\(\int (a+b x)^{10} (A+B x) \, dx\) [147]

Optimal result

Integrand size = 13, antiderivative size = 38 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {(A b-a B) (a+b x)^{11}}{11 b^2}+\frac {B (a+b x)^{12}}{12 b^2} \]

[Out]

1/11*(A*b-B*a)*(b*x+a)^11/b^2+1/12*B*(b*x+a)^12/b^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {(a+b x)^{11} (A b-a B)}{11 b^2}+\frac {B (a+b x)^{12}}{12 b^2} \]

[In]

Int[(a + b*x)^10*(A + B*x),x]

[Out]

((A*b - a*B)*(a + b*x)^11)/(11*b^2) + (B*(a + b*x)^12)/(12*b^2)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(A b-a B) (a+b x)^{10}}{b}+\frac {B (a+b x)^{11}}{b}\right ) \, dx \\ & = \frac {(A b-a B) (a+b x)^{11}}{11 b^2}+\frac {B (a+b x)^{12}}{12 b^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(198\) vs. \(2(38)=76\).

Time = 0.03 (sec) , antiderivative size = 198, normalized size of antiderivative = 5.21 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {1}{132} x \left (66 a^{10} (2 A+B x)+220 a^9 b x (3 A+2 B x)+495 a^8 b^2 x^2 (4 A+3 B x)+792 a^7 b^3 x^3 (5 A+4 B x)+924 a^6 b^4 x^4 (6 A+5 B x)+792 a^5 b^5 x^5 (7 A+6 B x)+495 a^4 b^6 x^6 (8 A+7 B x)+220 a^3 b^7 x^7 (9 A+8 B x)+66 a^2 b^8 x^8 (10 A+9 B x)+12 a b^9 x^9 (11 A+10 B x)+b^{10} x^{10} (12 A+11 B x)\right ) \]

[In]

Integrate[(a + b*x)^10*(A + B*x),x]

[Out]

(x*(66*a^10*(2*A + B*x) + 220*a^9*b*x*(3*A + 2*B*x) + 495*a^8*b^2*x^2*(4*A + 3*B*x) + 792*a^7*b^3*x^3*(5*A + 4
*B*x) + 924*a^6*b^4*x^4*(6*A + 5*B*x) + 792*a^5*b^5*x^5*(7*A + 6*B*x) + 495*a^4*b^6*x^6*(8*A + 7*B*x) + 220*a^
3*b^7*x^7*(9*A + 8*B*x) + 66*a^2*b^8*x^8*(10*A + 9*B*x) + 12*a*b^9*x^9*(11*A + 10*B*x) + b^10*x^10*(12*A + 11*
B*x)))/132

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(231\) vs. \(2(34)=68\).

Time = 0.40 (sec) , antiderivative size = 232, normalized size of antiderivative = 6.11

method	result	size
norman	\(\frac {b^{10} B \,x^{12}}{12}+\left (\frac {1}{11} b^{10} A +\frac {10}{11} a \,b^{9} B \right ) x^{11}+\left (a \,b^{9} A +\frac {9}{2} a^{2} b^{8} B \right ) x^{10}+\left (5 a^{2} b^{8} A +\frac {40}{3} a^{3} b^{7} B \right ) x^{9}+\left (15 a^{3} b^{7} A +\frac {105}{4} a^{4} b^{6} B \right ) x^{8}+\left (30 a^{4} b^{6} A +36 a^{5} b^{5} B \right ) x^{7}+\left (42 a^{5} b^{5} A +35 a^{6} b^{4} B \right ) x^{6}+\left (42 a^{6} b^{4} A +24 a^{7} b^{3} B \right ) x^{5}+\left (30 a^{7} b^{3} A +\frac {45}{4} a^{8} b^{2} B \right ) x^{4}+\left (15 a^{8} b^{2} A +\frac {10}{3} a^{9} b B \right ) x^{3}+\left (5 a^{9} b A +\frac {1}{2} a^{10} B \right ) x^{2}+a^{10} A x\)	\(232\)
default	\(\frac {b^{10} B \,x^{12}}{12}+\frac {\left (b^{10} A +10 a \,b^{9} B \right ) x^{11}}{11}+\frac {\left (10 a \,b^{9} A +45 a^{2} b^{8} B \right ) x^{10}}{10}+\frac {\left (45 a^{2} b^{8} A +120 a^{3} b^{7} B \right ) x^{9}}{9}+\frac {\left (120 a^{3} b^{7} A +210 a^{4} b^{6} B \right ) x^{8}}{8}+\frac {\left (210 a^{4} b^{6} A +252 a^{5} b^{5} B \right ) x^{7}}{7}+\frac {\left (252 a^{5} b^{5} A +210 a^{6} b^{4} B \right ) x^{6}}{6}+\frac {\left (210 a^{6} b^{4} A +120 a^{7} b^{3} B \right ) x^{5}}{5}+\frac {\left (120 a^{7} b^{3} A +45 a^{8} b^{2} B \right ) x^{4}}{4}+\frac {\left (45 a^{8} b^{2} A +10 a^{9} b B \right ) x^{3}}{3}+\frac {\left (10 a^{9} b A +a^{10} B \right ) x^{2}}{2}+a^{10} A x\)	\(241\)
gosper	\(\frac {1}{12} b^{10} B \,x^{12}+\frac {1}{11} x^{11} b^{10} A +\frac {10}{11} x^{11} a \,b^{9} B +x^{10} a \,b^{9} A +\frac {9}{2} x^{10} a^{2} b^{8} B +5 x^{9} a^{2} b^{8} A +\frac {40}{3} x^{9} a^{3} b^{7} B +15 x^{8} a^{3} b^{7} A +\frac {105}{4} x^{8} a^{4} b^{6} B +30 A \,a^{4} b^{6} x^{7}+36 B \,a^{5} b^{5} x^{7}+42 A \,a^{5} b^{5} x^{6}+35 B \,a^{6} b^{4} x^{6}+42 A \,a^{6} b^{4} x^{5}+24 B \,a^{7} b^{3} x^{5}+30 x^{4} a^{7} b^{3} A +\frac {45}{4} x^{4} a^{8} b^{2} B +15 x^{3} a^{8} b^{2} A +\frac {10}{3} x^{3} a^{9} b B +5 x^{2} a^{9} b A +\frac {1}{2} x^{2} a^{10} B +a^{10} A x\)	\(242\)
risch	\(\frac {1}{12} b^{10} B \,x^{12}+\frac {1}{11} x^{11} b^{10} A +\frac {10}{11} x^{11} a \,b^{9} B +x^{10} a \,b^{9} A +\frac {9}{2} x^{10} a^{2} b^{8} B +5 x^{9} a^{2} b^{8} A +\frac {40}{3} x^{9} a^{3} b^{7} B +15 x^{8} a^{3} b^{7} A +\frac {105}{4} x^{8} a^{4} b^{6} B +30 A \,a^{4} b^{6} x^{7}+36 B \,a^{5} b^{5} x^{7}+42 A \,a^{5} b^{5} x^{6}+35 B \,a^{6} b^{4} x^{6}+42 A \,a^{6} b^{4} x^{5}+24 B \,a^{7} b^{3} x^{5}+30 x^{4} a^{7} b^{3} A +\frac {45}{4} x^{4} a^{8} b^{2} B +15 x^{3} a^{8} b^{2} A +\frac {10}{3} x^{3} a^{9} b B +5 x^{2} a^{9} b A +\frac {1}{2} x^{2} a^{10} B +a^{10} A x\)	\(242\)
parallelrisch	\(\frac {1}{12} b^{10} B \,x^{12}+\frac {1}{11} x^{11} b^{10} A +\frac {10}{11} x^{11} a \,b^{9} B +x^{10} a \,b^{9} A +\frac {9}{2} x^{10} a^{2} b^{8} B +5 x^{9} a^{2} b^{8} A +\frac {40}{3} x^{9} a^{3} b^{7} B +15 x^{8} a^{3} b^{7} A +\frac {105}{4} x^{8} a^{4} b^{6} B +30 A \,a^{4} b^{6} x^{7}+36 B \,a^{5} b^{5} x^{7}+42 A \,a^{5} b^{5} x^{6}+35 B \,a^{6} b^{4} x^{6}+42 A \,a^{6} b^{4} x^{5}+24 B \,a^{7} b^{3} x^{5}+30 x^{4} a^{7} b^{3} A +\frac {45}{4} x^{4} a^{8} b^{2} B +15 x^{3} a^{8} b^{2} A +\frac {10}{3} x^{3} a^{9} b B +5 x^{2} a^{9} b A +\frac {1}{2} x^{2} a^{10} B +a^{10} A x\)	\(242\)

[In]

int((b*x+a)^10*(B*x+A),x,method=_RETURNVERBOSE)

[Out]

1/12*b^10*B*x^12+(1/11*b^10*A+10/11*a*b^9*B)*x^11+(a*b^9*A+9/2*a^2*b^8*B)*x^10+(5*a^2*b^8*A+40/3*a^3*b^7*B)*x^
9+(15*a^3*b^7*A+105/4*a^4*b^6*B)*x^8+(30*A*a^4*b^6+36*B*a^5*b^5)*x^7+(42*A*a^5*b^5+35*B*a^6*b^4)*x^6+(42*A*a^6
*b^4+24*B*a^7*b^3)*x^5+(30*a^7*b^3*A+45/4*a^8*b^2*B)*x^4+(15*a^8*b^2*A+10/3*a^9*b*B)*x^3+(5*a^9*b*A+1/2*a^10*B
)*x^2+a^10*A*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (34) = 68\).

Time = 0.22 (sec) , antiderivative size = 240, normalized size of antiderivative = 6.32 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {1}{12} \, B b^{10} x^{12} + A a^{10} x + \frac {1}{11} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{11} + \frac {1}{2} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{9} + \frac {15}{4} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{8} + 6 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{7} + 7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{6} + 6 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{5} + \frac {15}{4} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{2} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (32) = 64\).

Time = 0.05 (sec) , antiderivative size = 248, normalized size of antiderivative = 6.53 \[ \int (a+b x)^{10} (A+B x) \, dx=A a^{10} x + \frac {B b^{10} x^{12}}{12} + x^{11} \left (\frac {A b^{10}}{11} + \frac {10 B a b^{9}}{11}\right ) + x^{10} \left (A a b^{9} + \frac {9 B a^{2} b^{8}}{2}\right ) + x^{9} \cdot \left (5 A a^{2} b^{8} + \frac {40 B a^{3} b^{7}}{3}\right ) + x^{8} \cdot \left (15 A a^{3} b^{7} + \frac {105 B a^{4} b^{6}}{4}\right ) + x^{7} \cdot \left (30 A a^{4} b^{6} + 36 B a^{5} b^{5}\right ) + x^{6} \cdot \left (42 A a^{5} b^{5} + 35 B a^{6} b^{4}\right ) + x^{5} \cdot \left (42 A a^{6} b^{4} + 24 B a^{7} b^{3}\right ) + x^{4} \cdot \left (30 A a^{7} b^{3} + \frac {45 B a^{8} b^{2}}{4}\right ) + x^{3} \cdot \left (15 A a^{8} b^{2} + \frac {10 B a^{9} b}{3}\right ) + x^{2} \cdot \left (5 A a^{9} b + \frac {B a^{10}}{2}\right ) \]

[In]

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

[Out]

A*a**10*x + B*b**10*x**12/12 + x**11*(A*b**10/11 + 10*B*a*b**9/11) + x**10*(A*a*b**9 + 9*B*a**2*b**8/2) + x**9
*(5*A*a**2*b**8 + 40*B*a**3*b**7/3) + x**8*(15*A*a**3*b**7 + 105*B*a**4*b**6/4) + x**7*(30*A*a**4*b**6 + 36*B*
a**5*b**5) + x**6*(42*A*a**5*b**5 + 35*B*a**6*b**4) + x**5*(42*A*a**6*b**4 + 24*B*a**7*b**3) + x**4*(30*A*a**7
*b**3 + 45*B*a**8*b**2/4) + x**3*(15*A*a**8*b**2 + 10*B*a**9*b/3) + x**2*(5*A*a**9*b + B*a**10/2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (34) = 68\).

Time = 0.20 (sec) , antiderivative size = 240, normalized size of antiderivative = 6.32 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {1}{12} \, B b^{10} x^{12} + A a^{10} x + \frac {1}{11} \, {\left (10 \, B a b^{9} + A b^{10}\right )} x^{11} + \frac {1}{2} \, {\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{10} + \frac {5}{3} \, {\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{9} + \frac {15}{4} \, {\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{8} + 6 \, {\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{7} + 7 \, {\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{6} + 6 \, {\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{5} + \frac {15}{4} \, {\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{4} + \frac {5}{3} \, {\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{10} + 10 \, A a^{9} b\right )} x^{2} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (34) = 68\).

Time = 0.30 (sec) , antiderivative size = 241, normalized size of antiderivative = 6.34 \[ \int (a+b x)^{10} (A+B x) \, dx=\frac {1}{12} \, B b^{10} x^{12} + \frac {10}{11} \, B a b^{9} x^{11} + \frac {1}{11} \, A b^{10} x^{11} + \frac {9}{2} \, B a^{2} b^{8} x^{10} + A a b^{9} x^{10} + \frac {40}{3} \, B a^{3} b^{7} x^{9} + 5 \, A a^{2} b^{8} x^{9} + \frac {105}{4} \, B a^{4} b^{6} x^{8} + 15 \, A a^{3} b^{7} x^{8} + 36 \, B a^{5} b^{5} x^{7} + 30 \, A a^{4} b^{6} x^{7} + 35 \, B a^{6} b^{4} x^{6} + 42 \, A a^{5} b^{5} x^{6} + 24 \, B a^{7} b^{3} x^{5} + 42 \, A a^{6} b^{4} x^{5} + \frac {45}{4} \, B a^{8} b^{2} x^{4} + 30 \, A a^{7} b^{3} x^{4} + \frac {10}{3} \, B a^{9} b x^{3} + 15 \, A a^{8} b^{2} x^{3} + \frac {1}{2} \, B a^{10} x^{2} + 5 \, A a^{9} b x^{2} + A a^{10} x \]

[In]

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

[Out]

1/12*B*b^10*x^12 + 10/11*B*a*b^9*x^11 + 1/11*A*b^10*x^11 + 9/2*B*a^2*b^8*x^10 + A*a*b^9*x^10 + 40/3*B*a^3*b^7*
x^9 + 5*A*a^2*b^8*x^9 + 105/4*B*a^4*b^6*x^8 + 15*A*a^3*b^7*x^8 + 36*B*a^5*b^5*x^7 + 30*A*a^4*b^6*x^7 + 35*B*a^
6*b^4*x^6 + 42*A*a^5*b^5*x^6 + 24*B*a^7*b^3*x^5 + 42*A*a^6*b^4*x^5 + 45/4*B*a^8*b^2*x^4 + 30*A*a^7*b^3*x^4 + 1
0/3*B*a^9*b*x^3 + 15*A*a^8*b^2*x^3 + 1/2*B*a^10*x^2 + 5*A*a^9*b*x^2 + A*a^10*x

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 208, normalized size of antiderivative = 5.47 \[ \int (a+b x)^{10} (A+B x) \, dx=x^2\,\left (\frac {B\,a^{10}}{2}+5\,A\,b\,a^9\right )+x^{11}\,\left (\frac {A\,b^{10}}{11}+\frac {10\,B\,a\,b^9}{11}\right )+\frac {B\,b^{10}\,x^{12}}{12}+A\,a^{10}\,x+\frac {15\,a^7\,b^2\,x^4\,\left (8\,A\,b+3\,B\,a\right )}{4}+6\,a^6\,b^3\,x^5\,\left (7\,A\,b+4\,B\,a\right )+7\,a^5\,b^4\,x^6\,\left (6\,A\,b+5\,B\,a\right )+6\,a^4\,b^5\,x^7\,\left (5\,A\,b+6\,B\,a\right )+\frac {15\,a^3\,b^6\,x^8\,\left (4\,A\,b+7\,B\,a\right )}{4}+\frac {5\,a^2\,b^7\,x^9\,\left (3\,A\,b+8\,B\,a\right )}{3}+\frac {5\,a^8\,b\,x^3\,\left (9\,A\,b+2\,B\,a\right )}{3}+\frac {a\,b^8\,x^{10}\,\left (2\,A\,b+9\,B\,a\right )}{2} \]

[In]

int((A + B*x)*(a + b*x)^10,x)

[Out]

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