\(\int (a+\frac {b}{x})^3 \sqrt {x} \, dx\) [1661]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.01 (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 = {269, 45} \[ \int \left (a+\frac {b}{x}\right )^3 \sqrt {x} \, dx=\frac {2}{3} a^3 x^{3/2}+6 a^2 b \sqrt {x}-\frac {6 a b^2}{\sqrt {x}}-\frac {2 b^3}{3 x^{3/2}} \]

[In]

Int[(a + b/x)^3*Sqrt[x],x]

[Out]

(-2*b^3)/(3*x^(3/2)) - (6*a*b^2)/Sqrt[x] + 6*a^2*b*Sqrt[x] + (2*a^3*x^(3/2))/3

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 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.81 \[ \int \left (a+\frac {b}{x}\right )^3 \sqrt {x} \, dx=\frac {2 \left (-b^3-9 a b^2 x+9 a^2 b x^2+a^3 x^3\right )}{3 x^{3/2}} \]

[In]

Integrate[(a + b/x)^3*Sqrt[x],x]

[Out]

(2*(-b^3 - 9*a*b^2*x + 9*a^2*b*x^2 + a^3*x^3))/(3*x^(3/2))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74

method	result	size
gosper	\(\frac {-6 a \,b^{2} x -\frac {2}{3} b^{3}+\frac {2}{3} a^{3} x^{3}+6 a^{2} b \,x^{2}}{x^{\frac {3}{2}}}\)	\(35\)
trager	\(\frac {-6 a \,b^{2} x -\frac {2}{3} b^{3}+\frac {2}{3} a^{3} x^{3}+6 a^{2} b \,x^{2}}{x^{\frac {3}{2}}}\)	\(35\)
risch	\(\frac {-6 a \,b^{2} x -\frac {2}{3} b^{3}+\frac {2}{3} a^{3} x^{3}+6 a^{2} b \,x^{2}}{x^{\frac {3}{2}}}\)	\(35\)
derivativedivides	\(-\frac {2 b^{3}}{3 x^{\frac {3}{2}}}+\frac {2 a^{3} x^{\frac {3}{2}}}{3}-\frac {6 a \,b^{2}}{\sqrt {x}}+6 a^{2} b \sqrt {x}\)	\(36\)
default	\(-\frac {2 b^{3}}{3 x^{\frac {3}{2}}}+\frac {2 a^{3} x^{\frac {3}{2}}}{3}-\frac {6 a \,b^{2}}{\sqrt {x}}+6 a^{2} b \sqrt {x}\)	\(36\)

[In]

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

[Out]

2/3*(a^3*x^3+9*a^2*b*x^2-9*a*b^2*x-b^3)/x^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72 \[ \int \left (a+\frac {b}{x}\right )^3 \sqrt {x} \, dx=\frac {2 \, {\left (a^{3} x^{3} + 9 \, a^{2} b x^{2} - 9 \, a b^{2} x - b^{3}\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

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

[Out]

2/3*(a^3*x^3 + 9*a^2*b*x^2 - 9*a*b^2*x - b^3)/x^(3/2)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.98 \[ \int \left (a+\frac {b}{x}\right )^3 \sqrt {x} \, dx=\frac {2 a^{3} x^{\frac {3}{2}}}{3} + 6 a^{2} b \sqrt {x} - \frac {6 a b^{2}}{\sqrt {x}} - \frac {2 b^{3}}{3 x^{\frac {3}{2}}} \]

[In]

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

[Out]

2*a**3*x**(3/2)/3 + 6*a**2*b*sqrt(x) - 6*a*b**2/sqrt(x) - 2*b**3/(3*x**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.77 \[ \int \left (a+\frac {b}{x}\right )^3 \sqrt {x} \, dx=-\frac {6 \, a b^{2}}{\sqrt {x}} + \frac {2}{3} \, {\left (a^{3} + \frac {9 \, a^{2} b}{x}\right )} x^{\frac {3}{2}} - \frac {2 \, b^{3}}{3 \, x^{\frac {3}{2}}} \]

[In]

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

[Out]

-6*a*b^2/sqrt(x) + 2/3*(a^3 + 9*a^2*b/x)*x^(3/2) - 2/3*b^3/x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.72 \[ \int \left (a+\frac {b}{x}\right )^3 \sqrt {x} \, dx=\frac {2}{3} \, a^{3} x^{\frac {3}{2}} + 6 \, a^{2} b \sqrt {x} - \frac {2 \, {\left (9 \, a b^{2} x + b^{3}\right )}}{3 \, x^{\frac {3}{2}}} \]

[In]

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

[Out]

2/3*a^3*x^(3/2) + 6*a^2*b*sqrt(x) - 2/3*(9*a*b^2*x + b^3)/x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \left (a+\frac {b}{x}\right )^3 \sqrt {x} \, dx=-\frac {-2\,a^3\,x^3-18\,a^2\,b\,x^2+18\,a\,b^2\,x+2\,b^3}{3\,x^{3/2}} \]

[In]

int(x^(1/2)*(a + b/x)^3,x)

[Out]

-(2*b^3 - 2*a^3*x^3 - 18*a^2*b*x^2 + 18*a*b^2*x)/(3*x^(3/2))