\(\int (a+b \sqrt {x})^3 x^2 \, dx\) [2132]

Optimal result

Integrand size = 15, antiderivative size = 47 \[ \int \left (a+b \sqrt {x}\right )^3 x^2 \, dx=\frac {a^3 x^3}{3}+\frac {6}{7} a^2 b x^{7/2}+\frac {3}{4} a b^2 x^4+\frac {2}{9} b^3 x^{9/2} \]

[Out]

1/3*a^3*x^3+6/7*a^2*b*x^(7/2)+3/4*a*b^2*x^4+2/9*b^3*x^(9/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \left (a+b \sqrt {x}\right )^3 x^2 \, dx=\frac {a^3 x^3}{3}+\frac {6}{7} a^2 b x^{7/2}+\frac {3}{4} a b^2 x^4+\frac {2}{9} b^3 x^{9/2} \]

[In]

Int[(a + b*Sqrt[x])^3*x^2,x]

[Out]

(a^3*x^3)/3 + (6*a^2*b*x^(7/2))/7 + (3*a*b^2*x^4)/4 + (2*b^3*x^(9/2))/9

Rule 45

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])

Rule 272

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]]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \left (a+b \sqrt {x}\right )^3 x^2 \, dx=\frac {1}{252} \left (84 a^3 x^3+216 a^2 b x^{7/2}+189 a b^2 x^4+56 b^3 x^{9/2}\right ) \]

[In]

Integrate[(a + b*Sqrt[x])^3*x^2,x]

[Out]

(84*a^3*x^3 + 216*a^2*b*x^(7/2) + 189*a*b^2*x^4 + 56*b^3*x^(9/2))/252

Maple [A] (verified)

Time = 3.59 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77

method	result	size
derivativedivides	\(\frac {a^{3} x^{3}}{3}+\frac {6 a^{2} b \,x^{\frac {7}{2}}}{7}+\frac {3 a \,b^{2} x^{4}}{4}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\)	\(36\)
default	\(\frac {a^{3} x^{3}}{3}+\frac {6 a^{2} b \,x^{\frac {7}{2}}}{7}+\frac {3 a \,b^{2} x^{4}}{4}+\frac {2 b^{3} x^{\frac {9}{2}}}{9}\)	\(36\)
trager	\(\frac {a \left (9 b^{2} x^{3}+4 a^{2} x^{2}+9 b^{2} x^{2}+4 a^{2} x +9 b^{2} x +4 a^{2}+9 b^{2}\right ) \left (-1+x \right )}{12}+\frac {2 b \,x^{\frac {7}{2}} \left (7 b^{2} x +27 a^{2}\right )}{63}\)	\(73\)

[In]

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

[Out]

1/3*a^3*x^3+6/7*a^2*b*x^(7/2)+3/4*a*b^2*x^4+2/9*b^3*x^(9/2)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.87 \[ \int \left (a+b \sqrt {x}\right )^3 x^2 \, dx=\frac {3}{4} \, a b^{2} x^{4} + \frac {1}{3} \, a^{3} x^{3} + \frac {2}{63} \, {\left (7 \, b^{3} x^{4} + 27 \, a^{2} b x^{3}\right )} \sqrt {x} \]

[In]

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

[Out]

3/4*a*b^2*x^4 + 1/3*a^3*x^3 + 2/63*(7*b^3*x^4 + 27*a^2*b*x^3)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.94 \[ \int \left (a+b \sqrt {x}\right )^3 x^2 \, dx=\frac {a^{3} x^{3}}{3} + \frac {6 a^{2} b x^{\frac {7}{2}}}{7} + \frac {3 a b^{2} x^{4}}{4} + \frac {2 b^{3} x^{\frac {9}{2}}}{9} \]

[In]

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

[Out]

a**3*x**3/3 + 6*a**2*b*x**(7/2)/7 + 3*a*b**2*x**4/4 + 2*b**3*x**(9/2)/9

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (35) = 70\).

Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.09 \[ \int \left (a+b \sqrt {x}\right )^3 x^2 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{9}}{9 \, b^{6}} - \frac {5 \, {\left (b \sqrt {x} + a\right )}^{8} a}{4 \, b^{6}} + \frac {20 \, {\left (b \sqrt {x} + a\right )}^{7} a^{2}}{7 \, b^{6}} - \frac {10 \, {\left (b \sqrt {x} + a\right )}^{6} a^{3}}{3 \, b^{6}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{5} a^{4}}{b^{6}} - \frac {{\left (b \sqrt {x} + a\right )}^{4} a^{5}}{2 \, b^{6}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \left (a+b \sqrt {x}\right )^3 x^2 \, dx=\frac {2}{9} \, b^{3} x^{\frac {9}{2}} + \frac {3}{4} \, a b^{2} x^{4} + \frac {6}{7} \, a^{2} b x^{\frac {7}{2}} + \frac {1}{3} \, a^{3} x^{3} \]

[In]

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

[Out]

2/9*b^3*x^(9/2) + 3/4*a*b^2*x^4 + 6/7*a^2*b*x^(7/2) + 1/3*a^3*x^3

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \left (a+b \sqrt {x}\right )^3 x^2 \, dx=\frac {a^3\,x^3}{3}+\frac {2\,b^3\,x^{9/2}}{9}+\frac {3\,a\,b^2\,x^4}{4}+\frac {6\,a^2\,b\,x^{7/2}}{7} \]

[In]

int(x^2*(a + b*x^(1/2))^3,x)

[Out]

(a^3*x^3)/3 + (2*b^3*x^(9/2))/9 + (3*a*b^2*x^4)/4 + (6*a^2*b*x^(7/2))/7