3.771 \(\int \left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}} x^4 \, dx\)

Optimal. Leaf size=53 \[ \frac{x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (5 b c-2 a d)}{15 c^2}+\frac{a x^5 \left (c+\frac{d}{x^2}\right )^{3/2}}{5 c} \]

[Out]

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

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Rubi [A]  time = 0.106376, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x^3 \left (c+\frac{d}{x^2}\right )^{3/2} (5 b c-2 a d)}{15 c^2}+\frac{a x^5 \left (c+\frac{d}{x^2}\right )^{3/2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)*Sqrt[c + d/x^2]*x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.53265, size = 46, normalized size = 0.87 \[ \frac{a x^{5} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{5 c} - \frac{x^{3} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (2 a d - 5 b c\right )}{15 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*x**4*(c+d/x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0496418, size = 42, normalized size = 0.79 \[ \frac{x \sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (3 a c x^2-2 a d+5 b c\right )}{15 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)*Sqrt[c + d/x^2]*x^4,x]

[Out]

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

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Maple [A]  time = 0.007, size = 43, normalized size = 0.8 \[{\frac{x \left ( 3\,a{x}^{2}c-2\,ad+5\,bc \right ) \left ( c{x}^{2}+d \right ) }{15\,{c}^{2}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*x^4*(c+d/x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.4506, size = 74, normalized size = 1.4 \[ \frac{b{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{3}}{3 \, c} + \frac{{\left (3 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}} x^{5} - 5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} d x^{3}\right )} a}{15 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)*x^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.216349, size = 77, normalized size = 1.45 \[ \frac{{\left (3 \, a c^{2} x^{5} +{\left (5 \, b c^{2} + a c d\right )} x^{3} +{\left (5 \, b c d - 2 \, a d^{2}\right )} x\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)*x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.11268, size = 119, normalized size = 2.25 \[ \frac{a \sqrt{d} x^{4} \sqrt{\frac{c x^{2}}{d} + 1}}{5} + \frac{a d^{\frac{3}{2}} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c} - \frac{2 a d^{\frac{5}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{15 c^{2}} + \frac{b \sqrt{d} x^{2} \sqrt{\frac{c x^{2}}{d} + 1}}{3} + \frac{b d^{\frac{3}{2}} \sqrt{\frac{c x^{2}}{d} + 1}}{3 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*x**4*(c+d/x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216044, size = 99, normalized size = 1.87 \[ \frac{5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} b{\rm sign}\left (x\right ) + \frac{{\left (3 \,{\left (c x^{2} + d\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} d\right )} a{\rm sign}\left (x\right )}{c}}{15 \, c} - \frac{{\left (5 \, b c d^{\frac{3}{2}} - 2 \, a d^{\frac{5}{2}}\right )}{\rm sign}\left (x\right )}{15 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2)*x^4,x, algorithm="giac")

[Out]

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