3.1058 \(\int (a+b x)^6 (A+B x) (d+e x) \, dx\)

Optimal. Leaf size=75 \[ \frac {(a+b x)^8 (-2 a B e+A b e+b B d)}{8 b^3}+\frac {(a+b x)^7 (A b-a B) (b d-a e)}{7 b^3}+\frac {B e (a+b x)^9}{9 b^3} \]

[Out]

1/7*(A*b-B*a)*(-a*e+b*d)*(b*x+a)^7/b^3+1/8*(A*b*e-2*B*a*e+B*b*d)*(b*x+a)^8/b^3+1/9*B*e*(b*x+a)^9/b^3

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Rubi [A]  time = 0.20, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \frac {(a+b x)^8 (-2 a B e+A b e+b B d)}{8 b^3}+\frac {(a+b x)^7 (A b-a B) (b d-a e)}{7 b^3}+\frac {B e (a+b x)^9}{9 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((A*b - a*B)*(b*d - a*e)*(a + b*x)^7)/(7*b^3) + ((b*B*d + A*b*e - 2*a*B*e)*(a + b*x)^8)/(8*b^3) + (B*e*(a + b*
x)^9)/(9*b^3)

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int (a+b x)^6 (A+B x) (d+e x) \, dx &=\int \left (\frac {(A b-a B) (b d-a e) (a+b x)^6}{b^2}+\frac {(b B d+A b e-2 a B e) (a+b x)^7}{b^2}+\frac {B e (a+b x)^8}{b^2}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e) (a+b x)^7}{7 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^8}{8 b^3}+\frac {B e (a+b x)^9}{9 b^3}\\ \end {align*}

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Mathematica [B]  time = 0.14, size = 231, normalized size = 3.08 \[ \frac {1}{504} x \left (84 a^6 (3 A (2 d+e x)+B x (3 d+2 e x))+252 a^5 b x (A (6 d+4 e x)+B x (4 d+3 e x))+126 a^4 b^2 x^2 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+168 a^3 b^3 x^3 (3 A (5 d+4 e x)+2 B x (6 d+5 e x))+36 a^2 b^4 x^4 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+18 a b^5 x^5 (4 A (7 d+6 e x)+3 B x (8 d+7 e x))+b^6 x^6 (9 A (8 d+7 e x)+7 B x (9 d+8 e x))\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(84*a^6*(3*A*(2*d + e*x) + B*x*(3*d + 2*e*x)) + 126*a^4*b^2*x^2*(5*A*(4*d + 3*e*x) + 3*B*x*(5*d + 4*e*x)) +
252*a^5*b*x*(B*x*(4*d + 3*e*x) + A*(6*d + 4*e*x)) + 168*a^3*b^3*x^3*(3*A*(5*d + 4*e*x) + 2*B*x*(6*d + 5*e*x))
+ 36*a^2*b^4*x^4*(7*A*(6*d + 5*e*x) + 5*B*x*(7*d + 6*e*x)) + 18*a*b^5*x^5*(4*A*(7*d + 6*e*x) + 3*B*x*(8*d + 7
*e*x)) + b^6*x^6*(9*A*(8*d + 7*e*x) + 7*B*x*(9*d + 8*e*x))))/504

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fricas [B]  time = 0.74, size = 321, normalized size = 4.28 \[ \frac {1}{9} x^{9} e b^{6} B + \frac {1}{8} x^{8} d b^{6} B + \frac {3}{4} x^{8} e b^{5} a B + \frac {1}{8} x^{8} e b^{6} A + \frac {6}{7} x^{7} d b^{5} a B + \frac {15}{7} x^{7} e b^{4} a^{2} B + \frac {1}{7} x^{7} d b^{6} A + \frac {6}{7} x^{7} e b^{5} a A + \frac {5}{2} x^{6} d b^{4} a^{2} B + \frac {10}{3} x^{6} e b^{3} a^{3} B + x^{6} d b^{5} a A + \frac {5}{2} x^{6} e b^{4} a^{2} A + 4 x^{5} d b^{3} a^{3} B + 3 x^{5} e b^{2} a^{4} B + 3 x^{5} d b^{4} a^{2} A + 4 x^{5} e b^{3} a^{3} A + \frac {15}{4} x^{4} d b^{2} a^{4} B + \frac {3}{2} x^{4} e b a^{5} B + 5 x^{4} d b^{3} a^{3} A + \frac {15}{4} x^{4} e b^{2} a^{4} A + 2 x^{3} d b a^{5} B + \frac {1}{3} x^{3} e a^{6} B + 5 x^{3} d b^{2} a^{4} A + 2 x^{3} e b a^{5} A + \frac {1}{2} x^{2} d a^{6} B + 3 x^{2} d b a^{5} A + \frac {1}{2} x^{2} e a^{6} A + x d a^{6} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9*e*b^6*B + 1/8*x^8*d*b^6*B + 3/4*x^8*e*b^5*a*B + 1/8*x^8*e*b^6*A + 6/7*x^7*d*b^5*a*B + 15/7*x^7*e*b^4*a
^2*B + 1/7*x^7*d*b^6*A + 6/7*x^7*e*b^5*a*A + 5/2*x^6*d*b^4*a^2*B + 10/3*x^6*e*b^3*a^3*B + x^6*d*b^5*a*A + 5/2*
x^6*e*b^4*a^2*A + 4*x^5*d*b^3*a^3*B + 3*x^5*e*b^2*a^4*B + 3*x^5*d*b^4*a^2*A + 4*x^5*e*b^3*a^3*A + 15/4*x^4*d*b
^2*a^4*B + 3/2*x^4*e*b*a^5*B + 5*x^4*d*b^3*a^3*A + 15/4*x^4*e*b^2*a^4*A + 2*x^3*d*b*a^5*B + 1/3*x^3*e*a^6*B +
5*x^3*d*b^2*a^4*A + 2*x^3*e*b*a^5*A + 1/2*x^2*d*a^6*B + 3*x^2*d*b*a^5*A + 1/2*x^2*e*a^6*A + x*d*a^6*A

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giac [B]  time = 1.21, size = 335, normalized size = 4.47 \[ \frac {1}{9} \, B b^{6} x^{9} e + \frac {1}{8} \, B b^{6} d x^{8} + \frac {3}{4} \, B a b^{5} x^{8} e + \frac {1}{8} \, A b^{6} x^{8} e + \frac {6}{7} \, B a b^{5} d x^{7} + \frac {1}{7} \, A b^{6} d x^{7} + \frac {15}{7} \, B a^{2} b^{4} x^{7} e + \frac {6}{7} \, A a b^{5} x^{7} e + \frac {5}{2} \, B a^{2} b^{4} d x^{6} + A a b^{5} d x^{6} + \frac {10}{3} \, B a^{3} b^{3} x^{6} e + \frac {5}{2} \, A a^{2} b^{4} x^{6} e + 4 \, B a^{3} b^{3} d x^{5} + 3 \, A a^{2} b^{4} d x^{5} + 3 \, B a^{4} b^{2} x^{5} e + 4 \, A a^{3} b^{3} x^{5} e + \frac {15}{4} \, B a^{4} b^{2} d x^{4} + 5 \, A a^{3} b^{3} d x^{4} + \frac {3}{2} \, B a^{5} b x^{4} e + \frac {15}{4} \, A a^{4} b^{2} x^{4} e + 2 \, B a^{5} b d x^{3} + 5 \, A a^{4} b^{2} d x^{3} + \frac {1}{3} \, B a^{6} x^{3} e + 2 \, A a^{5} b x^{3} e + \frac {1}{2} \, B a^{6} d x^{2} + 3 \, A a^{5} b d x^{2} + \frac {1}{2} \, A a^{6} x^{2} e + A a^{6} d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^6*(B*x+A)*(e*x+d),x, algorithm="giac")

[Out]

1/9*B*b^6*x^9*e + 1/8*B*b^6*d*x^8 + 3/4*B*a*b^5*x^8*e + 1/8*A*b^6*x^8*e + 6/7*B*a*b^5*d*x^7 + 1/7*A*b^6*d*x^7
+ 15/7*B*a^2*b^4*x^7*e + 6/7*A*a*b^5*x^7*e + 5/2*B*a^2*b^4*d*x^6 + A*a*b^5*d*x^6 + 10/3*B*a^3*b^3*x^6*e + 5/2*
A*a^2*b^4*x^6*e + 4*B*a^3*b^3*d*x^5 + 3*A*a^2*b^4*d*x^5 + 3*B*a^4*b^2*x^5*e + 4*A*a^3*b^3*x^5*e + 15/4*B*a^4*b
^2*d*x^4 + 5*A*a^3*b^3*d*x^4 + 3/2*B*a^5*b*x^4*e + 15/4*A*a^4*b^2*x^4*e + 2*B*a^5*b*d*x^3 + 5*A*a^4*b^2*d*x^3
+ 1/3*B*a^6*x^3*e + 2*A*a^5*b*x^3*e + 1/2*B*a^6*d*x^2 + 3*A*a^5*b*d*x^2 + 1/2*A*a^6*x^2*e + A*a^6*d*x

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maple [B]  time = 0.00, size = 293, normalized size = 3.91 \[ \frac {B \,b^{6} e \,x^{9}}{9}+A \,a^{6} d x +\frac {\left (B \,b^{6} d +\left (b^{6} A +6 a \,b^{5} B \right ) e \right ) x^{8}}{8}+\frac {\left (\left (b^{6} A +6 a \,b^{5} B \right ) d +\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) e \right ) x^{7}}{7}+\frac {\left (\left (6 a \,b^{5} A +15 a^{2} b^{4} B \right ) d +\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) e \right ) x^{6}}{6}+\frac {\left (\left (15 a^{2} b^{4} A +20 a^{3} b^{3} B \right ) d +\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) e \right ) x^{5}}{5}+\frac {\left (\left (20 a^{3} b^{3} A +15 a^{4} b^{2} B \right ) d +\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) e \right ) x^{4}}{4}+\frac {\left (\left (15 a^{4} b^{2} A +6 a^{5} b B \right ) d +\left (6 a^{5} b A +a^{6} B \right ) e \right ) x^{3}}{3}+\frac {\left (A \,a^{6} e +\left (6 a^{5} b A +a^{6} B \right ) d \right ) x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^6*(B*x+A)*(e*x+d),x)

[Out]

1/9*b^6*B*e*x^9+1/8*((A*b^6+6*B*a*b^5)*e+b^6*B*d)*x^8+1/7*((6*A*a*b^5+15*B*a^2*b^4)*e+(A*b^6+6*B*a*b^5)*d)*x^7
+1/6*((15*A*a^2*b^4+20*B*a^3*b^3)*e+(6*A*a*b^5+15*B*a^2*b^4)*d)*x^6+1/5*((20*A*a^3*b^3+15*B*a^4*b^2)*e+(15*A*a
^2*b^4+20*B*a^3*b^3)*d)*x^5+1/4*((15*A*a^4*b^2+6*B*a^5*b)*e+(20*A*a^3*b^3+15*B*a^4*b^2)*d)*x^4+1/3*((6*A*a^5*b
+B*a^6)*e+(15*A*a^4*b^2+6*B*a^5*b)*d)*x^3+1/2*(a^6*A*e+(6*A*a^5*b+B*a^6)*d)*x^2+a^6*A*d*x

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maxima [B]  time = 0.54, size = 297, normalized size = 3.96 \[ \frac {1}{9} \, B b^{6} e x^{9} + A a^{6} d x + \frac {1}{8} \, {\left (B b^{6} d + {\left (6 \, B a b^{5} + A b^{6}\right )} e\right )} x^{8} + \frac {1}{7} \, {\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d + 3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e\right )} x^{7} + \frac {1}{6} \, {\left (3 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d + 5 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e\right )} x^{6} + {\left ({\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d + {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e\right )} x^{5} + \frac {1}{4} \, {\left (5 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d + 3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d + {\left (B a^{6} + 6 \, A a^{5} b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{6} e + {\left (B a^{6} + 6 \, A a^{5} b\right )} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.14, size = 257, normalized size = 3.43 \[ x^3\,\left (\frac {B\,a^6\,e}{3}+2\,A\,a^5\,b\,e+2\,B\,a^5\,b\,d+5\,A\,a^4\,b^2\,d\right )+x^7\,\left (\frac {A\,b^6\,d}{7}+\frac {6\,A\,a\,b^5\,e}{7}+\frac {6\,B\,a\,b^5\,d}{7}+\frac {15\,B\,a^2\,b^4\,e}{7}\right )+x^2\,\left (\frac {A\,a^6\,e}{2}+\frac {B\,a^6\,d}{2}+3\,A\,a^5\,b\,d\right )+x^8\,\left (\frac {A\,b^6\,e}{8}+\frac {B\,b^6\,d}{8}+\frac {3\,B\,a\,b^5\,e}{4}\right )+a^2\,b^2\,x^5\,\left (3\,A\,b^2\,d+3\,B\,a^2\,e+4\,A\,a\,b\,e+4\,B\,a\,b\,d\right )+A\,a^6\,d\,x+\frac {B\,b^6\,e\,x^9}{9}+\frac {a^3\,b\,x^4\,\left (20\,A\,b^2\,d+6\,B\,a^2\,e+15\,A\,a\,b\,e+15\,B\,a\,b\,d\right )}{4}+\frac {a\,b^3\,x^6\,\left (6\,A\,b^2\,d+20\,B\,a^2\,e+15\,A\,a\,b\,e+15\,B\,a\,b\,d\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^6*(d + e*x),x)

[Out]

x^3*((B*a^6*e)/3 + 2*A*a^5*b*e + 2*B*a^5*b*d + 5*A*a^4*b^2*d) + x^7*((A*b^6*d)/7 + (6*A*a*b^5*e)/7 + (6*B*a*b^
5*d)/7 + (15*B*a^2*b^4*e)/7) + x^2*((A*a^6*e)/2 + (B*a^6*d)/2 + 3*A*a^5*b*d) + x^8*((A*b^6*e)/8 + (B*b^6*d)/8
+ (3*B*a*b^5*e)/4) + a^2*b^2*x^5*(3*A*b^2*d + 3*B*a^2*e + 4*A*a*b*e + 4*B*a*b*d) + A*a^6*d*x + (B*b^6*e*x^9)/9
+ (a^3*b*x^4*(20*A*b^2*d + 6*B*a^2*e + 15*A*a*b*e + 15*B*a*b*d))/4 + (a*b^3*x^6*(6*A*b^2*d + 20*B*a^2*e + 15*
A*a*b*e + 15*B*a*b*d))/6

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sympy [B]  time = 0.12, size = 333, normalized size = 4.44 \[ A a^{6} d x + \frac {B b^{6} e x^{9}}{9} + x^{8} \left (\frac {A b^{6} e}{8} + \frac {3 B a b^{5} e}{4} + \frac {B b^{6} d}{8}\right ) + x^{7} \left (\frac {6 A a b^{5} e}{7} + \frac {A b^{6} d}{7} + \frac {15 B a^{2} b^{4} e}{7} + \frac {6 B a b^{5} d}{7}\right ) + x^{6} \left (\frac {5 A a^{2} b^{4} e}{2} + A a b^{5} d + \frac {10 B a^{3} b^{3} e}{3} + \frac {5 B a^{2} b^{4} d}{2}\right ) + x^{5} \left (4 A a^{3} b^{3} e + 3 A a^{2} b^{4} d + 3 B a^{4} b^{2} e + 4 B a^{3} b^{3} d\right ) + x^{4} \left (\frac {15 A a^{4} b^{2} e}{4} + 5 A a^{3} b^{3} d + \frac {3 B a^{5} b e}{2} + \frac {15 B a^{4} b^{2} d}{4}\right ) + x^{3} \left (2 A a^{5} b e + 5 A a^{4} b^{2} d + \frac {B a^{6} e}{3} + 2 B a^{5} b d\right ) + x^{2} \left (\frac {A a^{6} e}{2} + 3 A a^{5} b d + \frac {B a^{6} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**6*(B*x+A)*(e*x+d),x)

[Out]

A*a**6*d*x + B*b**6*e*x**9/9 + x**8*(A*b**6*e/8 + 3*B*a*b**5*e/4 + B*b**6*d/8) + x**7*(6*A*a*b**5*e/7 + A*b**6
*d/7 + 15*B*a**2*b**4*e/7 + 6*B*a*b**5*d/7) + x**6*(5*A*a**2*b**4*e/2 + A*a*b**5*d + 10*B*a**3*b**3*e/3 + 5*B*
a**2*b**4*d/2) + x**5*(4*A*a**3*b**3*e + 3*A*a**2*b**4*d + 3*B*a**4*b**2*e + 4*B*a**3*b**3*d) + x**4*(15*A*a**
4*b**2*e/4 + 5*A*a**3*b**3*d + 3*B*a**5*b*e/2 + 15*B*a**4*b**2*d/4) + x**3*(2*A*a**5*b*e + 5*A*a**4*b**2*d + B
*a**6*e/3 + 2*B*a**5*b*d) + x**2*(A*a**6*e/2 + 3*A*a**5*b*d + B*a**6*d/2)

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