3.1900 \(\int (a+\frac{b}{x^2})^{3/2} x^2 \, dx\)

Optimal. Leaf size=61 \[ -b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )+\frac{1}{3} x^3 \left (a+\frac{b}{x^2}\right )^{3/2}+b x \sqrt{a+\frac{b}{x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0330456, antiderivative size = 61, 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, 277, 217, 206} \[ -b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )+\frac{1}{3} x^3 \left (a+\frac{b}{x^2}\right )^{3/2}+b x \sqrt{a+\frac{b}{x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^(3/2)*x^2,x]

[Out]

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

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 277

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

Mathematica [A]  time = 0.0453037, size = 74, normalized size = 1.21 \[ \frac{x \sqrt{a+\frac{b}{x^2}} \left (\sqrt{a x^2+b} \left (a x^2+4 b\right )-3 b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x^2+b}}{\sqrt{b}}\right )\right )}{3 \sqrt{a x^2+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^(3/2)*x^2,x]

[Out]

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

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Maple [A]  time = 0.006, size = 78, normalized size = 1.3 \begin{align*} -{\frac{{x}^{3}}{3} \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 ) - \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}-3\,\sqrt{a{x}^{2}+b}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]

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

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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 = 1.65182, size = 311, normalized size = 5.1 \begin{align*} \left [\frac{1}{2} \, b^{\frac{3}{2}} \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) + \frac{1}{3} \,{\left (a x^{3} + 4 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}, \sqrt{-b} b \arctan \left (\frac{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}{a x^{2} + b}\right ) + \frac{1}{3} \,{\left (a x^{3} + 4 \, b x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}\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]

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

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Sympy [A]  time = 2.40664, size = 78, normalized size = 1.28 \begin{align*} \frac{a \sqrt{b} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3} + \frac{b^{\frac{3}{2}} \log{\left (\frac{a x^{2}}{b} \right )}}{2} - b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a x^{2}}{b} + 1} + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*sqrt(b)*x**2*sqrt(a*x**2/b + 1)/3 + 4*b**(3/2)*sqrt(a*x**2/b + 1)/3 + b**(3/2)*log(a*x**2/b)/2 - b**(3/2)*lo
g(sqrt(a*x**2/b + 1) + 1)

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Giac [A]  time = 1.52073, size = 119, normalized size = 1.95 \begin{align*} \frac{1}{3} \,{\left (\frac{3 \, b^{2} \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} +{\left (a x^{2} + b\right )}^{\frac{3}{2}} + 3 \, \sqrt{a x^{2} + b} b\right )} \mathrm{sgn}\left (x\right ) - \frac{{\left (3 \, b^{2} \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + 4 \, \sqrt{-b} b^{\frac{3}{2}}\right )} \mathrm{sgn}\left (x\right )}{3 \, \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, algorithm="giac")

[Out]

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