3.1765 \(\int \frac {(a+\frac {b}{x})^{3/2}}{\sqrt {x}} \, dx\)

Optimal. Leaf size=69 \[ 2 \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}-\frac {3 b \sqrt {a+\frac {b}{x}}}{\sqrt {x}}-3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {337, 277, 195, 217, 206} \[ 2 \sqrt {x} \left (a+\frac {b}{x}\right )^{3/2}-\frac {3 b \sqrt {a+\frac {b}{x}}}{\sqrt {x}}-3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)/Sqrt[x],x]

[Out]

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

Rule 195

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 206

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

Rule 217

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 277

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 337

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*} \int \frac {\left (a+\frac {b}{x}\right )^{3/2}}{\sqrt {x}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=2 \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}-(6 b) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {3 b \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+2 \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}-(3 a b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {3 b \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+2 \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}-(3 a b) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )\\ &=-\frac {3 b \sqrt {a+\frac {b}{x}}}{\sqrt {x}}+2 \left (a+\frac {b}{x}\right )^{3/2} \sqrt {x}-3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 52, normalized size = 0.75 \[ \frac {2 a \sqrt {x} \sqrt {a+\frac {b}{x}} \, _2F_1\left (-\frac {3}{2},-\frac {1}{2};\frac {1}{2};-\frac {b}{a x}\right )}{\sqrt {\frac {b}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)/Sqrt[x],x]

[Out]

(2*a*Sqrt[a + b/x]*Sqrt[x]*Hypergeometric2F1[-3/2, -1/2, 1/2, -(b/(a*x))])/Sqrt[1 + b/(a*x)]

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fricas [A]  time = 0.93, size = 130, normalized size = 1.88 \[ \left [\frac {3 \, a \sqrt {b} x \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (2 \, a x - b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{2 \, x}, \frac {3 \, a \sqrt {-b} x \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (2 \, a x - b\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.27, size = 56, normalized size = 0.81 \[ \frac {\frac {3 \, a^{2} b \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} + 2 \, \sqrt {a x + b} a^{2} - \frac {\sqrt {a x + b} a b}{x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 70, normalized size = 1.01 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (3 a b x \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )-2 \sqrt {a x +b}\, a \sqrt {b}\, x +\sqrt {a x +b}\, b^{\frac {3}{2}}\right )}{\sqrt {a x +b}\, \sqrt {b}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.33, size = 93, normalized size = 1.35 \[ \frac {3}{2} \, a \sqrt {b} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right ) + 2 \, \sqrt {a + \frac {b}{x}} a \sqrt {x} - \frac {\sqrt {a + \frac {b}{x}} a b \sqrt {x}}{{\left (a + \frac {b}{x}\right )} x - b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*a*sqrt(b)*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b))) + 2*sqrt(a + b/x)*a*sqr
t(x) - sqrt(a + b/x)*a*b*sqrt(x)/((a + b/x)*x - b)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^{3/2}}{\sqrt {x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(3/2)/x^(1/2),x)

[Out]

int((a + b/x)^(3/2)/x^(1/2), x)

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sympy [A]  time = 9.08, size = 92, normalized size = 1.33 \[ \frac {2 a^{\frac {3}{2}} \sqrt {x}}{\sqrt {1 + \frac {b}{a x}}} + \frac {\sqrt {a} b}{\sqrt {x} \sqrt {1 + \frac {b}{a x}}} - 3 a \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )} - \frac {b^{2}}{\sqrt {a} x^{\frac {3}{2}} \sqrt {1 + \frac {b}{a x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**(3/2)*sqrt(x)/sqrt(1 + b/(a*x)) + sqrt(a)*b/(sqrt(x)*sqrt(1 + b/(a*x))) - 3*a*sqrt(b)*asinh(sqrt(b)/(sqrt
(a)*sqrt(x))) - b**2/(sqrt(a)*x**(3/2)*sqrt(1 + b/(a*x)))

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