3.2121 \(\int (a+b \sqrt{x})^2 x^3 \, dx\)

Optimal. Leaf size=32 \[ \frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5} \]

[Out]

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

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Rubi [A]  time = 0.023469, antiderivative size = 32, 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} \[ \frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^2*x^3,x]

[Out]

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

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

Mathematica [A]  time = 0.0174577, size = 32, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{4}{9} a b x^{9/2}+\frac{b^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^2*x^3,x]

[Out]

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

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Maple [A]  time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{4}}{4}}+{\frac{4\,ab}{9}{x}^{{\frac{9}{2}}}}+{\frac{{b}^{2}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.961564, size = 178, normalized size = 5.56 \begin{align*} \frac{{\left (b \sqrt{x} + a\right )}^{10}}{5 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{9} a}{9 \, b^{8}} + \frac{21 \,{\left (b \sqrt{x} + a\right )}^{8} a^{2}}{4 \, b^{8}} - \frac{10 \,{\left (b \sqrt{x} + a\right )}^{7} a^{3}}{b^{8}} + \frac{35 \,{\left (b \sqrt{x} + a\right )}^{6} a^{4}}{3 \, b^{8}} - \frac{42 \,{\left (b \sqrt{x} + a\right )}^{5} a^{5}}{5 \, b^{8}} + \frac{7 \,{\left (b \sqrt{x} + a\right )}^{4} a^{6}}{2 \, b^{8}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3} 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))^2,x, algorithm="maxima")

[Out]

1/5*(b*sqrt(x) + a)^10/b^8 - 14/9*(b*sqrt(x) + a)^9*a/b^8 + 21/4*(b*sqrt(x) + a)^8*a^2/b^8 - 10*(b*sqrt(x) + a
)^7*a^3/b^8 + 35/3*(b*sqrt(x) + a)^6*a^4/b^8 - 42/5*(b*sqrt(x) + a)^5*a^5/b^8 + 7/2*(b*sqrt(x) + a)^4*a^6/b^8
- 2/3*(b*sqrt(x) + a)^3*a^7/b^8

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Fricas [A]  time = 1.45151, size = 61, normalized size = 1.91 \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{1}{4} \, a^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.744018, size = 27, normalized size = 0.84 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{4 a b x^{\frac{9}{2}}}{9} + \frac{b^{2} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1019, size = 32, normalized size = 1. \begin{align*} \frac{1}{5} \, b^{2} x^{5} + \frac{4}{9} \, a b x^{\frac{9}{2}} + \frac{1}{4} \, a^{2} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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