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

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

[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)/(12*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 = 0.17718, 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, Rules used = {76} \[ \frac{7 a^2 (a+b x)^{16} (3 A b-8 a B)}{16 b^9}-\frac{7 a^3 (a+b x)^{15} (A b-2 a B)}{3 b^9}+\frac{a^4 (a+b x)^{14} (5 A b-8 a B)}{2 b^9}-\frac{7 a^5 (a+b x)^{13} (3 A b-4 a B)}{13 b^9}+\frac{a^6 (a+b x)^{12} (7 A b-8 a B)}{12 b^9}-\frac{a^7 (a+b x)^{11} (A b-a B)}{11 b^9}+\frac{(a+b x)^{18} (A b-8 a B)}{18 b^9}-\frac{7 a (a+b x)^{17} (A b-4 a B)}{17 b^9}+\frac{B (a+b x)^{19}}{19 b^9} \]

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)/(12*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)

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^7 (a+b x)^{10} (A+B x) \, dx &=\int \left (\frac{a^7 (-A b+a B) (a+b x)^{10}}{b^8}-\frac{a^6 (-7 A b+8 a B) (a+b x)^{11}}{b^8}+\frac{7 a^5 (-3 A b+4 a B) (a+b x)^{12}}{b^8}-\frac{7 a^4 (-5 A b+8 a B) (a+b x)^{13}}{b^8}+\frac{35 a^3 (-A b+2 a B) (a+b x)^{14}}{b^8}-\frac{7 a^2 (-3 A b+8 a B) (a+b x)^{15}}{b^8}+\frac{7 a (-A b+4 a B) (a+b x)^{16}}{b^8}+\frac{(A b-8 a B) (a+b x)^{17}}{b^8}+\frac{B (a+b x)^{18}}{b^8}\right ) \, dx\\ &=-\frac{a^7 (A b-a B) (a+b x)^{11}}{11 b^9}+\frac{a^6 (7 A b-8 a B) (a+b x)^{12}}{12 b^9}-\frac{7 a^5 (3 A b-4 a B) (a+b x)^{13}}{13 b^9}+\frac{a^4 (5 A b-8 a B) (a+b x)^{14}}{2 b^9}-\frac{7 a^3 (A b-2 a B) (a+b x)^{15}}{3 b^9}+\frac{7 a^2 (3 A b-8 a B) (a+b x)^{16}}{16 b^9}-\frac{7 a (A b-4 a B) (a+b x)^{17}}{17 b^9}+\frac{(A b-8 a B) (a+b x)^{18}}{18 b^9}+\frac{B (a+b x)^{19}}{19 b^9}\\ \end{align*}

Mathematica [A]  time = 0.0291999, size = 227, normalized size = 1.06 \[ \frac{15}{16} a^2 b^7 x^{16} (8 a B+3 A b)+2 a^3 b^6 x^{15} (7 a B+4 A b)+3 a^4 b^5 x^{14} (6 a B+5 A b)+\frac{42}{13} a^5 b^4 x^{13} (5 a B+6 A b)+\frac{5}{2} a^6 b^3 x^{12} (4 a B+7 A b)+\frac{15}{11} a^7 b^2 x^{11} (3 a B+8 A b)+\frac{1}{2} a^8 b x^{10} (2 a B+9 A b)+\frac{1}{9} a^9 x^9 (a B+10 A b)+\frac{1}{8} a^{10} A x^8+\frac{1}{18} b^9 x^{18} (10 a B+A b)+\frac{5}{17} a b^8 x^{17} (9 a B+2 A b)+\frac{1}{19} b^{10} B x^{19} \]

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^1
1)/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 \begin{align*}{\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}} \end{align*}

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^15+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^13+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 = 1.03053, size = 328, normalized size = 1.53 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^10*(B*x+A),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 = 1.30444, size = 608, normalized size = 2.83 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^10*(B*x+A),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 = 0.122526, size = 262, normalized size = 1.22 \begin{align*} \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 ) \end{align*}

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*(10*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 [A]  time = 1.20169, size = 329, normalized size = 1.53 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^10*(B*x+A),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