3.2260 \(\int (a+b \sqrt{x}) x^m \, dx\)

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)

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Rubi [A]  time = 0.0083188, 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, Rules used = {14} \[ \frac{a x^{m+1}}{m+1}+\frac{2 b x^{m+\frac{3}{2}}}{2 m+3} \]

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)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \left (a+b \sqrt{x}\right ) x^m \, dx &=\int \left (a x^m+b x^{\frac{1}{2}+m}\right ) \, dx\\ &=\frac{a x^{1+m}}{1+m}+\frac{2 b x^{\frac{3}{2}+m}}{3+2 m}\\ \end{align*}

Mathematica [A]  time = 0.0220409, size = 30, normalized size = 1. \[ \frac{a x^{m+1}}{m+1}+\frac{2 b x^{m+\frac{3}{2}}}{2 m+3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])*x^m,x]

[Out]

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

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Maple [F]  time = 0.003, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a+b\sqrt{x} \right ) \, dx \end{align*}

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)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^(1/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2595, size = 85, normalized size = 2.83 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.11452, size = 49, normalized size = 1.63 \begin{align*} \frac{2 \, b x^{\frac{3}{2}} \sqrt{x}^{2 \, m}}{2 \, m + 3} + \frac{a x \sqrt{x}^{2 \, m}}{m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^(1/2)),x, algorithm="giac")

[Out]

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