3.431 \(\int \sqrt{x} (a+b x) \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{3} a x^{3/2}+\frac{2}{5} b x^{5/2} \]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(5/2))/5

_______________________________________________________________________________________

Rubi [A]  time = 0.0124579, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2}{3} a x^{3/2}+\frac{2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]*(a + b*x),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(5/2))/5

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.33767, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 b x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*x**(3/2)/3 + 2*b*x**(5/2)/5

_______________________________________________________________________________________

Mathematica [A]  time = 0.00488006, size = 17, normalized size = 0.81 \[ \frac{2}{15} x^{3/2} (5 a+3 b x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]*(a + b*x),x]

[Out]

(2*x^(3/2)*(5*a + 3*b*x))/15

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 14, normalized size = 0.7 \[{\frac{6\,bx+10\,a}{15}{x}^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*x^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34221, size = 18, normalized size = 0.86 \[ \frac{2}{5} \, b x^{\frac{5}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*b*x^(5/2) + 2/3*a*x^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.207749, size = 22, normalized size = 1.05 \[ \frac{2}{15} \,{\left (3 \, b x^{2} + 5 \, a x\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*b*x^2 + 5*a*x)*sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 1.6827, size = 19, normalized size = 0.9 \[ \frac{2 a x^{\frac{3}{2}}}{3} + \frac{2 b x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*x**(1/2),x)

[Out]

2*a*x**(3/2)/3 + 2*b*x**(5/2)/5

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.201202, size = 18, normalized size = 0.86 \[ \frac{2}{5} \, b x^{\frac{5}{2}} + \frac{2}{3} \, a x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*b*x^(5/2) + 2/3*a*x^(3/2)