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

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

[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)

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Rubi [A]  time = 0.54184, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{a^6 (a+b x)^{11} (A b-a B)}{11 b^8}-\frac{a^5 (a+b x)^{12} (6 A b-7 a B)}{12 b^8}+\frac{3 a^4 (a+b x)^{13} (5 A b-7 a B)}{13 b^8}-\frac{5 a^3 (a+b x)^{14} (4 A b-7 a B)}{14 b^8}+\frac{a^2 (a+b x)^{15} (3 A b-7 a B)}{3 b^8}+\frac{(a+b x)^{17} (A b-7 a B)}{17 b^8}-\frac{3 a (a+b x)^{16} (2 A b-7 a B)}{16 b^8}+\frac{B (a+b x)^{18}}{18 b^8} \]

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)

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Rubi in Sympy [A]  time = 90.7794, size = 185, normalized size = 0.97 \[ \frac{B \left (a + b x\right )^{18}}{18 b^{8}} + \frac{a^{6} \left (a + b x\right )^{11} \left (A b - B a\right )}{11 b^{8}} - \frac{a^{5} \left (a + b x\right )^{12} \left (6 A b - 7 B a\right )}{12 b^{8}} + \frac{3 a^{4} \left (a + b x\right )^{13} \left (5 A b - 7 B a\right )}{13 b^{8}} - \frac{5 a^{3} \left (a + b x\right )^{14} \left (4 A b - 7 B a\right )}{14 b^{8}} + \frac{a^{2} \left (a + b x\right )^{15} \left (3 A b - 7 B a\right )}{3 b^{8}} - \frac{3 a \left (a + b x\right )^{16} \left (2 A b - 7 B a\right )}{16 b^{8}} + \frac{\left (a + b x\right )^{17} \left (A b - 7 B a\right )}{17 b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(b*x+a)**10*(B*x+A),x)

[Out]

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

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Mathematica [A]  time = 0.0538563, size = 228, normalized size = 1.19 \[ \frac{1}{7} a^{10} A x^7+\frac{1}{8} a^9 x^8 (a B+10 A b)+\frac{5}{9} a^8 b x^9 (2 a B+9 A b)+\frac{3}{2} a^7 b^2 x^{10} (3 a B+8 A b)+\frac{30}{11} a^6 b^3 x^{11} (4 a B+7 A b)+\frac{7}{2} a^5 b^4 x^{12} (5 a B+6 A b)+\frac{42}{13} a^4 b^5 x^{13} (6 a B+5 A b)+\frac{15}{7} a^3 b^6 x^{14} (7 a B+4 A b)+a^2 b^7 x^{15} (8 a B+3 A b)+\frac{1}{17} b^9 x^{17} (10 a B+A b)+\frac{5}{16} a b^8 x^{16} (9 a B+2 A b)+\frac{1}{18} b^{10} B x^{18} \]

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^10)/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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Maple [A]  time = 0.003, size = 244, normalized size = 1.3 \[{\frac{{b}^{10}B{x}^{18}}{18}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{17}}{17}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{16}}{16}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{15}}{15}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{14}}{14}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{13}}{13}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{12}}{12}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{11}}{11}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{10}}{10}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{9}}{9}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{8}}{8}}+{\frac{{a}^{10}A{x}^{7}}{7}} \]

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^1
4+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^1
2+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 = 1.34128, size = 327, normalized size = 1.71 \[ \frac{1}{18} \, B b^{10} x^{18} + \frac{1}{7} \, A a^{10} x^{7} + \frac{1}{17} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{17} + \frac{5}{16} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{16} +{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{15} + \frac{15}{7} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{14} + \frac{42}{13} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{13} + \frac{7}{2} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{12} + \frac{30}{11} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{11} + \frac{3}{2} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{10} + \frac{5}{9} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^6,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^13 + 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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Fricas [A]  time = 0.180815, size = 1, normalized size = 0.01 \[ \frac{1}{18} x^{18} b^{10} B + \frac{10}{17} x^{17} b^{9} a B + \frac{1}{17} x^{17} b^{10} A + \frac{45}{16} x^{16} b^{8} a^{2} B + \frac{5}{8} x^{16} b^{9} a A + 8 x^{15} b^{7} a^{3} B + 3 x^{15} b^{8} a^{2} A + 15 x^{14} b^{6} a^{4} B + \frac{60}{7} x^{14} b^{7} a^{3} A + \frac{252}{13} x^{13} b^{5} a^{5} B + \frac{210}{13} x^{13} b^{6} a^{4} A + \frac{35}{2} x^{12} b^{4} a^{6} B + 21 x^{12} b^{5} a^{5} A + \frac{120}{11} x^{11} b^{3} a^{7} B + \frac{210}{11} x^{11} b^{4} a^{6} A + \frac{9}{2} x^{10} b^{2} a^{8} B + 12 x^{10} b^{3} a^{7} A + \frac{10}{9} x^{9} b a^{9} B + 5 x^{9} b^{2} a^{8} A + \frac{1}{8} x^{8} a^{10} B + \frac{5}{4} x^{8} b a^{9} A + \frac{1}{7} x^{7} a^{10} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^6,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 + 6
0/7*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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Sympy [A]  time = 0.252341, size = 264, normalized size = 1.38 \[ \frac{A a^{10} x^{7}}{7} + \frac{B b^{10} x^{18}}{18} + x^{17} \left (\frac{A b^{10}}{17} + \frac{10 B a b^{9}}{17}\right ) + x^{16} \left (\frac{5 A a b^{9}}{8} + \frac{45 B a^{2} b^{8}}{16}\right ) + x^{15} \left (3 A a^{2} b^{8} + 8 B a^{3} b^{7}\right ) + x^{14} \left (\frac{60 A a^{3} b^{7}}{7} + 15 B a^{4} b^{6}\right ) + x^{13} \left (\frac{210 A a^{4} b^{6}}{13} + \frac{252 B a^{5} b^{5}}{13}\right ) + x^{12} \left (21 A a^{5} b^{5} + \frac{35 B a^{6} b^{4}}{2}\right ) + x^{11} \left (\frac{210 A a^{6} b^{4}}{11} + \frac{120 B a^{7} b^{3}}{11}\right ) + x^{10} \left (12 A a^{7} b^{3} + \frac{9 B a^{8} b^{2}}{2}\right ) + x^{9} \left (5 A a^{8} b^{2} + \frac{10 B a^{9} b}{9}\right ) + x^{8} \left (\frac{5 A a^{9} b}{4} + \frac{B a^{10}}{8}\right ) \]

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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GIAC/XCAS [A]  time = 0.282318, size = 331, normalized size = 1.73 \[ \frac{1}{18} \, B b^{10} x^{18} + \frac{10}{17} \, B a b^{9} x^{17} + \frac{1}{17} \, A b^{10} x^{17} + \frac{45}{16} \, B a^{2} b^{8} x^{16} + \frac{5}{8} \, A a b^{9} x^{16} + 8 \, B a^{3} b^{7} x^{15} + 3 \, A a^{2} b^{8} x^{15} + 15 \, B a^{4} b^{6} x^{14} + \frac{60}{7} \, A a^{3} b^{7} x^{14} + \frac{252}{13} \, B a^{5} b^{5} x^{13} + \frac{210}{13} \, A a^{4} b^{6} x^{13} + \frac{35}{2} \, B a^{6} b^{4} x^{12} + 21 \, A a^{5} b^{5} x^{12} + \frac{120}{11} \, B a^{7} b^{3} x^{11} + \frac{210}{11} \, A a^{6} b^{4} x^{11} + \frac{9}{2} \, B a^{8} b^{2} x^{10} + 12 \, A a^{7} b^{3} x^{10} + \frac{10}{9} \, B a^{9} b x^{9} + 5 \, A a^{8} b^{2} x^{9} + \frac{1}{8} \, B a^{10} x^{8} + \frac{5}{4} \, A a^{9} b x^{8} + \frac{1}{7} \, A a^{10} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^6,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 + 6
0/7*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