∫ 1_____ (a+ b x)5∕2x7∕2 dx [1801]

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

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 = 0.03, 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 = {344, 294, 223, 212} \begin {gather*} -\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}} \end {gather*}

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])])

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]

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,

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]

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,

\begin {align*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{7/2}} \, dx &=-\left (2 \text {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 \text {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 \text {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 \text {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 = 10.14, size = 94, normalized size = 1.25 \begin {gather*} \frac {2 \left (\sqrt {b} \sqrt {x} (4 b+3 a x)-3 \sqrt {a} \sqrt {1+\frac {b}{a x}} x (b+a x) \sinh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} \sqrt {x}}\right )\right )}{3 b^{5/2} \sqrt {a+\frac {b}{x}} x (b+a x)} \end {gather*}

(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])]

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method	result	size

default	\(\frac {2 \sqrt {\frac {a x +b}{x}}\, \sqrt {x}\, \left (-3 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}\, a x +4 b^{\frac {3}{2}}+3 a x \sqrt {b}-3 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) b \sqrt {a x +b}\right )}{3 b^{\frac {5}{2}} \left (a x +b \right )^{2}}\)	\(85\)

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

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Maxima [A]
time = 0.49, size = 74, normalized size = 0.99 \begin {gather*} \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}}} \end {gather*}

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

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Fricas [A]
time = 0.37, size = 205, normalized size = 2.73 \begin {gather*} \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 ] \end {gather*}

[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 +

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 697 vs. \(2 (63) = 126\).
time = 31.27, size = 697, normalized size = 9.29 \begin {gather*} \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}}} \end {gather*}

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**

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Giac [A]
time = 0.61, size = 45, normalized size = 0.60 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {2 \, {\left (3 \, a x + 4 \, b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{2}} \end {gather*}

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

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

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3.19.1 \(\int \frac {1}{(a+\frac {b}{x})^{5/2} x^{7/2}} \, dx\) [1801]