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

Optimal. Leaf size=86 \[ \frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{5/2}}-\frac{2 b c-3 a d}{2 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^2}{2 c \sqrt{c+\frac{d}{x^2}}} \]

[Out]

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

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Rubi [A]  time = 0.189811, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(2 b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{5/2}}-\frac{2 b c-3 a d}{2 c^2 \sqrt{c+\frac{d}{x^2}}}+\frac{a x^2}{2 c \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.7331, size = 73, normalized size = 0.85 \[ \frac{a x^{2}}{2 c \sqrt{c + \frac{d}{x^{2}}}} + \frac{\frac{3 a d}{2} - b c}{c^{2} \sqrt{c + \frac{d}{x^{2}}}} - \frac{\left (\frac{3 a d}{2} - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.105913, size = 89, normalized size = 1.03 \[ \frac{\sqrt{c} x \left (a c x^2+3 a d-2 b c\right )+\sqrt{c x^2+d} (2 b c-3 a d) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{2 c^{5/2} x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 116, normalized size = 1.4 \[{\frac{c{x}^{2}+d}{2\,{x}^{3}} \left ({x}^{3}a{c}^{{\frac{7}{2}}}+3\,adx{c}^{5/2}-2\,xb{c}^{7/2}+2\,b\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{3}\sqrt{c{x}^{2}+d}-3\,ad\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2}\sqrt{c{x}^{2}+d} \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(c*x^2+d)*(x^3*a*c^(7/2)+3*a*d*x*c^(5/2)-2*x*b*c^(7/2)+2*b*ln(c^(1/2)*x+(c*x
^2+d)^(1/2))*c^3*(c*x^2+d)^(1/2)-3*a*d*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*c^2*(c*x^2+
d)^(1/2))/((c*x^2+d)/x^2)^(3/2)/x^3/c^(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)*x/(c + d/x^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238511, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, b c d - 3 \, a d^{2} +{\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{c} \log \left (2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \,{\left (a c^{2} x^{4} -{\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{4 \,{\left (c^{4} x^{2} + c^{3} d\right )}}, -\frac{{\left (2 \, b c d - 3 \, a d^{2} +{\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (a c^{2} x^{4} -{\left (2 \, b c^{2} - 3 \, a c d\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{2 \,{\left (c^{4} x^{2} + c^{3} d\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 16.3772, size = 264, normalized size = 3.07 \[ a \left (\frac{x^{3}}{2 c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + \frac{3 \sqrt{d} x}{2 c^{2} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{3 d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 c^{\frac{5}{2}}}\right ) + b \left (- \frac{2 c^{3} x^{2} \sqrt{1 + \frac{d}{c x^{2}}}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} - \frac{c^{3} x^{2} \log{\left (\frac{d}{c x^{2}} \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} + \frac{2 c^{3} x^{2} \log{\left (\sqrt{1 + \frac{d}{c x^{2}}} + 1 \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} - \frac{c^{2} d \log{\left (\frac{d}{c x^{2}} \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} + \frac{2 c^{2} d \log{\left (\sqrt{1 + \frac{d}{c x^{2}}} + 1 \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.249726, size = 182, normalized size = 2.12 \[ -\frac{1}{2} \, d{\left (\frac{{\left (2 \, b c - 3 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{c x^{2} + d}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{2} d} + \frac{2 \, b c^{2} - 2 \, a c d - \frac{2 \,{\left (c x^{2} + d\right )} b c}{x^{2}} + \frac{3 \,{\left (c x^{2} + d\right )} a d}{x^{2}}}{{\left (c \sqrt{\frac{c x^{2} + d}{x^{2}}} - \frac{{\left (c x^{2} + d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{x^{2}}\right )} c^{2} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*d*((2*b*c - 3*a*d)*arctan(sqrt((c*x^2 + d)/x^2)/sqrt(-c))/(sqrt(-c)*c^2*d)
+ (2*b*c^2 - 2*a*c*d - 2*(c*x^2 + d)*b*c/x^2 + 3*(c*x^2 + d)*a*d/x^2)/((c*sqrt((
c*x^2 + d)/x^2) - (c*x^2 + d)*sqrt((c*x^2 + d)/x^2)/x^2)*c^2*d))