*A*b^4*d*e^3 - 3*B*a^2*b^2*
e^4 - 2*A*a*b^3*e^4)*x^3 + 20*(55*B*b^4*d^3*e - 24*B*a*b^3*d^2*e^2 - 6*A*b^4*d^2
*e^2 - 9*B*a^2*b^2*d*e^3 - 6*A*a*b^3*d*e^3 - 4*B*a^3*b*e^4 - 6*A*a^2*b^2*e^4)*x^
2 + 5*(125*B*b^4*d^4 - 48*B*a*b^3*d^3*e - 12*A*b^4*d^3*e - 18*B*a^2*b^2*d^2*e^2
- 12*A*a*b^3*d^2*e^2 - 8*B*a^3*b*d*e^3 - 12*A*a^2*b^2*d*e^3 - 3*B*a^4*e^4 - 12*A
*a^3*b*e^4)*x + (137*B*b^4*d^5 - 48*B*a*b^3*d^4*e - 12*A*b^4*d^4*e - 18*B*a^2*b^
2*d^3*e^2 - 12*A*a*b^3*d^3*e^2 - 8*B*a^3*b*d^2*e^3 - 12*A*a^2*b^2*d^2*e^3 - 3*B*
a^4*d*e^4 - 12*A*a^3*b*d*e^4 - 12*A*a^4*e^5)*e^(-1))*e^(-5)/(x*e + d)^5