3.2142 \(\int (a+b \sqrt{x})^5 x^3 \, dx\)

Optimal. Leaf size=73 \[ \frac{20}{11} a^2 b^3 x^{11/2}+2 a^3 b^2 x^5+\frac{10}{9} a^4 b x^{9/2}+\frac{a^5 x^4}{4}+\frac{5}{6} a b^4 x^6+\frac{2}{13} b^5 x^{13/2} \]

[Out]

(a^5*x^4)/4 + (10*a^4*b*x^(9/2))/9 + 2*a^3*b^2*x^5 + (20*a^2*b^3*x^(11/2))/11 + (5*a*b^4*x^6)/6 + (2*b^5*x^(13
/2))/13

________________________________________________________________________________________

Rubi [A]  time = 0.0411744, antiderivative size = 73, 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} \[ \frac{20}{11} a^2 b^3 x^{11/2}+2 a^3 b^2 x^5+\frac{10}{9} a^4 b x^{9/2}+\frac{a^5 x^4}{4}+\frac{5}{6} a b^4 x^6+\frac{2}{13} b^5 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^5*x^3,x]

[Out]

(a^5*x^4)/4 + (10*a^4*b*x^(9/2))/9 + 2*a^3*b^2*x^5 + (20*a^2*b^3*x^(11/2))/11 + (5*a*b^4*x^6)/6 + (2*b^5*x^(13
/2))/13

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^3 \, dx &=2 \operatorname{Subst}\left (\int x^7 (a+b x)^5 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^5 x^7+5 a^4 b x^8+10 a^3 b^2 x^9+10 a^2 b^3 x^{10}+5 a b^4 x^{11}+b^5 x^{12}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^5 x^4}{4}+\frac{10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac{20}{11} a^2 b^3 x^{11/2}+\frac{5}{6} a b^4 x^6+\frac{2}{13} b^5 x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0270205, size = 73, normalized size = 1. \[ \frac{20}{11} a^2 b^3 x^{11/2}+2 a^3 b^2 x^5+\frac{10}{9} a^4 b x^{9/2}+\frac{a^5 x^4}{4}+\frac{5}{6} a b^4 x^6+\frac{2}{13} b^5 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^5*x^3,x]

[Out]

(a^5*x^4)/4 + (10*a^4*b*x^(9/2))/9 + 2*a^3*b^2*x^5 + (20*a^2*b^3*x^(11/2))/11 + (5*a*b^4*x^6)/6 + (2*b^5*x^(13
/2))/13

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{4}}{4}}+{\frac{10\,{a}^{4}b}{9}{x}^{{\frac{9}{2}}}}+2\,{a}^{3}{b}^{2}{x}^{5}+{\frac{20\,{a}^{2}{b}^{3}}{11}{x}^{{\frac{11}{2}}}}+{\frac{5\,a{b}^{4}{x}^{6}}{6}}+{\frac{2\,{b}^{5}}{13}{x}^{{\frac{13}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*x^(1/2))^5,x)

[Out]

1/4*a^5*x^4+10/9*a^4*b*x^(9/2)+2*a^3*b^2*x^5+20/11*a^2*b^3*x^(11/2)+5/6*a*b^4*x^6+2/13*b^5*x^(13/2)

________________________________________________________________________________________

Maxima [B]  time = 0.971637, size = 178, normalized size = 2.44 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{13}}{13 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{12} a}{6 \, b^{8}} + \frac{42 \,{\left (b \sqrt{x} + a\right )}^{11} a^{2}}{11 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{10} a^{3}}{b^{8}} + \frac{70 \,{\left (b \sqrt{x} + a\right )}^{9} a^{4}}{9 \, b^{8}} - \frac{21 \,{\left (b \sqrt{x} + a\right )}^{8} a^{5}}{4 \, b^{8}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{7} a^{6}}{b^{8}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a^{7}}{3 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

2/13*(b*sqrt(x) + a)^13/b^8 - 7/6*(b*sqrt(x) + a)^12*a/b^8 + 42/11*(b*sqrt(x) + a)^11*a^2/b^8 - 7*(b*sqrt(x) +
a)^10*a^3/b^8 + 70/9*(b*sqrt(x) + a)^9*a^4/b^8 - 21/4*(b*sqrt(x) + a)^8*a^5/b^8 + 2*(b*sqrt(x) + a)^7*a^6/b^8
- 1/3*(b*sqrt(x) + a)^6*a^7/b^8

________________________________________________________________________________________

Fricas [A]  time = 1.47321, size = 149, normalized size = 2.04 \begin{align*} \frac{5}{6} \, a b^{4} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac{1}{4} \, a^{5} x^{4} + \frac{2}{1287} \,{\left (99 \, b^{5} x^{6} + 1170 \, a^{2} b^{3} x^{5} + 715 \, a^{4} b x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

5/6*a*b^4*x^6 + 2*a^3*b^2*x^5 + 1/4*a^5*x^4 + 2/1287*(99*b^5*x^6 + 1170*a^2*b^3*x^5 + 715*a^4*b*x^4)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 2.93536, size = 71, normalized size = 0.97 \begin{align*} \frac{a^{5} x^{4}}{4} + \frac{10 a^{4} b x^{\frac{9}{2}}}{9} + 2 a^{3} b^{2} x^{5} + \frac{20 a^{2} b^{3} x^{\frac{11}{2}}}{11} + \frac{5 a b^{4} x^{6}}{6} + \frac{2 b^{5} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*x**(1/2))**5,x)

[Out]

a**5*x**4/4 + 10*a**4*b*x**(9/2)/9 + 2*a**3*b**2*x**5 + 20*a**2*b**3*x**(11/2)/11 + 5*a*b**4*x**6/6 + 2*b**5*x
**(13/2)/13

________________________________________________________________________________________

Giac [A]  time = 1.10895, size = 77, normalized size = 1.05 \begin{align*} \frac{2}{13} \, b^{5} x^{\frac{13}{2}} + \frac{5}{6} \, a b^{4} x^{6} + \frac{20}{11} \, a^{2} b^{3} x^{\frac{11}{2}} + 2 \, a^{3} b^{2} x^{5} + \frac{10}{9} \, a^{4} b x^{\frac{9}{2}} + \frac{1}{4} \, a^{5} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

2/13*b^5*x^(13/2) + 5/6*a*b^4*x^6 + 20/11*a^2*b^3*x^(11/2) + 2*a^3*b^2*x^5 + 10/9*a^4*b*x^(9/2) + 1/4*a^5*x^4