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

Optimal. Leaf size=46 \[ \frac{\left (a+b x^2\right )^{3/2} (A b-a B)}{3 b^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^2} \]

[Out]

((A*b - a*B)*(a + b*x^2)^(3/2))/(3*b^2) + (B*(a + b*x^2)^(5/2))/(5*b^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.100122, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\left (a+b x^2\right )^{3/2} (A b-a B)}{3 b^2}+\frac{B \left (a+b x^2\right )^{5/2}}{5 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*B)*(a + b*x^2)^(3/2))/(3*b^2) + (B*(a + b*x^2)^(5/2))/(5*b^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.1255, size = 37, normalized size = 0.8 \[ \frac{B \left (a + b x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \left (A b - B a\right )}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*(a + b*x**2)**(5/2)/(5*b**2) + (a + b*x**2)**(3/2)*(A*b - B*a)/(3*b**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0392104, size = 34, normalized size = 0.74 \[ \frac{\left (a+b x^2\right )^{3/2} \left (-2 a B+5 A b+3 b B x^2\right )}{15 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^2)^(3/2)*(5*A*b - 2*a*B + 3*b*B*x^2))/(15*b^2)

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 31, normalized size = 0.7 \[{\frac{3\,bB{x}^{2}+5\,Ab-2\,Ba}{15\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(b*x^2+a)^(3/2)*(3*B*b*x^2+5*A*b-2*B*a)/b^2

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.212523, size = 68, normalized size = 1.48 \[ \frac{{\left (3 \, B b^{2} x^{4} - 2 \, B a^{2} + 5 \, A a b +{\left (B a b + 5 \, A b^{2}\right )} x^{2}\right )} \sqrt{b x^{2} + a}}{15 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*B*b^2*x^4 - 2*B*a^2 + 5*A*a*b + (B*a*b + 5*A*b^2)*x^2)*sqrt(b*x^2 + a)/b
^2

_______________________________________________________________________________________

Sympy [A]  time = 0.982783, size = 110, normalized size = 2.39 \[ \begin{cases} \frac{A a \sqrt{a + b x^{2}}}{3 b} + \frac{A x^{2} \sqrt{a + b x^{2}}}{3} - \frac{2 B a^{2} \sqrt{a + b x^{2}}}{15 b^{2}} + \frac{B a x^{2} \sqrt{a + b x^{2}}}{15 b} + \frac{B x^{4} \sqrt{a + b x^{2}}}{5} & \text{for}\: b \neq 0 \\\sqrt{a} \left (\frac{A x^{2}}{2} + \frac{B x^{4}}{4}\right ) & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((A*a*sqrt(a + b*x**2)/(3*b) + A*x**2*sqrt(a + b*x**2)/3 - 2*B*a**2*sqr
t(a + b*x**2)/(15*b**2) + B*a*x**2*sqrt(a + b*x**2)/(15*b) + B*x**4*sqrt(a + b*x
**2)/5, Ne(b, 0)), (sqrt(a)*(A*x**2/2 + B*x**4/4), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.240444, size = 63, normalized size = 1.37 \[ \frac{5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} A + \frac{{\left (3 \,{\left (b x^{2} + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} a\right )} B}{b}}{15 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(5*(b*x^2 + a)^(3/2)*A + (3*(b*x^2 + a)^(5/2) - 5*(b*x^2 + a)^(3/2)*a)*B/b)
/b