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

Optimal. Leaf size=59 \[ \frac{b c-a d}{c d x \sqrt{c+\frac{d}{x^2}}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{d^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0340979, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {452, 335, 217, 206} \[ \frac{b c-a d}{c d x \sqrt{c+\frac{d}{x^2}}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{d^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 452

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

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

Mathematica [A]  time = 0.0318553, size = 71, normalized size = 1.2 \[ \frac{\sqrt{d} (b c-a d)-b c \sqrt{c x^2+d} \tanh ^{-1}\left (\frac{\sqrt{c x^2+d}}{\sqrt{d}}\right )}{c d^{3/2} x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 79, normalized size = 1.3 \begin{align*} -{\frac{c{x}^{2}+d}{{x}^{3}c} \left ( a{d}^{{\frac{5}{2}}}-{d}^{{\frac{3}{2}}}bc+\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) \sqrt{c{x}^{2}+d}bcd \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{d}^{-{\frac{5}{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^2,x)

[Out]

-(c*x^2+d)*(a*d^(5/2)-d^(3/2)*b*c+ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*(c*x^2+d)^(1/2)*b*c*d)/((c*x^2+d)/x^2)^(
3/2)/x^3/c/d^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.08589, size = 424, normalized size = 7.19 \begin{align*} \left [\frac{2 \,{\left (b c d - a d^{2}\right )} x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (b c^{2} x^{2} + b c d\right )} \sqrt{d} \log \left (-\frac{c x^{2} - 2 \, \sqrt{d} x \sqrt{\frac{c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right )}{2 \,{\left (c^{2} d^{2} x^{2} + c d^{3}\right )}}, \frac{{\left (b c d - a d^{2}\right )} x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (b c^{2} x^{2} + b c d\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d} x \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right )}{c^{2} d^{2} x^{2} + c d^{3}}\right ] \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^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 10.8989, size = 206, normalized size = 3.49 \begin{align*} - \frac{a}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + b \left (\frac{c d^{2} x^{2} \log{\left (\frac{c x^{2}}{d} \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} - \frac{2 c d^{2} x^{2} \log{\left (\sqrt{\frac{c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} + \frac{2 d^{3} \sqrt{\frac{c x^{2}}{d} + 1}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} + \frac{d^{3} \log{\left (\frac{c x^{2}}{d} \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} - \frac{2 d^{3} \log{\left (\sqrt{\frac{c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}}\right ) \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**2,x)

[Out]

-a/(c*sqrt(d)*sqrt(c*x**2/d + 1)) + b*(c*d**2*x**2*log(c*x**2/d)/(2*c*d**(7/2)*x**2 + 2*d**(9/2)) - 2*c*d**2*x
**2*log(sqrt(c*x**2/d + 1) + 1)/(2*c*d**(7/2)*x**2 + 2*d**(9/2)) + 2*d**3*sqrt(c*x**2/d + 1)/(2*c*d**(7/2)*x**
2 + 2*d**(9/2)) + d**3*log(c*x**2/d)/(2*c*d**(7/2)*x**2 + 2*d**(9/2)) - 2*d**3*log(sqrt(c*x**2/d + 1) + 1)/(2*
c*d**(7/2)*x**2 + 2*d**(9/2)))

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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}} x^{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^2,x, algorithm="giac")

[Out]

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