3.19.3 \(\int \frac {1}{(a+\frac {b}{x})^{5/2} x^{11/2}} \, dx\) [1803]

Optimal. Leaf size=129 \[ \frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 b^{9/2}} \]

[Out]

2/3/b/(a+b/x)^(3/2)/x^(7/2)-35/4*a^2*arctanh(b^(1/2)/(a+b/x)^(1/2)/x^(1/2))/b^(9/2)+14/3/b^2/x^(5/2)/(a+b/x)^(
1/2)-35/6*(a+b/x)^(1/2)/b^3/x^(3/2)+35/4*a*(a+b/x)^(1/2)/b^4/x^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {344, 294, 327, 223, 212} \begin {gather*} -\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {x} \sqrt {a+\frac {b}{x}}}\right )}{4 b^{9/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {14}{3 b^2 x^{5/2} \sqrt {a+\frac {b}{x}}}+\frac {2}{3 b x^{7/2} \left (a+\frac {b}{x}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*x^(11/2)),x]

[Out]

2/(3*b*(a + b/x)^(3/2)*x^(7/2)) + 14/(3*b^2*Sqrt[a + b/x]*x^(5/2)) - (35*Sqrt[a + b/x])/(6*b^3*x^(3/2)) + (35*
a*Sqrt[a + b/x])/(4*b^4*Sqrt[x]) - (35*a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x]*Sqrt[x])])/(4*b^(9/2))

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 294

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 327

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*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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*} \int \frac {1}{\left (a+\frac {b}{x}\right )^{5/2} x^{11/2}} \, dx &=-\left (2 \text {Subst}\left (\int \frac {x^8}{\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^{7/2}}-\frac {14 \text {Subst}\left (\int \frac {x^6}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{3 b}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {70 \text {Subst}\left (\int \frac {x^4}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{3 b^2}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {(35 a) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{2 b^3}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{4 b^4}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {\left (35 a^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 b^4}\\ &=\frac {2}{3 b \left (a+\frac {b}{x}\right )^{3/2} x^{7/2}}+\frac {14}{3 b^2 \sqrt {a+\frac {b}{x}} x^{5/2}}-\frac {35 \sqrt {a+\frac {b}{x}}}{6 b^3 x^{3/2}}+\frac {35 a \sqrt {a+\frac {b}{x}}}{4 b^4 \sqrt {x}}-\frac {35 a^2 \tanh ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x}} \sqrt {x}}\right )}{4 b^{9/2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 10.02, size = 56, normalized size = 0.43 \begin {gather*} -\frac {2 \sqrt {1+\frac {b}{a x}} \, _2F_1\left (\frac {5}{2},\frac {9}{2};\frac {11}{2};-\frac {b}{a x}\right )}{9 a^2 \sqrt {a+\frac {b}{x}} x^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*x^(11/2)),x]

[Out]

(-2*Sqrt[1 + b/(a*x)]*Hypergeometric2F1[5/2, 9/2, 11/2, -(b/(a*x))])/(9*a^2*Sqrt[a + b/x]*x^(9/2))

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Maple [A]
time = 0.07, size = 117, normalized size = 0.91

method	result	size
risch	\(\frac {\left (a x +b \right ) \left (11 a x -2 b \right )}{4 b^{4} x^{\frac {5}{2}} \sqrt {\frac {a x +b}{x}}}+\frac {a^{2} \left (-\frac {70 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {48}{\sqrt {a x +b}}+\frac {16 b}{3 \left (a x +b \right )^{\frac {3}{2}}}\right ) \sqrt {a x +b}}{8 b^{4} \sqrt {x}\, \sqrt {\frac {a x +b}{x}}}\)	\(100\)
default	\(-\frac {\sqrt {\frac {a x +b}{x}}\, \left (105 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) \sqrt {a x +b}\, a^{3} x^{3}+105 \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right ) a^{2} b \,x^{2} \sqrt {a x +b}-105 a^{3} x^{3} \sqrt {b}-140 b^{\frac {3}{2}} a^{2} x^{2}-21 b^{\frac {5}{2}} a x +6 b^{\frac {7}{2}}\right )}{12 x^{\frac {3}{2}} \left (a x +b \right )^{2} b^{\frac {9}{2}}}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x*b)^(5/2)/x^(11/2),x,method=_RETURNVERBOSE)

[Out]

-1/12*((a*x+b)/x)^(1/2)/x^(3/2)*(105*arctanh((a*x+b)^(1/2)/b^(1/2))*(a*x+b)^(1/2)*a^3*x^3+105*arctanh((a*x+b)^
(1/2)/b^(1/2))*a^2*b*x^2*(a*x+b)^(1/2)-105*a^3*x^3*b^(1/2)-140*b^(3/2)*a^2*x^2-21*b^(5/2)*a*x+6*b^(7/2))/(a*x+
b)^2/b^(9/2)

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Maxima [A]
time = 0.50, size = 163, normalized size = 1.26 \begin {gather*} \frac {105 \, {\left (a + \frac {b}{x}\right )}^{3} a^{2} x^{3} - 175 \, {\left (a + \frac {b}{x}\right )}^{2} a^{2} b x^{2} + 56 \, {\left (a + \frac {b}{x}\right )} a^{2} b^{2} x + 8 \, a^{2} b^{3}}{12 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {7}{2}} b^{4} x^{\frac {7}{2}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} b^{5} x^{\frac {5}{2}} + {\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} b^{6} x^{\frac {3}{2}}\right )}} + \frac {35 \, a^{2} \log \left (\frac {\sqrt {a + \frac {b}{x}} \sqrt {x} - \sqrt {b}}{\sqrt {a + \frac {b}{x}} \sqrt {x} + \sqrt {b}}\right )}{8 \, b^{\frac {9}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(11/2),x, algorithm="maxima")

[Out]

1/12*(105*(a + b/x)^3*a^2*x^3 - 175*(a + b/x)^2*a^2*b*x^2 + 56*(a + b/x)*a^2*b^2*x + 8*a^2*b^3)/((a + b/x)^(7/
2)*b^4*x^(7/2) - 2*(a + b/x)^(5/2)*b^5*x^(5/2) + (a + b/x)^(3/2)*b^6*x^(3/2)) + 35/8*a^2*log((sqrt(a + b/x)*sq
rt(x) - sqrt(b))/(sqrt(a + b/x)*sqrt(x) + sqrt(b)))/b^(9/2)

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Fricas [A]
time = 0.38, size = 283, normalized size = 2.19 \begin {gather*} \left [\frac {105 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{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 (105 \, a^{3} b x^{3} + 140 \, a^{2} b^{2} x^{2} + 21 \, a b^{3} x - 6 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{24 \, {\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )}}, \frac {105 \, {\left (a^{4} x^{4} + 2 \, a^{3} b x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{b}\right ) + {\left (105 \, a^{3} b x^{3} + 140 \, a^{2} b^{2} x^{2} + 21 \, a b^{3} x - 6 \, b^{4}\right )} \sqrt {x} \sqrt {\frac {a x + b}{x}}}{12 \, {\left (a^{2} b^{5} x^{4} + 2 \, a b^{6} x^{3} + b^{7} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(11/2),x, algorithm="fricas")

[Out]

[1/24*(105*(a^4*x^4 + 2*a^3*b*x^3 + a^2*b^2*x^2)*sqrt(b)*log((a*x - 2*sqrt(b)*sqrt(x)*sqrt((a*x + b)/x) + 2*b)
/x) + 2*(105*a^3*b*x^3 + 140*a^2*b^2*x^2 + 21*a*b^3*x - 6*b^4)*sqrt(x)*sqrt((a*x + b)/x))/(a^2*b^5*x^4 + 2*a*b
^6*x^3 + b^7*x^2), 1/12*(105*(a^4*x^4 + 2*a^3*b*x^3 + a^2*b^2*x^2)*sqrt(-b)*arctan(sqrt(-b)*sqrt(x)*sqrt((a*x
+ b)/x)/b) + (105*a^3*b*x^3 + 140*a^2*b^2*x^2 + 21*a*b^3*x - 6*b^4)*sqrt(x)*sqrt((a*x + b)/x))/(a^2*b^5*x^4 +
2*a*b^6*x^3 + b^7*x^2)]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)**(5/2)/x**(11/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4497 deep

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Giac [A]
time = 0.62, size = 93, normalized size = 0.72 \begin {gather*} \frac {35 \, a^{2} \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{4 \, \sqrt {-b} b^{4}} + \frac {2 \, {\left (9 \, {\left (a x + b\right )} a^{2} + a^{2} b\right )}}{3 \, {\left (a x + b\right )}^{\frac {3}{2}} b^{4}} + \frac {11 \, {\left (a x + b\right )}^{\frac {3}{2}} a^{2} - 13 \, \sqrt {a x + b} a^{2} b}{4 \, a^{2} b^{4} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/x^(11/2),x, algorithm="giac")

[Out]

35/4*a^2*arctan(sqrt(a*x + b)/sqrt(-b))/(sqrt(-b)*b^4) + 2/3*(9*(a*x + b)*a^2 + a^2*b)/((a*x + b)^(3/2)*b^4) +
1/4*(11*(a*x + b)^(3/2)*a^2 - 13*sqrt(a*x + b)*a^2*b)/(a^2*b^4*x^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^{11/2}\,{\left (a+\frac {b}{x}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(11/2)*(a + b/x)^(5/2)),x)

[Out]

int(1/(x^(11/2)*(a + b/x)^(5/2)), x)

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