\(\int (a+b \sqrt {x})^2 x^2 \, dx\) [2122]

Optimal result

Integrand size = 15, antiderivative size = 32 \[ \int \left (a+b \sqrt {x}\right )^2 x^2 \, dx=\frac {a^2 x^3}{3}+\frac {4}{7} a b x^{7/2}+\frac {b^2 x^4}{4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, 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} \[ \int \left (a+b \sqrt {x}\right )^2 x^2 \, dx=\frac {a^2 x^3}{3}+\frac {4}{7} a b x^{7/2}+\frac {b^2 x^4}{4} \]

[In]

Int[(a + b*Sqrt[x])^2*x^2,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \left (a+b \sqrt {x}\right )^2 x^2 \, dx=\frac {1}{84} x^3 \left (28 a^2+48 a b \sqrt {x}+21 b^2 x\right ) \]

[In]

Integrate[(a + b*Sqrt[x])^2*x^2,x]

[Out]

(x^3*(28*a^2 + 48*a*b*Sqrt[x] + 21*b^2*x))/84

Maple [A] (verified)

Time = 5.79 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {a^{2} x^{3}}{3}+\frac {4 a b \,x^{\frac {7}{2}}}{7}+\frac {b^{2} x^{4}}{4}\)	\(25\)
default	\(\frac {a^{2} x^{3}}{3}+\frac {4 a b \,x^{\frac {7}{2}}}{7}+\frac {b^{2} x^{4}}{4}\)	\(25\)
trager	\(\frac {\left (3 b^{2} x^{3}+4 a^{2} x^{2}+3 b^{2} x^{2}+4 a^{2} x +3 b^{2} x +4 a^{2}+3 b^{2}\right ) \left (-1+x \right )}{12}+\frac {4 a b \,x^{\frac {7}{2}}}{7}\)	\(61\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \left (a+b \sqrt {x}\right )^2 x^2 \, dx=\frac {1}{4} \, b^{2} x^{4} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {1}{3} \, a^{2} x^{3} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \left (a+b \sqrt {x}\right )^2 x^2 \, dx=\frac {a^{2} x^{3}}{3} + \frac {4 a b x^{\frac {7}{2}}}{7} + \frac {b^{2} x^{4}}{4} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (24) = 48\).

Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.06 \[ \int \left (a+b \sqrt {x}\right )^2 x^2 \, dx=\frac {{\left (b \sqrt {x} + a\right )}^{8}}{4 \, b^{6}} - \frac {10 \, {\left (b \sqrt {x} + a\right )}^{7} a}{7 \, b^{6}} + \frac {10 \, {\left (b \sqrt {x} + a\right )}^{6} a^{2}}{3 \, b^{6}} - \frac {4 \, {\left (b \sqrt {x} + a\right )}^{5} a^{3}}{b^{6}} + \frac {5 \, {\left (b \sqrt {x} + a\right )}^{4} a^{4}}{2 \, b^{6}} - \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3} a^{5}}{3 \, b^{6}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \left (a+b \sqrt {x}\right )^2 x^2 \, dx=\frac {1}{4} \, b^{2} x^{4} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {1}{3} \, a^{2} x^{3} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.82 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.75 \[ \int \left (a+b \sqrt {x}\right )^2 x^2 \, dx=\frac {a^2\,x^3}{3}+\frac {b^2\,x^4}{4}+\frac {4\,a\,b\,x^{7/2}}{7} \]

[In]

int(x^2*(a + b*x^(1/2))^2,x)

[Out]

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