3.1801 \(\int \frac {1}{(a+\frac {b}{x})^{5/2} x^{7/2}} \, dx\)

Optimal. Leaf size=75 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{5/2}}+\frac {2}{b^2 \sqrt {x} \sqrt {a+\frac {b}{x}}}+\frac {2}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {337, 288, 217, 206} \[ \frac {2}{b^2 \sqrt {x} \sqrt {a+\frac {b}{x}}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{b^{5/2}}+\frac {2}{3 b x^{3/2} \left (a+\frac {b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^(7/2)),x]

[Out]

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

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 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 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 {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b x^2\right )^{5/2}} \, dx,x,\frac {1}{\sqrt {x}}\right )\right )\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}-\frac {2 \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{b}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{b^2}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^2}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{3/2}}+\frac {2}{b^2 \sqrt {a+\frac {b}{x}} \sqrt {x}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{b^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 94, normalized size = 1.25 \[ \frac {2 \left (\sqrt {b} \sqrt {x} (3 a x+4 b)-3 \sqrt {a} x \sqrt {\frac {b}{a x}+1} (a x+b) \sinh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )\right )}{3 b^{5/2} x \sqrt {a+\frac {b}{x}} (a x+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^(7/2)),x]

[Out]

(2*(Sqrt[b]*Sqrt[x]*(4*b + 3*a*x) - 3*Sqrt[a]*Sqrt[1 + b/(a*x)]*x*(b + a*x)*ArcSinh[Sqrt[b]/(Sqrt[a]*Sqrt[x])]
))/(3*b^(5/2)*Sqrt[a + b/x]*x*(b + a*x))

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fricas [A]  time = 0.69, size = 205, normalized size = 2.73 \[ \left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {b} \log \left (\frac {a x - 2 \, \sqrt {b} \sqrt {x} \sqrt {\frac {a x + b}{x}} + 2 \, b}{x}\right ) + 2 \, {\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}}, \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (3 \, a b x + 4 \, b^{2}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}\right )}}{3 \, {\left (a^{2} b^{3} x^{2} + 2 \, a b^{4} x + b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(7/2),x, algorithm="fricas")

[Out]

[1/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(b)*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)/x) + 2*(3*a*b*
x + 4*b^2)*sqrt(x)*sqrt((a*x + b)/x))/(a^2*b^3*x^2 + 2*a*b^4*x + b^5), 2/3*(3*(a^2*x^2 + 2*a*b*x + b^2)*sqrt(-
b)*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x + b)/x)/b) + (3*a*b*x + 4*b^2)*sqrt(x)*sqrt((a*x + b)/x))/(a^2*b^3*x^2 +
2*a*b^4*x + b^5)]

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giac [A]  time = 0.19, size = 78, normalized size = 1.04 \[ \frac {2 \, \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} - \frac {2 \, {\left (3 \, \sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + 4 \, \sqrt {-b}\right )}}{3 \, \sqrt {-b} b^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, a x + 4 \, b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(7/2),x, algorithm="giac")

[Out]

2*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^2) - 2/3*(3*sqrt(b)*arctan(sqrt(b)/sqrt(-b)) + 4*sqrt(-b))/(sqrt(
-b)*b^(5/2)) + 2/3*(3*a*x + 4*b)/((a*x + b)^(3/2)*b^2)

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maple [A]  time = 0.02, size = 85, normalized size = 1.13 \[ -\frac {2 \sqrt {\frac {a x +b}{x}}\, \left (3 \sqrt {a x +b}\, a x \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )-3 a \sqrt {b}\, x +3 \sqrt {a x +b}\, b \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )-4 b^{\frac {3}{2}}\right ) \sqrt {x}}{3 \left (a x +b \right )^{2} b^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(5/2)/x^(7/2),x)

[Out]

-2/3*((a*x+b)/x)^(1/2)*x^(1/2)/b^(5/2)*(3*arctanh((a*x+b)^(1/2)/b^(1/2))*(a*x+b)^(1/2)*x*a+3*arctanh((a*x+b)^(
1/2)/b^(1/2))*b*(a*x+b)^(1/2)-3*a*b^(1/2)*x-4*b^(3/2))/(a*x+b)^2

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maxima [A]  time = 2.34, size = 74, normalized size = 0.99 \[ \frac {\log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{b^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (a + \frac {b}{x}\right )} x + b\right )}}{3 \, {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{2} x^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(7/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{7/2}\,{\left (a+\frac {b}{x}\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(7/2)*(a + b/x)^(5/2)),x)

[Out]

int(1/(x^(7/2)*(a + b/x)^(5/2)), x)

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sympy [B]  time = 51.16, size = 697, normalized size = 9.29 \[ \frac {3 a^{3} b^{4} x^{3} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {6 a^{3} b^{4} x^{3} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {6 a^{2} b^{5} x^{2} \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {9 a^{2} b^{5} x^{2} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {18 a^{2} b^{5} x^{2} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {14 a b^{6} x \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {9 a b^{6} x \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {18 a b^{6} x \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {8 b^{7} \sqrt {\frac {a x}{b} + 1}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} + \frac {3 b^{7} \log {\left (\frac {a x}{b} \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} - \frac {6 b^{7} \log {\left (\sqrt {\frac {a x}{b} + 1} + 1 \right )}}{3 a^{3} b^{\frac {13}{2}} x^{3} + 9 a^{2} b^{\frac {15}{2}} x^{2} + 9 a b^{\frac {17}{2}} x + 3 b^{\frac {19}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(5/2)/x**(7/2),x)

[Out]

3*a**3*b**4*x**3*log(a*x/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3*b**(19/2)) -
6*a**3*b**4*x**3*log(sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*x + 3
*b**(19/2)) + 6*a**2*b**5*x**2*sqrt(a*x/b + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*
x + 3*b**(19/2)) + 9*a**2*b**5*x**2*log(a*x/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/2)*
x + 3*b**(19/2)) - 18*a**2*b**5*x**2*log(sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 +
9*a*b**(17/2)*x + 3*b**(19/2)) + 14*a*b**6*x*sqrt(a*x/b + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 +
9*a*b**(17/2)*x + 3*b**(19/2)) + 9*a*b**6*x*log(a*x/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b
**(17/2)*x + 3*b**(19/2)) - 18*a*b**6*x*log(sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**
2 + 9*a*b**(17/2)*x + 3*b**(19/2)) + 8*b**7*sqrt(a*x/b + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9
*a*b**(17/2)*x + 3*b**(19/2)) + 3*b**7*log(a*x/b)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**(17/
2)*x + 3*b**(19/2)) - 6*b**7*log(sqrt(a*x/b + 1) + 1)/(3*a**3*b**(13/2)*x**3 + 9*a**2*b**(15/2)*x**2 + 9*a*b**
(17/2)*x + 3*b**(19/2))

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