3.2257 \(\int (a+b \sqrt{x})^4 x^m \, dx\)

Optimal. Leaf size=87 \[ \frac{6 a^2 b^2 x^{m+2}}{m+2}+\frac{8 a^3 b x^{m+\frac{3}{2}}}{2 m+3}+\frac{a^4 x^{m+1}}{m+1}+\frac{8 a b^3 x^{m+\frac{5}{2}}}{2 m+5}+\frac{b^4 x^{m+3}}{m+3} \]

[Out]

(a^4*x^(1 + m))/(1 + m) + (8*a^3*b*x^(3/2 + m))/(3 + 2*m) + (6*a^2*b^2*x^(2 + m))/(2 + m) + (8*a*b^3*x^(5/2 +
m))/(5 + 2*m) + (b^4*x^(3 + m))/(3 + m)

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Rubi [A]  time = 0.0428044, antiderivative size = 87, 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} \[ \frac{6 a^2 b^2 x^{m+2}}{m+2}+\frac{8 a^3 b x^{m+\frac{3}{2}}}{2 m+3}+\frac{a^4 x^{m+1}}{m+1}+\frac{8 a b^3 x^{m+\frac{5}{2}}}{2 m+5}+\frac{b^4 x^{m+3}}{m+3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^4*x^m,x]

[Out]

(a^4*x^(1 + m))/(1 + m) + (8*a^3*b*x^(3/2 + m))/(3 + 2*m) + (6*a^2*b^2*x^(2 + m))/(2 + m) + (8*a*b^3*x^(5/2 +
m))/(5 + 2*m) + (b^4*x^(3 + m))/(3 + 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 )^4 x^m \, dx &=\int \left (a^4 x^m+4 a^3 b x^{\frac{1}{2}+m}+6 a^2 b^2 x^{1+m}+4 a b^3 x^{\frac{3}{2}+m}+b^4 x^{2+m}\right ) \, dx\\ &=\frac{a^4 x^{1+m}}{1+m}+\frac{8 a^3 b x^{\frac{3}{2}+m}}{3+2 m}+\frac{6 a^2 b^2 x^{2+m}}{2+m}+\frac{8 a b^3 x^{\frac{5}{2}+m}}{5+2 m}+\frac{b^4 x^{3+m}}{3+m}\\ \end{align*}

Mathematica [A]  time = 0.0687767, size = 78, normalized size = 0.9 \[ x^{m+1} \left (\frac{6 a^2 b^2 x}{m+2}+\frac{8 a^3 b \sqrt{x}}{2 m+3}+\frac{a^4}{m+1}+\frac{8 a b^3 x^{3/2}}{2 m+5}+\frac{b^4 x^2}{m+3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^4*x^m,x]

[Out]

x^(1 + m)*(a^4/(1 + m) + (8*a^3*b*Sqrt[x])/(3 + 2*m) + (6*a^2*b^2*x)/(2 + m) + (8*a*b^3*x^(3/2))/(5 + 2*m) + (
b^4*x^2)/(3 + m))

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Maple [F]  time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( a+b\sqrt{x} \right ) ^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*(a+b*x^(1/2))^4,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))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.42253, size = 578, normalized size = 6.64 \begin{align*} \frac{{\left ({\left (4 \, b^{4} m^{4} + 28 \, b^{4} m^{3} + 71 \, b^{4} m^{2} + 77 \, b^{4} m + 30 \, b^{4}\right )} x^{3} + 6 \,{\left (4 \, a^{2} b^{2} m^{4} + 32 \, a^{2} b^{2} m^{3} + 91 \, a^{2} b^{2} m^{2} + 108 \, a^{2} b^{2} m + 45 \, a^{2} b^{2}\right )} x^{2} +{\left (4 \, a^{4} m^{4} + 36 \, a^{4} m^{3} + 119 \, a^{4} m^{2} + 171 \, a^{4} m + 90 \, a^{4}\right )} x + 8 \,{\left ({\left (2 \, a b^{3} m^{4} + 15 \, a b^{3} m^{3} + 40 \, a b^{3} m^{2} + 45 \, a b^{3} m + 18 \, a b^{3}\right )} x^{2} +{\left (2 \, a^{3} b m^{4} + 17 \, a^{3} b m^{3} + 52 \, a^{3} b m^{2} + 67 \, a^{3} b m + 30 \, a^{3} b\right )} x\right )} \sqrt{x}\right )} x^{m}}{4 \, m^{5} + 40 \, m^{4} + 155 \, m^{3} + 290 \, m^{2} + 261 \, m + 90} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^(1/2))^4,x, algorithm="fricas")

[Out]

((4*b^4*m^4 + 28*b^4*m^3 + 71*b^4*m^2 + 77*b^4*m + 30*b^4)*x^3 + 6*(4*a^2*b^2*m^4 + 32*a^2*b^2*m^3 + 91*a^2*b^
2*m^2 + 108*a^2*b^2*m + 45*a^2*b^2)*x^2 + (4*a^4*m^4 + 36*a^4*m^3 + 119*a^4*m^2 + 171*a^4*m + 90*a^4)*x + 8*((
2*a*b^3*m^4 + 15*a*b^3*m^3 + 40*a*b^3*m^2 + 45*a*b^3*m + 18*a*b^3)*x^2 + (2*a^3*b*m^4 + 17*a^3*b*m^3 + 52*a^3*
b*m^2 + 67*a^3*b*m + 30*a^3*b)*x)*sqrt(x))*x^m/(4*m^5 + 40*m^4 + 155*m^3 + 290*m^2 + 261*m + 90)

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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))**4,x)

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.14546, size = 143, normalized size = 1.64 \begin{align*} \frac{b^{4} x^{3} \sqrt{x}^{2 \, m}}{m + 3} + \frac{8 \, a b^{3} x^{\frac{5}{2}} \sqrt{x}^{2 \, m}}{2 \, m + 5} + \frac{6 \, a^{2} b^{2} x^{2} \sqrt{x}^{2 \, m}}{m + 2} + \frac{8 \, a^{3} b x^{\frac{3}{2}} \sqrt{x}^{2 \, m}}{2 \, m + 3} + \frac{a^{4} 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))^4,x, algorithm="giac")

[Out]

b^4*x^3*sqrt(x)^(2*m)/(m + 3) + 8*a*b^3*x^(5/2)*sqrt(x)^(2*m)/(2*m + 5) + 6*a^2*b^2*x^2*sqrt(x)^(2*m)/(m + 2)
+ 8*a^3*b*x^(3/2)*sqrt(x)^(2*m)/(2*m + 3) + a^4*x*sqrt(x)^(2*m)/(m + 1)