3.359 \(\int x^{7/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{9} a^3 A x^{9/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{21} b^2 x^{21/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2} \]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(
17/2))/17 + (2*b^2*(A*b + 3*a*B)*x^(21/2))/21 + (2*b^3*B*x^(25/2))/25

_______________________________________________________________________________________

Rubi [A]  time = 0.118122, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{2}{9} a^3 A x^{9/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{21} b^2 x^{21/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(7/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(
17/2))/17 + (2*b^2*(A*b + 3*a*B)*x^(21/2))/21 + (2*b^3*B*x^(25/2))/25

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.5911, size = 85, normalized size = 1. \[ \frac{2 A a^{3} x^{\frac{9}{2}}}{9} + \frac{2 B b^{3} x^{\frac{25}{2}}}{25} + \frac{2 a^{2} x^{\frac{13}{2}} \left (3 A b + B a\right )}{13} + \frac{6 a b x^{\frac{17}{2}} \left (A b + B a\right )}{17} + \frac{2 b^{2} x^{\frac{21}{2}} \left (A b + 3 B a\right )}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(7/2)*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

2*A*a**3*x**(9/2)/9 + 2*B*b**3*x**(25/2)/25 + 2*a**2*x**(13/2)*(3*A*b + B*a)/13
+ 6*a*b*x**(17/2)*(A*b + B*a)/17 + 2*b**2*x**(21/2)*(A*b + 3*B*a)/21

_______________________________________________________________________________________

Mathematica [A]  time = 0.0423958, size = 85, normalized size = 1. \[ \frac{2}{9} a^3 A x^{9/2}+\frac{2}{13} a^2 x^{13/2} (a B+3 A b)+\frac{2}{21} b^2 x^{21/2} (3 a B+A b)+\frac{6}{17} a b x^{17/2} (a B+A b)+\frac{2}{25} b^3 B x^{25/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(7/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*a^3*A*x^(9/2))/9 + (2*a^2*(3*A*b + a*B)*x^(13/2))/13 + (6*a*b*(A*b + a*B)*x^(
17/2))/17 + (2*b^2*(A*b + 3*a*B)*x^(21/2))/21 + (2*b^3*B*x^(25/2))/25

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 80, normalized size = 0.9 \[{\frac{27846\,B{b}^{3}{x}^{8}+33150\,{x}^{6}A{b}^{3}+99450\,{x}^{6}Ba{b}^{2}+122850\,{x}^{4}Aa{b}^{2}+122850\,{x}^{4}B{a}^{2}b+160650\,{x}^{2}A{a}^{2}b+53550\,{x}^{2}B{a}^{3}+77350\,A{a}^{3}}{348075}{x}^{{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(7/2)*(b*x^2+a)^3*(B*x^2+A),x)

[Out]

2/348075*x^(9/2)*(13923*B*b^3*x^8+16575*A*b^3*x^6+49725*B*a*b^2*x^6+61425*A*a*b^
2*x^4+61425*B*a^2*b*x^4+80325*A*a^2*b*x^2+26775*B*a^3*x^2+38675*A*a^3)

_______________________________________________________________________________________

Maxima [A]  time = 1.33881, size = 99, normalized size = 1.16 \[ \frac{2}{25} \, B b^{3} x^{\frac{25}{2}} + \frac{2}{21} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{21}{2}} + \frac{6}{17} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{17}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} + \frac{2}{13} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{13}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^3*x^(7/2),x, algorithm="maxima")

[Out]

2/25*B*b^3*x^(25/2) + 2/21*(3*B*a*b^2 + A*b^3)*x^(21/2) + 6/17*(B*a^2*b + A*a*b^
2)*x^(17/2) + 2/9*A*a^3*x^(9/2) + 2/13*(B*a^3 + 3*A*a^2*b)*x^(13/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.213486, size = 105, normalized size = 1.24 \[ \frac{2}{348075} \,{\left (13923 \, B b^{3} x^{12} + 16575 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{10} + 61425 \,{\left (B a^{2} b + A a b^{2}\right )} x^{8} + 38675 \, A a^{3} x^{4} + 26775 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{6}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^3*x^(7/2),x, algorithm="fricas")

[Out]

2/348075*(13923*B*b^3*x^12 + 16575*(3*B*a*b^2 + A*b^3)*x^10 + 61425*(B*a^2*b + A
*a*b^2)*x^8 + 38675*A*a^3*x^4 + 26775*(B*a^3 + 3*A*a^2*b)*x^6)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 123.375, size = 114, normalized size = 1.34 \[ \frac{2 A a^{3} x^{\frac{9}{2}}}{9} + \frac{6 A a^{2} b x^{\frac{13}{2}}}{13} + \frac{6 A a b^{2} x^{\frac{17}{2}}}{17} + \frac{2 A b^{3} x^{\frac{21}{2}}}{21} + \frac{2 B a^{3} x^{\frac{13}{2}}}{13} + \frac{6 B a^{2} b x^{\frac{17}{2}}}{17} + \frac{2 B a b^{2} x^{\frac{21}{2}}}{7} + \frac{2 B b^{3} x^{\frac{25}{2}}}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(7/2)*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

2*A*a**3*x**(9/2)/9 + 6*A*a**2*b*x**(13/2)/13 + 6*A*a*b**2*x**(17/2)/17 + 2*A*b*
*3*x**(21/2)/21 + 2*B*a**3*x**(13/2)/13 + 6*B*a**2*b*x**(17/2)/17 + 2*B*a*b**2*x
**(21/2)/7 + 2*B*b**3*x**(25/2)/25

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212349, size = 104, normalized size = 1.22 \[ \frac{2}{25} \, B b^{3} x^{\frac{25}{2}} + \frac{2}{7} \, B a b^{2} x^{\frac{21}{2}} + \frac{2}{21} \, A b^{3} x^{\frac{21}{2}} + \frac{6}{17} \, B a^{2} b x^{\frac{17}{2}} + \frac{6}{17} \, A a b^{2} x^{\frac{17}{2}} + \frac{2}{13} \, B a^{3} x^{\frac{13}{2}} + \frac{6}{13} \, A a^{2} b x^{\frac{13}{2}} + \frac{2}{9} \, A a^{3} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^3*x^(7/2),x, algorithm="giac")

[Out]

2/25*B*b^3*x^(25/2) + 2/7*B*a*b^2*x^(21/2) + 2/21*A*b^3*x^(21/2) + 6/17*B*a^2*b*
x^(17/2) + 6/17*A*a*b^2*x^(17/2) + 2/13*B*a^3*x^(13/2) + 6/13*A*a^2*b*x^(13/2) +
2/9*A*a^3*x^(9/2)