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

Optimal. Leaf size=52 \[ \frac{b c-a d}{c d \sqrt{c+\frac{d}{x^2}}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.148333, antiderivative size = 52, 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 \[ \frac{b c-a d}{c d \sqrt{c+\frac{d}{x^2}}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.1826, size = 42, normalized size = 0.81 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{c^{\frac{3}{2}}} - \frac{a d - b c}{c d \sqrt{c + \frac{d}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0669372, size = 75, normalized size = 1.44 \[ \frac{\sqrt{c} x (b c-a d)+a d \sqrt{c x^2+d} \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{c^{3/2} d x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 75, normalized size = 1.4 \[{\frac{c{x}^{2}+d}{d{x}^{3}} \left ( xb{c}^{{\frac{5}{2}}}-axd{c}^{{\frac{3}{2}}}+a\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) d\sqrt{c{x}^{2}+d}c \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x^2+d)*(x*b*c^(5/2)-a*x*d*c^(3/2)+a*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*d*(c*x^2+d)
^(1/2)*c)/((c*x^2+d)/x^2)^(3/2)/x^3/d/c^(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229632, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left (b c^{2} - a c d\right )} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (a c d x^{2} + a d^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right )}{2 \,{\left (c^{3} d x^{2} + c^{2} d^{2}\right )}}, \frac{{\left (b c^{2} - a c d\right )} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (a c d x^{2} + a d^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right )}{c^{3} d x^{2} + c^{2} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 15.6087, size = 218, normalized size = 4.19 \[ a \left (- \frac{2 c^{3} x^{2} \sqrt{1 + \frac{d}{c x^{2}}}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} - \frac{c^{3} x^{2} \log{\left (\frac{d}{c x^{2}} \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} + \frac{2 c^{3} x^{2} \log{\left (\sqrt{1 + \frac{d}{c x^{2}}} + 1 \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} - \frac{c^{2} d \log{\left (\frac{d}{c x^{2}} \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} + \frac{2 c^{2} d \log{\left (\sqrt{1 + \frac{d}{c x^{2}}} + 1 \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d}\right ) + b \left (\begin{cases} \frac{1}{d \sqrt{c + \frac{d}{x^{2}}}} & \text{for}\: d \neq 0 \\- \frac{1}{2 c^{\frac{3}{2}} x^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*(-2*c**3*x**2*sqrt(1 + d/(c*x**2))/(2*c**(9/2)*x**2 + 2*c**(7/2)*d) - c**3*x**
2*log(d/(c*x**2))/(2*c**(9/2)*x**2 + 2*c**(7/2)*d) + 2*c**3*x**2*log(sqrt(1 + d/
(c*x**2)) + 1)/(2*c**(9/2)*x**2 + 2*c**(7/2)*d) - c**2*d*log(d/(c*x**2))/(2*c**(
9/2)*x**2 + 2*c**(7/2)*d) + 2*c**2*d*log(sqrt(1 + d/(c*x**2)) + 1)/(2*c**(9/2)*x
**2 + 2*c**(7/2)*d)) + b*Piecewise((1/(d*sqrt(c + d/x**2)), Ne(d, 0)), (-1/(2*c*
*(3/2)*x**2), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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