3.2131 \(\int (a+b \sqrt{x})^3 x^3 \, dx\)

Optimal. Leaf size=47 \[ \frac{2}{3} a^2 b x^{9/2}+\frac{a^3 x^4}{4}+\frac{3}{5} a b^2 x^5+\frac{2}{11} b^3 x^{11/2} \]

[Out]

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

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Rubi [A]  time = 0.0283058, antiderivative size = 47, 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} \[ \frac{2}{3} a^2 b x^{9/2}+\frac{a^3 x^4}{4}+\frac{3}{5} a b^2 x^5+\frac{2}{11} b^3 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3*x^3,x]

[Out]

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

Mathematica [A]  time = 0.0188698, size = 47, normalized size = 1. \[ \frac{2}{3} a^2 b x^{9/2}+\frac{a^3 x^4}{4}+\frac{3}{5} a b^2 x^5+\frac{2}{11} b^3 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3*x^3,x]

[Out]

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

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Maple [A]  time = 0., size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{4}}{4}}+{\frac{2\,b{a}^{2}}{3}{x}^{{\frac{9}{2}}}}+{\frac{3\,{x}^{5}a{b}^{2}}{5}}+{\frac{2\,{b}^{3}}{11}{x}^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01894, size = 178, normalized size = 3.79 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{11}}{11 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{10} a}{5 \, b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{9} a^{2}}{3 \, b^{8}} - \frac{35 \,{\left (b \sqrt{x} + a\right )}^{8} a^{3}}{4 \, b^{8}} + \frac{10 \,{\left (b \sqrt{x} + a\right )}^{7} a^{4}}{b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{6} a^{5}}{b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{5} a^{6}}{5 \, b^{8}} - \frac{{\left (b \sqrt{x} + a\right )}^{4} a^{7}}{2 \, b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*(b*sqrt(x) + a)^11/b^8 - 7/5*(b*sqrt(x) + a)^10*a/b^8 + 14/3*(b*sqrt(x) + a)^9*a^2/b^8 - 35/4*(b*sqrt(x)
+ a)^8*a^3/b^8 + 10*(b*sqrt(x) + a)^7*a^4/b^8 - 7*(b*sqrt(x) + a)^6*a^5/b^8 + 14/5*(b*sqrt(x) + a)^5*a^6/b^8 -
1/2*(b*sqrt(x) + a)^4*a^7/b^8

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Fricas [A]  time = 1.4756, size = 96, normalized size = 2.04 \begin{align*} \frac{3}{5} \, a b^{2} x^{5} + \frac{1}{4} \, a^{3} x^{4} + \frac{2}{33} \,{\left (3 \, b^{3} x^{5} + 11 \, a^{2} 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))^3,x, algorithm="fricas")

[Out]

3/5*a*b^2*x^5 + 1/4*a^3*x^4 + 2/33*(3*b^3*x^5 + 11*a^2*b*x^4)*sqrt(x)

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Sympy [A]  time = 1.57005, size = 44, normalized size = 0.94 \begin{align*} \frac{a^{3} x^{4}}{4} + \frac{2 a^{2} b x^{\frac{9}{2}}}{3} + \frac{3 a b^{2} x^{5}}{5} + \frac{2 b^{3} x^{\frac{11}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**4/4 + 2*a**2*b*x**(9/2)/3 + 3*a*b**2*x**5/5 + 2*b**3*x**(11/2)/11

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Giac [A]  time = 1.12402, size = 47, normalized size = 1. \begin{align*} \frac{2}{11} \, b^{3} x^{\frac{11}{2}} + \frac{3}{5} \, a b^{2} x^{5} + \frac{2}{3} \, a^{2} b x^{\frac{9}{2}} + \frac{1}{4} \, a^{3} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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