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

Optimal. Leaf size=76 \[ a c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )-\frac{1}{3} a \left (c+\frac{d}{x^2}\right )^{3/2}-a c \sqrt{c+\frac{d}{x^2}}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d} \]

[Out]

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

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Rubi [A]  time = 0.174141, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ a c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )-\frac{1}{3} a \left (c+\frac{d}{x^2}\right )^{3/2}-a c \sqrt{c+\frac{d}{x^2}}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.6129, size = 65, normalized size = 0.86 \[ a c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )} - a c \sqrt{c + \frac{d}{x^{2}}} - \frac{a \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.20923, size = 92, normalized size = 1.21 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (\frac{15 a c^{3/2} x^5 \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{\sqrt{c x^2+d}}-5 a x^2 \left (4 c x^2+d\right )-\frac{3 b \left (c x^2+d\right )^2}{d}\right )}{15 x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.023, size = 142, normalized size = 1.9 \[{\frac{1}{15\,{x}^{2}{d}^{2}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{{\frac{3}{2}}} \left ( 15\,a{c}^{3/2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){x}^{5}{d}^{2}+10\,a{c}^{2}{x}^{6} \left ( c{x}^{2}+d \right ) ^{3/2}-10\,ac \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{4}+15\,a{c}^{2}{x}^{6}\sqrt{c{x}^{2}+d}d-5\,a \left ( c{x}^{2}+d \right ) ^{5/2}{x}^{2}d-3\,b \left ( c{x}^{2}+d \right ) ^{5/2}d \right ) \left ( c{x}^{2}+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*((c*x^2+d)/x^2)^(3/2)/x^2*(15*a*c^(3/2)*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*x^5*d
^2+10*a*c^2*x^6*(c*x^2+d)^(3/2)-10*a*c*(c*x^2+d)^(5/2)*x^4+15*a*c^2*x^6*(c*x^2+d
)^(1/2)*d-5*a*(c*x^2+d)^(5/2)*x^2*d-3*b*(c*x^2+d)^(5/2)*d)/(c*x^2+d)^(3/2)/d^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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248451, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a c^{\frac{3}{2}} d x^{4} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - d\right ) - 2 \,{\left ({\left (3 \, b c^{2} + 20 \, a c d\right )} x^{4} + 3 \, b d^{2} +{\left (6 \, b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{30 \, d x^{4}}, \frac{15 \, a \sqrt{-c} c d x^{4} \arctan \left (\frac{c}{\sqrt{-c} \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left ({\left (3 \, b c^{2} + 20 \, a c d\right )} x^{4} + 3 \, b d^{2} +{\left (6 \, b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \, d x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/30*(15*a*c^(3/2)*d*x^4*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d
) - 2*((3*b*c^2 + 20*a*c*d)*x^4 + 3*b*d^2 + (6*b*c*d + 5*a*d^2)*x^2)*sqrt((c*x^2
+ d)/x^2))/(d*x^4), 1/15*(15*a*sqrt(-c)*c*d*x^4*arctan(c/(sqrt(-c)*sqrt((c*x^2
+ d)/x^2))) - ((3*b*c^2 + 20*a*c*d)*x^4 + 3*b*d^2 + (6*b*c*d + 5*a*d^2)*x^2)*sqr
t((c*x^2 + d)/x^2))/(d*x^4)]

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Sympy [A]  time = 13.4561, size = 134, normalized size = 1.76 \[ a c^{2} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + \frac{d}{x^{2}} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + \frac{d}{x^{2}} \wedge - c < 0 \end{cases}\right ) - a c \sqrt{c + \frac{d}{x^{2}}} - \frac{a \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**2*Piecewise((-atan(sqrt(c + d/x**2)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sqr
t(c + d/x**2)/sqrt(c))/sqrt(c), (-c < 0) & (c < c + d/x**2)), (atanh(sqrt(c + d/
x**2)/sqrt(c))/sqrt(c), (-c < 0) & (c > c + d/x**2))) - a*c*sqrt(c + d/x**2) - a
*(c + d/x**2)**(3/2)/3 - b*(c + d/x**2)**(5/2)/(5*d)

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GIAC/XCAS [A]  time = 0.504302, size = 343, normalized size = 4.51 \[ -\frac{1}{2} \, a c^{\frac{3}{2}}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} b c^{\frac{5}{2}}{\rm sign}\left (x\right ) + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{3}{2}} d{\rm sign}\left (x\right ) - 90 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{3}{2}} d^{2}{\rm sign}\left (x\right ) + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{5}{2}} d^{2}{\rm sign}\left (x\right ) + 110 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{3}{2}} d^{3}{\rm sign}\left (x\right ) - 70 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{3}{2}} d^{4}{\rm sign}\left (x\right ) + 3 \, b c^{\frac{5}{2}} d^{4}{\rm sign}\left (x\right ) + 20 \, a c^{\frac{3}{2}} d^{5}{\rm sign}\left (x\right )\right )}}{15 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a*c^(3/2)*ln((sqrt(c)*x - sqrt(c*x^2 + d))^2)*sign(x) + 2/15*(15*(sqrt(c)*x
- sqrt(c*x^2 + d))^8*b*c^(5/2)*sign(x) + 30*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c
^(3/2)*d*sign(x) - 90*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(3/2)*d^2*sign(x) + 30
*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(5/2)*d^2*sign(x) + 110*(sqrt(c)*x - sqrt(c
*x^2 + d))^4*a*c^(3/2)*d^3*sign(x) - 70*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(3/2
)*d^4*sign(x) + 3*b*c^(5/2)*d^4*sign(x) + 20*a*c^(3/2)*d^5*sign(x))/((sqrt(c)*x
- sqrt(c*x^2 + d))^2 - d)^5