\(\int x^8 (a+b x^2)^5 (A+B x^2) \, dx\) [24]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{9} a^5 A x^9+\frac {1}{11} a^4 (5 A b+a B) x^{11}+\frac {5}{13} a^3 b (2 A b+a B) x^{13}+\frac {2}{3} a^2 b^2 (A b+a B) x^{15}+\frac {5}{17} a b^3 (A b+2 a B) x^{17}+\frac {1}{19} b^4 (A b+5 a B) x^{19}+\frac {1}{21} b^5 B x^{21} \]

[Out]

1/9*a^5*A*x^9+1/11*a^4*(5*A*b+B*a)*x^11+5/13*a^3*b*(2*A*b+B*a)*x^13+2/3*a^2*b^2*(A*b+B*a)*x^15+5/17*a*b^3*(A*b
+2*B*a)*x^17+1/19*b^4*(A*b+5*B*a)*x^19+1/21*b^5*B*x^21

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {459} \[ \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{9} a^5 A x^9+\frac {1}{11} a^4 x^{11} (a B+5 A b)+\frac {5}{13} a^3 b x^{13} (a B+2 A b)+\frac {2}{3} a^2 b^2 x^{15} (a B+A b)+\frac {1}{19} b^4 x^{19} (5 a B+A b)+\frac {5}{17} a b^3 x^{17} (2 a B+A b)+\frac {1}{21} b^5 B x^{21} \]

[In]

Int[x^8*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

(a^5*A*x^9)/9 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^13)/13 + (2*a^2*b^2*(A*b + a*B)*x^15)/3
+ (5*a*b^3*(A*b + 2*a*B)*x^17)/17 + (b^4*(A*b + 5*a*B)*x^19)/19 + (b^5*B*x^21)/21

Rule 459

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

Rubi steps \begin{align*} \text {integral}& = \int \left (a^5 A x^8+a^4 (5 A b+a B) x^{10}+5 a^3 b (2 A b+a B) x^{12}+10 a^2 b^2 (A b+a B) x^{14}+5 a b^3 (A b+2 a B) x^{16}+b^4 (A b+5 a B) x^{18}+b^5 B x^{20}\right ) \, dx \\ & = \frac {1}{9} a^5 A x^9+\frac {1}{11} a^4 (5 A b+a B) x^{11}+\frac {5}{13} a^3 b (2 A b+a B) x^{13}+\frac {2}{3} a^2 b^2 (A b+a B) x^{15}+\frac {5}{17} a b^3 (A b+2 a B) x^{17}+\frac {1}{19} b^4 (A b+5 a B) x^{19}+\frac {1}{21} b^5 B x^{21} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{9} a^5 A x^9+\frac {1}{11} a^4 (5 A b+a B) x^{11}+\frac {5}{13} a^3 b (2 A b+a B) x^{13}+\frac {2}{3} a^2 b^2 (A b+a B) x^{15}+\frac {5}{17} a b^3 (A b+2 a B) x^{17}+\frac {1}{19} b^4 (A b+5 a B) x^{19}+\frac {1}{21} b^5 B x^{21} \]

[In]

Integrate[x^8*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

(a^5*A*x^9)/9 + (a^4*(5*A*b + a*B)*x^11)/11 + (5*a^3*b*(2*A*b + a*B)*x^13)/13 + (2*a^2*b^2*(A*b + a*B)*x^15)/3
+ (5*a*b^3*(A*b + 2*a*B)*x^17)/17 + (b^4*(A*b + 5*a*B)*x^19)/19 + (b^5*B*x^21)/21

Maple [A] (verified)

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

method	result	size
norman	\(\frac {a^{5} A \,x^{9}}{9}+\left (\frac {5}{11} a^{4} b A +\frac {1}{11} a^{5} B \right ) x^{11}+\left (\frac {10}{13} a^{3} b^{2} A +\frac {5}{13} a^{4} b B \right ) x^{13}+\left (\frac {2}{3} a^{2} b^{3} A +\frac {2}{3} a^{3} b^{2} B \right ) x^{15}+\left (\frac {5}{17} a \,b^{4} A +\frac {10}{17} a^{2} b^{3} B \right ) x^{17}+\left (\frac {1}{19} b^{5} A +\frac {5}{19} a \,b^{4} B \right ) x^{19}+\frac {b^{5} B \,x^{21}}{21}\)	\(121\)
default	\(\frac {b^{5} B \,x^{21}}{21}+\frac {\left (b^{5} A +5 a \,b^{4} B \right ) x^{19}}{19}+\frac {\left (5 a \,b^{4} A +10 a^{2} b^{3} B \right ) x^{17}}{17}+\frac {\left (10 a^{2} b^{3} A +10 a^{3} b^{2} B \right ) x^{15}}{15}+\frac {\left (10 a^{3} b^{2} A +5 a^{4} b B \right ) x^{13}}{13}+\frac {\left (5 a^{4} b A +a^{5} B \right ) x^{11}}{11}+\frac {a^{5} A \,x^{9}}{9}\)	\(124\)
gosper	\(\frac {1}{9} a^{5} A \,x^{9}+\frac {5}{11} x^{11} a^{4} b A +\frac {1}{11} x^{11} a^{5} B +\frac {10}{13} x^{13} a^{3} b^{2} A +\frac {5}{13} x^{13} a^{4} b B +\frac {2}{3} x^{15} a^{2} b^{3} A +\frac {2}{3} x^{15} a^{3} b^{2} B +\frac {5}{17} x^{17} a \,b^{4} A +\frac {10}{17} x^{17} a^{2} b^{3} B +\frac {1}{19} x^{19} b^{5} A +\frac {5}{19} x^{19} a \,b^{4} B +\frac {1}{21} b^{5} B \,x^{21}\)	\(126\)
risch	\(\frac {1}{9} a^{5} A \,x^{9}+\frac {5}{11} x^{11} a^{4} b A +\frac {1}{11} x^{11} a^{5} B +\frac {10}{13} x^{13} a^{3} b^{2} A +\frac {5}{13} x^{13} a^{4} b B +\frac {2}{3} x^{15} a^{2} b^{3} A +\frac {2}{3} x^{15} a^{3} b^{2} B +\frac {5}{17} x^{17} a \,b^{4} A +\frac {10}{17} x^{17} a^{2} b^{3} B +\frac {1}{19} x^{19} b^{5} A +\frac {5}{19} x^{19} a \,b^{4} B +\frac {1}{21} b^{5} B \,x^{21}\)	\(126\)
parallelrisch	\(\frac {1}{9} a^{5} A \,x^{9}+\frac {5}{11} x^{11} a^{4} b A +\frac {1}{11} x^{11} a^{5} B +\frac {10}{13} x^{13} a^{3} b^{2} A +\frac {5}{13} x^{13} a^{4} b B +\frac {2}{3} x^{15} a^{2} b^{3} A +\frac {2}{3} x^{15} a^{3} b^{2} B +\frac {5}{17} x^{17} a \,b^{4} A +\frac {10}{17} x^{17} a^{2} b^{3} B +\frac {1}{19} x^{19} b^{5} A +\frac {5}{19} x^{19} a \,b^{4} B +\frac {1}{21} b^{5} B \,x^{21}\)	\(126\)

[In]

int(x^8*(b*x^2+a)^5*(B*x^2+A),x,method=_RETURNVERBOSE)

[Out]

1/9*a^5*A*x^9+(5/11*a^4*b*A+1/11*a^5*B)*x^11+(10/13*a^3*b^2*A+5/13*a^4*b*B)*x^13+(2/3*a^2*b^3*A+2/3*a^3*b^2*B)
*x^15+(5/17*a*b^4*A+10/17*a^2*b^3*B)*x^17+(1/19*b^5*A+5/19*a*b^4*B)*x^19+1/21*b^5*B*x^21

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{21} \, B b^{5} x^{21} + \frac {1}{19} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{19} + \frac {5}{17} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{17} + \frac {2}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{15} + \frac {1}{9} \, A a^{5} x^{9} + \frac {5}{13} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{13} + \frac {1}{11} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{11} \]

[In]

integrate(x^8*(b*x^2+a)^5*(B*x^2+A),x, algorithm="fricas")

[Out]

1/21*B*b^5*x^21 + 1/19*(5*B*a*b^4 + A*b^5)*x^19 + 5/17*(2*B*a^2*b^3 + A*a*b^4)*x^17 + 2/3*(B*a^3*b^2 + A*a^2*b
^3)*x^15 + 1/9*A*a^5*x^9 + 5/13*(B*a^4*b + 2*A*a^3*b^2)*x^13 + 1/11*(B*a^5 + 5*A*a^4*b)*x^11

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.18 \[ \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {A a^{5} x^{9}}{9} + \frac {B b^{5} x^{21}}{21} + x^{19} \left (\frac {A b^{5}}{19} + \frac {5 B a b^{4}}{19}\right ) + x^{17} \cdot \left (\frac {5 A a b^{4}}{17} + \frac {10 B a^{2} b^{3}}{17}\right ) + x^{15} \cdot \left (\frac {2 A a^{2} b^{3}}{3} + \frac {2 B a^{3} b^{2}}{3}\right ) + x^{13} \cdot \left (\frac {10 A a^{3} b^{2}}{13} + \frac {5 B a^{4} b}{13}\right ) + x^{11} \cdot \left (\frac {5 A a^{4} b}{11} + \frac {B a^{5}}{11}\right ) \]

[In]

integrate(x**8*(b*x**2+a)**5*(B*x**2+A),x)

[Out]

A*a**5*x**9/9 + B*b**5*x**21/21 + x**19*(A*b**5/19 + 5*B*a*b**4/19) + x**17*(5*A*a*b**4/17 + 10*B*a**2*b**3/17
) + x**15*(2*A*a**2*b**3/3 + 2*B*a**3*b**2/3) + x**13*(10*A*a**3*b**2/13 + 5*B*a**4*b/13) + x**11*(5*A*a**4*b/
11 + B*a**5/11)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{21} \, B b^{5} x^{21} + \frac {1}{19} \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{19} + \frac {5}{17} \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{17} + \frac {2}{3} \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{15} + \frac {1}{9} \, A a^{5} x^{9} + \frac {5}{13} \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{13} + \frac {1}{11} \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{11} \]

[In]

integrate(x^8*(b*x^2+a)^5*(B*x^2+A),x, algorithm="maxima")

[Out]

1/21*B*b^5*x^21 + 1/19*(5*B*a*b^4 + A*b^5)*x^19 + 5/17*(2*B*a^2*b^3 + A*a*b^4)*x^17 + 2/3*(B*a^3*b^2 + A*a^2*b
^3)*x^15 + 1/9*A*a^5*x^9 + 5/13*(B*a^4*b + 2*A*a^3*b^2)*x^13 + 1/11*(B*a^5 + 5*A*a^4*b)*x^11

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.07 \[ \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=\frac {1}{21} \, B b^{5} x^{21} + \frac {5}{19} \, B a b^{4} x^{19} + \frac {1}{19} \, A b^{5} x^{19} + \frac {10}{17} \, B a^{2} b^{3} x^{17} + \frac {5}{17} \, A a b^{4} x^{17} + \frac {2}{3} \, B a^{3} b^{2} x^{15} + \frac {2}{3} \, A a^{2} b^{3} x^{15} + \frac {5}{13} \, B a^{4} b x^{13} + \frac {10}{13} \, A a^{3} b^{2} x^{13} + \frac {1}{11} \, B a^{5} x^{11} + \frac {5}{11} \, A a^{4} b x^{11} + \frac {1}{9} \, A a^{5} x^{9} \]

[In]

integrate(x^8*(b*x^2+a)^5*(B*x^2+A),x, algorithm="giac")

[Out]

1/21*B*b^5*x^21 + 5/19*B*a*b^4*x^19 + 1/19*A*b^5*x^19 + 10/17*B*a^2*b^3*x^17 + 5/17*A*a*b^4*x^17 + 2/3*B*a^3*b
^2*x^15 + 2/3*A*a^2*b^3*x^15 + 5/13*B*a^4*b*x^13 + 10/13*A*a^3*b^2*x^13 + 1/11*B*a^5*x^11 + 5/11*A*a^4*b*x^11
+ 1/9*A*a^5*x^9

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx=x^{11}\,\left (\frac {B\,a^5}{11}+\frac {5\,A\,b\,a^4}{11}\right )+x^{19}\,\left (\frac {A\,b^5}{19}+\frac {5\,B\,a\,b^4}{19}\right )+\frac {A\,a^5\,x^9}{9}+\frac {B\,b^5\,x^{21}}{21}+\frac {2\,a^2\,b^2\,x^{15}\,\left (A\,b+B\,a\right )}{3}+\frac {5\,a^3\,b\,x^{13}\,\left (2\,A\,b+B\,a\right )}{13}+\frac {5\,a\,b^3\,x^{17}\,\left (A\,b+2\,B\,a\right )}{17} \]

[In]

int(x^8*(A + B*x^2)*(a + b*x^2)^5,x)

[Out]

x^11*((B*a^5)/11 + (5*A*a^4*b)/11) + x^19*((A*b^5)/19 + (5*B*a*b^4)/19) + (A*a^5*x^9)/9 + (B*b^5*x^21)/21 + (2
*a^2*b^2*x^15*(A*b + B*a))/3 + (5*a^3*b*x^13*(2*A*b + B*a))/13 + (5*a*b^3*x^17*(A*b + 2*B*a))/17