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

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

[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 = 0.238852, 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 \[ \frac{1}{9} a^5 A x^9+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{2}{3} a^2 b^2 x^{15} (a B+A b)+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{17} a b^3 x^{17} (2 a B+A b)+\frac{1}{21} b^5 B x^{21} \]

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

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Rubi in Sympy [A]  time = 28.3106, size = 114, normalized size = 0.97 \[ \frac{A a^{5} x^{9}}{9} + \frac{B b^{5} x^{21}}{21} + \frac{a^{4} x^{11} \left (5 A b + B a\right )}{11} + \frac{5 a^{3} b x^{13} \left (2 A b + B a\right )}{13} + \frac{2 a^{2} b^{2} x^{15} \left (A b + B a\right )}{3} + \frac{5 a b^{3} x^{17} \left (A b + 2 B a\right )}{17} + \frac{b^{4} x^{19} \left (A b + 5 B a\right )}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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 + a**4*x**11*(5*A*b + B*a)/11 + 5*a**3*b*x**13*(
2*A*b + B*a)/13 + 2*a**2*b**2*x**15*(A*b + B*a)/3 + 5*a*b**3*x**17*(A*b + 2*B*a)
/17 + b**4*x**19*(A*b + 5*B*a)/19

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Mathematica [A]  time = 0.0309702, size = 117, normalized size = 1. \[ \frac{1}{9} a^5 A x^9+\frac{1}{11} a^4 x^{11} (a B+5 A b)+\frac{5}{13} a^3 b x^{13} (a B+2 A b)+\frac{2}{3} a^2 b^2 x^{15} (a B+A b)+\frac{1}{19} b^4 x^{19} (5 a B+A b)+\frac{5}{17} a b^3 x^{17} (2 a B+A b)+\frac{1}{21} b^5 B x^{21} \]

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 = 0.003, size = 124, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{21}}{21}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{19}}{19}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{17}}{17}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{15}}{15}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{13}}{13}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{11}}{11}}+{\frac{{a}^{5}A{x}^{9}}{9}} \]

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 = 1.34997, size = 161, normalized size = 1.38 \[ \frac{1}{21} \, B b^{5} x^{21} + \frac{1}{19} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{19} + \frac{5}{17} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{17} + \frac{2}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{15} + \frac{1}{9} \, A a^{5} x^{9} + \frac{5}{13} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{13} + \frac{1}{11} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{11} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5*x^8,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 = 0.220797, size = 1, normalized size = 0.01 \[ \frac{1}{21} x^{21} b^{5} B + \frac{5}{19} x^{19} b^{4} a B + \frac{1}{19} x^{19} b^{5} A + \frac{10}{17} x^{17} b^{3} a^{2} B + \frac{5}{17} x^{17} b^{4} a A + \frac{2}{3} x^{15} b^{2} a^{3} B + \frac{2}{3} x^{15} b^{3} a^{2} A + \frac{5}{13} x^{13} b a^{4} B + \frac{10}{13} x^{13} b^{2} a^{3} A + \frac{1}{11} x^{11} a^{5} B + \frac{5}{11} x^{11} b a^{4} A + \frac{1}{9} x^{9} a^{5} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5*x^8,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 = 0.173894, size = 138, normalized size = 1.18 \[ \frac{A a^{5} x^{9}}{9} + \frac{B b^{5} x^{21}}{21} + x^{19} \left (\frac{A b^{5}}{19} + \frac{5 B a b^{4}}{19}\right ) + x^{17} \left (\frac{5 A a b^{4}}{17} + \frac{10 B a^{2} b^{3}}{17}\right ) + x^{15} \left (\frac{2 A a^{2} b^{3}}{3} + \frac{2 B a^{3} b^{2}}{3}\right ) + x^{13} \left (\frac{10 A a^{3} b^{2}}{13} + \frac{5 B a^{4} b}{13}\right ) + x^{11} \left (\frac{5 A a^{4} b}{11} + \frac{B a^{5}}{11}\right ) \]

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/XCAS [A]  time = 0.218383, size = 169, normalized size = 1.44 \[ \frac{1}{21} \, B b^{5} x^{21} + \frac{5}{19} \, B a b^{4} x^{19} + \frac{1}{19} \, A b^{5} x^{19} + \frac{10}{17} \, B a^{2} b^{3} x^{17} + \frac{5}{17} \, A a b^{4} x^{17} + \frac{2}{3} \, B a^{3} b^{2} x^{15} + \frac{2}{3} \, A a^{2} b^{3} x^{15} + \frac{5}{13} \, B a^{4} b x^{13} + \frac{10}{13} \, A a^{3} b^{2} x^{13} + \frac{1}{11} \, B a^{5} x^{11} + \frac{5}{11} \, A a^{4} b x^{11} + \frac{1}{9} \, A a^{5} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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