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

Optimal. Leaf size=21 \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2} \]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7

_______________________________________________________________________________________

Rubi [A]  time = 0.0139615, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a + b*x^2),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.98048, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 b x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*x**(3/2)/3 + 2*b*x**(7/2)/7

_______________________________________________________________________________________

Mathematica [A]  time = 0.00835316, size = 21, normalized size = 1. \[ \frac{2}{3} a x^{3/2}+\frac{2}{7} b x^{7/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(7/2))/7

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \[{\frac{6\,b{x}^{2}+14\,a}{21}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*x^(1/2),x)

[Out]

2/21*x^(3/2)*(3*b*x^2+7*a)

_______________________________________________________________________________________

Maxima [A]  time = 1.3426, size = 18, normalized size = 0.86 \[ \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*b*x^(7/2) + 2/3*a*x^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.228766, size = 22, normalized size = 1.05 \[ \frac{2}{21} \,{\left (3 \, b x^{3} + 7 \, a x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(3*b*x^3 + 7*a*x)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 1.48117, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 b x^{\frac{7}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*x**(3/2)/3 + 2*b*x**(7/2)/7

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.206018, size = 18, normalized size = 0.86 \[ \frac{2}{7} \, b x^{\frac{7}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/7*b*x^(7/2) + 2/3*a*x^(3/2)