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

Optimal. Leaf size=66 \[ \frac{a x^3 \left (c+\frac{d}{x^2}\right )^{3/2}}{3 c}+b x \sqrt{c+\frac{d}{x^2}}-b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right ) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.121181, antiderivative size = 66, 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 \[ \frac{a x^3 \left (c+\frac{d}{x^2}\right )^{3/2}}{3 c}+b x \sqrt{c+\frac{d}{x^2}}-b \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.8027, size = 56, normalized size = 0.85 \[ \frac{a x^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 c} - b \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )} + b x \sqrt{c + \frac{d}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.124374, size = 97, normalized size = 1.47 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (\sqrt{c x^2+d} \left (a c x^2+a d+3 b c\right )-3 b c \sqrt{d} \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )+3 b c \sqrt{d} \log (x)\right )}{3 c \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 83, normalized size = 1.3 \[ -{\frac{x}{3\,c}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 3\,\sqrt{d}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) bc-a \left ( c{x}^{2}+d \right ) ^{{\frac{3}{2}}}-3\,\sqrt{c{x}^{2}+d}bc \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.231912, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, b c \sqrt{d} \log \left (-\frac{c x^{2} - 2 \, \sqrt{d} x \sqrt{\frac{c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right ) + 2 \,{\left (a c x^{3} +{\left (3 \, b c + a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{6 \, c}, -\frac{3 \, b c \sqrt{-d} \arctan \left (\frac{d}{\sqrt{-d} x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (a c x^{3} +{\left (3 \, b c + a d\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{3 \, c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 4.30527, size = 107, normalized size = 1.62 \[ \frac{a \sqrt{d} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3} + \frac{a d^{\frac{3}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c} + \frac{b \sqrt{c} x}{\sqrt{1 + \frac{d}{c x^{2}}}} - b \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )} + \frac{b d}{\sqrt{c} x \sqrt{1 + \frac{d}{c x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217031, size = 157, normalized size = 2.38 \[ \frac{b d \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right ){\rm sign}\left (x\right )}{\sqrt{-d}} - \frac{{\left (3 \, b c d \arctan \left (\frac{\sqrt{d}}{\sqrt{-d}}\right ) + 3 \, b c \sqrt{-d} \sqrt{d} + a \sqrt{-d} d^{\frac{3}{2}}\right )}{\rm sign}\left (x\right )}{3 \, c \sqrt{-d}} + \frac{{\left (c x^{2} + d\right )}^{\frac{3}{2}} a c^{2}{\rm sign}\left (x\right ) + 3 \, \sqrt{c x^{2} + d} b c^{3}{\rm sign}\left (x\right )}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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