3.2258 \(\int (a+b \sqrt{x})^3 x^m \, dx\)

Optimal. Leaf size=70 \[ \frac{6 a^2 b x^{m+\frac{3}{2}}}{2 m+3}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+2}}{m+2}+\frac{2 b^3 x^{m+\frac{5}{2}}}{2 m+5} \]

[Out]

(a^3*x^(1 + m))/(1 + m) + (6*a^2*b*x^(3/2 + m))/(3 + 2*m) + (3*a*b^2*x^(2 + m))/(2 + m) + (2*b^3*x^(5/2 + m))/
(5 + 2*m)

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Rubi [A]  time = 0.0268059, antiderivative size = 70, 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} \[ \frac{6 a^2 b x^{m+\frac{3}{2}}}{2 m+3}+\frac{a^3 x^{m+1}}{m+1}+\frac{3 a b^2 x^{m+2}}{m+2}+\frac{2 b^3 x^{m+\frac{5}{2}}}{2 m+5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3*x^m,x]

[Out]

(a^3*x^(1 + m))/(1 + m) + (6*a^2*b*x^(3/2 + m))/(3 + 2*m) + (3*a*b^2*x^(2 + m))/(2 + m) + (2*b^3*x^(5/2 + m))/
(5 + 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 )^3 x^m \, dx &=\int \left (a^3 x^m+3 a^2 b x^{\frac{1}{2}+m}+3 a b^2 x^{1+m}+b^3 x^{\frac{3}{2}+m}\right ) \, dx\\ &=\frac{a^3 x^{1+m}}{1+m}+\frac{6 a^2 b x^{\frac{3}{2}+m}}{3+2 m}+\frac{3 a b^2 x^{2+m}}{2+m}+\frac{2 b^3 x^{\frac{5}{2}+m}}{5+2 m}\\ \end{align*}

Mathematica [A]  time = 0.0451151, size = 63, normalized size = 0.9 \[ x^{m+1} \left (\frac{6 a^2 b \sqrt{x}}{2 m+3}+\frac{a^3}{m+1}+\frac{3 a b^2 x}{m+2}+\frac{2 b^3 x^{3/2}}{2 m+5}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3*x^m,x]

[Out]

x^(1 + m)*(a^3/(1 + m) + (6*a^2*b*Sqrt[x])/(3 + 2*m) + (3*a*b^2*x)/(2 + m) + (2*b^3*x^(3/2))/(5 + 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 ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*(a+b*x^(1/2))^3,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))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.31799, size = 369, normalized size = 5.27 \begin{align*} \frac{{\left (3 \,{\left (4 \, a b^{2} m^{3} + 20 \, a b^{2} m^{2} + 31 \, a b^{2} m + 15 \, a b^{2}\right )} x^{2} +{\left (4 \, a^{3} m^{3} + 24 \, a^{3} m^{2} + 47 \, a^{3} m + 30 \, a^{3}\right )} x + 2 \,{\left ({\left (2 \, b^{3} m^{3} + 9 \, b^{3} m^{2} + 13 \, b^{3} m + 6 \, b^{3}\right )} x^{2} + 3 \,{\left (2 \, a^{2} b m^{3} + 11 \, a^{2} b m^{2} + 19 \, a^{2} b m + 10 \, a^{2} b\right )} x\right )} \sqrt{x}\right )} x^{m}}{4 \, m^{4} + 28 \, m^{3} + 71 \, m^{2} + 77 \, m + 30} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^(1/2))^3,x, algorithm="fricas")

[Out]

(3*(4*a*b^2*m^3 + 20*a*b^2*m^2 + 31*a*b^2*m + 15*a*b^2)*x^2 + (4*a^3*m^3 + 24*a^3*m^2 + 47*a^3*m + 30*a^3)*x +
2*((2*b^3*m^3 + 9*b^3*m^2 + 13*b^3*m + 6*b^3)*x^2 + 3*(2*a^2*b*m^3 + 11*a^2*b*m^2 + 19*a^2*b*m + 10*a^2*b)*x)
*sqrt(x))*x^m/(4*m^4 + 28*m^3 + 71*m^2 + 77*m + 30)

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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))**3,x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.11674, size = 113, normalized size = 1.61 \begin{align*} \frac{2 \, b^{3} x^{\frac{5}{2}} \sqrt{x}^{2 \, m}}{2 \, m + 5} + \frac{3 \, a b^{2} x^{2} \sqrt{x}^{2 \, m}}{m + 2} + \frac{6 \, a^{2} b x^{\frac{3}{2}} \sqrt{x}^{2 \, m}}{2 \, m + 3} + \frac{a^{3} 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))^3,x, algorithm="giac")

[Out]

2*b^3*x^(5/2)*sqrt(x)^(2*m)/(2*m + 5) + 3*a*b^2*x^2*sqrt(x)^(2*m)/(m + 2) + 6*a^2*b*x^(3/2)*sqrt(x)^(2*m)/(2*m
+ 3) + a^3*x*sqrt(x)^(2*m)/(m + 1)