3.1935 \(\int \frac{x^4}{\left (a+\frac{b}{x^2}\right )^{3/2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{16 b^2 x \sqrt{a+\frac{b}{x^2}}}{5 a^4}-\frac{8 b^2 x}{5 a^3 \sqrt{a+\frac{b}{x^2}}}-\frac{2 b x^3}{5 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}} \]

[Out]

(-8*b^2*x)/(5*a^3*Sqrt[a + b/x^2]) + (16*b^2*Sqrt[a + b/x^2]*x)/(5*a^4) - (2*b*x
^3)/(5*a^2*Sqrt[a + b/x^2]) + x^5/(5*a*Sqrt[a + b/x^2])

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Rubi [A]  time = 0.100294, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{16 b^2 x \sqrt{a+\frac{b}{x^2}}}{5 a^4}-\frac{8 b^2 x}{5 a^3 \sqrt{a+\frac{b}{x^2}}}-\frac{2 b x^3}{5 a^2 \sqrt{a+\frac{b}{x^2}}}+\frac{x^5}{5 a \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b/x^2)^(3/2),x]

[Out]

(-8*b^2*x)/(5*a^3*Sqrt[a + b/x^2]) + (16*b^2*Sqrt[a + b/x^2]*x)/(5*a^4) - (2*b*x
^3)/(5*a^2*Sqrt[a + b/x^2]) + x^5/(5*a*Sqrt[a + b/x^2])

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Rubi in Sympy [A]  time = 7.9117, size = 82, normalized size = 0.93 \[ \frac{x^{5}}{5 a \sqrt{a + \frac{b}{x^{2}}}} - \frac{2 b x^{3}}{5 a^{2} \sqrt{a + \frac{b}{x^{2}}}} - \frac{8 b^{2} x}{5 a^{3} \sqrt{a + \frac{b}{x^{2}}}} + \frac{16 b^{2} x \sqrt{a + \frac{b}{x^{2}}}}{5 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(a+b/x**2)**(3/2),x)

[Out]

x**5/(5*a*sqrt(a + b/x**2)) - 2*b*x**3/(5*a**2*sqrt(a + b/x**2)) - 8*b**2*x/(5*a
**3*sqrt(a + b/x**2)) + 16*b**2*x*sqrt(a + b/x**2)/(5*a**4)

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Mathematica [A]  time = 0.0338398, size = 52, normalized size = 0.59 \[ \frac{a^3 x^6-2 a^2 b x^4+8 a b^2 x^2+16 b^3}{5 a^4 x \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b/x^2)^(3/2),x]

[Out]

(16*b^3 + 8*a*b^2*x^2 - 2*a^2*b*x^4 + a^3*x^6)/(5*a^4*Sqrt[a + b/x^2]*x)

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Maple [A]  time = 0.009, size = 60, normalized size = 0.7 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ({a}^{3}{x}^{6}-2\,{a}^{2}b{x}^{4}+8\,a{b}^{2}{x}^{2}+16\,{b}^{3} \right ) }{5\,{a}^{4}{x}^{3}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b/x^2)^(3/2),x)

[Out]

1/5*(a*x^2+b)*(a^3*x^6-2*a^2*b*x^4+8*a*b^2*x^2+16*b^3)/a^4/x^3/((a*x^2+b)/x^2)^(
3/2)

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Maxima [A]  time = 1.44566, size = 93, normalized size = 1.06 \[ \frac{b^{3}}{\sqrt{a + \frac{b}{x^{2}}} a^{4} x} + \frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{5}{2}} x^{5} - 5 \,{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} b x^{3} + 15 \, \sqrt{a + \frac{b}{x^{2}}} b^{2} x}{5 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.239347, size = 84, normalized size = 0.95 \[ \frac{{\left (a^{3} x^{7} - 2 \, a^{2} b x^{5} + 8 \, a b^{2} x^{3} + 16 \, b^{3} x\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{5 \,{\left (a^{5} x^{2} + a^{4} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x^2)^(3/2),x, algorithm="fricas")

[Out]

1/5*(a^3*x^7 - 2*a^2*b*x^5 + 8*a*b^2*x^3 + 16*b^3*x)*sqrt((a*x^2 + b)/x^2)/(a^5*
x^2 + a^4*b)

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Sympy [A]  time = 5.09096, size = 337, normalized size = 3.83 \[ \frac{a^{5} b^{\frac{19}{2}} x^{10} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{5 a^{3} b^{\frac{23}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{30 a^{2} b^{\frac{25}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{40 a b^{\frac{27}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} + \frac{16 b^{\frac{29}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{5 a^{7} b^{9} x^{6} + 15 a^{6} b^{10} x^{4} + 15 a^{5} b^{11} x^{2} + 5 a^{4} b^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b/x**2)**(3/2),x)

[Out]

a**5*b**(19/2)*x**10*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 +
15*a**5*b**11*x**2 + 5*a**4*b**12) + 5*a**3*b**(23/2)*x**6*sqrt(a*x**2/b + 1)/(
5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*a**5*b**11*x**2 + 5*a**4*b**12) + 30*
a**2*b**(25/2)*x**4*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 +
15*a**5*b**11*x**2 + 5*a**4*b**12) + 40*a*b**(27/2)*x**2*sqrt(a*x**2/b + 1)/(5*a
**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*a**5*b**11*x**2 + 5*a**4*b**12) + 16*b**
(29/2)*sqrt(a*x**2/b + 1)/(5*a**7*b**9*x**6 + 15*a**6*b**10*x**4 + 15*a**5*b**11
*x**2 + 5*a**4*b**12)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x^2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^4/(a + b/x^2)^(3/2), x)