\(\int (a+\frac {b}{x})^{5/2} \sqrt {x} \, dx\) [1773]

Optimal result

Integrand size = 17, antiderivative size = 94 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=-\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \]

[Out]

2/3*(a+b/x)^(5/2)*x^(3/2)-5*a*b^(3/2)*arctanh(b^(1/2)/(a+b/x)^(1/2)/x^(1/2))-5*b^2*(a+b/x)^(1/2)/x^(1/2)+10/3*
b*(a+b/x)^(3/2)*x^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {344, 283, 201, 223, 212} \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=-5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )-\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {2}{3} x^{3/2} \left (a+\frac {b}{x}\right )^{5/2}+\frac {10}{3} b \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2} \]

[In]

Int[(a + b/x)^(5/2)*Sqrt[x],x]

[Out]

(-5*b^2*Sqrt[a + b/x])/Sqrt[x] + (10*b*(a + b/x)^(3/2)*Sqrt[x])/3 + (2*(a + b/x)^(5/2)*x^(3/2))/3 - 5*a*b^(3/2
)*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])]

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 344

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[-k/c, Subst[
Int[(a + b/(c^n*x^(k*n)))^p/x^(k*(m + 1) + 1), x], x, 1/(c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && ILtQ[n,
0] && FractionQ[m]

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/2}}{x^4} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = \frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-\frac {1}{3} (10 b) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = \frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-\left (10 b^2\right ) \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-\left (5 a b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right ) \\ & = -\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-\left (5 a b^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \\ & = -\frac {5 b^2 \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+\frac {10}{3} b \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}+\frac {2}{3} \left (a+\frac {b}{x}\right )^{5/2} x^{3/2}-5 a b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 6.90 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\frac {\sqrt {a+\frac {b}{x}} \sqrt {x} \left (\frac {\sqrt {b+a x} \left (-3 b^2+14 a b x+2 a^2 x^2\right )}{3 x}-5 a b^{3/2} \text {arctanh}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )\right )}{\sqrt {b+a x}} \]

[In]

Integrate[(a + b/x)^(5/2)*Sqrt[x],x]

[Out]

(Sqrt[a + b/x]*Sqrt[x]*((Sqrt[b + a*x]*(-3*b^2 + 14*a*b*x + 2*a^2*x^2))/(3*x) - 5*a*b^(3/2)*ArcTanh[Sqrt[b + a
*x]/Sqrt[b]]))/Sqrt[b + a*x]

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.87

method	result	size
risch	\(-\frac {b^{2} \sqrt {\frac {a x +b}{x}}}{\sqrt {x}}+\frac {a \left (\frac {4 \left (a x +b \right )^{\frac {3}{2}}}{3}+8 b \sqrt {a x +b}-10 b^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )\right ) \sqrt {\frac {a x +b}{x}}\, \sqrt {x}}{2 \sqrt {a x +b}}\)	\(82\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (-2 a^{2} x^{2} \sqrt {b}\, \sqrt {a x +b}+15 \,\operatorname {arctanh}\left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a \,b^{2} x -14 a \,b^{\frac {3}{2}} x \sqrt {a x +b}+3 b^{\frac {5}{2}} \sqrt {a x +b}\right )}{3 \sqrt {x}\, \sqrt {a x +b}\, \sqrt {b}}\)	\(91\)

[In]

int((a+b/x)^(5/2)*x^(1/2),x,method=_RETURNVERBOSE)

[Out]

-b^2/x^(1/2)*((a*x+b)/x)^(1/2)+1/2*a*(4/3*(a*x+b)^(3/2)+8*b*(a*x+b)^(1/2)-10*b^(3/2)*arctanh((a*x+b)^(1/2)/b^(
1/2)))*((a*x+b)/x)^(1/2)/(a*x+b)^(1/2)*x^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.64 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\left [\frac {15 \, a b^{\frac {3}{2}} x \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (2 \, a^{2} x^{2} + 14 \, a b x - 3 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{6 \, x}, \frac {15 \, a \sqrt {-b} b x \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (2 \, a^{2} x^{2} + 14 \, a b x - 3 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, x}\right ] \]

[In]

integrate((a+b/x)^(5/2)*x^(1/2),x, algorithm="fricas")

[Out]

[1/6*(15*a*b^(3/2)*x*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) + 2*(2*a^2*x^2 + 14*a*b*x - 3*b^
2)*sqrt(x)*sqrt((a*x + b)/x))/x, 1/3*(15*a*sqrt(-b)*b*x*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/b) + (2*a^2*
x^2 + 14*a*b*x - 3*b^2)*sqrt(x)*sqrt((a*x + b)/x))/x]

Sympy [A] (verification not implemented)

Time = 7.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\frac {2 a^{2} \sqrt {b} x \sqrt {\frac {a x}{b} + 1}}{3} + \frac {14 a b^{\frac {3}{2}} \sqrt {\frac {a x}{b} + 1}}{3} + \frac {5 a b^{\frac {3}{2}} \log {\left (\frac {a x}{b} \right )}}{2} - 5 a b^{\frac {3}{2}} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )} - \frac {b^{\frac {5}{2}} \sqrt {\frac {a x}{b} + 1}}{x} \]

[In]

integrate((a+b/x)**(5/2)*x**(1/2),x)

[Out]

2*a**2*sqrt(b)*x*sqrt(a*x/b + 1)/3 + 14*a*b**(3/2)*sqrt(a*x/b + 1)/3 + 5*a*b**(3/2)*log(a*x/b)/2 - 5*a*b**(3/2
)*log(sqrt(a*x/b + 1) + 1) - b**(5/2)*sqrt(a*x/b + 1)/x

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.18 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\frac {2}{3} \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a x^{\frac {3}{2}} + \frac {5}{2} \, a b^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right ) + 4 \, \sqrt {a + \frac {b}{x}} a b \sqrt {x} - \frac {\sqrt {a + \frac {b}{x}} a b^{2} \sqrt {x}}{{\left (a + \frac {b}{x}\right )} x - b} \]

[In]

integrate((a+b/x)^(5/2)*x^(1/2),x, algorithm="maxima")

[Out]

2/3*(a + b/x)^(3/2)*a*x^(3/2) + 5/2*a*b^(3/2)*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + s
qrt(b))) + 4*sqrt(a + b/x)*a*b*sqrt(x) - sqrt(a + b/x)*a*b^2*sqrt(x)/((a + b/x)*x - b)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.79 \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\frac {\frac {15 \, a^{2} b^{2} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + 2 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{2} + 12 \, \sqrt {a x + b} a^{2} b - \frac {3 \, \sqrt {a x + b} a b^{2}}{x}}{3 \, a} \]

[In]

integrate((a+b/x)^(5/2)*x^(1/2),x, algorithm="giac")

[Out]

1/3*(15*a^2*b^2*arctan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b) + 2*(a*x + b)^(3/2)*a^2 + 12*sqrt(a*x + b)*a^2*b - 3*s
qrt(a*x + b)*a*b^2/x)/a

Mupad [F(-1)]

Timed out. \[ \int \left (a+\frac {b}{x}\right )^{5/2} \sqrt {x} \, dx=\int \sqrt {x}\,{\left (a+\frac {b}{x}\right )}^{5/2} \,d x \]

[In]

int(x^(1/2)*(a + b/x)^(5/2),x)

[Out]

int(x^(1/2)*(a + b/x)^(5/2), x)