3.2259 \(\int (a+b \sqrt{x})^2 x^m \, dx\)

Optimal. Leaf size=47 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{4 a b x^{m+\frac{3}{2}}}{2 m+3}+\frac{b^2 x^{m+2}}{m+2} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (4*a*b*x^(3/2 + m))/(3 + 2*m) + (b^2*x^(2 + m))/(2 + m)

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Rubi [A]  time = 0.0173857, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ \frac{a^2 x^{m+1}}{m+1}+\frac{4 a b x^{m+\frac{3}{2}}}{2 m+3}+\frac{b^2 x^{m+2}}{m+2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^2*x^m,x]

[Out]

(a^2*x^(1 + m))/(1 + m) + (4*a*b*x^(3/2 + m))/(3 + 2*m) + (b^2*x^(2 + m))/(2 + m)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (a+b \sqrt{x}\right )^2 x^m \, dx &=\int \left (a^2 x^m+2 a b x^{\frac{1}{2}+m}+b^2 x^{1+m}\right ) \, dx\\ &=\frac{a^2 x^{1+m}}{1+m}+\frac{4 a b x^{\frac{3}{2}+m}}{3+2 m}+\frac{b^2 x^{2+m}}{2+m}\\ \end{align*}

Mathematica [A]  time = 0.0442697, size = 42, normalized size = 0.89 \[ x^{m+1} \left (\frac{a^2}{m+1}+\frac{4 a b \sqrt{x}}{2 m+3}+\frac{b^2 x}{m+2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^2*x^m,x]

[Out]

x^(1 + m)*(a^2/(1 + m) + (4*a*b*Sqrt[x])/(3 + 2*m) + (b^2*x)/(2 + m))

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Maple [F]  time = 0.012, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a+b\sqrt{x} \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.31299, size = 192, normalized size = 4.09 \begin{align*} \frac{{\left ({\left (2 \, b^{2} m^{2} + 5 \, b^{2} m + 3 \, b^{2}\right )} x^{2} + 4 \,{\left (a b m^{2} + 3 \, a b m + 2 \, a b\right )} x^{\frac{3}{2}} +{\left (2 \, a^{2} m^{2} + 7 \, a^{2} m + 6 \, a^{2}\right )} x\right )} x^{m}}{2 \, m^{3} + 9 \, m^{2} + 13 \, m + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*b^2*m^2 + 5*b^2*m + 3*b^2)*x^2 + 4*(a*b*m^2 + 3*a*b*m + 2*a*b)*x^(3/2) + (2*a^2*m^2 + 7*a^2*m + 6*a^2)*x)*
x^m/(2*m^3 + 9*m^2 + 13*m + 6)

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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))**2,x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.14146, size = 78, normalized size = 1.66 \begin{align*} \frac{b^{2} x^{2} \sqrt{x}^{2 \, m}}{m + 2} + \frac{4 \, a b x^{\frac{3}{2}} \sqrt{x}^{2 \, m}}{2 \, m + 3} + \frac{a^{2} 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))^2,x, algorithm="giac")

[Out]

b^2*x^2*sqrt(x)^(2*m)/(m + 2) + 4*a*b*x^(3/2)*sqrt(x)^(2*m)/(2*m + 3) + a^2*x*sqrt(x)^(2*m)/(m + 1)