3.2143 \(\int (a+b \sqrt{x})^5 x^2 \, dx\)

Optimal. Leaf size=72 \[ \frac{20}{9} a^2 b^3 x^{9/2}+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^4 b x^{7/2}+\frac{a^5 x^3}{3}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2} \]

[Out]

(a^5*x^3)/3 + (10*a^4*b*x^(7/2))/7 + (5*a^3*b^2*x^4)/2 + (20*a^2*b^3*x^(9/2))/9 + a*b^4*x^5 + (2*b^5*x^(11/2))
/11

________________________________________________________________________________________

Rubi [A]  time = 0.0402159, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{20}{9} a^2 b^3 x^{9/2}+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^4 b x^{7/2}+\frac{a^5 x^3}{3}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5*x^2,x]

[Out]

(a^5*x^3)/3 + (10*a^4*b*x^(7/2))/7 + (5*a^3*b^2*x^4)/2 + (20*a^2*b^3*x^(9/2))/9 + a*b^4*x^5 + (2*b^5*x^(11/2))
/11

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \left (a+b \sqrt{x}\right )^5 x^2 \, dx &=2 \operatorname{Subst}\left (\int x^5 (a+b x)^5 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^5 x^5+5 a^4 b x^6+10 a^3 b^2 x^7+10 a^2 b^3 x^8+5 a b^4 x^9+b^5 x^{10}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^5 x^3}{3}+\frac{10}{7} a^4 b x^{7/2}+\frac{5}{2} a^3 b^2 x^4+\frac{20}{9} a^2 b^3 x^{9/2}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2}\\ \end{align*}

Mathematica [A]  time = 0.0251665, size = 72, normalized size = 1. \[ \frac{20}{9} a^2 b^3 x^{9/2}+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^4 b x^{7/2}+\frac{a^5 x^3}{3}+a b^4 x^5+\frac{2}{11} b^5 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5*x^2,x]

[Out]

(a^5*x^3)/3 + (10*a^4*b*x^(7/2))/7 + (5*a^3*b^2*x^4)/2 + (20*a^2*b^3*x^(9/2))/9 + a*b^4*x^5 + (2*b^5*x^(11/2))
/11

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 57, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{3}}{3}}+{\frac{10\,{a}^{4}b}{7}{x}^{{\frac{7}{2}}}}+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{20\,{a}^{2}{b}^{3}}{9}{x}^{{\frac{9}{2}}}}+a{b}^{4}{x}^{5}+{\frac{2\,{b}^{5}}{11}{x}^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*x^(1/2))^5,x)

[Out]

1/3*a^5*x^3+10/7*a^4*b*x^(7/2)+5/2*a^3*b^2*x^4+20/9*a^2*b^3*x^(9/2)+a*b^4*x^5+2/11*b^5*x^(11/2)

________________________________________________________________________________________

Maxima [A]  time = 0.949297, size = 132, normalized size = 1.83 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11}}{11 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{10} a}{b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )}^{9} a^{2}}{9 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{8} a^{3}}{2 \, b^{6}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{7} a^{4}}{7 \, b^{6}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a^{5}}{3 \, b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

2/11*(b*sqrt(x) + a)^11/b^6 - (b*sqrt(x) + a)^10*a/b^6 + 20/9*(b*sqrt(x) + a)^9*a^2/b^6 - 5/2*(b*sqrt(x) + a)^
8*a^3/b^6 + 10/7*(b*sqrt(x) + a)^7*a^4/b^6 - 1/3*(b*sqrt(x) + a)^6*a^5/b^6

________________________________________________________________________________________

Fricas [A]  time = 1.47659, size = 143, normalized size = 1.99 \begin{align*} a b^{4} x^{5} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{1}{3} \, a^{5} x^{3} + \frac{2}{693} \,{\left (63 \, b^{5} x^{5} + 770 \, a^{2} b^{3} x^{4} + 495 \, a^{4} b x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

a*b^4*x^5 + 5/2*a^3*b^2*x^4 + 1/3*a^5*x^3 + 2/693*(63*b^5*x^5 + 770*a^2*b^3*x^4 + 495*a^4*b*x^3)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 2.90424, size = 70, normalized size = 0.97 \begin{align*} \frac{a^{5} x^{3}}{3} + \frac{10 a^{4} b x^{\frac{7}{2}}}{7} + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{20 a^{2} b^{3} x^{\frac{9}{2}}}{9} + a b^{4} x^{5} + \frac{2 b^{5} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**3/3 + 10*a**4*b*x**(7/2)/7 + 5*a**3*b**2*x**4/2 + 20*a**2*b**3*x**(9/2)/9 + a*b**4*x**5 + 2*b**5*x**(1
1/2)/11

________________________________________________________________________________________

Giac [A]  time = 1.12196, size = 76, normalized size = 1.06 \begin{align*} \frac{2}{11} \, b^{5} x^{\frac{11}{2}} + a b^{4} x^{5} + \frac{20}{9} \, a^{2} b^{3} x^{\frac{9}{2}} + \frac{5}{2} \, a^{3} b^{2} x^{4} + \frac{10}{7} \, a^{4} b x^{\frac{7}{2}} + \frac{1}{3} \, a^{5} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

2/11*b^5*x^(11/2) + a*b^4*x^5 + 20/9*a^2*b^3*x^(9/2) + 5/2*a^3*b^2*x^4 + 10/7*a^4*b*x^(7/2) + 1/3*a^5*x^3