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

Optimal. Leaf size=59 \[ -a \sqrt{c+\frac{d}{x^2}}+a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.148087, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -a \sqrt{c+\frac{d}{x^2}}+a \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.399, size = 49, normalized size = 0.83 \[ a \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )} - a \sqrt{c + \frac{d}{x^{2}}} - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.129236, size = 82, normalized size = 1.39 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (\frac{3 a \sqrt{c} x^3 \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{\sqrt{c x^2+d}}-3 a x^2-\frac{b c x^2}{d}-b\right )}{3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 101, normalized size = 1.7 \[{\frac{1}{3\,d{x}^{2}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 3\,a\sqrt{c}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){x}^{3}d+3\,ac{x}^{4}\sqrt{c{x}^{2}+d}-3\,a \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}-b \left ( c{x}^{2}+d \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*((c*x^2+d)/x^2)^(1/2)/x^2*(3*a*c^(1/2)*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*x^3*d+3
*a*c*x^4*(c*x^2+d)^(1/2)-3*a*(c*x^2+d)^(3/2)*x^2-b*(c*x^2+d)^(3/2))/(c*x^2+d)^(1
/2)/d

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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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23323, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, a \sqrt{c} d x^{2} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} - d\right ) - 2 \,{\left ({\left (b c + 3 \, a d\right )} x^{2} + b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{6 \, d x^{2}}, \frac{3 \, a \sqrt{-c} d x^{2} \arctan \left (\frac{c}{\sqrt{-c} \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left ({\left (b c + 3 \, a d\right )} x^{2} + b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \, d x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.52426, size = 117, normalized size = 1.98 \[ a c \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{- c}} \right )}}{\sqrt{- c}} & \text{for}\: - c > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: - c < 0 \wedge c < c + \frac{d}{x^{2}} \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{\sqrt{c}} & \text{for}\: c > c + \frac{d}{x^{2}} \wedge - c < 0 \end{cases}\right ) - a \sqrt{c + \frac{d}{x^{2}}} - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c*Piecewise((-atan(sqrt(c + d/x**2)/sqrt(-c))/sqrt(-c), -c > 0), (acoth(sqrt(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*sqrt(c + d/x**2) - b*(c +
d/x**2)**(3/2)/(3*d)

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GIAC/XCAS [A]  time = 0.326824, size = 220, normalized size = 3.73 \[ -\frac{1}{2} \, a \sqrt{c}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a \sqrt{c} d{\rm sign}\left (x\right ) - 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a \sqrt{c} d^{2}{\rm sign}\left (x\right ) + b c^{\frac{3}{2}} d^{2}{\rm sign}\left (x\right ) + 3 \, a \sqrt{c} d^{3}{\rm sign}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*a*sqrt(c)*ln((sqrt(c)*x - sqrt(c*x^2 + d))^2)*sign(x) + 2/3*(3*(sqrt(c)*x -
sqrt(c*x^2 + d))^4*b*c^(3/2)*sign(x) + 3*(sqrt(c)*x - sqrt(c*x^2 + d))^4*a*sqrt
(c)*d*sign(x) - 6*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*sqrt(c)*d^2*sign(x) + b*c^(3
/2)*d^2*sign(x) + 3*a*sqrt(c)*d^3*sign(x))/((sqrt(c)*x - sqrt(c*x^2 + d))^2 - d)
^3