\(\int x^2 (a+b x)^5 (A+B x) \, dx\) [122]

Optimal result

Integrand size = 16, antiderivative size = 87 \[ \int x^2 (a+b x)^5 (A+B x) \, dx=\frac {a^2 (A b-a B) (a+b x)^6}{6 b^4}-\frac {a (2 A b-3 a B) (a+b x)^7}{7 b^4}+\frac {(A b-3 a B) (a+b x)^8}{8 b^4}+\frac {B (a+b x)^9}{9 b^4} \]

[Out]

1/6*a^2*(A*b-B*a)*(b*x+a)^6/b^4-1/7*a*(2*A*b-3*B*a)*(b*x+a)^7/b^4+1/8*(A*b-3*B*a)*(b*x+a)^8/b^4+1/9*B*(b*x+a)^
9/b^4

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int x^2 (a+b x)^5 (A+B x) \, dx=\frac {a^2 (a+b x)^6 (A b-a B)}{6 b^4}+\frac {(a+b x)^8 (A b-3 a B)}{8 b^4}-\frac {a (a+b x)^7 (2 A b-3 a B)}{7 b^4}+\frac {B (a+b x)^9}{9 b^4} \]

[In]

Int[x^2*(a + b*x)^5*(A + B*x),x]

[Out]

(a^2*(A*b - a*B)*(a + b*x)^6)/(6*b^4) - (a*(2*A*b - 3*a*B)*(a + b*x)^7)/(7*b^4) + ((A*b - 3*a*B)*(a + b*x)^8)/
(8*b^4) + (B*(a + b*x)^9)/(9*b^4)

Rule 77

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*} \text {integral}& = \int \left (-\frac {a^2 (-A b+a B) (a+b x)^5}{b^3}+\frac {a (-2 A b+3 a B) (a+b x)^6}{b^3}+\frac {(A b-3 a B) (a+b x)^7}{b^3}+\frac {B (a+b x)^8}{b^3}\right ) \, dx \\ & = \frac {a^2 (A b-a B) (a+b x)^6}{6 b^4}-\frac {a (2 A b-3 a B) (a+b x)^7}{7 b^4}+\frac {(A b-3 a B) (a+b x)^8}{8 b^4}+\frac {B (a+b x)^9}{9 b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31 \[ \int x^2 (a+b x)^5 (A+B x) \, dx=\frac {1}{3} a^5 A x^3+\frac {1}{4} a^4 (5 A b+a B) x^4+a^3 b (2 A b+a B) x^5+\frac {5}{3} a^2 b^2 (A b+a B) x^6+\frac {5}{7} a b^3 (A b+2 a B) x^7+\frac {1}{8} b^4 (A b+5 a B) x^8+\frac {1}{9} b^5 B x^9 \]

[In]

Integrate[x^2*(a + b*x)^5*(A + B*x),x]

[Out]

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

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.38

method	result	size
norman	\(\frac {b^{5} B \,x^{9}}{9}+\left (\frac {1}{8} b^{5} A +\frac {5}{8} a \,b^{4} B \right ) x^{8}+\left (\frac {5}{7} a \,b^{4} A +\frac {10}{7} a^{2} b^{3} B \right ) x^{7}+\left (\frac {5}{3} a^{2} b^{3} A +\frac {5}{3} a^{3} b^{2} B \right ) x^{6}+\left (2 a^{3} b^{2} A +a^{4} b B \right ) x^{5}+\left (\frac {5}{4} a^{4} b A +\frac {1}{4} a^{5} B \right ) x^{4}+\frac {a^{5} A \,x^{3}}{3}\)	\(120\)
default	\(\frac {b^{5} B \,x^{9}}{9}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{8}}{8}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{7}}{7}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{6}}{6}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{5}}{5}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{4}}{4}+\frac {a^{5} A \,x^{3}}{3}\)	\(124\)
gosper	\(\frac {1}{9} b^{5} B \,x^{9}+\frac {1}{8} x^{8} b^{5} A +\frac {5}{8} x^{8} a \,b^{4} B +\frac {5}{7} x^{7} a \,b^{4} A +\frac {10}{7} x^{7} a^{2} b^{3} B +\frac {5}{3} x^{6} a^{2} b^{3} A +\frac {5}{3} x^{6} a^{3} b^{2} B +2 A \,a^{3} b^{2} x^{5}+B \,a^{4} b \,x^{5}+\frac {5}{4} x^{4} a^{4} b A +\frac {1}{4} x^{4} a^{5} B +\frac {1}{3} a^{5} A \,x^{3}\)	\(125\)
risch	\(\frac {1}{9} b^{5} B \,x^{9}+\frac {1}{8} x^{8} b^{5} A +\frac {5}{8} x^{8} a \,b^{4} B +\frac {5}{7} x^{7} a \,b^{4} A +\frac {10}{7} x^{7} a^{2} b^{3} B +\frac {5}{3} x^{6} a^{2} b^{3} A +\frac {5}{3} x^{6} a^{3} b^{2} B +2 A \,a^{3} b^{2} x^{5}+B \,a^{4} b \,x^{5}+\frac {5}{4} x^{4} a^{4} b A +\frac {1}{4} x^{4} a^{5} B +\frac {1}{3} a^{5} A \,x^{3}\)	\(125\)
parallelrisch	\(\frac {1}{9} b^{5} B \,x^{9}+\frac {1}{8} x^{8} b^{5} A +\frac {5}{8} x^{8} a \,b^{4} B +\frac {5}{7} x^{7} a \,b^{4} A +\frac {10}{7} x^{7} a^{2} b^{3} B +\frac {5}{3} x^{6} a^{2} b^{3} A +\frac {5}{3} x^{6} a^{3} b^{2} B +2 A \,a^{3} b^{2} x^{5}+B \,a^{4} b \,x^{5}+\frac {5}{4} x^{4} a^{4} b A +\frac {1}{4} x^{4} a^{5} B +\frac {1}{3} a^{5} A \,x^{3}\)	\(125\)

[In]

int(x^2*(b*x+a)^5*(B*x+A),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36 \[ \int x^2 (a+b x)^5 (A+B x) \, dx=\frac {1}{9} \, B b^{5} x^{9} + \frac {1}{3} \, A a^{5} x^{3} + \frac {1}{8} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + \frac {5}{7} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \]

[In]

integrate(x^2*(b*x+a)^5*(B*x+A),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.53 \[ \int x^2 (a+b x)^5 (A+B x) \, dx=\frac {A a^{5} x^{3}}{3} + \frac {B b^{5} x^{9}}{9} + x^{8} \left (\frac {A b^{5}}{8} + \frac {5 B a b^{4}}{8}\right ) + x^{7} \cdot \left (\frac {5 A a b^{4}}{7} + \frac {10 B a^{2} b^{3}}{7}\right ) + x^{6} \cdot \left (\frac {5 A a^{2} b^{3}}{3} + \frac {5 B a^{3} b^{2}}{3}\right ) + x^{5} \cdot \left (2 A a^{3} b^{2} + B a^{4} b\right ) + x^{4} \cdot \left (\frac {5 A a^{4} b}{4} + \frac {B a^{5}}{4}\right ) \]

[In]

integrate(x**2*(b*x+a)**5*(B*x+A),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36 \[ \int x^2 (a+b x)^5 (A+B x) \, dx=\frac {1}{9} \, B b^{5} x^{9} + \frac {1}{3} \, A a^{5} x^{3} + \frac {1}{8} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} + \frac {5}{7} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + \frac {5}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{4} \]

[In]

integrate(x^2*(b*x+a)^5*(B*x+A),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.43 \[ \int x^2 (a+b x)^5 (A+B x) \, dx=\frac {1}{9} \, B b^{5} x^{9} + \frac {5}{8} \, B a b^{4} x^{8} + \frac {1}{8} \, A b^{5} x^{8} + \frac {10}{7} \, B a^{2} b^{3} x^{7} + \frac {5}{7} \, A a b^{4} x^{7} + \frac {5}{3} \, B a^{3} b^{2} x^{6} + \frac {5}{3} \, A a^{2} b^{3} x^{6} + B a^{4} b x^{5} + 2 \, A a^{3} b^{2} x^{5} + \frac {1}{4} \, B a^{5} x^{4} + \frac {5}{4} \, A a^{4} b x^{4} + \frac {1}{3} \, A a^{5} x^{3} \]

[In]

integrate(x^2*(b*x+a)^5*(B*x+A),x, algorithm="giac")

[Out]

1/9*B*b^5*x^9 + 5/8*B*a*b^4*x^8 + 1/8*A*b^5*x^8 + 10/7*B*a^2*b^3*x^7 + 5/7*A*a*b^4*x^7 + 5/3*B*a^3*b^2*x^6 + 5
/3*A*a^2*b^3*x^6 + B*a^4*b*x^5 + 2*A*a^3*b^2*x^5 + 1/4*B*a^5*x^4 + 5/4*A*a^4*b*x^4 + 1/3*A*a^5*x^3

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.22 \[ \int x^2 (a+b x)^5 (A+B x) \, dx=x^4\,\left (\frac {B\,a^5}{4}+\frac {5\,A\,b\,a^4}{4}\right )+x^8\,\left (\frac {A\,b^5}{8}+\frac {5\,B\,a\,b^4}{8}\right )+\frac {A\,a^5\,x^3}{3}+\frac {B\,b^5\,x^9}{9}+\frac {5\,a^2\,b^2\,x^6\,\left (A\,b+B\,a\right )}{3}+a^3\,b\,x^5\,\left (2\,A\,b+B\,a\right )+\frac {5\,a\,b^3\,x^7\,\left (A\,b+2\,B\,a\right )}{7} \]

[In]

int(x^2*(A + B*x)*(a + b*x)^5,x)

[Out]

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