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

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.236516, 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 \[ \frac{1}{3} a^5 A x^3+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.4049, size = 114, normalized size = 0.97 \[ \frac{A a^{5} x^{3}}{3} + \frac{B b^{5} x^{15}}{15} + \frac{a^{4} x^{5} \left (5 A b + B a\right )}{5} + \frac{5 a^{3} b x^{7} \left (2 A b + B a\right )}{7} + \frac{10 a^{2} b^{2} x^{9} \left (A b + B a\right )}{9} + \frac{5 a b^{3} x^{11} \left (A b + 2 B a\right )}{11} + \frac{b^{4} x^{13} \left (A b + 5 B a\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0290692, size = 117, normalized size = 1. \[ \frac{1}{3} a^5 A x^3+\frac{1}{5} a^4 x^5 (a B+5 A b)+\frac{5}{7} a^3 b x^7 (a B+2 A b)+\frac{10}{9} a^2 b^2 x^9 (a B+A b)+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 124, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{15}}{15}}+{\frac{ \left ({b}^{5}A+5\,a{b}^{4}B \right ){x}^{13}}{13}}+{\frac{ \left ( 5\,a{b}^{4}A+10\,{a}^{2}{b}^{3}B \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,{a}^{2}{b}^{3}A+10\,{a}^{3}{b}^{2}B \right ){x}^{9}}{9}}+{\frac{ \left ( 10\,{a}^{3}{b}^{2}A+5\,{a}^{4}bB \right ){x}^{7}}{7}}+{\frac{ \left ( 5\,{a}^{4}bA+{a}^{5}B \right ){x}^{5}}{5}}+{\frac{{a}^{5}A{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34936, size = 161, normalized size = 1.38 \[ \frac{1}{15} \, B b^{5} x^{15} + \frac{1}{13} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{13} + \frac{5}{11} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{11} + \frac{10}{9} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + \frac{1}{3} \, A a^{5} x^{3} + \frac{5}{7} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{7} + \frac{1}{5} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.206406, size = 1, normalized size = 0.01 \[ \frac{1}{15} x^{15} b^{5} B + \frac{5}{13} x^{13} b^{4} a B + \frac{1}{13} x^{13} b^{5} A + \frac{10}{11} x^{11} b^{3} a^{2} B + \frac{5}{11} x^{11} b^{4} a A + \frac{10}{9} x^{9} b^{2} a^{3} B + \frac{10}{9} x^{9} b^{3} a^{2} A + \frac{5}{7} x^{7} b a^{4} B + \frac{10}{7} x^{7} b^{2} a^{3} A + \frac{1}{5} x^{5} a^{5} B + x^{5} b a^{4} A + \frac{1}{3} x^{3} a^{5} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*x^15*b^5*B + 5/13*x^13*b^4*a*B + 1/13*x^13*b^5*A + 10/11*x^11*b^3*a^2*B + 5
/11*x^11*b^4*a*A + 10/9*x^9*b^2*a^3*B + 10/9*x^9*b^3*a^2*A + 5/7*x^7*b*a^4*B + 1
0/7*x^7*b^2*a^3*A + 1/5*x^5*a^5*B + x^5*b*a^4*A + 1/3*x^3*a^5*A

_______________________________________________________________________________________

Sympy [A]  time = 0.171118, size = 134, normalized size = 1.15 \[ \frac{A a^{5} x^{3}}{3} + \frac{B b^{5} x^{15}}{15} + x^{13} \left (\frac{A b^{5}}{13} + \frac{5 B a b^{4}}{13}\right ) + x^{11} \left (\frac{5 A a b^{4}}{11} + \frac{10 B a^{2} b^{3}}{11}\right ) + x^{9} \left (\frac{10 A a^{2} b^{3}}{9} + \frac{10 B a^{3} b^{2}}{9}\right ) + x^{7} \left (\frac{10 A a^{3} b^{2}}{7} + \frac{5 B a^{4} b}{7}\right ) + x^{5} \left (A a^{4} b + \frac{B a^{5}}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.228103, size = 167, normalized size = 1.43 \[ \frac{1}{15} \, B b^{5} x^{15} + \frac{5}{13} \, B a b^{4} x^{13} + \frac{1}{13} \, A b^{5} x^{13} + \frac{10}{11} \, B a^{2} b^{3} x^{11} + \frac{5}{11} \, A a b^{4} x^{11} + \frac{10}{9} \, B a^{3} b^{2} x^{9} + \frac{10}{9} \, A a^{2} b^{3} x^{9} + \frac{5}{7} \, B a^{4} b x^{7} + \frac{10}{7} \, A a^{3} b^{2} x^{7} + \frac{1}{5} \, B a^{5} x^{5} + A a^{4} b x^{5} + \frac{1}{3} \, A a^{5} x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*B*b^5*x^15 + 5/13*B*a*b^4*x^13 + 1/13*A*b^5*x^13 + 10/11*B*a^2*b^3*x^11 + 5
/11*A*a*b^4*x^11 + 10/9*B*a^3*b^2*x^9 + 10/9*A*a^2*b^3*x^9 + 5/7*B*a^4*b*x^7 + 1
0/7*A*a^3*b^2*x^7 + 1/5*B*a^5*x^5 + A*a^4*b*x^5 + 1/3*A*a^5*x^3