3.1043 \(\int \frac{(a+b x)^6 (A+B x)}{d+e x} \, dx\)

Optimal. Leaf size=220 \[ -\frac{(b d-a e)^6 (B d-A e) \log (d+e x)}{e^8}+\frac{b x (b d-a e)^5 (B d-A e)}{e^7}-\frac{(a+b x)^2 (b d-a e)^4 (B d-A e)}{2 e^6}+\frac{(a+b x)^3 (b d-a e)^3 (B d-A e)}{3 e^5}-\frac{(a+b x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac{(a+b x)^5 (b d-a e) (B d-A e)}{5 e^3}-\frac{(a+b x)^6 (B d-A e)}{6 e^2}+\frac{B (a+b x)^7}{7 b e} \]

[Out]

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

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Rubi [A]  time = 0.409064, antiderivative size = 220, 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 \[ -\frac{(b d-a e)^6 (B d-A e) \log (d+e x)}{e^8}+\frac{b x (b d-a e)^5 (B d-A e)}{e^7}-\frac{(a+b x)^2 (b d-a e)^4 (B d-A e)}{2 e^6}+\frac{(a+b x)^3 (b d-a e)^3 (B d-A e)}{3 e^5}-\frac{(a+b x)^4 (b d-a e)^2 (B d-A e)}{4 e^4}+\frac{(a+b x)^5 (b d-a e) (B d-A e)}{5 e^3}-\frac{(a+b x)^6 (B d-A e)}{6 e^2}+\frac{B (a+b x)^7}{7 b e} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.543037, size = 501, normalized size = 2.28 \[ \frac{e x \left (420 a^6 B e^6+1260 a^5 b e^5 (2 A e-2 B d+B e x)+1050 a^4 b^2 e^4 \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+700 a^3 b^3 e^3 \left (2 A e \left (6 d^2-3 d e x+2 e^2 x^2\right )+B \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )\right )+105 a^2 b^4 e^2 \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+42 a b^5 e \left (A e \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )+B \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )\right )+b^6 \left (7 A e \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+B \left (420 d^6-210 d^5 e x+140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 (b d-a e)^6 (B d-A e) \log (d+e x)}{420 e^8} \]

Antiderivative was successfully verified.

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

[Out]

(e*x*(420*a^6*B*e^6 + 1260*a^5*b*e^5*(-2*B*d + 2*A*e + B*e*x) + 1050*a^4*b^2*e^4
*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 700*a^3*b^3*e^3*(2*A*e
*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + B*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^
3)) + 105*a^2*b^4*e^2*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B
*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)) + 42*a*b^5*
e*(A*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + B*(-
60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^
5)) + b^6*(7*A*e*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*
e^4*x^4 + 10*e^5*x^5) + B*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3
*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x^5 + 60*e^6*x^6))) - 420*(b*d - a*e)^6*(B*d -
A*e)*Log[d + e*x])/(420*e^8)

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Maple [B]  time = 0.011, size = 989, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/e*B*a^6*x+1/7/e*B*b^6*x^7+1/6/e*A*x^6*b^6+1/e*ln(e*x+d)*a^6*A-1/e^8*ln(e*x+d)*
b^6*B*d^7+20/3/e*A*x^3*a^3*b^3-1/4/e^4*B*x^4*b^6*d^3+5/e*B*x^3*a^4*b^2-1/3/e^4*A
*x^3*b^6*d^3+1/4/e^3*A*x^4*b^6*d^2+15/4/e*A*x^4*a^2*b^4+1/5/e^3*B*x^5*b^6*d^2+5/
e*B*x^4*a^3*b^3+3/e*B*x^2*a^5*b+1/3/e^5*B*x^3*b^6*d^4+1/2/e^5*A*x^2*b^6*d^4+15/2
/e*A*x^2*a^4*b^2+1/e^7*b^6*B*d^6*x+1/e*B*x^6*a*b^5+3/e*B*x^5*a^2*b^4-1/5/e^2*A*x
^5*b^6*d-20/3/e^2*B*x^3*a^3*b^3*d+5/e^3*B*x^3*a^2*b^4*d^2-2/e^4*B*x^3*a*b^5*d^3+
10/e^3*B*x^2*a^3*b^3*d^2-15/2/e^4*B*x^2*a^2*b^4*d^3+3/e^5*B*x^2*a*b^5*d^4-15/e^2
*A*a^4*b^2*d*x+20/e^3*A*a^3*b^3*d^2*x-15/e^4*A*a^2*b^4*d^3*x+6/e^5*A*a*b^5*d^4*x
-6/e^2*B*a^5*b*d*x+15/e^3*B*a^4*b^2*d^2*x-20/e^4*B*a^3*b^3*d^3*x+15/e^5*B*a^2*b^
4*d^4*x-6/e^6*B*a*b^5*d^5*x+15/2/e^3*A*x^2*a^2*b^4*d^2-3/e^4*A*x^2*a*b^5*d^3-15/
2/e^2*B*x^2*a^4*b^2*d-10/e^2*A*x^2*a^3*b^3*d-6/e^2*ln(e*x+d)*A*a^5*b*d+15/e^3*ln
(e*x+d)*A*a^4*b^2*d^2-20/e^4*ln(e*x+d)*A*a^3*b^3*d^3+15/e^5*ln(e*x+d)*A*a^2*b^4*
d^4-6/e^6*ln(e*x+d)*A*a*b^5*d^5+6/e^3*ln(e*x+d)*B*a^5*b*d^2-15/e^4*ln(e*x+d)*B*a
^4*b^2*d^3+20/e^5*ln(e*x+d)*B*a^3*b^3*d^4-15/e^6*ln(e*x+d)*B*a^2*b^4*d^5+6/e^7*l
n(e*x+d)*B*a*b^5*d^6-3/2/e^2*A*x^4*a*b^5*d-6/5/e^2*B*x^5*a*b^5*d-5/e^2*A*x^3*a^2
*b^4*d+2/e^3*A*x^3*a*b^5*d^2-15/4/e^2*B*x^4*a^2*b^4*d+3/2/e^3*B*x^4*a*b^5*d^2-1/
6/e^2*B*x^6*b^6*d-1/e^6*A*b^6*d^5*x+6/5/e*A*x^5*a*b^5-1/2/e^6*B*x^2*b^6*d^5+6/e*
A*a^5*b*x+1/e^7*ln(e*x+d)*A*b^6*d^6-1/e^2*ln(e*x+d)*B*a^6*d

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Maxima [A]  time = 1.38635, size = 1029, normalized size = 4.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/420*(60*B*b^6*e^6*x^7 - 70*(B*b^6*d*e^5 - (6*B*a*b^5 + A*b^6)*e^6)*x^6 + 84*(B
*b^6*d^2*e^4 - (6*B*a*b^5 + A*b^6)*d*e^5 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*e^6)*x^5
- 105*(B*b^6*d^3*e^3 - (6*B*a*b^5 + A*b^6)*d^2*e^4 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)
*d*e^5 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^6)*x^4 + 140*(B*b^6*d^4*e^2 - (6*B*a*b^
5 + A*b^6)*d^3*e^3 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^2*e^4 - 5*(4*B*a^3*b^3 + 3*A*
a^2*b^4)*d*e^5 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*e^6)*x^3 - 210*(B*b^6*d^5*e - (6*
B*a*b^5 + A*b^6)*d^4*e^2 + 3*(5*B*a^2*b^4 + 2*A*a*b^5)*d^3*e^3 - 5*(4*B*a^3*b^3
+ 3*A*a^2*b^4)*d^2*e^4 + 5*(3*B*a^4*b^2 + 4*A*a^3*b^3)*d*e^5 - 3*(2*B*a^5*b + 5*
A*a^4*b^2)*e^6)*x^2 + 420*(B*b^6*d^6 - (6*B*a*b^5 + A*b^6)*d^5*e + 3*(5*B*a^2*b^
4 + 2*A*a*b^5)*d^4*e^2 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*d^3*e^3 + 5*(3*B*a^4*b^2
+ 4*A*a^3*b^3)*d^2*e^4 - 3*(2*B*a^5*b + 5*A*a^4*b^2)*d*e^5 + (B*a^6 + 6*A*a^5*b)
*e^6)*x)/e^7 - (B*b^6*d^7 - A*a^6*e^7 - (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 - 5*(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 - 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)*log(e*x + d)/e^8

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Fricas [A]  time = 0.210198, size = 1030, normalized size = 4.68 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/420*(60*B*b^6*e^7*x^7 - 70*(B*b^6*d*e^6 - (6*B*a*b^5 + A*b^6)*e^7)*x^6 + 84*(B
*b^6*d^2*e^5 - (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
- 105*(B*b^6*d^3*e^4 - (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 - 5*(4*B*a^3*b^3 + 3*A*a^2*b^4)*e^7)*x^4 + 140*(B*b^6*d^4*e^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 - 5*(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 - 210*(B*b^6*d^5*e^2 - (
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 - 5*(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 - 3*(2*B*a^5*b +
5*A*a^4*b^2)*e^7)*x^2 + 420*(B*b^6*d^6*e - (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 + 5*(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 - 420*(B*b^6*d^7 - A*a^6*e^7 - (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 - 5*(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 - 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)*log(e*x + d))/e^8

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Sympy [A]  time = 8.34195, size = 709, normalized size = 3.22 \[ \frac{B b^{6} x^{7}}{7 e} + \frac{x^{6} \left (A b^{6} e + 6 B a b^{5} e - B b^{6} d\right )}{6 e^{2}} + \frac{x^{5} \left (6 A a b^{5} e^{2} - A b^{6} d e + 15 B a^{2} b^{4} e^{2} - 6 B a b^{5} d e + B b^{6} d^{2}\right )}{5 e^{3}} + \frac{x^{4} \left (15 A a^{2} b^{4} e^{3} - 6 A a b^{5} d e^{2} + A b^{6} d^{2} e + 20 B a^{3} b^{3} e^{3} - 15 B a^{2} b^{4} d e^{2} + 6 B a b^{5} d^{2} e - B b^{6} d^{3}\right )}{4 e^{4}} + \frac{x^{3} \left (20 A a^{3} b^{3} e^{4} - 15 A a^{2} b^{4} d e^{3} + 6 A a b^{5} d^{2} e^{2} - A b^{6} d^{3} e + 15 B a^{4} b^{2} e^{4} - 20 B a^{3} b^{3} d e^{3} + 15 B a^{2} b^{4} d^{2} e^{2} - 6 B a b^{5} d^{3} e + B b^{6} d^{4}\right )}{3 e^{5}} + \frac{x^{2} \left (15 A a^{4} b^{2} e^{5} - 20 A a^{3} b^{3} d e^{4} + 15 A a^{2} b^{4} d^{2} e^{3} - 6 A a b^{5} d^{3} e^{2} + A b^{6} d^{4} e + 6 B a^{5} b e^{5} - 15 B a^{4} b^{2} d e^{4} + 20 B a^{3} b^{3} d^{2} e^{3} - 15 B a^{2} b^{4} d^{3} e^{2} + 6 B a b^{5} d^{4} e - B b^{6} d^{5}\right )}{2 e^{6}} + \frac{x \left (6 A a^{5} b e^{6} - 15 A a^{4} b^{2} d e^{5} + 20 A a^{3} b^{3} d^{2} e^{4} - 15 A a^{2} b^{4} d^{3} e^{3} + 6 A a b^{5} d^{4} e^{2} - A b^{6} d^{5} e + B a^{6} e^{6} - 6 B a^{5} b d e^{5} + 15 B a^{4} b^{2} d^{2} e^{4} - 20 B a^{3} b^{3} d^{3} e^{3} + 15 B a^{2} b^{4} d^{4} e^{2} - 6 B a b^{5} d^{5} e + B b^{6} d^{6}\right )}{e^{7}} - \frac{\left (- A e + B d\right ) \left (a e - b d\right )^{6} \log{\left (d + e x \right )}}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*b**6*x**7/(7*e) + x**6*(A*b**6*e + 6*B*a*b**5*e - B*b**6*d)/(6*e**2) + x**5*(6
*A*a*b**5*e**2 - A*b**6*d*e + 15*B*a**2*b**4*e**2 - 6*B*a*b**5*d*e + B*b**6*d**2
)/(5*e**3) + x**4*(15*A*a**2*b**4*e**3 - 6*A*a*b**5*d*e**2 + A*b**6*d**2*e + 20*
B*a**3*b**3*e**3 - 15*B*a**2*b**4*d*e**2 + 6*B*a*b**5*d**2*e - B*b**6*d**3)/(4*e
**4) + x**3*(20*A*a**3*b**3*e**4 - 15*A*a**2*b**4*d*e**3 + 6*A*a*b**5*d**2*e**2
- A*b**6*d**3*e + 15*B*a**4*b**2*e**4 - 20*B*a**3*b**3*d*e**3 + 15*B*a**2*b**4*d
**2*e**2 - 6*B*a*b**5*d**3*e + B*b**6*d**4)/(3*e**5) + x**2*(15*A*a**4*b**2*e**5
- 20*A*a**3*b**3*d*e**4 + 15*A*a**2*b**4*d**2*e**3 - 6*A*a*b**5*d**3*e**2 + A*b
**6*d**4*e + 6*B*a**5*b*e**5 - 15*B*a**4*b**2*d*e**4 + 20*B*a**3*b**3*d**2*e**3
- 15*B*a**2*b**4*d**3*e**2 + 6*B*a*b**5*d**4*e - B*b**6*d**5)/(2*e**6) + x*(6*A*
a**5*b*e**6 - 15*A*a**4*b**2*d*e**5 + 20*A*a**3*b**3*d**2*e**4 - 15*A*a**2*b**4*
d**3*e**3 + 6*A*a*b**5*d**4*e**2 - A*b**6*d**5*e + B*a**6*e**6 - 6*B*a**5*b*d*e*
*5 + 15*B*a**4*b**2*d**2*e**4 - 20*B*a**3*b**3*d**3*e**3 + 15*B*a**2*b**4*d**4*e
**2 - 6*B*a*b**5*d**5*e + B*b**6*d**6)/e**7 - (-A*e + B*d)*(a*e - b*d)**6*log(d
+ e*x)/e**8

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GIAC/XCAS [A]  time = 0.223249, 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),x, algorithm="giac")

[Out]

Done