3.2020 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^{10}} \, dx\)

Optimal. Leaf size=59 \[ -\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]

[Out]

(-2*a^2*Sqrt[a + b/x^3])/(3*b^3) + (4*a*(a + b/x^3)^(3/2))/(9*b^3) - (2*(a + b/x
^3)^(5/2))/(15*b^3)

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Rubi [A]  time = 0.084263, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 a^2 \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{5/2}}{15 b^3}+\frac{4 a \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^3]*x^10),x]

[Out]

(-2*a^2*Sqrt[a + b/x^3])/(3*b^3) + (4*a*(a + b/x^3)^(3/2))/(9*b^3) - (2*(a + b/x
^3)^(5/2))/(15*b^3)

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Rubi in Sympy [A]  time = 10.5086, size = 54, normalized size = 0.92 \[ - \frac{2 a^{2} \sqrt{a + \frac{b}{x^{3}}}}{3 b^{3}} + \frac{4 a \left (a + \frac{b}{x^{3}}\right )^{\frac{3}{2}}}{9 b^{3}} - \frac{2 \left (a + \frac{b}{x^{3}}\right )^{\frac{5}{2}}}{15 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(a+b/x**3)**(1/2),x)

[Out]

-2*a**2*sqrt(a + b/x**3)/(3*b**3) + 4*a*(a + b/x**3)**(3/2)/(9*b**3) - 2*(a + b/
x**3)**(5/2)/(15*b**3)

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Mathematica [A]  time = 0.0458116, size = 42, normalized size = 0.71 \[ -\frac{2 \sqrt{a+\frac{b}{x^3}} \left (8 a^2 x^6-4 a b x^3+3 b^2\right )}{45 b^3 x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[a + b/x^3]*x^10),x]

[Out]

(-2*Sqrt[a + b/x^3]*(3*b^2 - 4*a*b*x^3 + 8*a^2*x^6))/(45*b^3*x^6)

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Maple [A]  time = 0.01, size = 50, normalized size = 0.9 \[ -{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 8\,{a}^{2}{x}^{6}-4\,ab{x}^{3}+3\,{b}^{2} \right ) }{45\,{b}^{3}{x}^{9}}{\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(a+b/x^3)^(1/2),x)

[Out]

-2/45*(a*x^3+b)*(8*a^2*x^6-4*a*b*x^3+3*b^2)/x^9/b^3/((a*x^3+b)/x^3)^(1/2)

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Maxima [A]  time = 1.46211, size = 63, normalized size = 1.07 \[ -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{5}{2}}}{15 \, b^{3}} + \frac{4 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} a}{9 \, b^{3}} - \frac{2 \, \sqrt{a + \frac{b}{x^{3}}} a^{2}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^10),x, algorithm="maxima")

[Out]

-2/15*(a + b/x^3)^(5/2)/b^3 + 4/9*(a + b/x^3)^(3/2)*a/b^3 - 2/3*sqrt(a + b/x^3)*
a^2/b^3

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Fricas [A]  time = 0.24519, size = 57, normalized size = 0.97 \[ -\frac{2 \,{\left (8 \, a^{2} x^{6} - 4 \, a b x^{3} + 3 \, b^{2}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{45 \, b^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^10),x, algorithm="fricas")

[Out]

-2/45*(8*a^2*x^6 - 4*a*b*x^3 + 3*b^2)*sqrt((a*x^3 + b)/x^3)/(b^3*x^6)

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Sympy [A]  time = 16.5594, size = 824, normalized size = 13.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**10/(a+b/x**3)**(1/2),x)

[Out]

-16*a**(15/2)*b**(9/2)*x**15*sqrt(a*x**3/b + 1)/(45*a**(11/2)*b**7*x**(33/2) + 1
35*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x**
(15/2)) - 40*a**(13/2)*b**(11/2)*x**12*sqrt(a*x**3/b + 1)/(45*a**(11/2)*b**7*x**
(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)
*b**10*x**(15/2)) - 30*a**(11/2)*b**(13/2)*x**9*sqrt(a*x**3/b + 1)/(45*a**(11/2)
*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45
*a**(5/2)*b**10*x**(15/2)) - 10*a**(9/2)*b**(15/2)*x**6*sqrt(a*x**3/b + 1)/(45*a
**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21
/2) + 45*a**(5/2)*b**10*x**(15/2)) - 10*a**(7/2)*b**(17/2)*x**3*sqrt(a*x**3/b +
1)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**
9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2)) - 6*a**(5/2)*b**(19/2)*sqrt(a*x**3/b
+ 1)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b
**9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2)) + 16*a**8*b**4*x**(33/2)/(45*a**(11
/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) +
45*a**(5/2)*b**10*x**(15/2)) + 48*a**7*b**5*x**(27/2)/(45*a**(11/2)*b**7*x**(33
/2) + 135*a**(9/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b*
*10*x**(15/2)) + 48*a**6*b**6*x**(21/2)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9
/2)*b**8*x**(27/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2))
+ 16*a**5*b**7*x**(15/2)/(45*a**(11/2)*b**7*x**(33/2) + 135*a**(9/2)*b**8*x**(27
/2) + 135*a**(7/2)*b**9*x**(21/2) + 45*a**(5/2)*b**10*x**(15/2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{3}}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^10),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^10), x)