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

Optimal. Leaf size=46 \[ \frac{\left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.120396, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.0012, size = 39, normalized size = 0.85 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5 d^{2}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0597789, size = 47, normalized size = 1.02 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (5 a d x^2-2 b c x^2+3 b d\right )}{15 d^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 48, normalized size = 1. \[ -{\frac{ \left ( 5\,ad{x}^{2}-2\,bc{x}^{2}+3\,bd \right ) \left ( c{x}^{2}+d \right ) }{15\,{d}^{2}{x}^{4}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*(c+d/x^2)^(1/2)/x^3,x)

[Out]

-1/15*((c*x^2+d)/x^2)^(1/2)*(5*a*d*x^2-2*b*c*x^2+3*b*d)*(c*x^2+d)/d^2/x^4

_______________________________________________________________________________________

Maxima [A]  time = 1.37597, size = 66, normalized size = 1.43 \[ -\frac{1}{15} \, b{\left (\frac{3 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{2}} - \frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{2}}\right )} - \frac{a{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/15*b*(3*(c + d/x^2)^(5/2)/d^2 - 5*(c + d/x^2)^(3/2)*c/d^2) - 1/3*a*(c + d/x^2
)^(3/2)/d

_______________________________________________________________________________________

Fricas [A]  time = 0.228014, size = 81, normalized size = 1.76 \[ \frac{{\left ({\left (2 \, b c^{2} - 5 \, a c d\right )} x^{4} - 3 \, b d^{2} -{\left (b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \, d^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*((2*b*c^2 - 5*a*c*d)*x^4 - 3*b*d^2 - (b*c*d + 5*a*d^2)*x^2)*sqrt((c*x^2 + d
)/x^2)/(d^2*x^4)

_______________________________________________________________________________________

Sympy [A]  time = 2.76533, size = 58, normalized size = 1.26 \[ - \frac{a \left (\begin{cases} \frac{\sqrt{c}}{x^{2}} & \text{for}\: d = 0 \\\frac{2 \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right )}{2} - \frac{b \left (- \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**3,x)

[Out]

-a*Piecewise((sqrt(c)/x**2, Eq(d, 0)), (2*(c + d/x**2)**(3/2)/(3*d), True))/2 -
b*(-c*(c + d/x**2)**(3/2)/3 + (c + d/x**2)**(5/2)/5)/d**2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.453602, size = 338, normalized size = 7.35 \[ \frac{2 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{5}{2}}{\rm sign}\left (x\right ) - 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{3}{2}} d{\rm sign}\left (x\right ) + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{5}{2}} d{\rm sign}\left (x\right ) + 20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{3}{2}} d^{2}{\rm sign}\left (x\right ) + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{5}{2}} d^{2}{\rm sign}\left (x\right ) - 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{3}{2}} d^{3}{\rm sign}\left (x\right ) - 2 \, b c^{\frac{5}{2}} d^{3}{\rm sign}\left (x\right ) + 5 \, a c^{\frac{3}{2}} d^{4}{\rm sign}\left (x\right )\right )}}{15 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(15*(sqrt(c)*x - sqrt(c*x^2 + d))^8*a*c^(3/2)*sign(x) + 30*(sqrt(c)*x - sqr
t(c*x^2 + d))^6*b*c^(5/2)*sign(x) - 30*(sqrt(c)*x - sqrt(c*x^2 + d))^6*a*c^(3/2)
*d*sign(x) + 10*(sqrt(c)*x - sqrt(c*x^2 + d))^4*b*c^(5/2)*d*sign(x) + 20*(sqrt(c
)*x - sqrt(c*x^2 + d))^4*a*c^(3/2)*d^2*sign(x) + 10*(sqrt(c)*x - sqrt(c*x^2 + d)
)^2*b*c^(5/2)*d^2*sign(x) - 10*(sqrt(c)*x - sqrt(c*x^2 + d))^2*a*c^(3/2)*d^3*sig
n(x) - 2*b*c^(5/2)*d^3*sign(x) + 5*a*c^(3/2)*d^4*sign(x))/((sqrt(c)*x - sqrt(c*x
^2 + d))^2 - d)^5