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

Optimal. Leaf size=123 \[ -\frac{c^2 (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{16 d^{5/2}}+\frac{c \sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{16 d^2 x}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{8 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3} \]

[Out]

((b*c - 2*a*d)*Sqrt[c + d/x^2])/(8*d*x^3) - (b*(c + d/x^2)^(3/2))/(6*d*x^3) + (c
*(b*c - 2*a*d)*Sqrt[c + d/x^2])/(16*d^2*x) - (c^2*(b*c - 2*a*d)*ArcTanh[Sqrt[d]/
(Sqrt[c + d/x^2]*x)])/(16*d^(5/2))

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Rubi [A]  time = 0.244556, 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 \[ -\frac{c^2 (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{16 d^{5/2}}+\frac{c \sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{16 d^2 x}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-2 a d)}{8 d x^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{6 d x^3} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - 2*a*d)*Sqrt[c + d/x^2])/(8*d*x^3) - (b*(c + d/x^2)^(3/2))/(6*d*x^3) + (c
*(b*c - 2*a*d)*Sqrt[c + d/x^2])/(16*d^2*x) - (c^2*(b*c - 2*a*d)*ArcTanh[Sqrt[d]/
(Sqrt[c + d/x^2]*x)])/(16*d^(5/2))

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Rubi in Sympy [A]  time = 18.3273, size = 107, normalized size = 0.87 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{6 d x^{3}} + \frac{c^{2} \left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{16 d^{\frac{5}{2}}} - \frac{c \sqrt{c + \frac{d}{x^{2}}} \left (2 a d - b c\right )}{16 d^{2} x} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (2 a d - b c\right )}{8 d x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**4,x)

[Out]

-b*(c + d/x**2)**(3/2)/(6*d*x**3) + c**2*(2*a*d - b*c)*atanh(sqrt(d)/(x*sqrt(c +
d/x**2)))/(16*d**(5/2)) - c*sqrt(c + d/x**2)*(2*a*d - b*c)/(16*d**2*x) - sqrt(c
+ d/x**2)*(2*a*d - b*c)/(8*d*x**3)

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Mathematica [A]  time = 0.309756, size = 146, normalized size = 1.19 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (\sqrt{d} \sqrt{c x^2+d} \left (6 a d x^2 \left (c x^2+2 d\right )+b \left (-3 c^2 x^4+2 c d x^2+8 d^2\right )\right )-3 c^2 x^6 \log (x) (b c-2 a d)+3 c^2 x^6 (b c-2 a d) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )\right )}{48 d^{5/2} x^5 \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d/x^2]*(Sqrt[d]*Sqrt[d + c*x^2]*(6*a*d*x^2*(2*d + c*x^2) + b*(8*d^2 +
2*c*d*x^2 - 3*c^2*x^4)) - 3*c^2*(b*c - 2*a*d)*x^6*Log[x] + 3*c^2*(b*c - 2*a*d)*
x^6*Log[d + Sqrt[d]*Sqrt[d + c*x^2]]))/(48*d^(5/2)*x^5*Sqrt[d + c*x^2])

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Maple [B]  time = 0.021, size = 232, normalized size = 1.9 \[ -{\frac{1}{48\,{x}^{5}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 6\,a{c}^{2}\sqrt{c{x}^{2}+d}{x}^{6}{d}^{7/2}-6\,ac \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{4}{d}^{7/2}-3\,b{c}^{3}\sqrt{c{x}^{2}+d}{x}^{6}{d}^{5/2}+12\,a \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}{d}^{9/2}+3\,b{c}^{2} \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{4}{d}^{5/2}-6\,bc \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}{d}^{7/2}-6\,a{c}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{6}{d}^{4}+3\,b{c}^{3}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{6}{d}^{3}+8\,b \left ( c{x}^{2}+d \right ) ^{3/2}{d}^{9/2} \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{d}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*(c+d/x^2)^(1/2)/x^4,x)

[Out]

-1/48*((c*x^2+d)/x^2)^(1/2)*(6*a*c^2*(c*x^2+d)^(1/2)*x^6*d^(7/2)-6*a*c*(c*x^2+d)
^(3/2)*x^4*d^(7/2)-3*b*c^3*(c*x^2+d)^(1/2)*x^6*d^(5/2)+12*a*(c*x^2+d)^(3/2)*x^2*
d^(9/2)+3*b*c^2*(c*x^2+d)^(3/2)*x^4*d^(5/2)-6*b*c*(c*x^2+d)^(3/2)*x^2*d^(7/2)-6*
a*c^2*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*x^6*d^4+3*b*c^3*ln(2*(d^(1/2)*(c*x^2+d
)^(1/2)+d)/x)*x^6*d^3+8*b*(c*x^2+d)^(3/2)*d^(9/2))/x^5/(c*x^2+d)^(1/2)/d^(11/2)

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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)*sqrt(c + d/x^2)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256401, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b c^{3} - 2 \, a c^{2} d\right )} \sqrt{d} x^{5} \log \left (-\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) - 2 \,{\left (3 \,{\left (b c^{2} d - 2 \, a c d^{2}\right )} x^{4} - 8 \, b d^{3} - 2 \,{\left (b c d^{2} + 6 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{96 \, d^{3} x^{5}}, \frac{3 \,{\left (b c^{3} - 2 \, a c^{2} d\right )} \sqrt{-d} x^{5} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (3 \,{\left (b c^{2} d - 2 \, a c d^{2}\right )} x^{4} - 8 \, b d^{3} - 2 \,{\left (b c d^{2} + 6 \, a d^{3}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{48 \, d^{3} x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/96*(3*(b*c^3 - 2*a*c^2*d)*sqrt(d)*x^5*log(-(2*d*x*sqrt((c*x^2 + d)/x^2) + (c
*x^2 + 2*d)*sqrt(d))/x^2) - 2*(3*(b*c^2*d - 2*a*c*d^2)*x^4 - 8*b*d^3 - 2*(b*c*d^
2 + 6*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2))/(d^3*x^5), 1/48*(3*(b*c^3 - 2*a*c^2*d)*
sqrt(-d)*x^5*arctan(sqrt(-d)/(x*sqrt((c*x^2 + d)/x^2))) + (3*(b*c^2*d - 2*a*c*d^
2)*x^4 - 8*b*d^3 - 2*(b*c*d^2 + 6*a*d^3)*x^2)*sqrt((c*x^2 + d)/x^2))/(d^3*x^5)]

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Sympy [A]  time = 16.7221, size = 226, normalized size = 1.84 \[ - \frac{a c^{\frac{3}{2}}}{8 d x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 a \sqrt{c}}{8 x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{a c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 d^{\frac{3}{2}}} - \frac{a d}{4 \sqrt{c} x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c^{\frac{5}{2}}}{16 d^{2} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c^{\frac{3}{2}}}{48 d x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{5 b \sqrt{c}}{24 x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{b c^{3} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{16 d^{\frac{5}{2}}} - \frac{b d}{6 \sqrt{c} x^{7} \sqrt{1 + \frac{d}{c x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**4,x)

[Out]

-a*c**(3/2)/(8*d*x*sqrt(1 + d/(c*x**2))) - 3*a*sqrt(c)/(8*x**3*sqrt(1 + d/(c*x**
2))) + a*c**2*asinh(sqrt(d)/(sqrt(c)*x))/(8*d**(3/2)) - a*d/(4*sqrt(c)*x**5*sqrt
(1 + d/(c*x**2))) + b*c**(5/2)/(16*d**2*x*sqrt(1 + d/(c*x**2))) + b*c**(3/2)/(48
*d*x**3*sqrt(1 + d/(c*x**2))) - 5*b*sqrt(c)/(24*x**5*sqrt(1 + d/(c*x**2))) - b*c
**3*asinh(sqrt(d)/(sqrt(c)*x))/(16*d**(5/2)) - b*d/(6*sqrt(c)*x**7*sqrt(1 + d/(c
*x**2)))

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GIAC/XCAS [A]  time = 0.257215, size = 207, normalized size = 1.68 \[ \frac{\frac{3 \,{\left (b c^{4}{\rm sign}\left (x\right ) - 2 \, a c^{3} d{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right )}{\sqrt{-d} d^{2}} + \frac{3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} b c^{4}{\rm sign}\left (x\right ) - 6 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} a c^{3} d{\rm sign}\left (x\right ) - 8 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b c^{4} d{\rm sign}\left (x\right ) - 3 \, \sqrt{c x^{2} + d} b c^{4} d^{2}{\rm sign}\left (x\right ) + 6 \, \sqrt{c x^{2} + d} a c^{3} d^{3}{\rm sign}\left (x\right )}{c^{3} d^{2} x^{6}}}{48 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*(3*(b*c^4*sign(x) - 2*a*c^3*d*sign(x))*arctan(sqrt(c*x^2 + d)/sqrt(-d))/(sq
rt(-d)*d^2) + (3*(c*x^2 + d)^(5/2)*b*c^4*sign(x) - 6*(c*x^2 + d)^(5/2)*a*c^3*d*s
ign(x) - 8*(c*x^2 + d)^(3/2)*b*c^4*d*sign(x) - 3*sqrt(c*x^2 + d)*b*c^4*d^2*sign(
x) + 6*sqrt(c*x^2 + d)*a*c^3*d^3*sign(x))/(c^3*d^2*x^6))/c