3.109 \(\int x^7 (a+b x)^{10} (A+B x) \, dx\)

Optimal. Leaf size=215 \[ -\frac{a^7 (a+b x)^{11} (A b-a B)}{11 b^9}+\frac{a^6 (a+b x)^{12} (7 A b-8 a B)}{12 b^9}-\frac{7 a^5 (a+b x)^{13} (3 A b-4 a B)}{13 b^9}+\frac{a^4 (a+b x)^{14} (5 A b-8 a B)}{2 b^9}-\frac{7 a^3 (a+b x)^{15} (A b-2 a B)}{3 b^9}+\frac{7 a^2 (a+b x)^{16} (3 A b-8 a B)}{16 b^9}+\frac{(a+b x)^{18} (A b-8 a B)}{18 b^9}-\frac{7 a (a+b x)^{17} (A b-4 a B)}{17 b^9}+\frac{B (a+b x)^{19}}{19 b^9} \]

[Out]

-(a^7*(A*b - a*B)*(a + b*x)^11)/(11*b^9) + (a^6*(7*A*b - 8*a*B)*(a + b*x)^12)/(1
2*b^9) - (7*a^5*(3*A*b - 4*a*B)*(a + b*x)^13)/(13*b^9) + (a^4*(5*A*b - 8*a*B)*(a
+ b*x)^14)/(2*b^9) - (7*a^3*(A*b - 2*a*B)*(a + b*x)^15)/(3*b^9) + (7*a^2*(3*A*b
- 8*a*B)*(a + b*x)^16)/(16*b^9) - (7*a*(A*b - 4*a*B)*(a + b*x)^17)/(17*b^9) + (
(A*b - 8*a*B)*(a + b*x)^18)/(18*b^9) + (B*(a + b*x)^19)/(19*b^9)

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Rubi [A]  time = 0.568197, antiderivative size = 215, 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 \[ -\frac{a^7 (a+b x)^{11} (A b-a B)}{11 b^9}+\frac{a^6 (a+b x)^{12} (7 A b-8 a B)}{12 b^9}-\frac{7 a^5 (a+b x)^{13} (3 A b-4 a B)}{13 b^9}+\frac{a^4 (a+b x)^{14} (5 A b-8 a B)}{2 b^9}-\frac{7 a^3 (a+b x)^{15} (A b-2 a B)}{3 b^9}+\frac{7 a^2 (a+b x)^{16} (3 A b-8 a B)}{16 b^9}+\frac{(a+b x)^{18} (A b-8 a B)}{18 b^9}-\frac{7 a (a+b x)^{17} (A b-4 a B)}{17 b^9}+\frac{B (a+b x)^{19}}{19 b^9} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x)^10*(A + B*x),x]

[Out]

-(a^7*(A*b - a*B)*(a + b*x)^11)/(11*b^9) + (a^6*(7*A*b - 8*a*B)*(a + b*x)^12)/(1
2*b^9) - (7*a^5*(3*A*b - 4*a*B)*(a + b*x)^13)/(13*b^9) + (a^4*(5*A*b - 8*a*B)*(a
+ b*x)^14)/(2*b^9) - (7*a^3*(A*b - 2*a*B)*(a + b*x)^15)/(3*b^9) + (7*a^2*(3*A*b
- 8*a*B)*(a + b*x)^16)/(16*b^9) - (7*a*(A*b - 4*a*B)*(a + b*x)^17)/(17*b^9) + (
(A*b - 8*a*B)*(a + b*x)^18)/(18*b^9) + (B*(a + b*x)^19)/(19*b^9)

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Rubi in Sympy [A]  time = 105.856, size = 209, normalized size = 0.97 \[ \frac{B \left (a + b x\right )^{19}}{19 b^{9}} - \frac{a^{7} \left (a + b x\right )^{11} \left (A b - B a\right )}{11 b^{9}} + \frac{a^{6} \left (a + b x\right )^{12} \left (7 A b - 8 B a\right )}{12 b^{9}} - \frac{7 a^{5} \left (a + b x\right )^{13} \left (3 A b - 4 B a\right )}{13 b^{9}} + \frac{a^{4} \left (a + b x\right )^{14} \left (5 A b - 8 B a\right )}{2 b^{9}} - \frac{7 a^{3} \left (a + b x\right )^{15} \left (A b - 2 B a\right )}{3 b^{9}} + \frac{7 a^{2} \left (a + b x\right )^{16} \left (3 A b - 8 B a\right )}{16 b^{9}} - \frac{7 a \left (a + b x\right )^{17} \left (A b - 4 B a\right )}{17 b^{9}} + \frac{\left (a + b x\right )^{18} \left (A b - 8 B a\right )}{18 b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x+a)**10*(B*x+A),x)

[Out]

B*(a + b*x)**19/(19*b**9) - a**7*(a + b*x)**11*(A*b - B*a)/(11*b**9) + a**6*(a +
b*x)**12*(7*A*b - 8*B*a)/(12*b**9) - 7*a**5*(a + b*x)**13*(3*A*b - 4*B*a)/(13*b
**9) + a**4*(a + b*x)**14*(5*A*b - 8*B*a)/(2*b**9) - 7*a**3*(a + b*x)**15*(A*b -
2*B*a)/(3*b**9) + 7*a**2*(a + b*x)**16*(3*A*b - 8*B*a)/(16*b**9) - 7*a*(a + b*x
)**17*(A*b - 4*B*a)/(17*b**9) + (a + b*x)**18*(A*b - 8*B*a)/(18*b**9)

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Mathematica [A]  time = 0.051708, size = 227, normalized size = 1.06 \[ \frac{1}{8} a^{10} A x^8+\frac{1}{9} a^9 x^9 (a B+10 A b)+\frac{1}{2} a^8 b x^{10} (2 a B+9 A b)+\frac{15}{11} a^7 b^2 x^{11} (3 a B+8 A b)+\frac{5}{2} a^6 b^3 x^{12} (4 a B+7 A b)+\frac{42}{13} a^5 b^4 x^{13} (5 a B+6 A b)+3 a^4 b^5 x^{14} (6 a B+5 A b)+2 a^3 b^6 x^{15} (7 a B+4 A b)+\frac{15}{16} a^2 b^7 x^{16} (8 a B+3 A b)+\frac{1}{18} b^9 x^{18} (10 a B+A b)+\frac{5}{17} a b^8 x^{17} (9 a B+2 A b)+\frac{1}{19} b^{10} B x^{19} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a + b*x)^10*(A + B*x),x]

[Out]

(a^10*A*x^8)/8 + (a^9*(10*A*b + a*B)*x^9)/9 + (a^8*b*(9*A*b + 2*a*B)*x^10)/2 + (
15*a^7*b^2*(8*A*b + 3*a*B)*x^11)/11 + (5*a^6*b^3*(7*A*b + 4*a*B)*x^12)/2 + (42*a
^5*b^4*(6*A*b + 5*a*B)*x^13)/13 + 3*a^4*b^5*(5*A*b + 6*a*B)*x^14 + 2*a^3*b^6*(4*
A*b + 7*a*B)*x^15 + (15*a^2*b^7*(3*A*b + 8*a*B)*x^16)/16 + (5*a*b^8*(2*A*b + 9*a
*B)*x^17)/17 + (b^9*(A*b + 10*a*B)*x^18)/18 + (b^10*B*x^19)/19

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Maple [A]  time = 0.001, size = 244, normalized size = 1.1 \[{\frac{{b}^{10}B{x}^{19}}{19}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{18}}{18}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{17}}{17}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{16}}{16}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{15}}{15}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{14}}{14}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{13}}{13}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{12}}{12}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{11}}{11}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{9}}{9}}+{\frac{{a}^{10}A{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x+a)^10*(B*x+A),x)

[Out]

1/19*b^10*B*x^19+1/18*(A*b^10+10*B*a*b^9)*x^18+1/17*(10*A*a*b^9+45*B*a^2*b^8)*x^
17+1/16*(45*A*a^2*b^8+120*B*a^3*b^7)*x^16+1/15*(120*A*a^3*b^7+210*B*a^4*b^6)*x^1
5+1/14*(210*A*a^4*b^6+252*B*a^5*b^5)*x^14+1/13*(252*A*a^5*b^5+210*B*a^6*b^4)*x^1
3+1/12*(210*A*a^6*b^4+120*B*a^7*b^3)*x^12+1/11*(120*A*a^7*b^3+45*B*a^8*b^2)*x^11
+1/10*(45*A*a^8*b^2+10*B*a^9*b)*x^10+1/9*(10*A*a^9*b+B*a^10)*x^9+1/8*a^10*A*x^8

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Maxima [A]  time = 1.36136, size = 328, normalized size = 1.53 \[ \frac{1}{19} \, B b^{10} x^{19} + \frac{1}{8} \, A a^{10} x^{8} + \frac{1}{18} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{18} + \frac{5}{17} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{17} + \frac{15}{16} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{16} + 2 \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{15} + 3 \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{14} + \frac{42}{13} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{13} + \frac{5}{2} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{12} + \frac{15}{11} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{11} + \frac{1}{2} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{10} + \frac{1}{9} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^7,x, algorithm="maxima")

[Out]

1/19*B*b^10*x^19 + 1/8*A*a^10*x^8 + 1/18*(10*B*a*b^9 + A*b^10)*x^18 + 5/17*(9*B*
a^2*b^8 + 2*A*a*b^9)*x^17 + 15/16*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^16 + 2*(7*B*a^4*
b^6 + 4*A*a^3*b^7)*x^15 + 3*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^14 + 42/13*(5*B*a^6*b^
4 + 6*A*a^5*b^5)*x^13 + 5/2*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^12 + 15/11*(3*B*a^8*b^
2 + 8*A*a^7*b^3)*x^11 + 1/2*(2*B*a^9*b + 9*A*a^8*b^2)*x^10 + 1/9*(B*a^10 + 10*A*
a^9*b)*x^9

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Fricas [A]  time = 0.177874, size = 1, normalized size = 0. \[ \frac{1}{19} x^{19} b^{10} B + \frac{5}{9} x^{18} b^{9} a B + \frac{1}{18} x^{18} b^{10} A + \frac{45}{17} x^{17} b^{8} a^{2} B + \frac{10}{17} x^{17} b^{9} a A + \frac{15}{2} x^{16} b^{7} a^{3} B + \frac{45}{16} x^{16} b^{8} a^{2} A + 14 x^{15} b^{6} a^{4} B + 8 x^{15} b^{7} a^{3} A + 18 x^{14} b^{5} a^{5} B + 15 x^{14} b^{6} a^{4} A + \frac{210}{13} x^{13} b^{4} a^{6} B + \frac{252}{13} x^{13} b^{5} a^{5} A + 10 x^{12} b^{3} a^{7} B + \frac{35}{2} x^{12} b^{4} a^{6} A + \frac{45}{11} x^{11} b^{2} a^{8} B + \frac{120}{11} x^{11} b^{3} a^{7} A + x^{10} b a^{9} B + \frac{9}{2} x^{10} b^{2} a^{8} A + \frac{1}{9} x^{9} a^{10} B + \frac{10}{9} x^{9} b a^{9} A + \frac{1}{8} x^{8} a^{10} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^7,x, algorithm="fricas")

[Out]

1/19*x^19*b^10*B + 5/9*x^18*b^9*a*B + 1/18*x^18*b^10*A + 45/17*x^17*b^8*a^2*B +
10/17*x^17*b^9*a*A + 15/2*x^16*b^7*a^3*B + 45/16*x^16*b^8*a^2*A + 14*x^15*b^6*a^
4*B + 8*x^15*b^7*a^3*A + 18*x^14*b^5*a^5*B + 15*x^14*b^6*a^4*A + 210/13*x^13*b^4
*a^6*B + 252/13*x^13*b^5*a^5*A + 10*x^12*b^3*a^7*B + 35/2*x^12*b^4*a^6*A + 45/11
*x^11*b^2*a^8*B + 120/11*x^11*b^3*a^7*A + x^10*b*a^9*B + 9/2*x^10*b^2*a^8*A + 1/
9*x^9*a^10*B + 10/9*x^9*b*a^9*A + 1/8*x^8*a^10*A

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Sympy [A]  time = 0.262629, size = 262, normalized size = 1.22 \[ \frac{A a^{10} x^{8}}{8} + \frac{B b^{10} x^{19}}{19} + x^{18} \left (\frac{A b^{10}}{18} + \frac{5 B a b^{9}}{9}\right ) + x^{17} \left (\frac{10 A a b^{9}}{17} + \frac{45 B a^{2} b^{8}}{17}\right ) + x^{16} \left (\frac{45 A a^{2} b^{8}}{16} + \frac{15 B a^{3} b^{7}}{2}\right ) + x^{15} \left (8 A a^{3} b^{7} + 14 B a^{4} b^{6}\right ) + x^{14} \left (15 A a^{4} b^{6} + 18 B a^{5} b^{5}\right ) + x^{13} \left (\frac{252 A a^{5} b^{5}}{13} + \frac{210 B a^{6} b^{4}}{13}\right ) + x^{12} \left (\frac{35 A a^{6} b^{4}}{2} + 10 B a^{7} b^{3}\right ) + x^{11} \left (\frac{120 A a^{7} b^{3}}{11} + \frac{45 B a^{8} b^{2}}{11}\right ) + x^{10} \left (\frac{9 A a^{8} b^{2}}{2} + B a^{9} b\right ) + x^{9} \left (\frac{10 A a^{9} b}{9} + \frac{B a^{10}}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*a**10*x**8/8 + B*b**10*x**19/19 + x**18*(A*b**10/18 + 5*B*a*b**9/9) + x**17*(1
0*A*a*b**9/17 + 45*B*a**2*b**8/17) + x**16*(45*A*a**2*b**8/16 + 15*B*a**3*b**7/2
) + x**15*(8*A*a**3*b**7 + 14*B*a**4*b**6) + x**14*(15*A*a**4*b**6 + 18*B*a**5*b
**5) + x**13*(252*A*a**5*b**5/13 + 210*B*a**6*b**4/13) + x**12*(35*A*a**6*b**4/2
+ 10*B*a**7*b**3) + x**11*(120*A*a**7*b**3/11 + 45*B*a**8*b**2/11) + x**10*(9*A
*a**8*b**2/2 + B*a**9*b) + x**9*(10*A*a**9*b/9 + B*a**10/9)

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GIAC/XCAS [A]  time = 0.273861, size = 329, normalized size = 1.53 \[ \frac{1}{19} \, B b^{10} x^{19} + \frac{5}{9} \, B a b^{9} x^{18} + \frac{1}{18} \, A b^{10} x^{18} + \frac{45}{17} \, B a^{2} b^{8} x^{17} + \frac{10}{17} \, A a b^{9} x^{17} + \frac{15}{2} \, B a^{3} b^{7} x^{16} + \frac{45}{16} \, A a^{2} b^{8} x^{16} + 14 \, B a^{4} b^{6} x^{15} + 8 \, A a^{3} b^{7} x^{15} + 18 \, B a^{5} b^{5} x^{14} + 15 \, A a^{4} b^{6} x^{14} + \frac{210}{13} \, B a^{6} b^{4} x^{13} + \frac{252}{13} \, A a^{5} b^{5} x^{13} + 10 \, B a^{7} b^{3} x^{12} + \frac{35}{2} \, A a^{6} b^{4} x^{12} + \frac{45}{11} \, B a^{8} b^{2} x^{11} + \frac{120}{11} \, A a^{7} b^{3} x^{11} + B a^{9} b x^{10} + \frac{9}{2} \, A a^{8} b^{2} x^{10} + \frac{1}{9} \, B a^{10} x^{9} + \frac{10}{9} \, A a^{9} b x^{9} + \frac{1}{8} \, A a^{10} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^7,x, algorithm="giac")

[Out]

1/19*B*b^10*x^19 + 5/9*B*a*b^9*x^18 + 1/18*A*b^10*x^18 + 45/17*B*a^2*b^8*x^17 +
10/17*A*a*b^9*x^17 + 15/2*B*a^3*b^7*x^16 + 45/16*A*a^2*b^8*x^16 + 14*B*a^4*b^6*x
^15 + 8*A*a^3*b^7*x^15 + 18*B*a^5*b^5*x^14 + 15*A*a^4*b^6*x^14 + 210/13*B*a^6*b^
4*x^13 + 252/13*A*a^5*b^5*x^13 + 10*B*a^7*b^3*x^12 + 35/2*A*a^6*b^4*x^12 + 45/11
*B*a^8*b^2*x^11 + 120/11*A*a^7*b^3*x^11 + B*a^9*b*x^10 + 9/2*A*a^8*b^2*x^10 + 1/
9*B*a^10*x^9 + 10/9*A*a^9*b*x^9 + 1/8*A*a^10*x^8