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

Optimal. Leaf size=42 \[ -\frac{b c-a d}{d^2 \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0354627, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {444, 43} \[ -\frac{b c-a d}{d^2 \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^3),x]

[Out]

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

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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

Mathematica [A]  time = 0.0184008, size = 36, normalized size = 0.86 \[ \frac{a d x^2-b \left (2 c x^2+d\right )}{d^2 x^2 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^3),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 46, normalized size = 1.1 \begin{align*}{\frac{ \left ( ad{x}^{2}-2\,bc{x}^{2}-bd \right ) \left ( c{x}^{2}+d \right ) }{{d}^{2}{x}^{4}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x^2)/(c+d/x^2)^(3/2)/x^3,x)

[Out]

(a*d*x^2-2*b*c*x^2-b*d)*(c*x^2+d)/((c*x^2+d)/x^2)^(3/2)/d^2/x^4

________________________________________________________________________________________

Maxima [A]  time = 0.927368, size = 62, normalized size = 1.48 \begin{align*} -b{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{d^{2}} + \frac{c}{\sqrt{c + \frac{d}{x^{2}}} d^{2}}\right )} + \frac{a}{\sqrt{c + \frac{d}{x^{2}}} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*(sqrt(c + d/x^2)/d^2 + c/(sqrt(c + d/x^2)*d^2)) + a/(sqrt(c + d/x^2)*d)

________________________________________________________________________________________

Fricas [A]  time = 1.47852, size = 92, normalized size = 2.19 \begin{align*} -\frac{{\left ({\left (2 \, b c - a d\right )} x^{2} + b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c d^{2} x^{2} + d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 3.39004, size = 68, normalized size = 1.62 \begin{align*} \begin{cases} \frac{a}{d \sqrt{c + \frac{d}{x^{2}}}} - \frac{2 b c}{d^{2} \sqrt{c + \frac{d}{x^{2}}}} - \frac{b}{d x^{2} \sqrt{c + \frac{d}{x^{2}}}} & \text{for}\: d \neq 0 \\\frac{- \frac{a}{2 x^{2}} - \frac{b}{4 x^{4}}}{c^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a/(d*sqrt(c + d/x**2)) - 2*b*c/(d**2*sqrt(c + d/x**2)) - b/(d*x**2*sqrt(c + d/x**2)), Ne(d, 0)), ((
-a/(2*x**2) - b/(4*x**4))/c**(3/2), True))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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