3.815 \(\int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2} x^4} \, dx\)

Optimal. Leaf size=92 \[ \frac{(3 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{2 d^{5/2}}-\frac{3 b c-2 a d}{2 d^2 x \sqrt{c+\frac{d}{x^2}}}-\frac{b}{2 d x^3 \sqrt{c+\frac{d}{x^2}}} \]

[Out]

-b/(2*d*Sqrt[c + d/x^2]*x^3) - (3*b*c - 2*a*d)/(2*d^2*Sqrt[c + d/x^2]*x) + ((3*b
*c - 2*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(2*d^(5/2))

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Rubi [A]  time = 0.170773, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{(3 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{2 d^{5/2}}-\frac{3 b c-2 a d}{2 d^2 x \sqrt{c+\frac{d}{x^2}}}-\frac{b}{2 d x^3 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^4),x]

[Out]

-b/(2*d*Sqrt[c + d/x^2]*x^3) - (3*b*c - 2*a*d)/(2*d^2*Sqrt[c + d/x^2]*x) + ((3*b
*c - 2*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(2*d^(5/2))

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Rubi in Sympy [A]  time = 14.2511, size = 80, normalized size = 0.87 \[ - \frac{b}{2 d x^{3} \sqrt{c + \frac{d}{x^{2}}}} + \frac{2 a d - 3 b c}{2 d^{2} x \sqrt{c + \frac{d}{x^{2}}}} - \frac{\left (2 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{2 d^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**4,x)

[Out]

-b/(2*d*x**3*sqrt(c + d/x**2)) + (2*a*d - 3*b*c)/(2*d**2*x*sqrt(c + d/x**2)) - (
2*a*d - 3*b*c)*atanh(sqrt(d)/(x*sqrt(c + d/x**2)))/(2*d**(5/2))

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Mathematica [A]  time = 0.164244, size = 119, normalized size = 1.29 \[ \frac{\sqrt{d} \left (2 a d x^2-b \left (3 c x^2+d\right )\right )+x^2 \log (x) \sqrt{c x^2+d} (2 a d-3 b c)+x^2 \sqrt{c x^2+d} (3 b c-2 a d) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )}{2 d^{5/2} x^3 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^4),x]

[Out]

(Sqrt[d]*(2*a*d*x^2 - b*(d + 3*c*x^2)) + (-3*b*c + 2*a*d)*x^2*Sqrt[d + c*x^2]*Lo
g[x] + (3*b*c - 2*a*d)*x^2*Sqrt[d + c*x^2]*Log[d + Sqrt[d]*Sqrt[d + c*x^2]])/(2*
d^(5/2)*Sqrt[c + d/x^2]*x^3)

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Maple [A]  time = 0.017, size = 134, normalized size = 1.5 \[{\frac{c{x}^{2}+d}{2\,{x}^{5}} \left ( 2\,a{d}^{7/2}{x}^{2}-3\,bc{d}^{5/2}{x}^{2}-2\,a\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){d}^{3}\sqrt{c{x}^{2}+d}{x}^{2}+3\,bc\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){d}^{2}\sqrt{c{x}^{2}+d}{x}^{2}-b{d}^{{\frac{7}{2}}} \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{d}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/(c+d/x^2)^(3/2)/x^4,x)

[Out]

1/2*(c*x^2+d)*(2*a*d^(7/2)*x^2-3*b*c*d^(5/2)*x^2-2*a*ln(2*(d^(1/2)*(c*x^2+d)^(1/
2)+d)/x)*d^3*(c*x^2+d)^(1/2)*x^2+3*b*c*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*d^2*(
c*x^2+d)^(1/2)*x^2-b*d^(7/2))/((c*x^2+d)/x^2)^(3/2)/x^5/d^(9/2)

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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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259195, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (3 \, b c^{2} - 2 \, a c d\right )} x^{3} +{\left (3 \, b c d - 2 \, a d^{2}\right )} x\right )} \sqrt{d} \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) + 2 \,{\left (b d^{2} +{\left (3 \, b c d - 2 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{4 \,{\left (c d^{3} x^{3} + d^{4} x\right )}}, -\frac{{\left ({\left (3 \, b c^{2} - 2 \, a c d\right )} x^{3} +{\left (3 \, b c d - 2 \, a d^{2}\right )} x\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (b d^{2} +{\left (3 \, b c d - 2 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{2 \,{\left (c d^{3} x^{3} + d^{4} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^4),x, algorithm="fricas")

[Out]

[-1/4*(((3*b*c^2 - 2*a*c*d)*x^3 + (3*b*c*d - 2*a*d^2)*x)*sqrt(d)*log((2*d*x*sqrt
((c*x^2 + d)/x^2) - (c*x^2 + 2*d)*sqrt(d))/x^2) + 2*(b*d^2 + (3*b*c*d - 2*a*d^2)
*x^2)*sqrt((c*x^2 + d)/x^2))/(c*d^3*x^3 + d^4*x), -1/2*(((3*b*c^2 - 2*a*c*d)*x^3
+ (3*b*c*d - 2*a*d^2)*x)*sqrt(-d)*arctan(sqrt(-d)/(x*sqrt((c*x^2 + d)/x^2))) +
(b*d^2 + (3*b*c*d - 2*a*d^2)*x^2)*sqrt((c*x^2 + d)/x^2))/(c*d^3*x^3 + d^4*x)]

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Sympy [A]  time = 30.6519, size = 262, normalized size = 2.85 \[ a \left (\frac{c d^{2} x^{2} \log{\left (\frac{c x^{2}}{d} \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} - \frac{2 c d^{2} x^{2} \log{\left (\sqrt{\frac{c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} + \frac{2 d^{3} \sqrt{\frac{c x^{2}}{d} + 1}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} + \frac{d^{3} \log{\left (\frac{c x^{2}}{d} \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} - \frac{2 d^{3} \log{\left (\sqrt{\frac{c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}}\right ) + b \left (- \frac{3 \sqrt{c}}{2 d^{2} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{3 c \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2 d^{\frac{5}{2}}} - \frac{1}{2 \sqrt{c} d x^{3} \sqrt{1 + \frac{d}{c x^{2}}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**4,x)

[Out]

a*(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*lo
g(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))) + b*(-3*sqrt(c)/(2*d**2*x*sqrt(1 + d/(c*x**2))) + 3*c*asinh(sqrt(d)/(sqrt(c
)*x))/(2*d**(5/2)) - 1/(2*sqrt(c)*d*x**3*sqrt(1 + d/(c*x**2))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^4),x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^4), x)