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

Optimal. Leaf size=112 \[ -\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (4 a d+b c)}{4 c x}-\frac{3 \sqrt{c+\frac{d}{x^2}} (4 a d+b c)}{8 x}-\frac{3 c (4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 \sqrt{d}}+\frac{a x \left (c+\frac{d}{x^2}\right )^{5/2}}{c} \]

[Out]

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

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Rubi [A]  time = 0.179567, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{\left (c+\frac{d}{x^2}\right )^{3/2} (4 a d+b c)}{4 c x}-\frac{3 \sqrt{c+\frac{d}{x^2}} (4 a d+b c)}{8 x}-\frac{3 c (4 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 \sqrt{d}}+\frac{a x \left (c+\frac{d}{x^2}\right )^{5/2}}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.356, size = 100, normalized size = 0.89 \[ \frac{a x \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{c} - \frac{3 c \left (4 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{8 \sqrt{d}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (\frac{3 a d}{2} + \frac{3 b c}{8}\right )}{x} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (4 a d + b c\right )}{4 c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.194194, size = 129, normalized size = 1.15 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (3 c x^4 \log (x) (4 a d+b c)-\sqrt{d} \sqrt{c x^2+d} \left (-8 a c x^4+4 a d x^2+5 b c x^2+2 b d\right )-3 c x^4 (4 a d+b c) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )\right )}{8 \sqrt{d} x^3 \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 227, normalized size = 2. \[{\frac{1}{8\,x} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 12\,ac\sqrt{c{x}^{2}+d}{x}^{4}{d}^{5/2}+4\,ac \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{4}{d}^{3/2}-4\,a \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{2}{d}^{3/2}+3\,b{c}^{2}\sqrt{c{x}^{2}+d}{x}^{4}{d}^{3/2}+b{c}^{2} \left ( c{x}^{2}+d \right ) ^{{\frac{3}{2}}}{x}^{4}\sqrt{d}-bc \left ( c{x}^{2}+d \right ) ^{{\frac{5}{2}}}{x}^{2}\sqrt{d}-12\,ac\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{3}-3\,b{c}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{2}-2\,b \left ( c{x}^{2}+d \right ) ^{5/2}{d}^{3/2} \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}{d}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*((c*x^2+d)/x^2)^(3/2)*(12*a*c*(c*x^2+d)^(1/2)*x^4*d^(5/2)+4*a*c*(c*x^2+d)^(3
/2)*x^4*d^(3/2)-4*a*(c*x^2+d)^(5/2)*x^2*d^(3/2)+3*b*c^2*(c*x^2+d)^(1/2)*x^4*d^(3
/2)+b*c^2*(c*x^2+d)^(3/2)*x^4*d^(1/2)-b*c*(c*x^2+d)^(5/2)*x^2*d^(1/2)-12*a*c*ln(
2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*x^4*d^3-3*b*c^2*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d
)/x)*x^4*d^2-2*b*(c*x^2+d)^(5/2)*d^(3/2))/x/(c*x^2+d)^(3/2)/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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23879, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b c^{2} + 4 \, a c d\right )} \sqrt{d} x^{3} \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) + 2 \,{\left (8 \, a c d x^{4} - 2 \, b d^{2} -{\left (5 \, b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, d x^{3}}, \frac{3 \,{\left (b c^{2} + 4 \, a c d\right )} \sqrt{-d} x^{3} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (8 \, a c d x^{4} - 2 \, b d^{2} -{\left (5 \, b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, d x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 16.0014, size = 216, normalized size = 1.93 \[ \frac{a c^{\frac{3}{2}} x}{\sqrt{1 + \frac{d}{c x^{2}}}} - \frac{a \sqrt{c} d \sqrt{1 + \frac{d}{c x^{2}}}}{2 x} + \frac{a \sqrt{c} d}{x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 a c \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2} - \frac{b c^{\frac{3}{2}} \sqrt{1 + \frac{d}{c x^{2}}}}{2 x} - \frac{b c^{\frac{3}{2}}}{8 x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 b \sqrt{c} d}{8 x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 \sqrt{d}} - \frac{b d^{2}}{4 \sqrt{c} x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.253282, size = 196, normalized size = 1.75 \[ \frac{8 \, \sqrt{c x^{2} + d} a c^{2}{\rm sign}\left (x\right ) + \frac{3 \,{\left (b c^{3}{\rm sign}\left (x\right ) + 4 \, a c^{2} d{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right )}{\sqrt{-d}} - \frac{5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b c^{3}{\rm sign}\left (x\right ) + 4 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} a c^{2} d{\rm sign}\left (x\right ) - 3 \, \sqrt{c x^{2} + d} b c^{3} d{\rm sign}\left (x\right ) - 4 \, \sqrt{c x^{2} + d} a c^{2} d^{2}{\rm sign}\left (x\right )}{c^{2} x^{4}}}{8 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(8*sqrt(c*x^2 + d)*a*c^2*sign(x) + 3*(b*c^3*sign(x) + 4*a*c^2*d*sign(x))*arc
tan(sqrt(c*x^2 + d)/sqrt(-d))/sqrt(-d) - (5*(c*x^2 + d)^(3/2)*b*c^3*sign(x) + 4*
(c*x^2 + d)^(3/2)*a*c^2*d*sign(x) - 3*sqrt(c*x^2 + d)*b*c^3*d*sign(x) - 4*sqrt(c
*x^2 + d)*a*c^2*d^2*sign(x))/(c^2*x^4))/c