3.144 \(\int x^3 (a+b x)^{10} (A+B x) \, dx\)

Optimal. Leaf size=112 \[ \frac{a^2 (a+b x)^{12} (3 A b-4 a B)}{12 b^5}-\frac{a^3 (a+b x)^{11} (A b-a B)}{11 b^5}+\frac{(a+b x)^{14} (A b-4 a B)}{14 b^5}-\frac{3 a (a+b x)^{13} (A b-2 a B)}{13 b^5}+\frac{B (a+b x)^{15}}{15 b^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0981898, antiderivative size = 112, 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} \[ \frac{a^2 (a+b x)^{12} (3 A b-4 a B)}{12 b^5}-\frac{a^3 (a+b x)^{11} (A b-a B)}{11 b^5}+\frac{(a+b x)^{14} (A b-4 a B)}{14 b^5}-\frac{3 a (a+b x)^{13} (A b-2 a B)}{13 b^5}+\frac{B (a+b x)^{15}}{15 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x)^10*(A + B*x),x]

[Out]

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

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

Mathematica [B]  time = 0.0313981, size = 231, normalized size = 2.06 \[ \frac{5}{4} a^2 b^7 x^{12} (8 a B+3 A b)+\frac{30}{11} a^3 b^6 x^{11} (7 a B+4 A b)+\frac{21}{5} a^4 b^5 x^{10} (6 a B+5 A b)+\frac{14}{3} a^5 b^4 x^9 (5 a B+6 A b)+\frac{15}{4} a^6 b^3 x^8 (4 a B+7 A b)+\frac{15}{7} a^7 b^2 x^7 (3 a B+8 A b)+\frac{5}{6} a^8 b x^6 (2 a B+9 A b)+\frac{1}{5} a^9 x^5 (a B+10 A b)+\frac{1}{4} a^{10} A x^4+\frac{1}{14} b^9 x^{14} (10 a B+A b)+\frac{5}{13} a b^8 x^{13} (9 a B+2 A b)+\frac{1}{15} b^{10} B x^{15} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x)^10*(A + B*x),x]

[Out]

(a^10*A*x^4)/4 + (a^9*(10*A*b + a*B)*x^5)/5 + (5*a^8*b*(9*A*b + 2*a*B)*x^6)/6 + (15*a^7*b^2*(8*A*b + 3*a*B)*x^
7)/7 + (15*a^6*b^3*(7*A*b + 4*a*B)*x^8)/4 + (14*a^5*b^4*(6*A*b + 5*a*B)*x^9)/3 + (21*a^4*b^5*(5*A*b + 6*a*B)*x
^10)/5 + (30*a^3*b^6*(4*A*b + 7*a*B)*x^11)/11 + (5*a^2*b^7*(3*A*b + 8*a*B)*x^12)/4 + (5*a*b^8*(2*A*b + 9*a*B)*
x^13)/13 + (b^9*(A*b + 10*a*B)*x^14)/14 + (b^10*B*x^15)/15

________________________________________________________________________________________

Maple [B]  time = 0.001, size = 244, normalized size = 2.2 \begin{align*}{\frac{{b}^{10}B{x}^{15}}{15}}+{\frac{ \left ({b}^{10}A+10\,a{b}^{9}B \right ){x}^{14}}{14}}+{\frac{ \left ( 10\,a{b}^{9}A+45\,{a}^{2}{b}^{8}B \right ){x}^{13}}{13}}+{\frac{ \left ( 45\,{a}^{2}{b}^{8}A+120\,{a}^{3}{b}^{7}B \right ){x}^{12}}{12}}+{\frac{ \left ( 120\,{a}^{3}{b}^{7}A+210\,{a}^{4}{b}^{6}B \right ){x}^{11}}{11}}+{\frac{ \left ( 210\,{a}^{4}{b}^{6}A+252\,{a}^{5}{b}^{5}B \right ){x}^{10}}{10}}+{\frac{ \left ( 252\,{a}^{5}{b}^{5}A+210\,{a}^{6}{b}^{4}B \right ){x}^{9}}{9}}+{\frac{ \left ( 210\,{a}^{6}{b}^{4}A+120\,{a}^{7}{b}^{3}B \right ){x}^{8}}{8}}+{\frac{ \left ( 120\,{a}^{7}{b}^{3}A+45\,{a}^{8}{b}^{2}B \right ){x}^{7}}{7}}+{\frac{ \left ( 45\,{a}^{8}{b}^{2}A+10\,{a}^{9}bB \right ){x}^{6}}{6}}+{\frac{ \left ( 10\,{a}^{9}bA+{a}^{10}B \right ){x}^{5}}{5}}+{\frac{{a}^{10}A{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x+a)^10*(B*x+A),x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.02183, size = 328, normalized size = 2.93 \begin{align*} \frac{1}{15} \, B b^{10} x^{15} + \frac{1}{4} \, A a^{10} x^{4} + \frac{1}{14} \,{\left (10 \, B a b^{9} + A b^{10}\right )} x^{14} + \frac{5}{13} \,{\left (9 \, B a^{2} b^{8} + 2 \, A a b^{9}\right )} x^{13} + \frac{5}{4} \,{\left (8 \, B a^{3} b^{7} + 3 \, A a^{2} b^{8}\right )} x^{12} + \frac{30}{11} \,{\left (7 \, B a^{4} b^{6} + 4 \, A a^{3} b^{7}\right )} x^{11} + \frac{21}{5} \,{\left (6 \, B a^{5} b^{5} + 5 \, A a^{4} b^{6}\right )} x^{10} + \frac{14}{3} \,{\left (5 \, B a^{6} b^{4} + 6 \, A a^{5} b^{5}\right )} x^{9} + \frac{15}{4} \,{\left (4 \, B a^{7} b^{3} + 7 \, A a^{6} b^{4}\right )} x^{8} + \frac{15}{7} \,{\left (3 \, B a^{8} b^{2} + 8 \, A a^{7} b^{3}\right )} x^{7} + \frac{5}{6} \,{\left (2 \, B a^{9} b + 9 \, A a^{8} b^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B a^{10} + 10 \, A a^{9} b\right )} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^10*(B*x+A),x, algorithm="maxima")

[Out]

1/15*B*b^10*x^15 + 1/4*A*a^10*x^4 + 1/14*(10*B*a*b^9 + A*b^10)*x^14 + 5/13*(9*B*a^2*b^8 + 2*A*a*b^9)*x^13 + 5/
4*(8*B*a^3*b^7 + 3*A*a^2*b^8)*x^12 + 30/11*(7*B*a^4*b^6 + 4*A*a^3*b^7)*x^11 + 21/5*(6*B*a^5*b^5 + 5*A*a^4*b^6)
*x^10 + 14/3*(5*B*a^6*b^4 + 6*A*a^5*b^5)*x^9 + 15/4*(4*B*a^7*b^3 + 7*A*a^6*b^4)*x^8 + 15/7*(3*B*a^8*b^2 + 8*A*
a^7*b^3)*x^7 + 5/6*(2*B*a^9*b + 9*A*a^8*b^2)*x^6 + 1/5*(B*a^10 + 10*A*a^9*b)*x^5

________________________________________________________________________________________

Fricas [B]  time = 1.3222, size = 602, normalized size = 5.38 \begin{align*} \frac{1}{15} x^{15} b^{10} B + \frac{5}{7} x^{14} b^{9} a B + \frac{1}{14} x^{14} b^{10} A + \frac{45}{13} x^{13} b^{8} a^{2} B + \frac{10}{13} x^{13} b^{9} a A + 10 x^{12} b^{7} a^{3} B + \frac{15}{4} x^{12} b^{8} a^{2} A + \frac{210}{11} x^{11} b^{6} a^{4} B + \frac{120}{11} x^{11} b^{7} a^{3} A + \frac{126}{5} x^{10} b^{5} a^{5} B + 21 x^{10} b^{6} a^{4} A + \frac{70}{3} x^{9} b^{4} a^{6} B + 28 x^{9} b^{5} a^{5} A + 15 x^{8} b^{3} a^{7} B + \frac{105}{4} x^{8} b^{4} a^{6} A + \frac{45}{7} x^{7} b^{2} a^{8} B + \frac{120}{7} x^{7} b^{3} a^{7} A + \frac{5}{3} x^{6} b a^{9} B + \frac{15}{2} x^{6} b^{2} a^{8} A + \frac{1}{5} x^{5} a^{10} B + 2 x^{5} b a^{9} A + \frac{1}{4} x^{4} a^{10} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^10*(B*x+A),x, algorithm="fricas")

[Out]

1/15*x^15*b^10*B + 5/7*x^14*b^9*a*B + 1/14*x^14*b^10*A + 45/13*x^13*b^8*a^2*B + 10/13*x^13*b^9*a*A + 10*x^12*b
^7*a^3*B + 15/4*x^12*b^8*a^2*A + 210/11*x^11*b^6*a^4*B + 120/11*x^11*b^7*a^3*A + 126/5*x^10*b^5*a^5*B + 21*x^1
0*b^6*a^4*A + 70/3*x^9*b^4*a^6*B + 28*x^9*b^5*a^5*A + 15*x^8*b^3*a^7*B + 105/4*x^8*b^4*a^6*A + 45/7*x^7*b^2*a^
8*B + 120/7*x^7*b^3*a^7*A + 5/3*x^6*b*a^9*B + 15/2*x^6*b^2*a^8*A + 1/5*x^5*a^10*B + 2*x^5*b*a^9*A + 1/4*x^4*a^
10*A

________________________________________________________________________________________

Sympy [B]  time = 0.117245, size = 265, normalized size = 2.37 \begin{align*} \frac{A a^{10} x^{4}}{4} + \frac{B b^{10} x^{15}}{15} + x^{14} \left (\frac{A b^{10}}{14} + \frac{5 B a b^{9}}{7}\right ) + x^{13} \left (\frac{10 A a b^{9}}{13} + \frac{45 B a^{2} b^{8}}{13}\right ) + x^{12} \left (\frac{15 A a^{2} b^{8}}{4} + 10 B a^{3} b^{7}\right ) + x^{11} \left (\frac{120 A a^{3} b^{7}}{11} + \frac{210 B a^{4} b^{6}}{11}\right ) + x^{10} \left (21 A a^{4} b^{6} + \frac{126 B a^{5} b^{5}}{5}\right ) + x^{9} \left (28 A a^{5} b^{5} + \frac{70 B a^{6} b^{4}}{3}\right ) + x^{8} \left (\frac{105 A a^{6} b^{4}}{4} + 15 B a^{7} b^{3}\right ) + x^{7} \left (\frac{120 A a^{7} b^{3}}{7} + \frac{45 B a^{8} b^{2}}{7}\right ) + x^{6} \left (\frac{15 A a^{8} b^{2}}{2} + \frac{5 B a^{9} b}{3}\right ) + x^{5} \left (2 A a^{9} b + \frac{B a^{10}}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x+a)**10*(B*x+A),x)

[Out]

A*a**10*x**4/4 + B*b**10*x**15/15 + x**14*(A*b**10/14 + 5*B*a*b**9/7) + x**13*(10*A*a*b**9/13 + 45*B*a**2*b**8
/13) + x**12*(15*A*a**2*b**8/4 + 10*B*a**3*b**7) + x**11*(120*A*a**3*b**7/11 + 210*B*a**4*b**6/11) + x**10*(21
*A*a**4*b**6 + 126*B*a**5*b**5/5) + x**9*(28*A*a**5*b**5 + 70*B*a**6*b**4/3) + x**8*(105*A*a**6*b**4/4 + 15*B*
a**7*b**3) + x**7*(120*A*a**7*b**3/7 + 45*B*a**8*b**2/7) + x**6*(15*A*a**8*b**2/2 + 5*B*a**9*b/3) + x**5*(2*A*
a**9*b + B*a**10/5)

________________________________________________________________________________________

Giac [B]  time = 1.21622, size = 331, normalized size = 2.96 \begin{align*} \frac{1}{15} \, B b^{10} x^{15} + \frac{5}{7} \, B a b^{9} x^{14} + \frac{1}{14} \, A b^{10} x^{14} + \frac{45}{13} \, B a^{2} b^{8} x^{13} + \frac{10}{13} \, A a b^{9} x^{13} + 10 \, B a^{3} b^{7} x^{12} + \frac{15}{4} \, A a^{2} b^{8} x^{12} + \frac{210}{11} \, B a^{4} b^{6} x^{11} + \frac{120}{11} \, A a^{3} b^{7} x^{11} + \frac{126}{5} \, B a^{5} b^{5} x^{10} + 21 \, A a^{4} b^{6} x^{10} + \frac{70}{3} \, B a^{6} b^{4} x^{9} + 28 \, A a^{5} b^{5} x^{9} + 15 \, B a^{7} b^{3} x^{8} + \frac{105}{4} \, A a^{6} b^{4} x^{8} + \frac{45}{7} \, B a^{8} b^{2} x^{7} + \frac{120}{7} \, A a^{7} b^{3} x^{7} + \frac{5}{3} \, B a^{9} b x^{6} + \frac{15}{2} \, A a^{8} b^{2} x^{6} + \frac{1}{5} \, B a^{10} x^{5} + 2 \, A a^{9} b x^{5} + \frac{1}{4} \, A a^{10} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x+a)^10*(B*x+A),x, algorithm="giac")

[Out]

1/15*B*b^10*x^15 + 5/7*B*a*b^9*x^14 + 1/14*A*b^10*x^14 + 45/13*B*a^2*b^8*x^13 + 10/13*A*a*b^9*x^13 + 10*B*a^3*
b^7*x^12 + 15/4*A*a^2*b^8*x^12 + 210/11*B*a^4*b^6*x^11 + 120/11*A*a^3*b^7*x^11 + 126/5*B*a^5*b^5*x^10 + 21*A*a
^4*b^6*x^10 + 70/3*B*a^6*b^4*x^9 + 28*A*a^5*b^5*x^9 + 15*B*a^7*b^3*x^8 + 105/4*A*a^6*b^4*x^8 + 45/7*B*a^8*b^2*
x^7 + 120/7*A*a^7*b^3*x^7 + 5/3*B*a^9*b*x^6 + 15/2*A*a^8*b^2*x^6 + 1/5*B*a^10*x^5 + 2*A*a^9*b*x^5 + 1/4*A*a^10
*x^4