3.112 \(\int x^4 (a+b x)^{10} (A+B x) \, dx\)

Optimal. Leaf size=139 \[ \frac{a^4 (a+b x)^{11} (A b-a B)}{11 b^6}-\frac{a^3 (a+b x)^{12} (4 A b-5 a B)}{12 b^6}+\frac{2 a^2 (a+b x)^{13} (3 A b-5 a B)}{13 b^6}+\frac{(a+b x)^{15} (A b-5 a B)}{15 b^6}-\frac{a (a+b x)^{14} (2 A b-5 a B)}{7 b^6}+\frac{B (a+b x)^{16}}{16 b^6} \]

[Out]

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

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Rubi [A]  time = 0.456742, antiderivative size = 139, 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 \[ \frac{a^4 (a+b x)^{11} (A b-a B)}{11 b^6}-\frac{a^3 (a+b x)^{12} (4 A b-5 a B)}{12 b^6}+\frac{2 a^2 (a+b x)^{13} (3 A b-5 a B)}{13 b^6}+\frac{(a+b x)^{15} (A b-5 a B)}{15 b^6}-\frac{a (a+b x)^{14} (2 A b-5 a B)}{7 b^6}+\frac{B (a+b x)^{16}}{16 b^6} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x)^10*(A + B*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 71.3857, size = 131, normalized size = 0.94 \[ \frac{B \left (a + b x\right )^{16}}{16 b^{6}} + \frac{a^{4} \left (a + b x\right )^{11} \left (A b - B a\right )}{11 b^{6}} - \frac{a^{3} \left (a + b x\right )^{12} \left (4 A b - 5 B a\right )}{12 b^{6}} + \frac{2 a^{2} \left (a + b x\right )^{13} \left (3 A b - 5 B a\right )}{13 b^{6}} - \frac{a \left (a + b x\right )^{14} \left (2 A b - 5 B a\right )}{7 b^{6}} + \frac{\left (a + b x\right )^{15} \left (A b - 5 B a\right )}{15 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x+a)**10*(B*x+A),x)

[Out]

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

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Mathematica [A]  time = 0.0495033, size = 231, normalized size = 1.66 \[ \frac{1}{5} a^{10} A x^5+\frac{1}{6} a^9 x^6 (a B+10 A b)+\frac{5}{7} a^8 b x^7 (2 a B+9 A b)+\frac{15}{8} a^7 b^2 x^8 (3 a B+8 A b)+\frac{10}{3} a^6 b^3 x^9 (4 a B+7 A b)+\frac{21}{5} a^5 b^4 x^{10} (5 a B+6 A b)+\frac{42}{11} a^4 b^5 x^{11} (6 a B+5 A b)+\frac{5}{2} a^3 b^6 x^{12} (7 a B+4 A b)+\frac{15}{13} a^2 b^7 x^{13} (8 a B+3 A b)+\frac{1}{15} b^9 x^{15} (10 a B+A b)+\frac{5}{14} a b^8 x^{14} (9 a B+2 A b)+\frac{1}{16} b^{10} B x^{16} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*(a + b*x)^10*(A + B*x),x]

[Out]

(a^10*A*x^5)/5 + (a^9*(10*A*b + a*B)*x^6)/6 + (5*a^8*b*(9*A*b + 2*a*B)*x^7)/7 +
(15*a^7*b^2*(8*A*b + 3*a*B)*x^8)/8 + (10*a^6*b^3*(7*A*b + 4*a*B)*x^9)/3 + (21*a^
5*b^4*(6*A*b + 5*a*B)*x^10)/5 + (42*a^4*b^5*(5*A*b + 6*a*B)*x^11)/11 + (5*a^3*b^
6*(4*A*b + 7*a*B)*x^12)/2 + (15*a^2*b^7*(3*A*b + 8*a*B)*x^13)/13 + (5*a*b^8*(2*A
*b + 9*a*B)*x^14)/14 + (b^9*(A*b + 10*a*B)*x^15)/15 + (b^10*B*x^16)/16

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Maple [A]  time = 0.001, size = 244, normalized size = 1.8 \[{\frac{{b}^{10}B{x}^{16}}{16}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{15}}{15}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{14}}{14}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{13}}{13}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{12}}{12}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{11}}{11}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{10}}{10}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{9}}{9}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{8}}{8}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{7}}{7}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{6}}{6}}+{\frac{{a}^{10}A{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x+a)^10*(B*x+A),x)

[Out]

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

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Maxima [A]  time = 1.34861, size = 328, normalized size = 2.36 \[ \frac{1}{16} \, B b^{10} x^{16} + \frac{1}{5} \, A a^{10} x^{5} + \frac{1}{15} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{15} + \frac{5}{14} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{14} + \frac{15}{13} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{13} + \frac{5}{2} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{12} + \frac{42}{11} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{11} + \frac{21}{5} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{10} + \frac{10}{3} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{9} + \frac{15}{8} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{8} + \frac{5}{7} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^4,x, algorithm="maxima")

[Out]

1/16*B*b^10*x^16 + 1/5*A*a^10*x^5 + 1/15*(10*B*a*b^9 + A*b^10)*x^15 + 5/14*(9*B*
a^2*b^8 + 2*A*a*b^9)*x^14 + 15/13*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^13 + 5/2*(7*B*a^
4*b^6 + 4*A*a^3*b^7)*x^12 + 42/11*(6*B*a^5*b^5 + 5*A*a^4*b^6)*x^11 + 21/5*(5*B*a
^6*b^4 + 6*A*a^5*b^5)*x^10 + 10/3*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^9 + 15/8*(3*B*a^
8*b^2 + 8*A*a^7*b^3)*x^8 + 5/7*(2*B*a^9*b + 9*A*a^8*b^2)*x^7 + 1/6*(B*a^10 + 10*
A*a^9*b)*x^6

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Fricas [A]  time = 0.182999, size = 1, normalized size = 0.01 \[ \frac{1}{16} x^{16} b^{10} B + \frac{2}{3} x^{15} b^{9} a B + \frac{1}{15} x^{15} b^{10} A + \frac{45}{14} x^{14} b^{8} a^{2} B + \frac{5}{7} x^{14} b^{9} a A + \frac{120}{13} x^{13} b^{7} a^{3} B + \frac{45}{13} x^{13} b^{8} a^{2} A + \frac{35}{2} x^{12} b^{6} a^{4} B + 10 x^{12} b^{7} a^{3} A + \frac{252}{11} x^{11} b^{5} a^{5} B + \frac{210}{11} x^{11} b^{6} a^{4} A + 21 x^{10} b^{4} a^{6} B + \frac{126}{5} x^{10} b^{5} a^{5} A + \frac{40}{3} x^{9} b^{3} a^{7} B + \frac{70}{3} x^{9} b^{4} a^{6} A + \frac{45}{8} x^{8} b^{2} a^{8} B + 15 x^{8} b^{3} a^{7} A + \frac{10}{7} x^{7} b a^{9} B + \frac{45}{7} x^{7} b^{2} a^{8} A + \frac{1}{6} x^{6} a^{10} B + \frac{5}{3} x^{6} b a^{9} A + \frac{1}{5} x^{5} a^{10} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^4,x, algorithm="fricas")

[Out]

1/16*x^16*b^10*B + 2/3*x^15*b^9*a*B + 1/15*x^15*b^10*A + 45/14*x^14*b^8*a^2*B +
5/7*x^14*b^9*a*A + 120/13*x^13*b^7*a^3*B + 45/13*x^13*b^8*a^2*A + 35/2*x^12*b^6*
a^4*B + 10*x^12*b^7*a^3*A + 252/11*x^11*b^5*a^5*B + 210/11*x^11*b^6*a^4*A + 21*x
^10*b^4*a^6*B + 126/5*x^10*b^5*a^5*A + 40/3*x^9*b^3*a^7*B + 70/3*x^9*b^4*a^6*A +
45/8*x^8*b^2*a^8*B + 15*x^8*b^3*a^7*A + 10/7*x^7*b*a^9*B + 45/7*x^7*b^2*a^8*A +
1/6*x^6*a^10*B + 5/3*x^6*b*a^9*A + 1/5*x^5*a^10*A

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Sympy [A]  time = 0.252673, size = 269, normalized size = 1.94 \[ \frac{A a^{10} x^{5}}{5} + \frac{B b^{10} x^{16}}{16} + x^{15} \left (\frac{A b^{10}}{15} + \frac{2 B a b^{9}}{3}\right ) + x^{14} \left (\frac{5 A a b^{9}}{7} + \frac{45 B a^{2} b^{8}}{14}\right ) + x^{13} \left (\frac{45 A a^{2} b^{8}}{13} + \frac{120 B a^{3} b^{7}}{13}\right ) + x^{12} \left (10 A a^{3} b^{7} + \frac{35 B a^{4} b^{6}}{2}\right ) + x^{11} \left (\frac{210 A a^{4} b^{6}}{11} + \frac{252 B a^{5} b^{5}}{11}\right ) + x^{10} \left (\frac{126 A a^{5} b^{5}}{5} + 21 B a^{6} b^{4}\right ) + x^{9} \left (\frac{70 A a^{6} b^{4}}{3} + \frac{40 B a^{7} b^{3}}{3}\right ) + x^{8} \left (15 A a^{7} b^{3} + \frac{45 B a^{8} b^{2}}{8}\right ) + x^{7} \left (\frac{45 A a^{8} b^{2}}{7} + \frac{10 B a^{9} b}{7}\right ) + x^{6} \left (\frac{5 A a^{9} b}{3} + \frac{B a^{10}}{6}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x+a)**10*(B*x+A),x)

[Out]

A*a**10*x**5/5 + B*b**10*x**16/16 + x**15*(A*b**10/15 + 2*B*a*b**9/3) + x**14*(5
*A*a*b**9/7 + 45*B*a**2*b**8/14) + x**13*(45*A*a**2*b**8/13 + 120*B*a**3*b**7/13
) + x**12*(10*A*a**3*b**7 + 35*B*a**4*b**6/2) + x**11*(210*A*a**4*b**6/11 + 252*
B*a**5*b**5/11) + x**10*(126*A*a**5*b**5/5 + 21*B*a**6*b**4) + x**9*(70*A*a**6*b
**4/3 + 40*B*a**7*b**3/3) + x**8*(15*A*a**7*b**3 + 45*B*a**8*b**2/8) + x**7*(45*
A*a**8*b**2/7 + 10*B*a**9*b/7) + x**6*(5*A*a**9*b/3 + B*a**10/6)

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GIAC/XCAS [A]  time = 0.299109, size = 331, normalized size = 2.38 \[ \frac{1}{16} \, B b^{10} x^{16} + \frac{2}{3} \, B a b^{9} x^{15} + \frac{1}{15} \, A b^{10} x^{15} + \frac{45}{14} \, B a^{2} b^{8} x^{14} + \frac{5}{7} \, A a b^{9} x^{14} + \frac{120}{13} \, B a^{3} b^{7} x^{13} + \frac{45}{13} \, A a^{2} b^{8} x^{13} + \frac{35}{2} \, B a^{4} b^{6} x^{12} + 10 \, A a^{3} b^{7} x^{12} + \frac{252}{11} \, B a^{5} b^{5} x^{11} + \frac{210}{11} \, A a^{4} b^{6} x^{11} + 21 \, B a^{6} b^{4} x^{10} + \frac{126}{5} \, A a^{5} b^{5} x^{10} + \frac{40}{3} \, B a^{7} b^{3} x^{9} + \frac{70}{3} \, A a^{6} b^{4} x^{9} + \frac{45}{8} \, B a^{8} b^{2} x^{8} + 15 \, A a^{7} b^{3} x^{8} + \frac{10}{7} \, B a^{9} b x^{7} + \frac{45}{7} \, A a^{8} b^{2} x^{7} + \frac{1}{6} \, B a^{10} x^{6} + \frac{5}{3} \, A a^{9} b x^{6} + \frac{1}{5} \, A a^{10} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^10*x^4,x, algorithm="giac")

[Out]

1/16*B*b^10*x^16 + 2/3*B*a*b^9*x^15 + 1/15*A*b^10*x^15 + 45/14*B*a^2*b^8*x^14 +
5/7*A*a*b^9*x^14 + 120/13*B*a^3*b^7*x^13 + 45/13*A*a^2*b^8*x^13 + 35/2*B*a^4*b^6
*x^12 + 10*A*a^3*b^7*x^12 + 252/11*B*a^5*b^5*x^11 + 210/11*A*a^4*b^6*x^11 + 21*B
*a^6*b^4*x^10 + 126/5*A*a^5*b^5*x^10 + 40/3*B*a^7*b^3*x^9 + 70/3*A*a^6*b^4*x^9 +
45/8*B*a^8*b^2*x^8 + 15*A*a^7*b^3*x^8 + 10/7*B*a^9*b*x^7 + 45/7*A*a^8*b^2*x^7 +
1/6*B*a^10*x^6 + 5/3*A*a^9*b*x^6 + 1/5*A*a^10*x^5