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3.1045 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^3} \, dx\)

Optimal. Leaf size=276 \[ -\frac{b^5 (d+e x)^4 (-6 a B e-A b e+7 b B d)}{4 e^8}+\frac{b^4 (d+e x)^3 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8}-\frac{5 b^3 (d+e x)^2 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8}+\frac{5 b^2 x (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^7}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 (d+e x)}+\frac{(b d-a e)^6 (B d-A e)}{2 e^8 (d+e x)^2}-\frac{3 b (b d-a e)^4 \log (d+e x) (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac{b^6 B (d+e x)^5}{5 e^8} \]

[Out]

(5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*x)/e^7 + ((b*d - a*e)^6*(B*d
- A*e))/(2*e^8*(d + e*x)^2) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(e^8*(
d + e*x)) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^2)/(2*e
^8) + (b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^3)/e^8 - (b^5*(7*
b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^4)/(4*e^8) + (b^6*B*(d + e*x)^5)/(5*e^8) - (3
*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*Log[d + e*x])/e^8

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Rubi [A]  time = 1.31269, antiderivative size = 276, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{b^5 (d+e x)^4 (-6 a B e-A b e+7 b B d)}{4 e^8}+\frac{b^4 (d+e x)^3 (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^8}-\frac{5 b^3 (d+e x)^2 (b d-a e)^2 (-4 a B e-3 A b e+7 b B d)}{2 e^8}+\frac{5 b^2 x (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^7}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{e^8 (d+e x)}+\frac{(b d-a e)^6 (B d-A e)}{2 e^8 (d+e x)^2}-\frac{3 b (b d-a e)^4 \log (d+e x) (-2 a B e-5 A b e+7 b B d)}{e^8}+\frac{b^6 B (d+e x)^5}{5 e^8} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^6*(A + B*x))/(d + e*x)^3,x]

[Out]

(5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e)*x)/e^7 + ((b*d - a*e)^6*(B*d
- A*e))/(2*e^8*(d + e*x)^2) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(e^8*(
d + e*x)) - (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*(d + e*x)^2)/(2*e
^8) + (b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*(d + e*x)^3)/e^8 - (b^5*(7*
b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^4)/(4*e^8) + (b^6*B*(d + e*x)^5)/(5*e^8) - (3
*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e)*Log[d + e*x])/e^8

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B b^{6} \left (d + e x\right )^{5}}{5 e^{8}} + \frac{b^{5} \left (d + e x\right )^{4} \left (A b e + 6 B a e - 7 B b d\right )}{4 e^{8}} + \frac{b^{4} \left (d + e x\right )^{3} \left (a e - b d\right ) \left (2 A b e + 5 B a e - 7 B b d\right )}{e^{8}} + \frac{5 b^{3} \left (d + e x\right )^{2} \left (a e - b d\right )^{2} \left (3 A b e + 4 B a e - 7 B b d\right )}{2 e^{8}} + \frac{3 b \left (a e - b d\right )^{4} \left (5 A b e + 2 B a e - 7 B b d\right ) \log{\left (d + e x \right )}}{e^{8}} + \frac{5 \left (a e - b d\right )^{3} \left (4 A b e + 3 B a e - 7 B b d\right ) \int b^{2}\, dx}{e^{7}} - \frac{\left (a e - b d\right )^{5} \left (6 A b e + B a e - 7 B b d\right )}{e^{8} \left (d + e x\right )} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{6}}{2 e^{8} \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6*(B*x+A)/(e*x+d)**3,x)

[Out]

B*b**6*(d + e*x)**5/(5*e**8) + b**5*(d + e*x)**4*(A*b*e + 6*B*a*e - 7*B*b*d)/(4*
e**8) + b**4*(d + e*x)**3*(a*e - b*d)*(2*A*b*e + 5*B*a*e - 7*B*b*d)/e**8 + 5*b**
3*(d + e*x)**2*(a*e - b*d)**2*(3*A*b*e + 4*B*a*e - 7*B*b*d)/(2*e**8) + 3*b*(a*e
- b*d)**4*(5*A*b*e + 2*B*a*e - 7*B*b*d)*log(d + e*x)/e**8 + 5*(a*e - b*d)**3*(4*
A*b*e + 3*B*a*e - 7*B*b*d)*Integral(b**2, x)/e**7 - (a*e - b*d)**5*(6*A*b*e + B*
a*e - 7*B*b*d)/(e**8*(d + e*x)) - (A*e - B*d)*(a*e - b*d)**6/(2*e**8*(d + e*x)**
2)

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Mathematica [A]  time = 0.3415, size = 352, normalized size = 1.28 \[ \frac{-20 b^4 e^3 x^3 \left (-5 a^2 B e^2-2 a b e (A e-3 B d)+b^2 d (A e-2 B d)\right )+10 b^3 e^2 x^2 \left (20 a^3 B e^3+15 a^2 b e^2 (A e-3 B d)+18 a b^2 d e (2 B d-A e)+2 b^3 d^2 (3 A e-5 B d)\right )-20 b^2 e x \left (-15 a^4 B e^4-20 a^3 b e^3 (A e-3 B d)+45 a^2 b^2 d e^2 (A e-2 B d)+12 a b^3 d^2 e (5 B d-3 A e)-5 b^4 d^3 (3 B d-2 A e)\right )+5 b^5 e^4 x^4 (6 a B e+A b e-3 b B d)-\frac{20 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{d+e x}+\frac{10 (b d-a e)^6 (B d-A e)}{(d+e x)^2}-60 b (b d-a e)^4 \log (d+e x) (-2 a B e-5 A b e+7 b B d)+4 b^6 B e^5 x^5}{20 e^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^3,x]

[Out]

(-20*b^2*e*(-15*a^4*B*e^4 + 12*a*b^3*d^2*e*(5*B*d - 3*A*e) - 5*b^4*d^3*(3*B*d -
2*A*e) - 20*a^3*b*e^3*(-3*B*d + A*e) + 45*a^2*b^2*d*e^2*(-2*B*d + A*e))*x + 10*b
^3*e^2*(20*a^3*B*e^3 + 18*a*b^2*d*e*(2*B*d - A*e) + 15*a^2*b*e^2*(-3*B*d + A*e)
+ 2*b^3*d^2*(-5*B*d + 3*A*e))*x^2 - 20*b^4*e^3*(-5*a^2*B*e^2 - 2*a*b*e*(-3*B*d +
 A*e) + b^2*d*(-2*B*d + A*e))*x^3 + 5*b^5*e^4*(-3*b*B*d + A*b*e + 6*a*B*e)*x^4 +
 4*b^6*B*e^5*x^5 + (10*(b*d - a*e)^6*(B*d - A*e))/(d + e*x)^2 - (20*(b*d - a*e)^
5*(7*b*B*d - 6*A*b*e - a*B*e))/(d + e*x) - 60*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e
 - 2*a*B*e)*Log[d + e*x])/(20*e^8)

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Maple [B]  time = 0.025, size = 1101, normalized size = 4. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6*(B*x+A)/(e*x+d)^3,x)

[Out]

-1/e^2/(e*x+d)*B*a^6-1/2/e/(e*x+d)^2*a^6*A+1/4*b^6/e^3*A*x^4+1/5*b^6/e^3*B*x^5-7
/e^8/(e*x+d)*b^6*B*d^6-1/2/e^7/(e*x+d)^2*A*b^6*d^6+1/2/e^2/(e*x+d)^2*B*d*a^6+1/2
/e^8/(e*x+d)^2*b^6*B*d^7+2*b^5/e^3*A*x^3*a-b^6/e^4*A*x^3*d+15/2/e^6/(e*x+d)^2*B*
a^2*b^4*d^5-3/e^7/(e*x+d)^2*B*a*b^5*d^6+90*b^4/e^5*ln(e*x+d)*A*a^2*d^2-60*b^5/e^
6*ln(e*x+d)*A*a*d^3-45*b^2/e^4*ln(e*x+d)*B*a^4*d+120*b^3/e^5*ln(e*x+d)*B*a^3*d^2
-150*b^4/e^6*ln(e*x+d)*B*a^2*d^3+90*b^5/e^7*ln(e*x+d)*B*a*d^4+30/e^3/(e*x+d)*A*a
^4*b^2*d-9*b^5/e^4*A*x^2*a*d-45/2*b^4/e^4*B*x^2*a^2*d-60/e^4/(e*x+d)*A*a^3*b^3*d
^2+60/e^5/(e*x+d)*A*a^2*b^4*d^3-30/e^6/(e*x+d)*A*a*b^5*d^4+12/e^3/(e*x+d)*B*a^5*
b*d-45/e^4/(e*x+d)*B*a^4*b^2*d^2+80/e^5/(e*x+d)*B*a^3*b^3*d^3-75/e^6/(e*x+d)*B*a
^2*b^4*d^4+36/e^7/(e*x+d)*B*a*b^5*d^5+3/e^2/(e*x+d)^2*A*d*a^5*b-15/2/e^3/(e*x+d)
^2*A*d^2*a^4*b^2+10/e^4/(e*x+d)^2*A*a^3*b^3*d^3-15/2/e^5/(e*x+d)^2*A*a^2*b^4*d^4
+3/e^6/(e*x+d)^2*A*a*b^5*d^5-3/e^3/(e*x+d)^2*B*d^2*a^5*b+15/2/e^4/(e*x+d)^2*B*a^
4*b^2*d^3-10/e^5/(e*x+d)^2*B*a^3*b^3*d^4+18*b^5/e^5*B*x^2*a*d^2-45*b^4/e^4*A*a^2
*d*x+36*b^5/e^5*A*a*d^2*x-60*b^3/e^4*B*a^3*d*x+90*b^4/e^5*B*a^2*d^2*x-60*b^5/e^6
*B*a*d^3*x-6*b^5/e^4*B*x^3*a*d-60*b^3/e^4*ln(e*x+d)*A*a^3*d+5*b^4/e^3*B*x^3*a^2+
2*b^6/e^5*B*x^3*d^2+15/2*b^4/e^3*A*x^2*a^2+3*b^6/e^5*A*x^2*d^2+3/2*b^5/e^3*B*x^4
*a+15*b^2/e^3*ln(e*x+d)*A*a^4+15*b^6/e^7*ln(e*x+d)*A*d^4+6*b/e^3*ln(e*x+d)*B*a^5
-21*b^6/e^8*ln(e*x+d)*B*d^5+20*b^3/e^3*A*a^3*x-10*b^6/e^6*A*d^3*x+15*b^2/e^3*B*a
^4*x+15*b^6/e^7*B*d^4*x-3/4*b^6/e^4*B*x^4*d+10*b^3/e^3*B*x^2*a^3-5*b^6/e^6*B*x^2
*d^3+6/e^7/(e*x+d)*A*b^6*d^5-6/e^2/(e*x+d)*A*a^5*b

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Maxima [A]  time = 1.38605, size = 1052, normalized size = 3.81 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^3,x, algorithm="maxima")

[Out]

-1/2*(13*B*b^6*d^7 + A*a^6*e^7 - 11*(6*B*a*b^5 + A*b^6)*d^6*e + 27*(5*B*a^2*b^4
+ 2*A*a*b^5)*d^5*e^2 - 35*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 25*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d^3*e^4 - 9*(2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (B*a^6 + 6*A*a^5*
b)*d*e^6 + 2*(7*B*b^6*d^6*e - 6*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 15*(5*B*a^2*b^4 +
2*A*a*b^5)*d^4*e^3 - 20*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 + 15*(3*B*a^4*b^2 +
4*A*a^3*b^3)*d^2*e^5 - 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 + (B*a^6 + 6*A*a^5*b)*e
^7)*x)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8) + 1/20*(4*B*b^6*e^4*x^5 - 5*(3*B*b^6*d*e
^3 - (6*B*a*b^5 + A*b^6)*e^4)*x^4 + 20*(2*B*b^6*d^2*e^2 - (6*B*a*b^5 + A*b^6)*d*
e^3 + (5*B*a^2*b^4 + 2*A*a*b^5)*e^4)*x^3 - 10*(10*B*b^6*d^3*e - 6*(6*B*a*b^5 + A
*b^6)*d^2*e^2 + 9*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^3 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4
)*e^4)*x^2 + 20*(15*B*b^6*d^4 - 10*(6*B*a*b^5 + A*b^6)*d^3*e + 18*(5*B*a^2*b^4 +
 2*A*a*b^5)*d^2*e^2 - 15*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^3 + 5*(3*B*a^4*b^2 + 4*
A*a^3*b^3)*e^4)*x)/e^7 - 3*(7*B*b^6*d^5 - 5*(6*B*a*b^5 + A*b^6)*d^4*e + 10*(5*B*
a^2*b^4 + 2*A*a*b^5)*d^3*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^3 + 5*(3*B*a
^4*b^2 + 4*A*a^3*b^3)*d*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*e^5)*log(e*x + d)/e^8

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Fricas [A]  time = 0.213403, size = 1589, normalized size = 5.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^3,x, algorithm="fricas")

[Out]

1/20*(4*B*b^6*e^7*x^7 - 130*B*b^6*d^7 - 10*A*a^6*e^7 + 110*(6*B*a*b^5 + A*b^6)*d
^6*e - 270*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 350*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d
^4*e^3 - 250*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 90*(2*B*a^5*b + 5*A*a^4*b^2)*
d^2*e^5 - 10*(B*a^6 + 6*A*a^5*b)*d*e^6 - (7*B*b^6*d*e^6 - 5*(6*B*a*b^5 + A*b^6)*
e^7)*x^6 + 2*(7*B*b^6*d^2*e^5 - 5*(6*B*a*b^5 + A*b^6)*d*e^6 + 10*(5*B*a^2*b^4 +
2*A*a*b^5)*e^7)*x^5 - 5*(7*B*b^6*d^3*e^4 - 5*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 10*(5
*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 20*(7*
B*b^6*d^4*e^3 - 5*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2
*e^5 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)
*x^3 + 10*(50*B*b^6*d^5*e^2 - 34*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 63*(5*B*a^2*b^4 +
 2*A*a*b^5)*d^3*e^4 - 55*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 20*(3*B*a^4*b^2 +
 4*A*a^3*b^3)*d*e^6)*x^2 + 20*(8*B*b^6*d^6*e - 4*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 3
*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 + 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^4 - 10*
(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 + 6*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^6 - (B*a^6
 + 6*A*a^5*b)*e^7)*x - 60*(7*B*b^6*d^7 - 5*(6*B*a*b^5 + A*b^6)*d^6*e + 10*(5*B*a
^2*b^4 + 2*A*a*b^5)*d^5*e^2 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 5*(3*B*a^
4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - (2*B*a^5*b + 5*A*a^4*b^2)*d^2*e^5 + (7*B*b^6*d^5*
e^2 - 5*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 10*
(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 - (2*B
*a^5*b + 5*A*a^4*b^2)*e^7)*x^2 + 2*(7*B*b^6*d^6*e - 5*(6*B*a*b^5 + A*b^6)*d^5*e^
2 + 10*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 10*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^
4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^2*e^5 - (2*B*a^5*b + 5*A*a^4*b^2)*d*e^6)*x)*
log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)

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Sympy [A]  time = 74.4923, size = 802, normalized size = 2.91 \[ \frac{B b^{6} x^{5}}{5 e^{3}} + \frac{3 b \left (a e - b d\right )^{4} \left (5 A b e + 2 B a e - 7 B b d\right ) \log{\left (d + e x \right )}}{e^{8}} - \frac{A a^{6} e^{7} + 6 A a^{5} b d e^{6} - 45 A a^{4} b^{2} d^{2} e^{5} + 100 A a^{3} b^{3} d^{3} e^{4} - 105 A a^{2} b^{4} d^{4} e^{3} + 54 A a b^{5} d^{5} e^{2} - 11 A b^{6} d^{6} e + B a^{6} d e^{6} - 18 B a^{5} b d^{2} e^{5} + 75 B a^{4} b^{2} d^{3} e^{4} - 140 B a^{3} b^{3} d^{4} e^{3} + 135 B a^{2} b^{4} d^{5} e^{2} - 66 B a b^{5} d^{6} e + 13 B b^{6} d^{7} + x \left (12 A a^{5} b e^{7} - 60 A a^{4} b^{2} d e^{6} + 120 A a^{3} b^{3} d^{2} e^{5} - 120 A a^{2} b^{4} d^{3} e^{4} + 60 A a b^{5} d^{4} e^{3} - 12 A b^{6} d^{5} e^{2} + 2 B a^{6} e^{7} - 24 B a^{5} b d e^{6} + 90 B a^{4} b^{2} d^{2} e^{5} - 160 B a^{3} b^{3} d^{3} e^{4} + 150 B a^{2} b^{4} d^{4} e^{3} - 72 B a b^{5} d^{5} e^{2} + 14 B b^{6} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} + \frac{x^{4} \left (A b^{6} e + 6 B a b^{5} e - 3 B b^{6} d\right )}{4 e^{4}} + \frac{x^{3} \left (2 A a b^{5} e^{2} - A b^{6} d e + 5 B a^{2} b^{4} e^{2} - 6 B a b^{5} d e + 2 B b^{6} d^{2}\right )}{e^{5}} + \frac{x^{2} \left (15 A a^{2} b^{4} e^{3} - 18 A a b^{5} d e^{2} + 6 A b^{6} d^{2} e + 20 B a^{3} b^{3} e^{3} - 45 B a^{2} b^{4} d e^{2} + 36 B a b^{5} d^{2} e - 10 B b^{6} d^{3}\right )}{2 e^{6}} + \frac{x \left (20 A a^{3} b^{3} e^{4} - 45 A a^{2} b^{4} d e^{3} + 36 A a b^{5} d^{2} e^{2} - 10 A b^{6} d^{3} e + 15 B a^{4} b^{2} e^{4} - 60 B a^{3} b^{3} d e^{3} + 90 B a^{2} b^{4} d^{2} e^{2} - 60 B a b^{5} d^{3} e + 15 B b^{6} d^{4}\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**6*(B*x+A)/(e*x+d)**3,x)

[Out]

B*b**6*x**5/(5*e**3) + 3*b*(a*e - b*d)**4*(5*A*b*e + 2*B*a*e - 7*B*b*d)*log(d +
e*x)/e**8 - (A*a**6*e**7 + 6*A*a**5*b*d*e**6 - 45*A*a**4*b**2*d**2*e**5 + 100*A*
a**3*b**3*d**3*e**4 - 105*A*a**2*b**4*d**4*e**3 + 54*A*a*b**5*d**5*e**2 - 11*A*b
**6*d**6*e + B*a**6*d*e**6 - 18*B*a**5*b*d**2*e**5 + 75*B*a**4*b**2*d**3*e**4 -
140*B*a**3*b**3*d**4*e**3 + 135*B*a**2*b**4*d**5*e**2 - 66*B*a*b**5*d**6*e + 13*
B*b**6*d**7 + x*(12*A*a**5*b*e**7 - 60*A*a**4*b**2*d*e**6 + 120*A*a**3*b**3*d**2
*e**5 - 120*A*a**2*b**4*d**3*e**4 + 60*A*a*b**5*d**4*e**3 - 12*A*b**6*d**5*e**2
+ 2*B*a**6*e**7 - 24*B*a**5*b*d*e**6 + 90*B*a**4*b**2*d**2*e**5 - 160*B*a**3*b**
3*d**3*e**4 + 150*B*a**2*b**4*d**4*e**3 - 72*B*a*b**5*d**5*e**2 + 14*B*b**6*d**6
*e))/(2*d**2*e**8 + 4*d*e**9*x + 2*e**10*x**2) + x**4*(A*b**6*e + 6*B*a*b**5*e -
 3*B*b**6*d)/(4*e**4) + x**3*(2*A*a*b**5*e**2 - A*b**6*d*e + 5*B*a**2*b**4*e**2
- 6*B*a*b**5*d*e + 2*B*b**6*d**2)/e**5 + x**2*(15*A*a**2*b**4*e**3 - 18*A*a*b**5
*d*e**2 + 6*A*b**6*d**2*e + 20*B*a**3*b**3*e**3 - 45*B*a**2*b**4*d*e**2 + 36*B*a
*b**5*d**2*e - 10*B*b**6*d**3)/(2*e**6) + x*(20*A*a**3*b**3*e**4 - 45*A*a**2*b**
4*d*e**3 + 36*A*a*b**5*d**2*e**2 - 10*A*b**6*d**3*e + 15*B*a**4*b**2*e**4 - 60*B
*a**3*b**3*d*e**3 + 90*B*a**2*b**4*d**2*e**2 - 60*B*a*b**5*d**3*e + 15*B*b**6*d*
*4)/e**7

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GIAC/XCAS [A]  time = 0.233065, size = 1094, normalized size = 3.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6/(e*x + d)^3,x, algorithm="giac")

[Out]

-3*(7*B*b^6*d^5 - 30*B*a*b^5*d^4*e - 5*A*b^6*d^4*e + 50*B*a^2*b^4*d^3*e^2 + 20*A
*a*b^5*d^3*e^2 - 40*B*a^3*b^3*d^2*e^3 - 30*A*a^2*b^4*d^2*e^3 + 15*B*a^4*b^2*d*e^
4 + 20*A*a^3*b^3*d*e^4 - 2*B*a^5*b*e^5 - 5*A*a^4*b^2*e^5)*e^(-8)*ln(abs(x*e + d)
) + 1/20*(4*B*b^6*x^5*e^12 - 15*B*b^6*d*x^4*e^11 + 40*B*b^6*d^2*x^3*e^10 - 100*B
*b^6*d^3*x^2*e^9 + 300*B*b^6*d^4*x*e^8 + 30*B*a*b^5*x^4*e^12 + 5*A*b^6*x^4*e^12
- 120*B*a*b^5*d*x^3*e^11 - 20*A*b^6*d*x^3*e^11 + 360*B*a*b^5*d^2*x^2*e^10 + 60*A
*b^6*d^2*x^2*e^10 - 1200*B*a*b^5*d^3*x*e^9 - 200*A*b^6*d^3*x*e^9 + 100*B*a^2*b^4
*x^3*e^12 + 40*A*a*b^5*x^3*e^12 - 450*B*a^2*b^4*d*x^2*e^11 - 180*A*a*b^5*d*x^2*e
^11 + 1800*B*a^2*b^4*d^2*x*e^10 + 720*A*a*b^5*d^2*x*e^10 + 200*B*a^3*b^3*x^2*e^1
2 + 150*A*a^2*b^4*x^2*e^12 - 1200*B*a^3*b^3*d*x*e^11 - 900*A*a^2*b^4*d*x*e^11 +
300*B*a^4*b^2*x*e^12 + 400*A*a^3*b^3*x*e^12)*e^(-15) - 1/2*(13*B*b^6*d^7 - 66*B*
a*b^5*d^6*e - 11*A*b^6*d^6*e + 135*B*a^2*b^4*d^5*e^2 + 54*A*a*b^5*d^5*e^2 - 140*
B*a^3*b^3*d^4*e^3 - 105*A*a^2*b^4*d^4*e^3 + 75*B*a^4*b^2*d^3*e^4 + 100*A*a^3*b^3
*d^3*e^4 - 18*B*a^5*b*d^2*e^5 - 45*A*a^4*b^2*d^2*e^5 + B*a^6*d*e^6 + 6*A*a^5*b*d
*e^6 + A*a^6*e^7 + 2*(7*B*b^6*d^6*e - 36*B*a*b^5*d^5*e^2 - 6*A*b^6*d^5*e^2 + 75*
B*a^2*b^4*d^4*e^3 + 30*A*a*b^5*d^4*e^3 - 80*B*a^3*b^3*d^3*e^4 - 60*A*a^2*b^4*d^3
*e^4 + 45*B*a^4*b^2*d^2*e^5 + 60*A*a^3*b^3*d^2*e^5 - 12*B*a^5*b*d*e^6 - 30*A*a^4
*b^2*d*e^6 + B*a^6*e^7 + 6*A*a^5*b*e^7)*x)*e^(-8)/(x*e + d)^2

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