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

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]

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

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Rubi [A]  time = 0.14, antiderivative size = 163, 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} \[ \frac {5 a^2 (a+b x)^{14} (A b-2 a B)}{7 b^7}-\frac {5 a^3 (a+b x)^{13} (2 A b-3 a B)}{13 b^7}+\frac {a^4 (a+b x)^{12} (5 A b-6 a B)}{12 b^7}-\frac {a^5 (a+b x)^{11} (A b-a B)}{11 b^7}+\frac {(a+b x)^{16} (A b-6 a B)}{16 b^7}-\frac {a (a+b x)^{15} (A b-3 a B)}{3 b^7}+\frac {B (a+b x)^{17}}{17 b^7} \]

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)/(12*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)

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

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Mathematica [A]  time = 0.04, size = 229, normalized size = 1.40 \[ \frac {1}{6} a^{10} A x^6+\frac {1}{7} a^9 x^7 (a B+10 A b)+\frac {5}{8} a^8 b x^8 (2 a B+9 A b)+\frac {5}{3} a^7 b^2 x^9 (3 a B+8 A b)+3 a^6 b^3 x^{10} (4 a B+7 A b)+\frac {42}{11} a^5 b^4 x^{11} (5 a B+6 A b)+\frac {7}{2} a^4 b^5 x^{12} (6 a B+5 A b)+\frac {30}{13} a^3 b^6 x^{13} (7 a B+4 A b)+\frac {15}{14} a^2 b^7 x^{14} (8 a B+3 A b)+\frac {1}{16} b^9 x^{16} (10 a B+A b)+\frac {1}{3} a b^8 x^{15} (9 a B+2 A b)+\frac {1}{17} b^{10} B x^{17} \]

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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fricas [A]  time = 0.61, size = 245, normalized size = 1.50 \[ \frac {1}{17} x^{17} b^{10} B + \frac {5}{8} x^{16} b^{9} a B + \frac {1}{16} x^{16} b^{10} A + 3 x^{15} b^{8} a^{2} B + \frac {2}{3} x^{15} b^{9} a A + \frac {60}{7} x^{14} b^{7} a^{3} B + \frac {45}{14} x^{14} b^{8} a^{2} A + \frac {210}{13} x^{13} b^{6} a^{4} B + \frac {120}{13} x^{13} b^{7} a^{3} A + 21 x^{12} b^{5} a^{5} B + \frac {35}{2} x^{12} b^{6} a^{4} A + \frac {210}{11} x^{11} b^{4} a^{6} B + \frac {252}{11} x^{11} b^{5} a^{5} A + 12 x^{10} b^{3} a^{7} B + 21 x^{10} b^{4} a^{6} A + 5 x^{9} b^{2} a^{8} B + \frac {40}{3} x^{9} b^{3} a^{7} A + \frac {5}{4} x^{8} b a^{9} B + \frac {45}{8} x^{8} b^{2} a^{8} A + \frac {1}{7} x^{7} a^{10} B + \frac {10}{7} x^{7} b a^{9} A + \frac {1}{6} x^{6} a^{10} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^10*(B*x+A),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^11*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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giac [A]  time = 1.32, size = 245, normalized size = 1.50 \[ \frac {1}{17} \, B b^{10} x^{17} + \frac {5}{8} \, B a b^{9} x^{16} + \frac {1}{16} \, A b^{10} x^{16} + 3 \, B a^{2} b^{8} x^{15} + \frac {2}{3} \, A a b^{9} x^{15} + \frac {60}{7} \, B a^{3} b^{7} x^{14} + \frac {45}{14} \, A a^{2} b^{8} x^{14} + \frac {210}{13} \, B a^{4} b^{6} x^{13} + \frac {120}{13} \, A a^{3} b^{7} x^{13} + 21 \, B a^{5} b^{5} x^{12} + \frac {35}{2} \, A a^{4} b^{6} x^{12} + \frac {210}{11} \, B a^{6} b^{4} x^{11} + \frac {252}{11} \, A a^{5} b^{5} x^{11} + 12 \, B a^{7} b^{3} x^{10} + 21 \, A a^{6} b^{4} x^{10} + 5 \, B a^{8} b^{2} x^{9} + \frac {40}{3} \, A a^{7} b^{3} x^{9} + \frac {5}{4} \, B a^{9} b x^{8} + \frac {45}{8} \, A a^{8} b^{2} x^{8} + \frac {1}{7} \, B a^{10} x^{7} + \frac {10}{7} \, A a^{9} b x^{7} + \frac {1}{6} \, A a^{10} x^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^10*(B*x+A),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^13 + 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

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maple [A]  time = 0.00, size = 244, normalized size = 1.50 \[ \frac {B \,b^{10} x^{17}}{17}+\frac {A \,a^{10} x^{6}}{6}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{16}}{16}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{15}}{15}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{14}}{14}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{13}}{13}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{12}}{12}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{11}}{11}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{10}}{10}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{9}}{9}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{8}}{8}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{7}}{7} \]

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^13+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^11+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 = 1.17, size = 243, normalized size = 1.49 \[ \frac {1}{17} \, B b^{10} x^{17} + \frac {1}{6} \, A a^{10} x^{6} + \frac {1}{16} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{16} + \frac {1}{3} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{15} + \frac {15}{14} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{14} + \frac {30}{13} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{13} + \frac {7}{2} \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{12} + \frac {42}{11} \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{11} + 3 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{10} + \frac {5}{3} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{9} + \frac {5}{8} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{8} + \frac {1}{7} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(b*x+a)^10*(B*x+A),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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mupad [B]  time = 0.10, size = 211, normalized size = 1.29 \[ x^7\,\left (\frac {B\,a^{10}}{7}+\frac {10\,A\,b\,a^9}{7}\right )+x^{16}\,\left (\frac {A\,b^{10}}{16}+\frac {5\,B\,a\,b^9}{8}\right )+\frac {A\,a^{10}\,x^6}{6}+\frac {B\,b^{10}\,x^{17}}{17}+\frac {5\,a^7\,b^2\,x^9\,\left (8\,A\,b+3\,B\,a\right )}{3}+3\,a^6\,b^3\,x^{10}\,\left (7\,A\,b+4\,B\,a\right )+\frac {42\,a^5\,b^4\,x^{11}\,\left (6\,A\,b+5\,B\,a\right )}{11}+\frac {7\,a^4\,b^5\,x^{12}\,\left (5\,A\,b+6\,B\,a\right )}{2}+\frac {30\,a^3\,b^6\,x^{13}\,\left (4\,A\,b+7\,B\,a\right )}{13}+\frac {15\,a^2\,b^7\,x^{14}\,\left (3\,A\,b+8\,B\,a\right )}{14}+\frac {5\,a^8\,b\,x^8\,\left (9\,A\,b+2\,B\,a\right )}{8}+\frac {a\,b^8\,x^{15}\,\left (2\,A\,b+9\,B\,a\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(A + B*x)*(a + b*x)^10,x)

[Out]

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

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sympy [A]  time = 0.15, 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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