3.1047 \(\int \frac{(a+b x)^6 (A+B x)}{(d+e x)^5} \, dx\)
Optimal. Leaf size=279 \[ -\frac{b^5 (d+e x)^2 (-6 a B e-A b e+7 b B d)}{2 e^8}+\frac{3 b^4 x (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^7}-\frac{5 b^3 (b d-a e)^2 \log (d+e x) (-4 a B e-3 A b e+7 b B d)}{e^8}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8 (d+e x)}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{2 e^8 (d+e x)^2}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8 (d+e x)^3}+\frac{(b d-a e)^6 (B d-A e)}{4 e^8 (d+e x)^4}+\frac{b^6 B (d+e x)^3}{3 e^8} \]
[Out]
(3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*x)/e^7 + ((b*d - a*e)^6*(B*d -
A*e))/(4*e^8*(d + e*x)^4) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(3*e^8*(
d + e*x)^3) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(2*e^8*(d + e*x)
^2) - (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e))/(e^8*(d + e*x)) - (b^5
*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^2)/(2*e^8) + (b^6*B*(d + e*x)^3)/(3*e^8)
- (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*Log[d + e*x])/e^8
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Rubi [A] time = 1.11175, antiderivative size = 279, 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
\[ -\frac{b^5 (d+e x)^2 (-6 a B e-A b e+7 b B d)}{2 e^8}+\frac{3 b^4 x (b d-a e) (-5 a B e-2 A b e+7 b B d)}{e^7}-\frac{5 b^3 (b d-a e)^2 \log (d+e x) (-4 a B e-3 A b e+7 b B d)}{e^8}-\frac{5 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{e^8 (d+e x)}+\frac{3 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{2 e^8 (d+e x)^2}-\frac{(b d-a e)^5 (-a B e-6 A b e+7 b B d)}{3 e^8 (d+e x)^3}+\frac{(b d-a e)^6 (B d-A e)}{4 e^8 (d+e x)^4}+\frac{b^6 B (d+e x)^3}{3 e^8} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^6*(A + B*x))/(d + e*x)^5,x]
[Out]
(3*b^4*(b*d - a*e)*(7*b*B*d - 2*A*b*e - 5*a*B*e)*x)/e^7 + ((b*d - a*e)^6*(B*d -
A*e))/(4*e^8*(d + e*x)^4) - ((b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(3*e^8*(
d + e*x)^3) + (3*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(2*e^8*(d + e*x)
^2) - (5*b^2*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e))/(e^8*(d + e*x)) - (b^5
*(7*b*B*d - A*b*e - 6*a*B*e)*(d + e*x)^2)/(2*e^8) + (b^6*B*(d + e*x)^3)/(3*e^8)
- (5*b^3*(b*d - a*e)^2*(7*b*B*d - 3*A*b*e - 4*a*B*e)*Log[d + e*x])/e^8
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Rubi in Sympy [A] time = 129.116, size = 284, normalized size = 1.02 \[ \frac{B b^{6} \left (d + e x\right )^{3}}{3 e^{8}} + \frac{b^{5} \left (d + e x\right )^{2} \left (A b e + 6 B a e - 7 B b d\right )}{2 e^{8}} + \frac{3 b^{4} x \left (a e - b d\right ) \left (2 A b e + 5 B a e - 7 B b d\right )}{e^{7}} + \frac{5 b^{3} \left (a e - b d\right )^{2} \left (3 A b e + 4 B a e - 7 B b d\right ) \log{\left (d + e x \right )}}{e^{8}} - \frac{5 b^{2} \left (a e - b d\right )^{3} \left (4 A b e + 3 B a e - 7 B b d\right )}{e^{8} \left (d + e x\right )} - \frac{3 b \left (a e - b d\right )^{4} \left (5 A b e + 2 B a e - 7 B b d\right )}{2 e^{8} \left (d + e x\right )^{2}} - \frac{\left (a e - b d\right )^{5} \left (6 A b e + B a e - 7 B b d\right )}{3 e^{8} \left (d + e x\right )^{3}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{6}}{4 e^{8} \left (d + e x\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**6*(B*x+A)/(e*x+d)**5,x)
[Out]
B*b**6*(d + e*x)**3/(3*e**8) + b**5*(d + e*x)**2*(A*b*e + 6*B*a*e - 7*B*b*d)/(2*
e**8) + 3*b**4*x*(a*e - b*d)*(2*A*b*e + 5*B*a*e - 7*B*b*d)/e**7 + 5*b**3*(a*e -
b*d)**2*(3*A*b*e + 4*B*a*e - 7*B*b*d)*log(d + e*x)/e**8 - 5*b**2*(a*e - b*d)**3*
(4*A*b*e + 3*B*a*e - 7*B*b*d)/(e**8*(d + e*x)) - 3*b*(a*e - b*d)**4*(5*A*b*e + 2
*B*a*e - 7*B*b*d)/(2*e**8*(d + e*x)**2) - (a*e - b*d)**5*(6*A*b*e + B*a*e - 7*B*
b*d)/(3*e**8*(d + e*x)**3) - (A*e - B*d)*(a*e - b*d)**6/(4*e**8*(d + e*x)**4)
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Mathematica [A] time = 0.376862, size = 263, normalized size = 0.94 \[ \frac{-12 b^4 e x \left (-15 a^2 B e^2-6 a b e (A e-5 B d)+5 b^2 d (A e-3 B d)\right )+6 b^5 e^2 x^2 (6 a B e+A b e-5 b B d)-60 b^3 (b d-a e)^2 \log (d+e x) (-4 a B e-3 A b e+7 b B d)-\frac{60 b^2 (b d-a e)^3 (-3 a B e-4 A b e+7 b B d)}{d+e x}+\frac{18 b (b d-a e)^4 (-2 a B e-5 A b e+7 b B d)}{(d+e x)^2}-\frac{4 (b d-a e)^5 (-a B e-6 A b e+7 b B d)}{(d+e x)^3}+\frac{3 (b d-a e)^6 (B d-A e)}{(d+e x)^4}+4 b^6 B e^3 x^3}{12 e^8} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^6*(A + B*x))/(d + e*x)^5,x]
[Out]
(-12*b^4*e*(-15*a^2*B*e^2 - 6*a*b*e*(-5*B*d + A*e) + 5*b^2*d*(-3*B*d + A*e))*x +
6*b^5*e^2*(-5*b*B*d + A*b*e + 6*a*B*e)*x^2 + 4*b^6*B*e^3*x^3 + (3*(b*d - a*e)^6
*(B*d - A*e))/(d + e*x)^4 - (4*(b*d - a*e)^5*(7*b*B*d - 6*A*b*e - a*B*e))/(d + e
*x)^3 + (18*b*(b*d - a*e)^4*(7*b*B*d - 5*A*b*e - 2*a*B*e))/(d + e*x)^2 - (60*b^2
*(b*d - a*e)^3*(7*b*B*d - 4*A*b*e - 3*a*B*e))/(d + e*x) - 60*b^3*(b*d - a*e)^2*(
7*b*B*d - 3*A*b*e - 4*a*B*e)*Log[d + e*x])/(12*e^8)
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Maple [B] time = 0.024, size = 1177, 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)^6*(B*x+A)/(e*x+d)^5,x)
[Out]
15*b^4/e^5*B*a^2*x+15*b^6/e^7*B*d^2*x+15*b^4/e^5*ln(e*x+d)*A*a^2+15*b^6/e^7*ln(e
*x+d)*A*d^2+20*b^3/e^5*ln(e*x+d)*B*a^3-35*b^6/e^8*ln(e*x+d)*B*d^3-2/e^2/(e*x+d)^
3*A*a^5*b+2/e^7/(e*x+d)^3*A*b^6*d^5-7/3/e^8/(e*x+d)^3*b^6*B*d^6-20*b^3/e^4/(e*x+
d)*A*a^3+20*b^6/e^7/(e*x+d)*A*d^3-15*b^2/e^4/(e*x+d)*B*a^4-35*b^6/e^8/(e*x+d)*B*
d^4-15/2*b^2/e^3/(e*x+d)^2*A*a^4-15/2*b^6/e^7/(e*x+d)^2*A*d^4-3*b/e^3/(e*x+d)^2*
B*a^5+21/2*b^6/e^8/(e*x+d)^2*B*d^5-1/4/e^7/(e*x+d)^4*A*b^6*d^6+1/4/e^2/(e*x+d)^4
*B*d*a^6+1/4/e^8/(e*x+d)^4*b^6*B*d^7+3*b^5/e^5*B*x^2*a-5/2*b^6/e^6*B*x^2*d-1/4/e
/(e*x+d)^4*a^6*A+1/3*b^6/e^5*B*x^3+1/2*b^6/e^5*A*x^2-1/3/e^2/(e*x+d)^3*B*a^6+6*b
^5/e^5*A*a*x-5*b^6/e^6*A*d*x-30*b^5/e^6*ln(e*x+d)*A*d*a-75*b^4/e^6*ln(e*x+d)*B*a
^2*d+90*b^5/e^7*ln(e*x+d)*B*d^2*a+10/e^3/(e*x+d)^3*A*a^4*b^2*d-20/e^4/(e*x+d)^3*
A*a^3*b^3*d^2+20/e^5/(e*x+d)^3*A*a^2*b^4*d^3-10/e^6/(e*x+d)^3*A*a*b^5*d^4+4/e^3/
(e*x+d)^3*B*a^5*b*d-15/e^4/(e*x+d)^3*B*a^4*b^2*d^2+80/3/e^5/(e*x+d)^3*B*a^3*b^3*
d^3-25/e^6/(e*x+d)^3*B*a^2*b^4*d^4-30*b^5/e^6*B*a*d*x+30*b^3/e^4/(e*x+d)^2*A*a^3
*d-45*b^4/e^5/(e*x+d)^2*A*a^2*d^2+30*b^5/e^6/(e*x+d)^2*A*a*d^3+45/2*b^2/e^4/(e*x
+d)^2*B*a^4*d-60*b^3/e^5/(e*x+d)^2*B*a^3*d^2+75*b^4/e^6/(e*x+d)^2*B*a^2*d^3-45*b
^5/e^7/(e*x+d)^2*B*a*d^4+3/2/e^2/(e*x+d)^4*A*d*a^5*b-15/4/e^3/(e*x+d)^4*A*d^2*a^
4*b^2+5/e^4/(e*x+d)^4*A*d^3*a^3*b^3-15/4/e^5/(e*x+d)^4*A*d^4*a^2*b^4+3/2/e^6/(e*
x+d)^4*A*a*b^5*d^5-3/2/e^3/(e*x+d)^4*B*d^2*a^5*b+15/4/e^4/(e*x+d)^4*B*d^3*a^4*b^
2-5/e^5/(e*x+d)^4*B*d^4*a^3*b^3+15/4/e^6/(e*x+d)^4*B*a^2*b^4*d^5-3/2/e^7/(e*x+d)
^4*B*a*b^5*d^6+12/e^7/(e*x+d)^3*B*a*b^5*d^5+60*b^4/e^5/(e*x+d)*A*a^2*d-60*b^5/e^
6/(e*x+d)*A*a*d^2+80*b^3/e^5/(e*x+d)*B*a^3*d-150*b^4/e^6/(e*x+d)*B*a^2*d^2+120*b
^5/e^7/(e*x+d)*B*a*d^3
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Maxima [A] time = 1.42267, size = 1081, normalized size = 3.87 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^6/(e*x + d)^5,x, algorithm="maxima")
[Out]
-1/12*(319*B*b^6*d^7 + 3*A*a^6*e^7 - 171*(6*B*a*b^5 + A*b^6)*d^6*e + 231*(5*B*a^
2*b^4 + 2*A*a*b^5)*d^5*e^2 - 125*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + 15*(3*B*a
^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 + 3*(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 + 60*(7*B*b^6*d^4*e^3 - 4*(6*B*a*b^5 + A*b^6)*d^3*e^4 + 6*(5*B*a
^2*b^4 + 2*A*a*b^5)*d^2*e^5 - 4*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d*e^6 + (3*B*a^4*b^2
+ 4*A*a^3*b^3)*e^7)*x^3 + 18*(63*B*b^6*d^5*e^2 - 35*(6*B*a*b^5 + A*b^6)*d^4*e^3
+ 50*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 30*(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 + 4*
(259*B*b^6*d^6*e - 141*(6*B*a*b^5 + A*b^6)*d^5*e^2 + 195*(5*B*a^2*b^4 + 2*A*a*b^
5)*d^4*e^3 - 110*(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 + 3*(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^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8) + 1/6*(2*B*b^
6*e^2*x^3 - 3*(5*B*b^6*d*e - (6*B*a*b^5 + A*b^6)*e^2)*x^2 + 6*(15*B*b^6*d^2 - 5*
(6*B*a*b^5 + A*b^6)*d*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^2)*x)/e^7 - 5*(7*B*b^6*d
^3 - 3*(6*B*a*b^5 + A*b^6)*d^2*e + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^2 - (4*B*a^3*
b^3 + 3*A*a^2*b^4)*e^3)*log(e*x + d)/e^8
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Fricas [A] time = 0.218999, size = 1650, normalized size = 5.91 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^6/(e*x + d)^5,x, algorithm="fricas")
[Out]
1/12*(4*B*b^6*e^7*x^7 - 319*B*b^6*d^7 - 3*A*a^6*e^7 + 171*(6*B*a*b^5 + A*b^6)*d^
6*e - 231*(5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 + 125*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^
4*e^3 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d^3*e^4 - 3*(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*e^6 - 3*(6*B*a*b^5 + A*b^6)*e^7)
*x^6 + 12*(7*B*b^6*d^2*e^5 - 3*(6*B*a*b^5 + A*b^6)*d*e^6 + 3*(5*B*a^2*b^4 + 2*A*
a*b^5)*e^7)*x^5 + 4*(139*B*b^6*d^3*e^4 - 51*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 36*(5*
B*a^2*b^4 + 2*A*a*b^5)*d*e^6)*x^4 + 4*(136*B*b^6*d^4*e^3 - 24*(6*B*a*b^5 + A*b^6
)*d^3*e^4 - 36*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 + 60*(4*B*a^3*b^3 + 3*A*a^2*b^4
)*d*e^6 - 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^7)*x^3 - 6*(74*B*b^6*d^5*e^2 - 66*(6*
B*a*b^5 + A*b^6)*d^4*e^3 + 126*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^4 - 90*(4*B*a^3*b
^3 + 3*A*a^2*b^4)*d^2*e^5 + 15*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^6 + 3*(2*B*a^5*b
+ 5*A*a^4*b^2)*e^7)*x^2 - 4*(214*B*b^6*d^6*e - 126*(6*B*a*b^5 + A*b^6)*d^5*e^2 +
186*(5*B*a^2*b^4 + 2*A*a*b^5)*d^4*e^3 - 110*(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 + 3*(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 - 3*(6*B*a*b^5 + A*b^6)*d^6*e + 3*(
5*B*a^2*b^4 + 2*A*a*b^5)*d^5*e^2 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^4*e^3 + (7*B*b^
6*d^3*e^4 - 3*(6*B*a*b^5 + A*b^6)*d^2*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d*e^6 -
(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 4*(7*B*b^6*d^4*e^3 - 3*(6*B*a*b^5 + A*b^6
)*d^3*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^5 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*d*
e^6)*x^3 + 6*(7*B*b^6*d^5*e^2 - 3*(6*B*a*b^5 + A*b^6)*d^4*e^3 + 3*(5*B*a^2*b^4 +
2*A*a*b^5)*d^3*e^4 - (4*B*a^3*b^3 + 3*A*a^2*b^4)*d^2*e^5)*x^2 + 4*(7*B*b^6*d^6*
e - 3*(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 - (4*B*a
^3*b^3 + 3*A*a^2*b^4)*d^3*e^4)*x)*log(e*x + d))/(e^12*x^4 + 4*d*e^11*x^3 + 6*d^2
*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8)
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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*x+a)**6*(B*x+A)/(e*x+d)**5,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.229964, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^6/(e*x + d)^5,x, algorithm="giac")
[Out]
Done