3.1904 \(\int \frac{(a+\frac{b}{x^2})^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=71 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^2}}}{8 x}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{4 x} \]

[Out]

(-3*a*Sqrt[a + b/x^2])/(8*x) - (a + b/x^2)^(3/2)/(4*x) - (3*a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(8*Sqrt[
b])

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Rubi [A]  time = 0.0277239, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {335, 195, 217, 206} \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{8 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^2}}}{8 x}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{4 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^(3/2)/x^2,x]

[Out]

(-3*a*Sqrt[a + b/x^2])/(8*x) - (a + b/x^2)^(3/2)/(4*x) - (3*a^2*ArcTanh[Sqrt[b]/(Sqrt[a + b/x^2]*x)])/(8*Sqrt[
b])

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

Mathematica [A]  time = 0.0347009, size = 85, normalized size = 1.2 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (3 a^2 x^4 \sqrt{\frac{a x^2}{b}+1} \tanh ^{-1}\left (\sqrt{\frac{a x^2}{b}+1}\right )+5 a^2 x^4+7 a b x^2+2 b^2\right )}{8 x^3 \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^(3/2)/x^2,x]

[Out]

-(Sqrt[a + b/x^2]*(2*b^2 + 7*a*b*x^2 + 5*a^2*x^4 + 3*a^2*x^4*Sqrt[1 + (a*x^2)/b]*ArcTanh[Sqrt[1 + (a*x^2)/b]])
)/(8*x^3*(b + a*x^2))

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Maple [B]  time = 0.007, size = 125, normalized size = 1.8 \begin{align*} -{\frac{1}{8\,{b}^{2}x} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{b}^{3/2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{4}{a}^{2}- \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}{x}^{4}{a}^{2}+ \left ( a{x}^{2}+b \right ) ^{{\frac{5}{2}}}{x}^{2}a-3\,\sqrt{a{x}^{2}+b}{x}^{4}{a}^{2}b+2\, \left ( a{x}^{2}+b \right ) ^{5/2}b \right ) \left ( a{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*((a*x^2+b)/x^2)^(3/2)/x*(3*b^(3/2)*ln(2*(b^(1/2)*(a*x^2+b)^(1/2)+b)/x)*x^4*a^2-(a*x^2+b)^(3/2)*x^4*a^2+(a
*x^2+b)^(5/2)*x^2*a-3*(a*x^2+b)^(1/2)*x^4*a^2*b+2*(a*x^2+b)^(5/2)*b)/(a*x^2+b)^(3/2)/b^2

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.55882, size = 371, normalized size = 5.23 \begin{align*} \left [\frac{3 \, a^{2} \sqrt{b} x^{3} \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 (5 \, a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{16 \, b x^{3}}, \frac{3 \, a^{2} \sqrt{-b} x^{3} \arctan \left (\frac{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) -{\left (5 \, a b x^{2} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{8 \, b x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(3*a^2*sqrt(b)*x^3*log(-(a*x^2 - 2*sqrt(b)*x*sqrt((a*x^2 + b)/x^2) + 2*b)/x^2) - 2*(5*a*b*x^2 + 2*b^2)*s
qrt((a*x^2 + b)/x^2))/(b*x^3), 1/8*(3*a^2*sqrt(-b)*x^3*arctan(sqrt(-b)*x*sqrt((a*x^2 + b)/x^2)/(a*x^2 + b)) -
(5*a*b*x^2 + 2*b^2)*sqrt((a*x^2 + b)/x^2))/(b*x^3)]

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Sympy [A]  time = 3.10074, size = 71, normalized size = 1. \begin{align*} - \frac{5 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{2}}}}{8 x} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{2}}}}{4 x^{3}} - \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{8 \sqrt{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*a**(3/2)*sqrt(1 + b/(a*x**2))/(8*x) - sqrt(a)*b*sqrt(1 + b/(a*x**2))/(4*x**3) - 3*a**2*asinh(sqrt(b)/(sqrt(
a)*x))/(8*sqrt(b))

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Giac [A]  time = 1.23633, size = 85, normalized size = 1.2 \begin{align*} \frac{1}{8} \, a^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (a x^{2} + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x^{2} + b} b}{a^{2} x^{4}}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*a^2*(3*arctan(sqrt(a*x^2 + b)/sqrt(-b))/sqrt(-b) - (5*(a*x^2 + b)^(3/2) - 3*sqrt(a*x^2 + b)*b)/(a^2*x^4))*
sgn(x)