3.2.27 \(\int x^6 (a+b x)^{10} (A+B x) \, dx\)

Optimal. Leaf size=191 \[ \frac {a^6 (a+b x)^{11} (A b-a B)}{11 b^8}-\frac {a^5 (a+b x)^{12} (6 A b-7 a B)}{12 b^8}+\frac {3 a^4 (a+b x)^{13} (5 A b-7 a B)}{13 b^8}-\frac {5 a^3 (a+b x)^{14} (4 A b-7 a B)}{14 b^8}+\frac {a^2 (a+b x)^{15} (3 A b-7 a B)}{3 b^8}+\frac {(a+b x)^{17} (A b-7 a B)}{17 b^8}-\frac {3 a (a+b x)^{16} (2 A b-7 a B)}{16 b^8}+\frac {B (a+b x)^{18}}{18 b^8} \]

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Rubi [A]  time = 0.16, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {76} \begin {gather*} \frac {a^2 (a+b x)^{15} (3 A b-7 a B)}{3 b^8}-\frac {5 a^3 (a+b x)^{14} (4 A b-7 a B)}{14 b^8}+\frac {3 a^4 (a+b x)^{13} (5 A b-7 a B)}{13 b^8}-\frac {a^5 (a+b x)^{12} (6 A b-7 a B)}{12 b^8}+\frac {a^6 (a+b x)^{11} (A b-a B)}{11 b^8}+\frac {(a+b x)^{17} (A b-7 a B)}{17 b^8}-\frac {3 a (a+b x)^{16} (2 A b-7 a B)}{16 b^8}+\frac {B (a+b x)^{18}}{18 b^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*(a + b*x)^10*(A + B*x),x]

[Out]

(a^6*(A*b - a*B)*(a + b*x)^11)/(11*b^8) - (a^5*(6*A*b - 7*a*B)*(a + b*x)^12)/(12*b^8) + (3*a^4*(5*A*b - 7*a*B)
*(a + b*x)^13)/(13*b^8) - (5*a^3*(4*A*b - 7*a*B)*(a + b*x)^14)/(14*b^8) + (a^2*(3*A*b - 7*a*B)*(a + b*x)^15)/(
3*b^8) - (3*a*(2*A*b - 7*a*B)*(a + b*x)^16)/(16*b^8) + ((A*b - 7*a*B)*(a + b*x)^17)/(17*b^8) + (B*(a + b*x)^18
)/(18*b^8)

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin {align*} \int x^6 (a+b x)^{10} (A+B x) \, dx &=\int \left (-\frac {a^6 (-A b+a B) (a+b x)^{10}}{b^7}+\frac {a^5 (-6 A b+7 a B) (a+b x)^{11}}{b^7}-\frac {3 a^4 (-5 A b+7 a B) (a+b x)^{12}}{b^7}+\frac {5 a^3 (-4 A b+7 a B) (a+b x)^{13}}{b^7}-\frac {5 a^2 (-3 A b+7 a B) (a+b x)^{14}}{b^7}+\frac {3 a (-2 A b+7 a B) (a+b x)^{15}}{b^7}+\frac {(A b-7 a B) (a+b x)^{16}}{b^7}+\frac {B (a+b x)^{17}}{b^7}\right ) \, dx\\ &=\frac {a^6 (A b-a B) (a+b x)^{11}}{11 b^8}-\frac {a^5 (6 A b-7 a B) (a+b x)^{12}}{12 b^8}+\frac {3 a^4 (5 A b-7 a B) (a+b x)^{13}}{13 b^8}-\frac {5 a^3 (4 A b-7 a B) (a+b x)^{14}}{14 b^8}+\frac {a^2 (3 A b-7 a B) (a+b x)^{15}}{3 b^8}-\frac {3 a (2 A b-7 a B) (a+b x)^{16}}{16 b^8}+\frac {(A b-7 a B) (a+b x)^{17}}{17 b^8}+\frac {B (a+b x)^{18}}{18 b^8}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 228, normalized size = 1.19 \begin {gather*} \frac {1}{7} a^{10} A x^7+\frac {1}{8} a^9 x^8 (a B+10 A b)+\frac {5}{9} a^8 b x^9 (2 a B+9 A b)+\frac {3}{2} a^7 b^2 x^{10} (3 a B+8 A b)+\frac {30}{11} a^6 b^3 x^{11} (4 a B+7 A b)+\frac {7}{2} a^5 b^4 x^{12} (5 a B+6 A b)+\frac {42}{13} a^4 b^5 x^{13} (6 a B+5 A b)+\frac {15}{7} a^3 b^6 x^{14} (7 a B+4 A b)+a^2 b^7 x^{15} (8 a B+3 A b)+\frac {1}{17} b^9 x^{17} (10 a B+A b)+\frac {5}{16} a b^8 x^{16} (9 a B+2 A b)+\frac {1}{18} b^{10} B x^{18} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(a + b*x)^10*(A + B*x),x]

[Out]

(a^10*A*x^7)/7 + (a^9*(10*A*b + a*B)*x^8)/8 + (5*a^8*b*(9*A*b + 2*a*B)*x^9)/9 + (3*a^7*b^2*(8*A*b + 3*a*B)*x^1
0)/2 + (30*a^6*b^3*(7*A*b + 4*a*B)*x^11)/11 + (7*a^5*b^4*(6*A*b + 5*a*B)*x^12)/2 + (42*a^4*b^5*(5*A*b + 6*a*B)
*x^13)/13 + (15*a^3*b^6*(4*A*b + 7*a*B)*x^14)/7 + a^2*b^7*(3*A*b + 8*a*B)*x^15 + (5*a*b^8*(2*A*b + 9*a*B)*x^16
)/16 + (b^9*(A*b + 10*a*B)*x^17)/17 + (b^10*B*x^18)/18

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^6 (a+b x)^{10} (A+B x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^6*(a + b*x)^10*(A + B*x),x]

[Out]

IntegrateAlgebraic[x^6*(a + b*x)^10*(A + B*x), x]

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fricas [A]  time = 1.05, size = 245, normalized size = 1.28 \begin {gather*} \frac {1}{18} x^{18} b^{10} B + \frac {10}{17} x^{17} b^{9} a B + \frac {1}{17} x^{17} b^{10} A + \frac {45}{16} x^{16} b^{8} a^{2} B + \frac {5}{8} x^{16} b^{9} a A + 8 x^{15} b^{7} a^{3} B + 3 x^{15} b^{8} a^{2} A + 15 x^{14} b^{6} a^{4} B + \frac {60}{7} x^{14} b^{7} a^{3} A + \frac {252}{13} x^{13} b^{5} a^{5} B + \frac {210}{13} x^{13} b^{6} a^{4} A + \frac {35}{2} x^{12} b^{4} a^{6} B + 21 x^{12} b^{5} a^{5} A + \frac {120}{11} x^{11} b^{3} a^{7} B + \frac {210}{11} x^{11} b^{4} a^{6} A + \frac {9}{2} x^{10} b^{2} a^{8} B + 12 x^{10} b^{3} a^{7} A + \frac {10}{9} x^{9} b a^{9} B + 5 x^{9} b^{2} a^{8} A + \frac {1}{8} x^{8} a^{10} B + \frac {5}{4} x^{8} b a^{9} A + \frac {1}{7} x^{7} a^{10} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^10*(B*x+A),x, algorithm="fricas")

[Out]

1/18*x^18*b^10*B + 10/17*x^17*b^9*a*B + 1/17*x^17*b^10*A + 45/16*x^16*b^8*a^2*B + 5/8*x^16*b^9*a*A + 8*x^15*b^
7*a^3*B + 3*x^15*b^8*a^2*A + 15*x^14*b^6*a^4*B + 60/7*x^14*b^7*a^3*A + 252/13*x^13*b^5*a^5*B + 210/13*x^13*b^6
*a^4*A + 35/2*x^12*b^4*a^6*B + 21*x^12*b^5*a^5*A + 120/11*x^11*b^3*a^7*B + 210/11*x^11*b^4*a^6*A + 9/2*x^10*b^
2*a^8*B + 12*x^10*b^3*a^7*A + 10/9*x^9*b*a^9*B + 5*x^9*b^2*a^8*A + 1/8*x^8*a^10*B + 5/4*x^8*b*a^9*A + 1/7*x^7*
a^10*A

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giac [A]  time = 1.21, size = 245, normalized size = 1.28 \begin {gather*} \frac {1}{18} \, B b^{10} x^{18} + \frac {10}{17} \, B a b^{9} x^{17} + \frac {1}{17} \, A b^{10} x^{17} + \frac {45}{16} \, B a^{2} b^{8} x^{16} + \frac {5}{8} \, A a b^{9} x^{16} + 8 \, B a^{3} b^{7} x^{15} + 3 \, A a^{2} b^{8} x^{15} + 15 \, B a^{4} b^{6} x^{14} + \frac {60}{7} \, A a^{3} b^{7} x^{14} + \frac {252}{13} \, B a^{5} b^{5} x^{13} + \frac {210}{13} \, A a^{4} b^{6} x^{13} + \frac {35}{2} \, B a^{6} b^{4} x^{12} + 21 \, A a^{5} b^{5} x^{12} + \frac {120}{11} \, B a^{7} b^{3} x^{11} + \frac {210}{11} \, A a^{6} b^{4} x^{11} + \frac {9}{2} \, B a^{8} b^{2} x^{10} + 12 \, A a^{7} b^{3} x^{10} + \frac {10}{9} \, B a^{9} b x^{9} + 5 \, A a^{8} b^{2} x^{9} + \frac {1}{8} \, B a^{10} x^{8} + \frac {5}{4} \, A a^{9} b x^{8} + \frac {1}{7} \, A a^{10} x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^10*(B*x+A),x, algorithm="giac")

[Out]

1/18*B*b^10*x^18 + 10/17*B*a*b^9*x^17 + 1/17*A*b^10*x^17 + 45/16*B*a^2*b^8*x^16 + 5/8*A*a*b^9*x^16 + 8*B*a^3*b
^7*x^15 + 3*A*a^2*b^8*x^15 + 15*B*a^4*b^6*x^14 + 60/7*A*a^3*b^7*x^14 + 252/13*B*a^5*b^5*x^13 + 210/13*A*a^4*b^
6*x^13 + 35/2*B*a^6*b^4*x^12 + 21*A*a^5*b^5*x^12 + 120/11*B*a^7*b^3*x^11 + 210/11*A*a^6*b^4*x^11 + 9/2*B*a^8*b
^2*x^10 + 12*A*a^7*b^3*x^10 + 10/9*B*a^9*b*x^9 + 5*A*a^8*b^2*x^9 + 1/8*B*a^10*x^8 + 5/4*A*a^9*b*x^8 + 1/7*A*a^
10*x^7

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maple [A]  time = 0.00, size = 244, normalized size = 1.28 \begin {gather*} \frac {B \,b^{10} x^{18}}{18}+\frac {A \,a^{10} x^{7}}{7}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{17}}{17}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{16}}{16}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{15}}{15}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{14}}{14}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{13}}{13}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{12}}{12}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{11}}{11}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{10}}{10}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{9}}{9}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(b*x+a)^10*(B*x+A),x)

[Out]

1/18*b^10*B*x^18+1/17*(A*b^10+10*B*a*b^9)*x^17+1/16*(10*A*a*b^9+45*B*a^2*b^8)*x^16+1/15*(45*A*a^2*b^8+120*B*a^
3*b^7)*x^15+1/14*(120*A*a^3*b^7+210*B*a^4*b^6)*x^14+1/13*(210*A*a^4*b^6+252*B*a^5*b^5)*x^13+1/12*(252*A*a^5*b^
5+210*B*a^6*b^4)*x^12+1/11*(210*A*a^6*b^4+120*B*a^7*b^3)*x^11+1/10*(120*A*a^7*b^3+45*B*a^8*b^2)*x^10+1/9*(45*A
*a^8*b^2+10*B*a^9*b)*x^9+1/8*(10*A*a^9*b+B*a^10)*x^8+1/7*a^10*A*x^7

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maxima [A]  time = 1.11, size = 242, normalized size = 1.27 \begin {gather*} \frac {1}{18} \, B b^{10} x^{18} + \frac {1}{7} \, A a^{10} x^{7} + \frac {1}{17} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{17} + \frac {5}{16} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{16} + {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{15} + \frac {15}{7} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{14} + \frac {42}{13} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{13} + \frac {7}{2} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{12} + \frac {30}{11} \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{11} + \frac {3}{2} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{10} + \frac {5}{9} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{9} + \frac {1}{8} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(b*x+a)^10*(B*x+A),x, algorithm="maxima")

[Out]

1/18*B*b^10*x^18 + 1/7*A*a^10*x^7 + 1/17*(10*B*a*b^9 + A*b^10)*x^17 + 5/16*(9*B*a^2*b^8 + 2*A*a*b^9)*x^16 + (8
*B*a^3*b^7 + 3*A*a^2*b^8)*x^15 + 15/7*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^14 + 42/13*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^1
3 + 7/2*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^12 + 30/11*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^11 + 3/2*(3*B*a^8*b^2 + 8*A*a^7
*b^3)*x^10 + 5/9*(2*B*a^9*b + 9*A*a^8*b^2)*x^9 + 1/8*(B*a^10 + 10*A*a^9*b)*x^8

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mupad [B]  time = 0.11, size = 210, normalized size = 1.10 \begin {gather*} x^8\,\left (\frac {B\,a^{10}}{8}+\frac {5\,A\,b\,a^9}{4}\right )+x^{17}\,\left (\frac {A\,b^{10}}{17}+\frac {10\,B\,a\,b^9}{17}\right )+\frac {A\,a^{10}\,x^7}{7}+\frac {B\,b^{10}\,x^{18}}{18}+\frac {3\,a^7\,b^2\,x^{10}\,\left (8\,A\,b+3\,B\,a\right )}{2}+\frac {30\,a^6\,b^3\,x^{11}\,\left (7\,A\,b+4\,B\,a\right )}{11}+\frac {7\,a^5\,b^4\,x^{12}\,\left (6\,A\,b+5\,B\,a\right )}{2}+\frac {42\,a^4\,b^5\,x^{13}\,\left (5\,A\,b+6\,B\,a\right )}{13}+\frac {15\,a^3\,b^6\,x^{14}\,\left (4\,A\,b+7\,B\,a\right )}{7}+a^2\,b^7\,x^{15}\,\left (3\,A\,b+8\,B\,a\right )+\frac {5\,a^8\,b\,x^9\,\left (9\,A\,b+2\,B\,a\right )}{9}+\frac {5\,a\,b^8\,x^{16}\,\left (2\,A\,b+9\,B\,a\right )}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(A + B*x)*(a + b*x)^10,x)

[Out]

x^8*((B*a^10)/8 + (5*A*a^9*b)/4) + x^17*((A*b^10)/17 + (10*B*a*b^9)/17) + (A*a^10*x^7)/7 + (B*b^10*x^18)/18 +
(3*a^7*b^2*x^10*(8*A*b + 3*B*a))/2 + (30*a^6*b^3*x^11*(7*A*b + 4*B*a))/11 + (7*a^5*b^4*x^12*(6*A*b + 5*B*a))/2
+ (42*a^4*b^5*x^13*(5*A*b + 6*B*a))/13 + (15*a^3*b^6*x^14*(4*A*b + 7*B*a))/7 + a^2*b^7*x^15*(3*A*b + 8*B*a) +
(5*a^8*b*x^9*(9*A*b + 2*B*a))/9 + (5*a*b^8*x^16*(2*A*b + 9*B*a))/16

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sympy [A]  time = 0.20, size = 264, normalized size = 1.38 \begin {gather*} \frac {A a^{10} x^{7}}{7} + \frac {B b^{10} x^{18}}{18} + x^{17} \left (\frac {A b^{10}}{17} + \frac {10 B a b^{9}}{17}\right ) + x^{16} \left (\frac {5 A a b^{9}}{8} + \frac {45 B a^{2} b^{8}}{16}\right ) + x^{15} \left (3 A a^{2} b^{8} + 8 B a^{3} b^{7}\right ) + x^{14} \left (\frac {60 A a^{3} b^{7}}{7} + 15 B a^{4} b^{6}\right ) + x^{13} \left (\frac {210 A a^{4} b^{6}}{13} + \frac {252 B a^{5} b^{5}}{13}\right ) + x^{12} \left (21 A a^{5} b^{5} + \frac {35 B a^{6} b^{4}}{2}\right ) + x^{11} \left (\frac {210 A a^{6} b^{4}}{11} + \frac {120 B a^{7} b^{3}}{11}\right ) + x^{10} \left (12 A a^{7} b^{3} + \frac {9 B a^{8} b^{2}}{2}\right ) + x^{9} \left (5 A a^{8} b^{2} + \frac {10 B a^{9} b}{9}\right ) + x^{8} \left (\frac {5 A a^{9} b}{4} + \frac {B a^{10}}{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(b*x+a)**10*(B*x+A),x)

[Out]

A*a**10*x**7/7 + B*b**10*x**18/18 + x**17*(A*b**10/17 + 10*B*a*b**9/17) + x**16*(5*A*a*b**9/8 + 45*B*a**2*b**8
/16) + x**15*(3*A*a**2*b**8 + 8*B*a**3*b**7) + x**14*(60*A*a**3*b**7/7 + 15*B*a**4*b**6) + x**13*(210*A*a**4*b
**6/13 + 252*B*a**5*b**5/13) + x**12*(21*A*a**5*b**5 + 35*B*a**6*b**4/2) + x**11*(210*A*a**6*b**4/11 + 120*B*a
**7*b**3/11) + x**10*(12*A*a**7*b**3 + 9*B*a**8*b**2/2) + x**9*(5*A*a**8*b**2 + 10*B*a**9*b/9) + x**8*(5*A*a**
9*b/4 + B*a**10/8)

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