3.26 \(\int x^6 \left (a+b x^2\right )^5 \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=117 \[ \frac{1}{7} a^5 A x^7+\frac{1}{9} a^4 x^9 (a B+5 A b)+\frac{5}{11} a^3 b x^{11} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{1}{3} a b^3 x^{15} (2 a B+A b)+\frac{1}{19} b^5 B x^{19} \]

[Out]

(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^9)/9 + (5*a^3*b*(2*A*b + a*B)*x^11)/11 + (1
0*a^2*b^2*(A*b + a*B)*x^13)/13 + (a*b^3*(A*b + 2*a*B)*x^15)/3 + (b^4*(A*b + 5*a*
B)*x^17)/17 + (b^5*B*x^19)/19

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Rubi [A]  time = 0.236656, 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 \[ \frac{1}{7} a^5 A x^7+\frac{1}{9} a^4 x^9 (a B+5 A b)+\frac{5}{11} a^3 b x^{11} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{1}{3} a b^3 x^{15} (2 a B+A b)+\frac{1}{19} b^5 B x^{19} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^9)/9 + (5*a^3*b*(2*A*b + a*B)*x^11)/11 + (1
0*a^2*b^2*(A*b + a*B)*x^13)/13 + (a*b^3*(A*b + 2*a*B)*x^15)/3 + (b^4*(A*b + 5*a*
B)*x^17)/17 + (b^5*B*x^19)/19

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Rubi in Sympy [A]  time = 27.3275, size = 112, normalized size = 0.96 \[ \frac{A a^{5} x^{7}}{7} + \frac{B b^{5} x^{19}}{19} + \frac{a^{4} x^{9} \left (5 A b + B a\right )}{9} + \frac{5 a^{3} b x^{11} \left (2 A b + B a\right )}{11} + \frac{10 a^{2} b^{2} x^{13} \left (A b + B a\right )}{13} + \frac{a b^{3} x^{15} \left (A b + 2 B a\right )}{3} + \frac{b^{4} x^{17} \left (A b + 5 B a\right )}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(b*x**2+a)**5*(B*x**2+A),x)

[Out]

A*a**5*x**7/7 + B*b**5*x**19/19 + a**4*x**9*(5*A*b + B*a)/9 + 5*a**3*b*x**11*(2*
A*b + B*a)/11 + 10*a**2*b**2*x**13*(A*b + B*a)/13 + a*b**3*x**15*(A*b + 2*B*a)/3
+ b**4*x**17*(A*b + 5*B*a)/17

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Mathematica [A]  time = 0.0276702, size = 117, normalized size = 1. \[ \frac{1}{7} a^5 A x^7+\frac{1}{9} a^4 x^9 (a B+5 A b)+\frac{5}{11} a^3 b x^{11} (a B+2 A b)+\frac{10}{13} a^2 b^2 x^{13} (a B+A b)+\frac{1}{17} b^4 x^{17} (5 a B+A b)+\frac{1}{3} a b^3 x^{15} (2 a B+A b)+\frac{1}{19} b^5 B x^{19} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6*(a + b*x^2)^5*(A + B*x^2),x]

[Out]

(a^5*A*x^7)/7 + (a^4*(5*A*b + a*B)*x^9)/9 + (5*a^3*b*(2*A*b + a*B)*x^11)/11 + (1
0*a^2*b^2*(A*b + a*B)*x^13)/13 + (a*b^3*(A*b + 2*a*B)*x^15)/3 + (b^4*(A*b + 5*a*
B)*x^17)/17 + (b^5*B*x^19)/19

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Maple [A]  time = 0.001, size = 124, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{19}}{19}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{17}}{17}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{15}}{15}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{13}}{13}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{11}}{11}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{9}}{9}}+{\frac{{a}^{5}A{x}^{7}}{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x^2+a)^5*(B*x^2+A),x)

[Out]

1/19*b^5*B*x^19+1/17*(A*b^5+5*B*a*b^4)*x^17+1/15*(5*A*a*b^4+10*B*a^2*b^3)*x^15+1
/13*(10*A*a^2*b^3+10*B*a^3*b^2)*x^13+1/11*(10*A*a^3*b^2+5*B*a^4*b)*x^11+1/9*(5*A
*a^4*b+B*a^5)*x^9+1/7*a^5*A*x^7

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Maxima [A]  time = 1.35183, size = 161, normalized size = 1.38 \[ \frac{1}{19} \, B b^{5} x^{19} + \frac{1}{17} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{17} + \frac{1}{3} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{15} + \frac{10}{13} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{13} + \frac{1}{7} \, A a^{5} x^{7} + \frac{5}{11} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{11} + \frac{1}{9} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5*x^6,x, algorithm="maxima")

[Out]

1/19*B*b^5*x^19 + 1/17*(5*B*a*b^4 + A*b^5)*x^17 + 1/3*(2*B*a^2*b^3 + A*a*b^4)*x^
15 + 10/13*(B*a^3*b^2 + A*a^2*b^3)*x^13 + 1/7*A*a^5*x^7 + 5/11*(B*a^4*b + 2*A*a^
3*b^2)*x^11 + 1/9*(B*a^5 + 5*A*a^4*b)*x^9

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Fricas [A]  time = 0.212281, size = 1, normalized size = 0.01 \[ \frac{1}{19} x^{19} b^{5} B + \frac{5}{17} x^{17} b^{4} a B + \frac{1}{17} x^{17} b^{5} A + \frac{2}{3} x^{15} b^{3} a^{2} B + \frac{1}{3} x^{15} b^{4} a A + \frac{10}{13} x^{13} b^{2} a^{3} B + \frac{10}{13} x^{13} b^{3} a^{2} A + \frac{5}{11} x^{11} b a^{4} B + \frac{10}{11} x^{11} b^{2} a^{3} A + \frac{1}{9} x^{9} a^{5} B + \frac{5}{9} x^{9} b a^{4} A + \frac{1}{7} x^{7} a^{5} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5*x^6,x, algorithm="fricas")

[Out]

1/19*x^19*b^5*B + 5/17*x^17*b^4*a*B + 1/17*x^17*b^5*A + 2/3*x^15*b^3*a^2*B + 1/3
*x^15*b^4*a*A + 10/13*x^13*b^2*a^3*B + 10/13*x^13*b^3*a^2*A + 5/11*x^11*b*a^4*B
+ 10/11*x^11*b^2*a^3*A + 1/9*x^9*a^5*B + 5/9*x^9*b*a^4*A + 1/7*x^7*a^5*A

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Sympy [A]  time = 0.178191, size = 136, normalized size = 1.16 \[ \frac{A a^{5} x^{7}}{7} + \frac{B b^{5} x^{19}}{19} + x^{17} \left (\frac{A b^{5}}{17} + \frac{5 B a b^{4}}{17}\right ) + x^{15} \left (\frac{A a b^{4}}{3} + \frac{2 B a^{2} b^{3}}{3}\right ) + x^{13} \left (\frac{10 A a^{2} b^{3}}{13} + \frac{10 B a^{3} b^{2}}{13}\right ) + x^{11} \left (\frac{10 A a^{3} b^{2}}{11} + \frac{5 B a^{4} b}{11}\right ) + x^{9} \left (\frac{5 A a^{4} b}{9} + \frac{B a^{5}}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(b*x**2+a)**5*(B*x**2+A),x)

[Out]

A*a**5*x**7/7 + B*b**5*x**19/19 + x**17*(A*b**5/17 + 5*B*a*b**4/17) + x**15*(A*a
*b**4/3 + 2*B*a**2*b**3/3) + x**13*(10*A*a**2*b**3/13 + 10*B*a**3*b**2/13) + x**
11*(10*A*a**3*b**2/11 + 5*B*a**4*b/11) + x**9*(5*A*a**4*b/9 + B*a**5/9)

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GIAC/XCAS [A]  time = 0.223134, size = 169, normalized size = 1.44 \[ \frac{1}{19} \, B b^{5} x^{19} + \frac{5}{17} \, B a b^{4} x^{17} + \frac{1}{17} \, A b^{5} x^{17} + \frac{2}{3} \, B a^{2} b^{3} x^{15} + \frac{1}{3} \, A a b^{4} x^{15} + \frac{10}{13} \, B a^{3} b^{2} x^{13} + \frac{10}{13} \, A a^{2} b^{3} x^{13} + \frac{5}{11} \, B a^{4} b x^{11} + \frac{10}{11} \, A a^{3} b^{2} x^{11} + \frac{1}{9} \, B a^{5} x^{9} + \frac{5}{9} \, A a^{4} b x^{9} + \frac{1}{7} \, A a^{5} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^5*x^6,x, algorithm="giac")

[Out]

1/19*B*b^5*x^19 + 5/17*B*a*b^4*x^17 + 1/17*A*b^5*x^17 + 2/3*B*a^2*b^3*x^15 + 1/3
*A*a*b^4*x^15 + 10/13*B*a^3*b^2*x^13 + 10/13*A*a^2*b^3*x^13 + 5/11*B*a^4*b*x^11
+ 10/11*A*a^3*b^2*x^11 + 1/9*B*a^5*x^9 + 5/9*A*a^4*b*x^9 + 1/7*A*a^5*x^7