3.2133 \(\int (a+b \sqrt{x})^3 x \, dx\)

Optimal. Leaf size=44 \[ \frac{6}{5} a^2 b x^{5/2}+\frac{a^3 x^2}{2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2} \]

[Out]

(a^3*x^2)/2 + (6*a^2*b*x^(5/2))/5 + a*b^2*x^3 + (2*b^3*x^(7/2))/7

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Rubi [A]  time = 0.0245843, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{6}{5} a^2 b x^{5/2}+\frac{a^3 x^2}{2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3*x,x]

[Out]

(a^3*x^2)/2 + (6*a^2*b*x^(5/2))/5 + a*b^2*x^3 + (2*b^3*x^(7/2))/7

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

Mathematica [A]  time = 0.0157025, size = 44, normalized size = 1. \[ \frac{6}{5} a^2 b x^{5/2}+\frac{a^3 x^2}{2}+a b^2 x^3+\frac{2}{7} b^3 x^{7/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3*x,x]

[Out]

(a^3*x^2)/2 + (6*a^2*b*x^(5/2))/5 + a*b^2*x^3 + (2*b^3*x^(7/2))/7

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Maple [A]  time = 0.001, size = 35, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}{a}^{3}}{2}}+{\frac{6\,b{a}^{2}}{5}{x}^{{\frac{5}{2}}}}+{x}^{3}a{b}^{2}+{\frac{2\,{b}^{3}}{7}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2*a^3+6/5*a^2*b*x^(5/2)+x^3*a*b^2+2/7*b^3*x^(7/2)

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Maxima [A]  time = 0.972826, size = 86, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (b \sqrt{x} + a\right )}^{7}}{7 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{6} a}{b^{4}} + \frac{6 \,{\left (b \sqrt{x} + a\right )}^{5} a^{2}}{5 \, b^{4}} - \frac{{\left (b \sqrt{x} + a\right )}^{4} a^{3}}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(b*sqrt(x) + a)^7/b^4 - (b*sqrt(x) + a)^6*a/b^4 + 6/5*(b*sqrt(x) + a)^5*a^2/b^4 - 1/2*(b*sqrt(x) + a)^4*a^
3/b^4

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Fricas [A]  time = 1.45211, size = 90, normalized size = 2.05 \begin{align*} a b^{2} x^{3} + \frac{1}{2} \, a^{3} x^{2} + \frac{2}{35} \,{\left (5 \, b^{3} x^{3} + 21 \, a^{2} b x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*b^2*x^3 + 1/2*a^3*x^2 + 2/35*(5*b^3*x^3 + 21*a^2*b*x^2)*sqrt(x)

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Sympy [A]  time = 1.50303, size = 41, normalized size = 0.93 \begin{align*} \frac{a^{3} x^{2}}{2} + \frac{6 a^{2} b x^{\frac{5}{2}}}{5} + a b^{2} x^{3} + \frac{2 b^{3} x^{\frac{7}{2}}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*x**2/2 + 6*a**2*b*x**(5/2)/5 + a*b**2*x**3 + 2*b**3*x**(7/2)/7

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Giac [A]  time = 1.10067, size = 46, normalized size = 1.05 \begin{align*} \frac{2}{7} \, b^{3} x^{\frac{7}{2}} + a b^{2} x^{3} + \frac{6}{5} \, a^{2} b x^{\frac{5}{2}} + \frac{1}{2} \, a^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*b^3*x^(7/2) + a*b^2*x^3 + 6/5*a^2*b*x^(5/2) + 1/2*a^3*x^2