3.360 \(\int x^{5/2} \left (a+b x^2\right )^3 \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=85 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2} \]

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (2*a*b*(A*b + a*B)*x^(
15/2))/5 + (2*b^2*(A*b + 3*a*B)*x^(19/2))/19 + (2*b^3*B*x^(23/2))/23

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Rubi [A]  time = 0.116085, 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 \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (2*a*b*(A*b + a*B)*x^(
15/2))/5 + (2*b^2*(A*b + 3*a*B)*x^(19/2))/19 + (2*b^3*B*x^(23/2))/23

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Rubi in Sympy [A]  time = 16.528, size = 85, normalized size = 1. \[ \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{2 B b^{3} x^{\frac{23}{2}}}{23} + \frac{2 a^{2} x^{\frac{11}{2}} \left (3 A b + B a\right )}{11} + \frac{2 a b x^{\frac{15}{2}} \left (A b + B a\right )}{5} + \frac{2 b^{2} x^{\frac{19}{2}} \left (A b + 3 B a\right )}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

2*A*a**3*x**(7/2)/7 + 2*B*b**3*x**(23/2)/23 + 2*a**2*x**(11/2)*(3*A*b + B*a)/11
+ 2*a*b*x**(15/2)*(A*b + B*a)/5 + 2*b**2*x**(19/2)*(A*b + 3*B*a)/19

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Mathematica [A]  time = 0.0380789, size = 85, normalized size = 1. \[ \frac{2}{7} a^3 A x^{7/2}+\frac{2}{11} a^2 x^{11/2} (a B+3 A b)+\frac{2}{19} b^2 x^{19/2} (3 a B+A b)+\frac{2}{5} a b x^{15/2} (a B+A b)+\frac{2}{23} b^3 B x^{23/2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(a + b*x^2)^3*(A + B*x^2),x]

[Out]

(2*a^3*A*x^(7/2))/7 + (2*a^2*(3*A*b + a*B)*x^(11/2))/11 + (2*a*b*(A*b + a*B)*x^(
15/2))/5 + (2*b^2*(A*b + 3*a*B)*x^(19/2))/19 + (2*b^3*B*x^(23/2))/23

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Maple [A]  time = 0.009, size = 80, normalized size = 0.9 \[{\frac{14630\,B{b}^{3}{x}^{8}+17710\,{x}^{6}A{b}^{3}+53130\,{x}^{6}Ba{b}^{2}+67298\,{x}^{4}Aa{b}^{2}+67298\,{x}^{4}B{a}^{2}b+91770\,{x}^{2}A{a}^{2}b+30590\,{x}^{2}B{a}^{3}+48070\,A{a}^{3}}{168245}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x^2+a)^3*(B*x^2+A),x)

[Out]

2/168245*x^(7/2)*(7315*B*b^3*x^8+8855*A*b^3*x^6+26565*B*a*b^2*x^6+33649*A*a*b^2*
x^4+33649*B*a^2*b*x^4+45885*A*a^2*b*x^2+15295*B*a^3*x^2+24035*A*a^3)

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Maxima [A]  time = 1.35314, size = 99, normalized size = 1.16 \[ \frac{2}{23} \, B b^{3} x^{\frac{23}{2}} + \frac{2}{19} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{19}{2}} + \frac{2}{5} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{15}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} + \frac{2}{11} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{11}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^3*x^(5/2),x, algorithm="maxima")

[Out]

2/23*B*b^3*x^(23/2) + 2/19*(3*B*a*b^2 + A*b^3)*x^(19/2) + 2/5*(B*a^2*b + A*a*b^2
)*x^(15/2) + 2/7*A*a^3*x^(7/2) + 2/11*(B*a^3 + 3*A*a^2*b)*x^(11/2)

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Fricas [A]  time = 0.213378, size = 105, normalized size = 1.24 \[ \frac{2}{168245} \,{\left (7315 \, B b^{3} x^{11} + 8855 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{9} + 33649 \,{\left (B a^{2} b + A a b^{2}\right )} x^{7} + 24035 \, A a^{3} x^{3} + 15295 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^3*x^(5/2),x, algorithm="fricas")

[Out]

2/168245*(7315*B*b^3*x^11 + 8855*(3*B*a*b^2 + A*b^3)*x^9 + 33649*(B*a^2*b + A*a*
b^2)*x^7 + 24035*A*a^3*x^3 + 15295*(B*a^3 + 3*A*a^2*b)*x^5)*sqrt(x)

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Sympy [A]  time = 71.5639, size = 114, normalized size = 1.34 \[ \frac{2 A a^{3} x^{\frac{7}{2}}}{7} + \frac{6 A a^{2} b x^{\frac{11}{2}}}{11} + \frac{2 A a b^{2} x^{\frac{15}{2}}}{5} + \frac{2 A b^{3} x^{\frac{19}{2}}}{19} + \frac{2 B a^{3} x^{\frac{11}{2}}}{11} + \frac{2 B a^{2} b x^{\frac{15}{2}}}{5} + \frac{6 B a b^{2} x^{\frac{19}{2}}}{19} + \frac{2 B b^{3} x^{\frac{23}{2}}}{23} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(b*x**2+a)**3*(B*x**2+A),x)

[Out]

2*A*a**3*x**(7/2)/7 + 6*A*a**2*b*x**(11/2)/11 + 2*A*a*b**2*x**(15/2)/5 + 2*A*b**
3*x**(19/2)/19 + 2*B*a**3*x**(11/2)/11 + 2*B*a**2*b*x**(15/2)/5 + 6*B*a*b**2*x**
(19/2)/19 + 2*B*b**3*x**(23/2)/23

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GIAC/XCAS [A]  time = 0.214408, size = 104, normalized size = 1.22 \[ \frac{2}{23} \, B b^{3} x^{\frac{23}{2}} + \frac{6}{19} \, B a b^{2} x^{\frac{19}{2}} + \frac{2}{19} \, A b^{3} x^{\frac{19}{2}} + \frac{2}{5} \, B a^{2} b x^{\frac{15}{2}} + \frac{2}{5} \, A a b^{2} x^{\frac{15}{2}} + \frac{2}{11} \, B a^{3} x^{\frac{11}{2}} + \frac{6}{11} \, A a^{2} b x^{\frac{11}{2}} + \frac{2}{7} \, A a^{3} x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)^3*x^(5/2),x, algorithm="giac")

[Out]

2/23*B*b^3*x^(23/2) + 6/19*B*a*b^2*x^(19/2) + 2/19*A*b^3*x^(19/2) + 2/5*B*a^2*b*
x^(15/2) + 2/5*A*a*b^2*x^(15/2) + 2/11*B*a^3*x^(11/2) + 6/11*A*a^2*b*x^(11/2) +
2/7*A*a^3*x^(7/2)