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

Optimal. Leaf size=43 \[ \frac{\sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^2} \]

[Out]

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

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Rubi [A]  time = 0.0356831, antiderivative size = 43, 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} \[ \frac{\sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0251518, size = 39, normalized size = 0.91 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (3 a d x^2+b \left (d-2 c x^2\right )\right )}{3 d^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 47, normalized size = 1.1 \begin{align*} -{\frac{ \left ( 3\,ad{x}^{2}-2\,bc{x}^{2}+bd \right ) \left ( c{x}^{2}+d \right ) }{3\,{d}^{2}{x}^{4}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(3*a*d*x^2-2*b*c*x^2+b*d)*(c*x^2+d)/((c*x^2+d)/x^2)^(1/2)/d^2/x^4

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Maxima [A]  time = 0.929946, size = 65, normalized size = 1.51 \begin{align*} -\frac{1}{3} \, b{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{d^{2}} - \frac{3 \, \sqrt{c + \frac{d}{x^{2}}} c}{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)/x^3/(c+d/x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.32556, size = 88, normalized size = 2.05 \begin{align*} \frac{{\left ({\left (2 \, b c - 3 \, a d\right )} x^{2} - b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \, d^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 5.26124, size = 139, normalized size = 3.23 \begin{align*} - \frac{\begin{cases} \frac{\frac{a}{x^{2}} + \frac{b}{2 x^{4}}}{\sqrt{c}} & \text{for}\: d = 0 \\- \frac{\frac{2 a c}{\sqrt{c + \frac{d}{x^{2}}}} + 2 a \left (- \frac{c}{\sqrt{c + \frac{d}{x^{2}}}} - \sqrt{c + \frac{d}{x^{2}}}\right ) + \frac{2 b c \left (- \frac{c}{\sqrt{c + \frac{d}{x^{2}}}} - \sqrt{c + \frac{d}{x^{2}}}\right )}{d} + \frac{2 b \left (\frac{c^{2}}{\sqrt{c + \frac{d}{x^{2}}}} + 2 c \sqrt{c + \frac{d}{x^{2}}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3}\right )}{d}}{d} & \text{otherwise} \end{cases}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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