3.352 \(\int x^{5/2} \left (a+b x^2\right )^2 \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=63 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2} \]

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*b*(A*b + 2*a*B)*x^(15
/2))/15 + (2*b^2*B*x^(19/2))/19

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Rubi [A]  time = 0.0876206, antiderivative size = 63, 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 \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*b*(A*b + 2*a*B)*x^(15
/2))/15 + (2*b^2*B*x^(19/2))/19

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Rubi in Sympy [A]  time = 12.6446, size = 63, normalized size = 1. \[ \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{2 B b^{2} x^{\frac{19}{2}}}{19} + \frac{2 a x^{\frac{11}{2}} \left (2 A b + B a\right )}{11} + \frac{2 b x^{\frac{15}{2}} \left (A b + 2 B a\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(7/2)/7 + 2*B*b**2*x**(19/2)/19 + 2*a*x**(11/2)*(2*A*b + B*a)/11 + 2
*b*x**(15/2)*(A*b + 2*B*a)/15

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Mathematica [A]  time = 0.0304358, size = 63, normalized size = 1. \[ \frac{2}{7} a^2 A x^{7/2}+\frac{2}{15} b x^{15/2} (2 a B+A b)+\frac{2}{11} a x^{11/2} (a B+2 A b)+\frac{2}{19} b^2 B x^{19/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*A*x^(7/2))/7 + (2*a*(2*A*b + a*B)*x^(11/2))/11 + (2*b*(A*b + 2*a*B)*x^(15
/2))/15 + (2*b^2*B*x^(19/2))/19

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Maple [A]  time = 0.009, size = 56, normalized size = 0.9 \[{\frac{2310\,{b}^{2}B{x}^{6}+2926\,A{b}^{2}{x}^{4}+5852\,{x}^{4}abB+7980\,aAb{x}^{2}+3990\,{x}^{2}{a}^{2}B+6270\,{a}^{2}A}{21945}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*x^(7/2)*(1155*B*b^2*x^6+1463*A*b^2*x^4+2926*B*a*b*x^4+3990*A*a*b*x^2+199
5*B*a^2*x^2+3135*A*a^2)

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Maxima [A]  time = 1.37332, size = 69, normalized size = 1.1 \[ \frac{2}{19} \, B b^{2} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{15}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*B*b^2*x^(19/2) + 2/15*(2*B*a*b + A*b^2)*x^(15/2) + 2/7*A*a^2*x^(7/2) + 2/11
*(B*a^2 + 2*A*a*b)*x^(11/2)

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Fricas [A]  time = 0.211977, size = 76, normalized size = 1.21 \[ \frac{2}{21945} \,{\left (1155 \, B b^{2} x^{9} + 1463 \,{\left (2 \, B a b + A b^{2}\right )} x^{7} + 3135 \, A a^{2} x^{3} + 1995 \,{\left (B a^{2} + 2 \, A a b\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21945*(1155*B*b^2*x^9 + 1463*(2*B*a*b + A*b^2)*x^7 + 3135*A*a^2*x^3 + 1995*(B*
a^2 + 2*A*a*b)*x^5)*sqrt(x)

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Sympy [A]  time = 37.5169, size = 80, normalized size = 1.27 \[ \frac{2 A a^{2} x^{\frac{7}{2}}}{7} + \frac{4 A a b x^{\frac{11}{2}}}{11} + \frac{2 A b^{2} x^{\frac{15}{2}}}{15} + \frac{2 B a^{2} x^{\frac{11}{2}}}{11} + \frac{4 B a b x^{\frac{15}{2}}}{15} + \frac{2 B b^{2} x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(7/2)/7 + 4*A*a*b*x**(11/2)/11 + 2*A*b**2*x**(15/2)/15 + 2*B*a**2*x*
*(11/2)/11 + 4*B*a*b*x**(15/2)/15 + 2*B*b**2*x**(19/2)/19

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GIAC/XCAS [A]  time = 0.215108, size = 72, normalized size = 1.14 \[ \frac{2}{19} \, B b^{2} x^{\frac{19}{2}} + \frac{4}{15} \, B a b x^{\frac{15}{2}} + \frac{2}{15} \, A b^{2} x^{\frac{15}{2}} + \frac{2}{11} \, B a^{2} x^{\frac{11}{2}} + \frac{4}{11} \, A a b x^{\frac{11}{2}} + \frac{2}{7} \, A a^{2} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/19*B*b^2*x^(19/2) + 4/15*B*a*b*x^(15/2) + 2/15*A*b^2*x^(15/2) + 2/11*B*a^2*x^(
11/2) + 4/11*A*a*b*x^(11/2) + 2/7*A*a^2*x^(7/2)