3.111 \(\int x^5 (a+b x)^{10} (A+B x) \, dx\)

Optimal. Leaf size=163 \[ -\frac{a^5 (a+b x)^{11} (A b-a B)}{11 b^7}+\frac{a^4 (a+b x)^{12} (5 A b-6 a B)}{12 b^7}-\frac{5 a^3 (a+b x)^{13} (2 A b-3 a B)}{13 b^7}+\frac{5 a^2 (a+b x)^{14} (A b-2 a B)}{7 b^7}+\frac{(a+b x)^{16} (A b-6 a B)}{16 b^7}-\frac{a (a+b x)^{15} (A b-3 a B)}{3 b^7}+\frac{B (a+b x)^{17}}{17 b^7} \]

[Out]

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

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Rubi [A]  time = 0.481367, antiderivative size = 163, 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 \[ -\frac{a^5 (a+b x)^{11} (A b-a B)}{11 b^7}+\frac{a^4 (a+b x)^{12} (5 A b-6 a B)}{12 b^7}-\frac{5 a^3 (a+b x)^{13} (2 A b-3 a B)}{13 b^7}+\frac{5 a^2 (a+b x)^{14} (A b-2 a B)}{7 b^7}+\frac{(a+b x)^{16} (A b-6 a B)}{16 b^7}-\frac{a (a+b x)^{15} (A b-3 a B)}{3 b^7}+\frac{B (a+b x)^{17}}{17 b^7} \]

Antiderivative was successfully verified.

[In]  Int[x^5*(a + b*x)^10*(A + B*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 81.417, size = 155, normalized size = 0.95 \[ \frac{B \left (a + b x\right )^{17}}{17 b^{7}} - \frac{a^{5} \left (a + b x\right )^{11} \left (A b - B a\right )}{11 b^{7}} + \frac{a^{4} \left (a + b x\right )^{12} \left (5 A b - 6 B a\right )}{12 b^{7}} - \frac{5 a^{3} \left (a + b x\right )^{13} \left (2 A b - 3 B a\right )}{13 b^{7}} + \frac{5 a^{2} \left (a + b x\right )^{14} \left (A b - 2 B a\right )}{7 b^{7}} - \frac{a \left (a + b x\right )^{15} \left (A b - 3 B a\right )}{3 b^{7}} + \frac{\left (a + b x\right )^{16} \left (A b - 6 B a\right )}{16 b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x+a)**10*(B*x+A),x)

[Out]

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

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Mathematica [A]  time = 0.0509554, size = 229, normalized size = 1.4 \[ \frac{1}{6} a^{10} A x^6+\frac{1}{7} a^9 x^7 (a B+10 A b)+\frac{5}{8} a^8 b x^8 (2 a B+9 A b)+\frac{5}{3} a^7 b^2 x^9 (3 a B+8 A b)+3 a^6 b^3 x^{10} (4 a B+7 A b)+\frac{42}{11} a^5 b^4 x^{11} (5 a B+6 A b)+\frac{7}{2} a^4 b^5 x^{12} (6 a B+5 A b)+\frac{30}{13} a^3 b^6 x^{13} (7 a B+4 A b)+\frac{15}{14} a^2 b^7 x^{14} (8 a B+3 A b)+\frac{1}{16} b^9 x^{16} (10 a B+A b)+\frac{1}{3} a b^8 x^{15} (9 a B+2 A b)+\frac{1}{17} b^{10} B x^{17} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*(a + b*x)^10*(A + B*x),x]

[Out]

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

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Maple [A]  time = 0.002, size = 244, normalized size = 1.5 \[{\frac{{b}^{10}B{x}^{17}}{17}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{16}}{16}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{15}}{15}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{14}}{14}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{13}}{13}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{12}}{12}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{11}}{11}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{10}}{10}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{9}}{9}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{8}}{8}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{7}}{7}}+{\frac{{a}^{10}A{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x+a)^10*(B*x+A),x)

[Out]

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

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Maxima [A]  time = 1.35542, size = 328, normalized size = 2.01 \[ \frac{1}{17} \, B b^{10} x^{17} + \frac{1}{6} \, A a^{10} x^{6} + \frac{1}{16} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{16} + \frac{1}{3} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{15} + \frac{15}{14} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{14} + \frac{30}{13} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{13} + \frac{7}{2} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{12} + \frac{42}{11} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{11} + 3 \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{10} + \frac{5}{3} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{9} + \frac{5}{8} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{8} + \frac{1}{7} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.17962, size = 1, normalized size = 0.01 \[ \frac{1}{17} x^{17} b^{10} B + \frac{5}{8} x^{16} b^{9} a B + \frac{1}{16} x^{16} b^{10} A + 3 x^{15} b^{8} a^{2} B + \frac{2}{3} x^{15} b^{9} a A + \frac{60}{7} x^{14} b^{7} a^{3} B + \frac{45}{14} x^{14} b^{8} a^{2} A + \frac{210}{13} x^{13} b^{6} a^{4} B + \frac{120}{13} x^{13} b^{7} a^{3} A + 21 x^{12} b^{5} a^{5} B + \frac{35}{2} x^{12} b^{6} a^{4} A + \frac{210}{11} x^{11} b^{4} a^{6} B + \frac{252}{11} x^{11} b^{5} a^{5} A + 12 x^{10} b^{3} a^{7} B + 21 x^{10} b^{4} a^{6} A + 5 x^{9} b^{2} a^{8} B + \frac{40}{3} x^{9} b^{3} a^{7} A + \frac{5}{4} x^{8} b a^{9} B + \frac{45}{8} x^{8} b^{2} a^{8} A + \frac{1}{7} x^{7} a^{10} B + \frac{10}{7} x^{7} b a^{9} A + \frac{1}{6} x^{6} a^{10} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^5,x, algorithm="fricas")

[Out]

1/17*x^17*b^10*B + 5/8*x^16*b^9*a*B + 1/16*x^16*b^10*A + 3*x^15*b^8*a^2*B + 2/3*
x^15*b^9*a*A + 60/7*x^14*b^7*a^3*B + 45/14*x^14*b^8*a^2*A + 210/13*x^13*b^6*a^4*
B + 120/13*x^13*b^7*a^3*A + 21*x^12*b^5*a^5*B + 35/2*x^12*b^6*a^4*A + 210/11*x^1
1*b^4*a^6*B + 252/11*x^11*b^5*a^5*A + 12*x^10*b^3*a^7*B + 21*x^10*b^4*a^6*A + 5*
x^9*b^2*a^8*B + 40/3*x^9*b^3*a^7*A + 5/4*x^8*b*a^9*B + 45/8*x^8*b^2*a^8*A + 1/7*
x^7*a^10*B + 10/7*x^7*b*a^9*A + 1/6*x^6*a^10*A

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Sympy [A]  time = 0.290734, size = 265, normalized size = 1.63 \[ \frac{A a^{10} x^{6}}{6} + \frac{B b^{10} x^{17}}{17} + x^{16} \left (\frac{A b^{10}}{16} + \frac{5 B a b^{9}}{8}\right ) + x^{15} \left (\frac{2 A a b^{9}}{3} + 3 B a^{2} b^{8}\right ) + x^{14} \left (\frac{45 A a^{2} b^{8}}{14} + \frac{60 B a^{3} b^{7}}{7}\right ) + x^{13} \left (\frac{120 A a^{3} b^{7}}{13} + \frac{210 B a^{4} b^{6}}{13}\right ) + x^{12} \left (\frac{35 A a^{4} b^{6}}{2} + 21 B a^{5} b^{5}\right ) + x^{11} \left (\frac{252 A a^{5} b^{5}}{11} + \frac{210 B a^{6} b^{4}}{11}\right ) + x^{10} \left (21 A a^{6} b^{4} + 12 B a^{7} b^{3}\right ) + x^{9} \left (\frac{40 A a^{7} b^{3}}{3} + 5 B a^{8} b^{2}\right ) + x^{8} \left (\frac{45 A a^{8} b^{2}}{8} + \frac{5 B a^{9} b}{4}\right ) + x^{7} \left (\frac{10 A a^{9} b}{7} + \frac{B a^{10}}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x+a)**10*(B*x+A),x)

[Out]

A*a**10*x**6/6 + B*b**10*x**17/17 + x**16*(A*b**10/16 + 5*B*a*b**9/8) + x**15*(2
*A*a*b**9/3 + 3*B*a**2*b**8) + x**14*(45*A*a**2*b**8/14 + 60*B*a**3*b**7/7) + x*
*13*(120*A*a**3*b**7/13 + 210*B*a**4*b**6/13) + x**12*(35*A*a**4*b**6/2 + 21*B*a
**5*b**5) + x**11*(252*A*a**5*b**5/11 + 210*B*a**6*b**4/11) + x**10*(21*A*a**6*b
**4 + 12*B*a**7*b**3) + x**9*(40*A*a**7*b**3/3 + 5*B*a**8*b**2) + x**8*(45*A*a**
8*b**2/8 + 5*B*a**9*b/4) + x**7*(10*A*a**9*b/7 + B*a**10/7)

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GIAC/XCAS [A]  time = 0.255159, size = 331, normalized size = 2.03 \[ \frac{1}{17} \, B b^{10} x^{17} + \frac{5}{8} \, B a b^{9} x^{16} + \frac{1}{16} \, A b^{10} x^{16} + 3 \, B a^{2} b^{8} x^{15} + \frac{2}{3} \, A a b^{9} x^{15} + \frac{60}{7} \, B a^{3} b^{7} x^{14} + \frac{45}{14} \, A a^{2} b^{8} x^{14} + \frac{210}{13} \, B a^{4} b^{6} x^{13} + \frac{120}{13} \, A a^{3} b^{7} x^{13} + 21 \, B a^{5} b^{5} x^{12} + \frac{35}{2} \, A a^{4} b^{6} x^{12} + \frac{210}{11} \, B a^{6} b^{4} x^{11} + \frac{252}{11} \, A a^{5} b^{5} x^{11} + 12 \, B a^{7} b^{3} x^{10} + 21 \, A a^{6} b^{4} x^{10} + 5 \, B a^{8} b^{2} x^{9} + \frac{40}{3} \, A a^{7} b^{3} x^{9} + \frac{5}{4} \, B a^{9} b x^{8} + \frac{45}{8} \, A a^{8} b^{2} x^{8} + \frac{1}{7} \, B a^{10} x^{7} + \frac{10}{7} \, A a^{9} b x^{7} + \frac{1}{6} \, A a^{10} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^5,x, algorithm="giac")

[Out]

1/17*B*b^10*x^17 + 5/8*B*a*b^9*x^16 + 1/16*A*b^10*x^16 + 3*B*a^2*b^8*x^15 + 2/3*
A*a*b^9*x^15 + 60/7*B*a^3*b^7*x^14 + 45/14*A*a^2*b^8*x^14 + 210/13*B*a^4*b^6*x^1
3 + 120/13*A*a^3*b^7*x^13 + 21*B*a^5*b^5*x^12 + 35/2*A*a^4*b^6*x^12 + 210/11*B*a
^6*b^4*x^11 + 252/11*A*a^5*b^5*x^11 + 12*B*a^7*b^3*x^10 + 21*A*a^6*b^4*x^10 + 5*
B*a^8*b^2*x^9 + 40/3*A*a^7*b^3*x^9 + 5/4*B*a^9*b*x^8 + 45/8*A*a^8*b^2*x^8 + 1/7*
B*a^10*x^7 + 10/7*A*a^9*b*x^7 + 1/6*A*a^10*x^6