3.1949 \(\int \frac{1}{(a+\frac{b}{x^2})^{5/2} x^7} \, dx\)

Optimal. Leaf size=55 \[ \frac{a^2}{3 b^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{2 a}{b^3 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0293698, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a^2}{3 b^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{2 a}{b^3 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^(5/2)*x^7),x]

[Out]

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

Rule 266

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^{5/2} x^7} \, dx &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{5/2}} \, dx,x,\frac{1}{x^2}\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{5/2}}-\frac{2 a}{b^2 (a+b x)^{3/2}}+\frac{1}{b^2 \sqrt{a+b x}}\right ) \, dx,x,\frac{1}{x^2}\right )\right )\\ &=\frac{a^2}{3 b^3 \left (a+\frac{b}{x^2}\right )^{3/2}}-\frac{2 a}{b^3 \sqrt{a+\frac{b}{x^2}}}-\frac{\sqrt{a+\frac{b}{x^2}}}{b^3}\\ \end{align*}

Mathematica [A]  time = 0.0188686, size = 48, normalized size = 0.87 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (8 a^2 x^4+12 a b x^2+3 b^2\right )}{3 b^3 \left (a x^2+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^(5/2)*x^7),x]

[Out]

-(Sqrt[a + b/x^2]*(3*b^2 + 12*a*b*x^2 + 8*a^2*x^4))/(3*b^3*(b + a*x^2)^2)

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Maple [A]  time = 0.004, size = 50, normalized size = 0.9 \begin{align*} -{\frac{ \left ( a{x}^{2}+b \right ) \left ( 8\,{a}^{2}{x}^{4}+12\,ab{x}^{2}+3\,{b}^{2} \right ) }{3\,{b}^{3}{x}^{6}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^(5/2)/x^7,x)

[Out]

-1/3*(a*x^2+b)*(8*a^2*x^4+12*a*b*x^2+3*b^2)/x^6/b^3/((a*x^2+b)/x^2)^(5/2)

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Maxima [A]  time = 1.00109, size = 63, normalized size = 1.15 \begin{align*} -\frac{\sqrt{a + \frac{b}{x^{2}}}}{b^{3}} - \frac{2 \, a}{\sqrt{a + \frac{b}{x^{2}}} b^{3}} + \frac{a^{2}}{3 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.52117, size = 128, normalized size = 2.33 \begin{align*} -\frac{{\left (8 \, a^{2} x^{4} + 12 \, a b x^{2} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a^{2} b^{3} x^{4} + 2 \, a b^{4} x^{2} + b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^7,x, algorithm="fricas")

[Out]

-1/3*(8*a^2*x^4 + 12*a*b*x^2 + 3*b^2)*sqrt((a*x^2 + b)/x^2)/(a^2*b^3*x^4 + 2*a*b^4*x^2 + b^5)

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Sympy [A]  time = 11.2919, size = 153, normalized size = 2.78 \begin{align*} \begin{cases} - \frac{8 a^{2} x^{4}}{3 a b^{3} x^{4} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{4} x^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{12 a b x^{2}}{3 a b^{3} x^{4} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{4} x^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{3 b^{2}}{3 a b^{3} x^{4} \sqrt{a + \frac{b}{x^{2}}} + 3 b^{4} x^{2} \sqrt{a + \frac{b}{x^{2}}}} & \text{for}\: b \neq 0 \\- \frac{1}{6 a^{\frac{5}{2}} x^{6}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**(5/2)/x**7,x)

[Out]

Piecewise((-8*a**2*x**4/(3*a*b**3*x**4*sqrt(a + b/x**2) + 3*b**4*x**2*sqrt(a + b/x**2)) - 12*a*b*x**2/(3*a*b**
3*x**4*sqrt(a + b/x**2) + 3*b**4*x**2*sqrt(a + b/x**2)) - 3*b**2/(3*a*b**3*x**4*sqrt(a + b/x**2) + 3*b**4*x**2
*sqrt(a + b/x**2)), Ne(b, 0)), (-1/(6*a**(5/2)*x**6), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(5/2)/x^7,x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(5/2)*x^7), x)