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

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 = 0.122652, 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 \[ \frac{b c-a d}{c d x \sqrt{c+\frac{d}{x^2}}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{d^{3/2}} \]

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)

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Rubi in Sympy [A]  time = 12.0733, size = 48, normalized size = 0.81 \[ - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{d^{\frac{3}{2}}} - \frac{a d - b c}{c d x \sqrt{c + \frac{d}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.107482, size = 89, normalized size = 1.51 \[ \frac{\sqrt{d} (b c-a d)+b c \log (x) \sqrt{c x^2+d}-b c \sqrt{c x^2+d} \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )}{c d^{3/2} x \sqrt{c+\frac{d}{x^2}}} \]

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

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Maple [A]  time = 0.016, size = 79, normalized size = 1.3 \[ -{\frac{c{x}^{2}+d}{c{x}^{3}} \left ( a{d}^{{\frac{5}{2}}}-bc{d}^{{\frac{3}{2}}}+b\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) c\sqrt{c{x}^{2}+d}d \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{d}^{-{\frac{5}{2}}}} \]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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 = 0.255286, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left (b c d - a d^{2}\right )} x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (b c^{2} x^{2} + b c d\right )} \sqrt{d} \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right )}{2 \,{\left (c^{2} d^{2} x^{2} + c d^{3}\right )}}, \frac{{\left (b c d - a d^{2}\right )} x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (b c^{2} x^{2} + b c d\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right )}{c^{2} d^{2} x^{2} + c d^{3}}\right ] \]

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)*lo
g((2*d*x*sqrt((c*x^2 + d)/x^2) - (c*x^2 + 2*d)*sqrt(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)*arc
tan(sqrt(-d)/(x*sqrt((c*x^2 + d)/x^2))))/(c^2*d^2*x^2 + c*d^3)]

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Sympy [A]  time = 18.2554, size = 206, normalized size = 3.49 \[ - \frac{a}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + b \left (\frac{c d^{2} x^{2} \log{\left (\frac{c x^{2}}{d} \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} - \frac{2 c d^{2} x^{2} \log{\left (\sqrt{\frac{c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} + \frac{2 d^{3} \sqrt{\frac{c x^{2}}{d} + 1}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} + \frac{d^{3} \log{\left (\frac{c x^{2}}{d} \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} - \frac{2 d^{3} \log{\left (\sqrt{\frac{c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}}\right ) \]

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

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)