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

Optimal. Leaf size=45 \[ \frac{a x}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b c-2 a d}{c^2 x \sqrt{c+\frac{d}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.0299469, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {375, 453, 191} \[ \frac{a x}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b c-2 a d}{c^2 x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/(c + d/x^2)^(3/2),x]

[Out]

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

Rule 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 453

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1))/(a*e*(m + 1)), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

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

Mathematica [A]  time = 0.0227865, size = 33, normalized size = 0.73 \[ \frac{a c x^2+2 a d-b c}{c^2 x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 43, normalized size = 1. \begin{align*}{\frac{ \left ( a{x}^{2}c+2\,ad-bc \right ) \left ( c{x}^{2}+d \right ) }{{x}^{3}{c}^{2}} \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)

[Out]

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

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Maxima [A]  time = 0.943443, size = 72, normalized size = 1.6 \begin{align*} a{\left (\frac{\sqrt{c + \frac{d}{x^{2}}} x}{c^{2}} + \frac{d}{\sqrt{c + \frac{d}{x^{2}}} c^{2} x}\right )} - \frac{b}{\sqrt{c + \frac{d}{x^{2}}} c 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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.28034, size = 93, normalized size = 2.07 \begin{align*} \frac{{\left (a c x^{3} -{\left (b c - 2 \, a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c^{3} x^{2} + c^{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, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.42385, size = 65, normalized size = 1.44 \begin{align*} a \left (\frac{x^{2}}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{2 \sqrt{d}}{c^{2} \sqrt{\frac{c x^{2}}{d} + 1}}\right ) - \frac{b}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} \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)

[Out]

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

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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}}}\,{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, algorithm="giac")

[Out]

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