3.1775 \(\int \frac {(a+\frac {b}{x})^{5/2}}{x^{3/2}} \, dx\)

Optimal. Leaf size=100 \[ -\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{8 \sqrt {b}}-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{8 \sqrt {x}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{12 \sqrt {x}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{3 \sqrt {x}} \]

[Out]

-5/8*a^3*arctanh(b^(1/2)/(a+b/x)^(1/2)/x^(1/2))/b^(1/2)-5/12*a*(a+b/x)^(3/2)/x^(1/2)-1/3*(a+b/x)^(5/2)/x^(1/2)
-5/8*a^2*(a+b/x)^(1/2)/x^(1/2)

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Rubi [A]  time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {337, 195, 217, 206} \[ -\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{8 \sqrt {x}}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{8 \sqrt {b}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{12 \sqrt {x}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{3 \sqrt {x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)/x^(3/2),x]

[Out]

(-5*a^2*Sqrt[a + b/x])/(8*Sqrt[x]) - (5*a*(a + b/x)^(3/2))/(12*Sqrt[x]) - (a + b/x)^(5/2)/(3*Sqrt[x]) - (5*a^3
*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(8*Sqrt[b])

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 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 )^{5/2}}{x^{3/2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \left (a+b x^2\right )^{5/2} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{3 \sqrt {x}}-\frac {1}{3} (5 a) \operatorname {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{12 \sqrt {x}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{3 \sqrt {x}}-\frac {1}{4} \left (5 a^2\right ) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{8 \sqrt {x}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{12 \sqrt {x}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{3 \sqrt {x}}-\frac {1}{8} \left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{8 \sqrt {x}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{12 \sqrt {x}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{3 \sqrt {x}}-\frac {1}{8} \left (5 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )\\ &=-\frac {5 a^2 \sqrt {a+\frac {b}{x}}}{8 \sqrt {x}}-\frac {5 a \left (a+\frac {b}{x}\right )^{3/2}}{12 \sqrt {x}}-\frac {\left (a+\frac {b}{x}\right )^{5/2}}{3 \sqrt {x}}-\frac {5 a^3 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{8 \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 85, normalized size = 0.85 \[ \frac {1}{24} \sqrt {a+\frac {b}{x}} \left (-\frac {15 a^{5/2} \sinh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )}{\sqrt {b} \sqrt {\frac {b}{a x}+1}}-\frac {33 a^2 x^2+26 a b x+8 b^2}{x^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)/x^(3/2),x]

[Out]

(Sqrt[a + b/x]*(-((8*b^2 + 26*a*b*x + 33*a^2*x^2)/x^(5/2)) - (15*a^(5/2)*ArcSinh[Sqrt[b]/(Sqrt[a]*Sqrt[x])])/(
Sqrt[b]*Sqrt[1 + b/(a*x)])))/24

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fricas [A]  time = 0.69, size = 174, normalized size = 1.74 \[ \left [\frac {15 \, a^{3} \sqrt {b} x^{3} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) - 2 \, {\left (33 \, a^{2} b x^{2} + 26 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{48 \, b x^{3}}, \frac {15 \, a^{3} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) - {\left (33 \, a^{2} b x^{2} + 26 \, a b^{2} x + 8 \, b^{3}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, b x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)/x^(3/2),x, algorithm="fricas")

[Out]

[1/48*(15*a^3*sqrt(b)*x^3*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) - 2*(33*a^2*b*x^2 + 26*a*b^
2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x))/(b*x^3), 1/24*(15*a^3*sqrt(-b)*x^3*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x +
b)/x)/b) - (33*a^2*b*x^2 + 26*a*b^2*x + 8*b^3)*sqrt(x)*sqrt((a*x + b)/x))/(b*x^3)]

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giac [A]  time = 0.34, size = 79, normalized size = 0.79 \[ \frac {\frac {15 \, a^{4} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {33 \, {\left (a x + b\right )}^{\frac {5}{2}} a^{4} - 40 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{4} b + 15 \, \sqrt {a x + b} a^{4} b^{2}}{a^{3} x^{3}}}{24 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)/x^(3/2),x, algorithm="giac")

[Out]

1/24*(15*a^4*arctan(sqrt(a*x + b)/sqrt(-b))/sqrt(-b) - (33*(a*x + b)^(5/2)*a^4 - 40*(a*x + b)^(3/2)*a^4*b + 15
*sqrt(a*x + b)*a^4*b^2)/(a^3*x^3))/a

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maple [A]  time = 0.02, size = 92, normalized size = 0.92 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (15 a^{3} x^{3} \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )+33 \sqrt {a x +b}\, a^{2} \sqrt {b}\, x^{2}+26 \sqrt {a x +b}\, a \,b^{\frac {3}{2}} x +8 \sqrt {a x +b}\, b^{\frac {5}{2}}\right )}{24 \sqrt {a x +b}\, \sqrt {b}\, x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^(5/2)/x^(3/2),x)

[Out]

-1/24*((a*x+b)/x)^(1/2)/x^(5/2)*(15*arctanh((a*x+b)^(1/2)/b^(1/2))*x^3*a^3+33*(a*x+b)^(1/2)*a^2*b^(1/2)*x^2+26
*(a*x+b)^(1/2)*a*b^(3/2)*x+8*(a*x+b)^(1/2)*b^(5/2))/(a*x+b)^(1/2)/b^(1/2)

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maxima [B]  time = 2.48, size = 156, normalized size = 1.56 \[ \frac {5 \, a^{3} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{16 \, \sqrt {b}} - \frac {33 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} a^{3} x^{\frac {5}{2}} - 40 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} a^{3} b x^{\frac {3}{2}} + 15 \, \sqrt {a + \frac {b}{x}} a^{3} b^{2} \sqrt {x}}{24 \, {\left ({\left (a + \frac {b}{x}\right )}^{3} x^{3} - 3 \, {\left (a + \frac {b}{x}\right )}^{2} b x^{2} + 3 \, {\left (a + \frac {b}{x}\right )} b^{2} x - b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(5/2)/x^(3/2),x, algorithm="maxima")

[Out]

5/16*a^3*log((sqrt(a + b/x)*sqrt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b)))/sqrt(b) - 1/24*(33*(a + b/x)
^(5/2)*a^3*x^(5/2) - 40*(a + b/x)^(3/2)*a^3*b*x^(3/2) + 15*sqrt(a + b/x)*a^3*b^2*sqrt(x))/((a + b/x)^3*x^3 - 3
*(a + b/x)^2*b*x^2 + 3*(a + b/x)*b^2*x - b^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {b}{x}\right )}^{5/2}}{x^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^(5/2)/x^(3/2),x)

[Out]

int((a + b/x)^(5/2)/x^(3/2), x)

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sympy [A]  time = 14.40, size = 104, normalized size = 1.04 \[ - \frac {11 a^{\frac {5}{2}} \sqrt {1 + \frac {b}{a x}}}{8 \sqrt {x}} - \frac {13 a^{\frac {3}{2}} b \sqrt {1 + \frac {b}{a x}}}{12 x^{\frac {3}{2}}} - \frac {\sqrt {a} b^{2} \sqrt {1 + \frac {b}{a x}}}{3 x^{\frac {5}{2}}} - \frac {5 a^{3} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}} \right )}}{8 \sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(5/2)/x**(3/2),x)

[Out]

-11*a**(5/2)*sqrt(1 + b/(a*x))/(8*sqrt(x)) - 13*a**(3/2)*b*sqrt(1 + b/(a*x))/(12*x**(3/2)) - sqrt(a)*b**2*sqrt
(1 + b/(a*x))/(3*x**(5/2)) - 5*a**3*asinh(sqrt(b)/(sqrt(a)*sqrt(x)))/(8*sqrt(b))

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