\(\int x^9 (a+b x^2)^5 (A+B x^2) \, dx\) [23]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 (5 A b+a B) x^{12}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{18} a b^3 (A b+2 a B) x^{18}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{22} b^5 B x^{22} \]

[Out]

1/10*a^5*A*x^10+1/12*a^4*(5*A*b+B*a)*x^12+5/14*a^3*b*(2*A*b+B*a)*x^14+5/8*a^2*b^2*(A*b+B*a)*x^16+5/18*a*b^3*(A
*b+2*B*a)*x^18+1/20*b^4*(A*b+5*B*a)*x^20+1/22*b^5*B*x^22

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77} \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 x^{12} (a B+5 A b)+\frac {5}{14} a^3 b x^{14} (a B+2 A b)+\frac {5}{8} a^2 b^2 x^{16} (a B+A b)+\frac {1}{20} b^4 x^{20} (5 a B+A b)+\frac {5}{18} a b^3 x^{18} (2 a B+A b)+\frac {1}{22} b^5 B x^{22} \]

[In]

Int[x^9*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

(a^5*A*x^10)/10 + (a^4*(5*A*b + a*B)*x^12)/12 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (5*a^2*b^2*(A*b + a*B)*x^16)
/8 + (5*a*b^3*(A*b + 2*a*B)*x^18)/18 + (b^4*(A*b + 5*a*B)*x^20)/20 + (b^5*B*x^22)/22

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])

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^4 (a+b x)^5 (A+B x) \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (a^5 A x^4+a^4 (5 A b+a B) x^5+5 a^3 b (2 A b+a B) x^6+10 a^2 b^2 (A b+a B) x^7+5 a b^3 (A b+2 a B) x^8+b^4 (A b+5 a B) x^9+b^5 B x^{10}\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 (5 A b+a B) x^{12}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{18} a b^3 (A b+2 a B) x^{18}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{22} b^5 B x^{22} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{10} a^5 A x^{10}+\frac {1}{12} a^4 (5 A b+a B) x^{12}+\frac {5}{14} a^3 b (2 A b+a B) x^{14}+\frac {5}{8} a^2 b^2 (A b+a B) x^{16}+\frac {5}{18} a b^3 (A b+2 a B) x^{18}+\frac {1}{20} b^4 (A b+5 a B) x^{20}+\frac {1}{22} b^5 B x^{22} \]

[In]

Integrate[x^9*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

(a^5*A*x^10)/10 + (a^4*(5*A*b + a*B)*x^12)/12 + (5*a^3*b*(2*A*b + a*B)*x^14)/14 + (5*a^2*b^2*(A*b + a*B)*x^16)
/8 + (5*a*b^3*(A*b + 2*a*B)*x^18)/18 + (b^4*(A*b + 5*a*B)*x^20)/20 + (b^5*B*x^22)/22

Maple [A] (verified)

Time = 2.54 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03

method	result	size
norman	\(\frac {a^{5} A \,x^{10}}{10}+\left (\frac {5}{12} a^{4} b A +\frac {1}{12} a^{5} B \right ) x^{12}+\left (\frac {5}{7} a^{3} b^{2} A +\frac {5}{14} a^{4} b B \right ) x^{14}+\left (\frac {5}{8} a^{2} b^{3} A +\frac {5}{8} a^{3} b^{2} B \right ) x^{16}+\left (\frac {5}{18} a \,b^{4} A +\frac {5}{9} a^{2} b^{3} B \right ) x^{18}+\left (\frac {1}{20} b^{5} A +\frac {1}{4} a \,b^{4} B \right ) x^{20}+\frac {b^{5} B \,x^{22}}{22}\)	\(121\)
default	\(\frac {b^{5} B \,x^{22}}{22}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{20}}{20}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{18}}{18}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{16}}{16}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{14}}{14}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{12}}{12}+\frac {a^{5} A \,x^{10}}{10}\)	\(124\)
gosper	\(\frac {1}{10} a^{5} A \,x^{10}+\frac {5}{12} x^{12} a^{4} b A +\frac {1}{12} x^{12} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{18} x^{18} a \,b^{4} A +\frac {5}{9} x^{18} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\)	\(126\)
risch	\(\frac {1}{10} a^{5} A \,x^{10}+\frac {5}{12} x^{12} a^{4} b A +\frac {1}{12} x^{12} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{18} x^{18} a \,b^{4} A +\frac {5}{9} x^{18} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\)	\(126\)
parallelrisch	\(\frac {1}{10} a^{5} A \,x^{10}+\frac {5}{12} x^{12} a^{4} b A +\frac {1}{12} x^{12} a^{5} B +\frac {5}{7} x^{14} a^{3} b^{2} A +\frac {5}{14} x^{14} a^{4} b B +\frac {5}{8} x^{16} a^{2} b^{3} A +\frac {5}{8} x^{16} a^{3} b^{2} B +\frac {5}{18} x^{18} a \,b^{4} A +\frac {5}{9} x^{18} a^{2} b^{3} B +\frac {1}{20} x^{20} b^{5} A +\frac {1}{4} x^{20} a \,b^{4} B +\frac {1}{22} b^{5} B \,x^{22}\)	\(126\)

[In]

int(x^9*(b*x^2+a)^5*(B*x^2+A),x,method=_RETURNVERBOSE)

[Out]

1/10*a^5*A*x^10+(5/12*a^4*b*A+1/12*a^5*B)*x^12+(5/7*a^3*b^2*A+5/14*a^4*b*B)*x^14+(5/8*a^2*b^3*A+5/8*a^3*b^2*B)
*x^16+(5/18*a*b^4*A+5/9*a^2*b^3*B)*x^18+(1/20*b^5*A+1/4*a*b^4*B)*x^20+1/22*b^5*B*x^22

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{22} \, B b^{5} x^{22} + \frac {1}{20} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{20} + \frac {5}{18} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac {5}{8} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac {1}{10} \, A a^{5} x^{10} + \frac {5}{14} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac {1}{12} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \]

[In]

integrate(x^9*(b*x^2+a)^5*(B*x^2+A),x, algorithm="fricas")

[Out]

1/22*B*b^5*x^22 + 1/20*(5*B*a*b^4 + A*b^5)*x^20 + 5/18*(2*B*a^2*b^3 + A*a*b^4)*x^18 + 5/8*(B*a^3*b^2 + A*a^2*b
^3)*x^16 + 1/10*A*a^5*x^10 + 5/14*(B*a^4*b + 2*A*a^3*b^2)*x^14 + 1/12*(B*a^5 + 5*A*a^4*b)*x^12

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.16 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {A a^{5} x^{10}}{10} + \frac {B b^{5} x^{22}}{22} + x^{20} \left (\frac {A b^{5}}{20} + \frac {B a b^{4}}{4}\right ) + x^{18} \cdot \left (\frac {5 A a b^{4}}{18} + \frac {5 B a^{2} b^{3}}{9}\right ) + x^{16} \cdot \left (\frac {5 A a^{2} b^{3}}{8} + \frac {5 B a^{3} b^{2}}{8}\right ) + x^{14} \cdot \left (\frac {5 A a^{3} b^{2}}{7} + \frac {5 B a^{4} b}{14}\right ) + x^{12} \cdot \left (\frac {5 A a^{4} b}{12} + \frac {B a^{5}}{12}\right ) \]

[In]

integrate(x**9*(b*x**2+a)**5*(B*x**2+A),x)

[Out]

A*a**5*x**10/10 + B*b**5*x**22/22 + x**20*(A*b**5/20 + B*a*b**4/4) + x**18*(5*A*a*b**4/18 + 5*B*a**2*b**3/9) +
x**16*(5*A*a**2*b**3/8 + 5*B*a**3*b**2/8) + x**14*(5*A*a**3*b**2/7 + 5*B*a**4*b/14) + x**12*(5*A*a**4*b/12 +
B*a**5/12)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{22} \, B b^{5} x^{22} + \frac {1}{20} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{20} + \frac {5}{18} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{18} + \frac {5}{8} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{16} + \frac {1}{10} \, A a^{5} x^{10} + \frac {5}{14} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{14} + \frac {1}{12} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{12} \]

[In]

integrate(x^9*(b*x^2+a)^5*(B*x^2+A),x, algorithm="maxima")

[Out]

1/22*B*b^5*x^22 + 1/20*(5*B*a*b^4 + A*b^5)*x^20 + 5/18*(2*B*a^2*b^3 + A*a*b^4)*x^18 + 5/8*(B*a^3*b^2 + A*a^2*b
^3)*x^16 + 1/10*A*a^5*x^10 + 5/14*(B*a^4*b + 2*A*a^3*b^2)*x^14 + 1/12*(B*a^5 + 5*A*a^4*b)*x^12

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{22} \, B b^{5} x^{22} + \frac {1}{4} \, B a b^{4} x^{20} + \frac {1}{20} \, A b^{5} x^{20} + \frac {5}{9} \, B a^{2} b^{3} x^{18} + \frac {5}{18} \, A a b^{4} x^{18} + \frac {5}{8} \, B a^{3} b^{2} x^{16} + \frac {5}{8} \, A a^{2} b^{3} x^{16} + \frac {5}{14} \, B a^{4} b x^{14} + \frac {5}{7} \, A a^{3} b^{2} x^{14} + \frac {1}{12} \, B a^{5} x^{12} + \frac {5}{12} \, A a^{4} b x^{12} + \frac {1}{10} \, A a^{5} x^{10} \]

[In]

integrate(x^9*(b*x^2+a)^5*(B*x^2+A),x, algorithm="giac")

[Out]

1/22*B*b^5*x^22 + 1/4*B*a*b^4*x^20 + 1/20*A*b^5*x^20 + 5/9*B*a^2*b^3*x^18 + 5/18*A*a*b^4*x^18 + 5/8*B*a^3*b^2*
x^16 + 5/8*A*a^2*b^3*x^16 + 5/14*B*a^4*b*x^14 + 5/7*A*a^3*b^2*x^14 + 1/12*B*a^5*x^12 + 5/12*A*a^4*b*x^12 + 1/1
0*A*a^5*x^10

Mupad [B] (verification not implemented)

Time = 4.96 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^9 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=x^{12}\,\left (\frac {B\,a^5}{12}+\frac {5\,A\,b\,a^4}{12}\right )+x^{20}\,\left (\frac {A\,b^5}{20}+\frac {B\,a\,b^4}{4}\right )+\frac {A\,a^5\,x^{10}}{10}+\frac {B\,b^5\,x^{22}}{22}+\frac {5\,a^2\,b^2\,x^{16}\,\left (A\,b+B\,a\right )}{8}+\frac {5\,a^3\,b\,x^{14}\,\left (2\,A\,b+B\,a\right )}{14}+\frac {5\,a\,b^3\,x^{18}\,\left (A\,b+2\,B\,a\right )}{18} \]

[In]

int(x^9*(A + B*x^2)*(a + b*x^2)^5,x)

[Out]

x^12*((B*a^5)/12 + (5*A*a^4*b)/12) + x^20*((A*b^5)/20 + (B*a*b^4)/4) + (A*a^5*x^10)/10 + (B*b^5*x^22)/22 + (5*
a^2*b^2*x^16*(A*b + B*a))/8 + (5*a^3*b*x^14*(2*A*b + B*a))/14 + (5*a*b^3*x^18*(A*b + 2*B*a))/18