3.346 \(\int \sqrt{x} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{7} x^{7/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{11} b B x^{11/2} \]

[Out]

(2*a*A*x^(3/2))/3 + (2*(A*b + a*B)*x^(7/2))/7 + (2*b*B*x^(11/2))/11

_______________________________________________________________________________________

Rubi [A]  time = 0.0470413, antiderivative size = 39, 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 \[ \frac{2}{7} x^{7/2} (a B+A b)+\frac{2}{3} a A x^{3/2}+\frac{2}{11} b B x^{11/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*A*x^(3/2))/3 + (2*(A*b + a*B)*x^(7/2))/7 + (2*b*B*x^(11/2))/11

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.1917, size = 41, normalized size = 1.05 \[ \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{11}{2}}}{11} + x^{\frac{7}{2}} \left (\frac{2 A b}{7} + \frac{2 B a}{7}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)*(B*x**2+A)*x**(1/2),x)

[Out]

2*A*a*x**(3/2)/3 + 2*B*b*x**(11/2)/11 + x**(7/2)*(2*A*b/7 + 2*B*a/7)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0169348, size = 33, normalized size = 0.85 \[ \frac{2}{231} x^{3/2} \left (33 x^2 (a B+A b)+77 a A+21 b B x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(3/2)*(77*a*A + 33*(A*b + a*B)*x^2 + 21*b*B*x^4))/231

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \[{\frac{42\,bB{x}^{4}+66\,A{x}^{2}b+66\,B{x}^{2}a+154\,Aa}{231}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(B*x^2+A)*x^(1/2),x)

[Out]

2/231*x^(3/2)*(21*B*b*x^4+33*A*b*x^2+33*B*a*x^2+77*A*a)

_______________________________________________________________________________________

Maxima [A]  time = 1.35088, size = 36, normalized size = 0.92 \[ \frac{2}{11} \, B b x^{\frac{11}{2}} + \frac{2}{7} \,{\left (B a + A b\right )} x^{\frac{7}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*B*b*x^(11/2) + 2/7*(B*a + A*b)*x^(7/2) + 2/3*A*a*x^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.220071, size = 41, normalized size = 1.05 \[ \frac{2}{231} \,{\left (21 \, B b x^{5} + 33 \,{\left (B a + A b\right )} x^{3} + 77 \, A a x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/231*(21*B*b*x^5 + 33*(B*a + A*b)*x^3 + 77*A*a*x)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 2.64564, size = 37, normalized size = 0.95 \[ \frac{2 A a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{11}{2}}}{11} + \frac{2 x^{\frac{7}{2}} \left (A b + B a\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)*(B*x**2+A)*x**(1/2),x)

[Out]

2*A*a*x**(3/2)/3 + 2*B*b*x**(11/2)/11 + 2*x**(7/2)*(A*b + B*a)/7

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.250886, size = 39, normalized size = 1. \[ \frac{2}{11} \, B b x^{\frac{11}{2}} + \frac{2}{7} \, B a x^{\frac{7}{2}} + \frac{2}{7} \, A b x^{\frac{7}{2}} + \frac{2}{3} \, A a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/11*B*b*x^(11/2) + 2/7*B*a*x^(7/2) + 2/7*A*b*x^(7/2) + 2/3*A*a*x^(3/2)