3.2258 \(\int \left (a+b \sqrt{x}\right ) x^m \, dx\)

Optimal. Leaf size=30 \[ \frac{a x^{m+1}}{m+1}+\frac{2 b x^{m+\frac{3}{2}}}{2 m+3} \]

[Out]

(a*x^(1 + m))/(1 + m) + (2*b*x^(3/2 + m))/(3 + 2*m)

_______________________________________________________________________________________

Rubi [A]  time = 0.0242294, antiderivative size = 30, 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 \[ \frac{a x^{m+1}}{m+1}+\frac{2 b x^{m+\frac{3}{2}}}{2 m+3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])*x^m,x]

[Out]

(a*x^(1 + m))/(1 + m) + (2*b*x^(3/2 + m))/(3 + 2*m)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.46858, size = 24, normalized size = 0.8 \[ \frac{a x^{m + 1}}{m + 1} + \frac{2 b x^{m + \frac{3}{2}}}{2 m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*x**(m + 1)/(m + 1) + 2*b*x**(m + 3/2)/(2*m + 3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0412618, size = 28, normalized size = 0.93 \[ x^m \left (\frac{a x}{m+1}+\frac{2 b x^{3/2}}{2 m+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

x^m*((a*x)/(1 + m) + (2*b*x^(3/2))/(3 + 2*m))

_______________________________________________________________________________________

Maple [F]  time = 0.007, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( a+b\sqrt{x} \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^(1/2)),x)

[Out]

int(x^m*(a+b*x^(1/2)),x)

_______________________________________________________________________________________

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*sqrt(x) + a)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.254054, size = 50, normalized size = 1.67 \[ \frac{{\left (2 \,{\left (b m + b\right )} x^{\frac{3}{2}} +{\left (2 \, a m + 3 \, a\right )} x\right )} x^{m}}{2 \, m^{2} + 5 \, m + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269757, size = 51, normalized size = 1.7 \[ \frac{2 \, b x^{\frac{3}{2}} e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{2 \, m + 3} + \frac{a x e^{\left (2 \, m{\rm ln}\left (\sqrt{x}\right )\right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*b*x^(3/2)*e^(2*m*ln(sqrt(x)))/(2*m + 3) + a*x*e^(2*m*ln(sqrt(x)))/(m + 1)