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

Optimal. Leaf size=117 \[ \frac{2}{3} a^2 b^2 x^{15} (a B+A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{1}{9} a^5 A x^9+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{17} a b^3 x^{17} (2 a B+A b)+\frac{1}{21} b^5 B x^{21} \]

[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

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Rubi [A]  time = 0.092316, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {448} \[ \frac{2}{3} a^2 b^2 x^{15} (a B+A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{1}{9} a^5 A x^9+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{17} a b^3 x^{17} (2 a B+A b)+\frac{1}{21} b^5 B x^{21} \]

Antiderivative was successfully verified.

[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 448

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*} \int x^8 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx &=\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]  time = 0.0169401, size = 117, normalized size = 1. \[ \frac{2}{3} a^2 b^2 x^{15} (a B+A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{1}{9} a^5 A x^9+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{17} a b^3 x^{17} (2 a B+A b)+\frac{1}{21} b^5 B x^{21} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.001, size = 124, normalized size = 1.1 \begin{align*}{\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}bB \right ){x}^{13}}{13}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{11}}{11}}+{\frac{{a}^{5}A{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.977337, size = 161, normalized size = 1.38 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.36036, size = 320, normalized size = 2.74 \begin{align*} \frac{1}{21} x^{21} b^{5} B + \frac{5}{19} x^{19} b^{4} a B + \frac{1}{19} x^{19} b^{5} A + \frac{10}{17} x^{17} b^{3} a^{2} B + \frac{5}{17} x^{17} b^{4} a A + \frac{2}{3} x^{15} b^{2} a^{3} B + \frac{2}{3} x^{15} b^{3} a^{2} A + \frac{5}{13} x^{13} b a^{4} B + \frac{10}{13} x^{13} b^{2} a^{3} A + \frac{1}{11} x^{11} a^{5} B + \frac{5}{11} x^{11} b a^{4} A + \frac{1}{9} x^{9} a^{5} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.082476, size = 138, normalized size = 1.18 \begin{align*} \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} \left (\frac{5 A a b^{4}}{17} + \frac{10 B a^{2} b^{3}}{17}\right ) + x^{15} \left (\frac{2 A a^{2} b^{3}}{3} + \frac{2 B a^{3} b^{2}}{3}\right ) + x^{13} \left (\frac{10 A a^{3} b^{2}}{13} + \frac{5 B a^{4} b}{13}\right ) + x^{11} \left (\frac{5 A a^{4} b}{11} + \frac{B a^{5}}{11}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.19136, size = 169, normalized size = 1.44 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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