3.1942 \(\int \frac{1}{(a+\frac{b}{x^2})^{3/2} x^6} \, dx\)

Optimal. Leaf size=71 \[ -\frac{3 \sqrt{a+\frac{b}{x^2}}}{2 b^2 x}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{5/2}}+\frac{1}{b x^3 \sqrt{a+\frac{b}{x^2}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0339055, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {335, 288, 321, 217, 206} \[ -\frac{3 \sqrt{a+\frac{b}{x^2}}}{2 b^2 x}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{5/2}}+\frac{1}{b x^3 \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(3/2)*x^6),x]

[Out]

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

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

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 321

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 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 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])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^6} \, dx &=-\operatorname{Subst}\left (\int \frac{x^4}{\left (a+b x^2\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{b \sqrt{a+\frac{b}{x^2}} x^3}-\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )}{b}\\ &=\frac{1}{b \sqrt{a+\frac{b}{x^2}} x^3}-\frac{3 \sqrt{a+\frac{b}{x^2}}}{2 b^2 x}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\frac{1}{x}\right )}{2 b^2}\\ &=\frac{1}{b \sqrt{a+\frac{b}{x^2}} x^3}-\frac{3 \sqrt{a+\frac{b}{x^2}}}{2 b^2 x}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{b}{x^2}} x}\right )}{2 b^2}\\ &=\frac{1}{b \sqrt{a+\frac{b}{x^2}} x^3}-\frac{3 \sqrt{a+\frac{b}{x^2}}}{2 b^2 x}+\frac{3 a \tanh ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a+\frac{b}{x^2}} x}\right )}{2 b^{5/2}}\\ \end{align*}

Mathematica [C]  time = 0.0112985, size = 38, normalized size = 0.54 \[ -\frac{a \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{a x^2}{b}+1\right )}{b^2 x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^(3/2)*x^6),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 81, normalized size = 1.1 \begin{align*}{\frac{a{x}^{2}+b}{2\,{x}^{5}} \left ( 3\,\sqrt{a{x}^{2}+b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{2}ab-3\,{b}^{3/2}{x}^{2}a-{b}^{{\frac{5}{2}}} \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^(3/2)/x^6,x)

[Out]

1/2*(a*x^2+b)*(3*(a*x^2+b)^(1/2)*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*x^2*a*b-3*b^(3/2)*x^2*a-b^(5/2))/((a*x^2+
b)/x^2)^(3/2)/x^5/b^(7/2)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(3/2)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.61334, size = 425, normalized size = 5.99 \begin{align*} \left [\frac{3 \,{\left (a^{2} x^{3} + a b x\right )} \sqrt{b} \log \left (-\frac{a x^{2} + 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \,{\left (3 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \,{\left (a b^{3} x^{3} + b^{4} x\right )}}, -\frac{3 \,{\left (a^{2} x^{3} + a b x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) +{\left (3 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \,{\left (a b^{3} x^{3} + b^{4} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(3/2)/x^6,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 4.13408, size = 73, normalized size = 1.03 \begin{align*} - \frac{3 \sqrt{a}}{2 b^{2} x \sqrt{1 + \frac{b}{a x^{2}}}} + \frac{3 a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{2 b^{\frac{5}{2}}} - \frac{1}{2 \sqrt{a} b x^{3} \sqrt{1 + \frac{b}{a x^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**(3/2)/x**6,x)

[Out]

-3*sqrt(a)/(2*b**2*x*sqrt(1 + b/(a*x**2))) + 3*a*asinh(sqrt(b)/(sqrt(a)*x))/(2*b**(5/2)) - 1/(2*sqrt(a)*b*x**3
*sqrt(1 + b/(a*x**2)))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(3/2)/x^6,x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(3/2)*x^6), x)