3.797 \(\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x^5} \, dx\)

Optimal. Leaf size=72 \[ \frac{\left (c+\frac{d}{x^2}\right )^{3/2} (2 b c-a d)}{3 d^3}-\frac{c \sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^3} \]

[Out]

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

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Rubi [A]  time = 0.178179, antiderivative size = 72, 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 \[ \frac{\left (c+\frac{d}{x^2}\right )^{3/2} (2 b c-a d)}{3 d^3}-\frac{c \sqrt{c+\frac{d}{x^2}} (b c-a d)}{d^3}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/(Sqrt[c + d/x^2]*x^5),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.4303, size = 61, normalized size = 0.85 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5 d^{3}} + \frac{c \sqrt{c + \frac{d}{x^{2}}} \left (a d - b c\right )}{d^{3}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - 2 b c\right )}{3 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/x**5/(c+d/x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0798956, size = 60, normalized size = 0.83 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (b \left (-8 c^2 x^4+4 c d x^2-3 d^2\right )-5 a d x^2 \left (d-2 c x^2\right )\right )}{15 d^3 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x^5),x]

[Out]

(Sqrt[c + d/x^2]*(-5*a*d*x^2*(d - 2*c*x^2) + b*(-3*d^2 + 4*c*d*x^2 - 8*c^2*x^4))
)/(15*d^3*x^4)

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Maple [A]  time = 0.012, size = 70, normalized size = 1. \[{\frac{ \left ( 10\,acd{x}^{4}-8\,b{c}^{2}{x}^{4}-5\,a{d}^{2}{x}^{2}+4\,bcd{x}^{2}-3\,b{d}^{2} \right ) \left ( c{x}^{2}+d \right ) }{15\,{d}^{3}{x}^{6}}{\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/x^5/(c+d/x^2)^(1/2),x)

[Out]

1/15*(10*a*c*d*x^4-8*b*c^2*x^4-5*a*d^2*x^2+4*b*c*d*x^2-3*b*d^2)*(c*x^2+d)/((c*x^
2+d)/x^2)^(1/2)/d^3/x^6

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Maxima [A]  time = 1.39216, size = 112, normalized size = 1.56 \[ -\frac{1}{15} \, b{\left (\frac{3 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{3}} - \frac{10 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{3}} + \frac{15 \, \sqrt{c + \frac{d}{x^{2}}} c^{2}}{d^{3}}\right )} - \frac{1}{3} \, a{\left (\frac{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{d^{2}} - \frac{3 \, \sqrt{c + \frac{d}{x^{2}}} c}{d^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(sqrt(c + d/x^2)*x^5),x, algorithm="maxima")

[Out]

-1/15*b*(3*(c + d/x^2)^(5/2)/d^3 - 10*(c + d/x^2)^(3/2)*c/d^3 + 15*sqrt(c + d/x^
2)*c^2/d^3) - 1/3*a*((c + d/x^2)^(3/2)/d^2 - 3*sqrt(c + d/x^2)*c/d^2)

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Fricas [A]  time = 0.225838, size = 84, normalized size = 1.17 \[ -\frac{{\left (2 \,{\left (4 \, b c^{2} - 5 \, a c d\right )} x^{4} + 3 \, b d^{2} -{\left (4 \, b c d - 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \, d^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(sqrt(c + d/x^2)*x^5),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.67736, size = 206, normalized size = 2.86 \[ - \frac{\begin{cases} \frac{\frac{a}{2 x^{4}} + \frac{b}{3 x^{6}}}{\sqrt{c}} & \text{for}\: d = 0 \\- \frac{\frac{2 a c \left (- \frac{c}{\sqrt{c + \frac{d}{x^{2}}}} - \sqrt{c + \frac{d}{x^{2}}}\right )}{d} + \frac{2 a \left (\frac{c^{2}}{\sqrt{c + \frac{d}{x^{2}}}} + 2 c \sqrt{c + \frac{d}{x^{2}}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3}\right )}{d} + \frac{2 b c \left (\frac{c^{2}}{\sqrt{c + \frac{d}{x^{2}}}} + 2 c \sqrt{c + \frac{d}{x^{2}}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3}\right )}{d^{2}} + \frac{2 b \left (- \frac{c^{3}}{\sqrt{c + \frac{d}{x^{2}}}} - 3 c^{2} \sqrt{c + \frac{d}{x^{2}}} + c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d^{2}}}{d} & \text{otherwise} \end{cases}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/x**5/(c+d/x**2)**(1/2),x)

[Out]

-Piecewise(((a/(2*x**4) + b/(3*x**6))/sqrt(c), Eq(d, 0)), (-(2*a*c*(-c/sqrt(c +
d/x**2) - sqrt(c + d/x**2))/d + 2*a*(c**2/sqrt(c + d/x**2) + 2*c*sqrt(c + d/x**2
) - (c + d/x**2)**(3/2)/3)/d + 2*b*c*(c**2/sqrt(c + d/x**2) + 2*c*sqrt(c + d/x**
2) - (c + d/x**2)**(3/2)/3)/d**2 + 2*b*(-c**3/sqrt(c + d/x**2) - 3*c**2*sqrt(c +
d/x**2) + c*(c + d/x**2)**(3/2) - (c + d/x**2)**(5/2)/5)/d**2)/d, True))/2

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{\sqrt{c + \frac{d}{x^{2}}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(sqrt(c + d/x^2)*x^5),x, algorithm="giac")

[Out]

integrate((a + b/x^2)/(sqrt(c + d/x^2)*x^5), x)