3.816 \(\int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2} x^6} \, dx\)

Optimal. Leaf size=123 \[ -\frac{3 c (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{7/2}}+\frac{3 \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{8 d^3 x}-\frac{5 b c-4 a d}{4 d^2 x^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b}{4 d x^5 \sqrt{c+\frac{d}{x^2}}} \]

[Out]

-b/(4*d*Sqrt[c + d/x^2]*x^5) - (5*b*c - 4*a*d)/(4*d^2*Sqrt[c + d/x^2]*x^3) + (3*
(5*b*c - 4*a*d)*Sqrt[c + d/x^2])/(8*d^3*x) - (3*c*(5*b*c - 4*a*d)*ArcTanh[Sqrt[d
]/(Sqrt[c + d/x^2]*x)])/(8*d^(7/2))

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Rubi [A]  time = 0.224599, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ -\frac{3 c (5 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{7/2}}+\frac{3 \sqrt{c+\frac{d}{x^2}} (5 b c-4 a d)}{8 d^3 x}-\frac{5 b c-4 a d}{4 d^2 x^3 \sqrt{c+\frac{d}{x^2}}}-\frac{b}{4 d x^5 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x]

[Out]

-b/(4*d*Sqrt[c + d/x^2]*x^5) - (5*b*c - 4*a*d)/(4*d^2*Sqrt[c + d/x^2]*x^3) + (3*
(5*b*c - 4*a*d)*Sqrt[c + d/x^2])/(8*d^3*x) - (3*c*(5*b*c - 4*a*d)*ArcTanh[Sqrt[d
]/(Sqrt[c + d/x^2]*x)])/(8*d^(7/2))

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Rubi in Sympy [A]  time = 18.7608, size = 114, normalized size = 0.93 \[ - \frac{b}{4 d x^{5} \sqrt{c + \frac{d}{x^{2}}}} + \frac{3 c \left (4 a d - 5 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{8 d^{\frac{7}{2}}} + \frac{4 a d - 5 b c}{4 d^{2} x^{3} \sqrt{c + \frac{d}{x^{2}}}} - \frac{3 \sqrt{c + \frac{d}{x^{2}}} \left (4 a d - 5 b c\right )}{8 d^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**6,x)

[Out]

-b/(4*d*x**5*sqrt(c + d/x**2)) + 3*c*(4*a*d - 5*b*c)*atanh(sqrt(d)/(x*sqrt(c + d
/x**2)))/(8*d**(7/2)) + (4*a*d - 5*b*c)/(4*d**2*x**3*sqrt(c + d/x**2)) - 3*sqrt(
c + d/x**2)*(4*a*d - 5*b*c)/(8*d**3*x)

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Mathematica [A]  time = 0.216183, size = 143, normalized size = 1.16 \[ \frac{\sqrt{d} \left (b \left (15 c^2 x^4+5 c d x^2-2 d^2\right )-4 a d x^2 \left (3 c x^2+d\right )\right )+3 c x^4 \log (x) \sqrt{c x^2+d} (5 b c-4 a d)-3 c x^4 \sqrt{c x^2+d} (5 b c-4 a d) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )}{8 d^{7/2} x^5 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x]

[Out]

(Sqrt[d]*(-4*a*d*x^2*(d + 3*c*x^2) + b*(-2*d^2 + 5*c*d*x^2 + 15*c^2*x^4)) + 3*c*
(5*b*c - 4*a*d)*x^4*Sqrt[d + c*x^2]*Log[x] - 3*c*(5*b*c - 4*a*d)*x^4*Sqrt[d + c*
x^2]*Log[d + Sqrt[d]*Sqrt[d + c*x^2]])/(8*d^(7/2)*Sqrt[c + d/x^2]*x^5)

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Maple [A]  time = 0.019, size = 159, normalized size = 1.3 \[ -{\frac{c{x}^{2}+d}{8\,{x}^{7}} \left ( 12\,ac{d}^{9/2}{x}^{4}-15\,b{c}^{2}{x}^{4}{d}^{7/2}-12\,ac\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){d}^{4}{x}^{4}\sqrt{c{x}^{2}+d}+4\,a{d}^{11/2}{x}^{2}-5\,bc{d}^{9/2}{x}^{2}+15\,b{c}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){d}^{3}{x}^{4}\sqrt{c{x}^{2}+d}+2\,b{d}^{11/2} \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{d}^{-{\frac{13}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/(c+d/x^2)^(3/2)/x^6,x)

[Out]

-1/8*(c*x^2+d)*(12*a*c*d^(9/2)*x^4-15*b*c^2*x^4*d^(7/2)-12*a*c*ln(2*(d^(1/2)*(c*
x^2+d)^(1/2)+d)/x)*d^4*x^4*(c*x^2+d)^(1/2)+4*a*d^(11/2)*x^2-5*b*c*d^(9/2)*x^2+15
*b*c^2*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*d^3*x^4*(c*x^2+d)^(1/2)+2*b*d^(11/2))
/((c*x^2+d)/x^2)^(3/2)/x^7/d^(13/2)

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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^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.254471, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (5 \, b c^{3} - 4 \, a c^{2} d\right )} x^{5} +{\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{3}\right )} \sqrt{d} \log \left (-\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{4} - 2 \, b d^{3} +{\left (5 \, b c d^{2} - 4 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \,{\left (c d^{4} x^{5} + d^{5} x^{3}\right )}}, \frac{3 \,{\left ({\left (5 \, b c^{3} - 4 \, a c^{2} d\right )} x^{5} +{\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{3}\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (3 \,{\left (5 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{4} - 2 \, b d^{3} +{\left (5 \, b c d^{2} - 4 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \,{\left (c d^{4} x^{5} + d^{5} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x, algorithm="fricas")

[Out]

[-1/16*(3*((5*b*c^3 - 4*a*c^2*d)*x^5 + (5*b*c^2*d - 4*a*c*d^2)*x^3)*sqrt(d)*log(
-(2*d*x*sqrt((c*x^2 + d)/x^2) + (c*x^2 + 2*d)*sqrt(d))/x^2) - 2*(3*(5*b*c^2*d -
4*a*c*d^2)*x^4 - 2*b*d^3 + (5*b*c*d^2 - 4*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2))/(c*
d^4*x^5 + d^5*x^3), 1/8*(3*((5*b*c^3 - 4*a*c^2*d)*x^5 + (5*b*c^2*d - 4*a*c*d^2)*
x^3)*sqrt(-d)*arctan(sqrt(-d)/(x*sqrt((c*x^2 + d)/x^2))) + (3*(5*b*c^2*d - 4*a*c
*d^2)*x^4 - 2*b*d^3 + (5*b*c*d^2 - 4*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2))/(c*d^4*x
^5 + d^5*x^3)]

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Sympy [A]  time = 53.3011, size = 180, normalized size = 1.46 \[ a \left (- \frac{3 \sqrt{c}}{2 d^{2} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{3 c \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2 d^{\frac{5}{2}}} - \frac{1}{2 \sqrt{c} d x^{3} \sqrt{1 + \frac{d}{c x^{2}}}}\right ) + b \left (\frac{15 c^{\frac{3}{2}}}{8 d^{3} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{5 \sqrt{c}}{8 d^{2} x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{15 c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 d^{\frac{7}{2}}} - \frac{1}{4 \sqrt{c} d x^{5} \sqrt{1 + \frac{d}{c x^{2}}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**6,x)

[Out]

a*(-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)))) + b*(15*c**(3/2)/(8*d**
3*x*sqrt(1 + d/(c*x**2))) + 5*sqrt(c)/(8*d**2*x**3*sqrt(1 + d/(c*x**2))) - 15*c*
*2*asinh(sqrt(d)/(sqrt(c)*x))/(8*d**(7/2)) - 1/(4*sqrt(c)*d*x**5*sqrt(1 + d/(c*x
**2))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^6),x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^6), x)